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4 TYPES OF VOR CHECKS
Your airplanes VOR received must be checked every 30 days for IFR Operations and there are multiple ways pilot's can check their VORs. How are they performed? What do you need to annotate?
Here's what you need to know.
VOR Checks:
VOR Receivers are required to be checked every 30 days for IFR Flight Operations. However, it is also important for VFR Pilot’s to check their aircraft’s VOR Receivers.
What to Write (SLED)
Signature (of pilot performing the check)
Location (of the check)
Error (amount of error detected during check)
Date (of the check)
VOT (VOR Test Facility)
A VOT is coded to emit the 360 Radial in all directions around the facility.
This means the airplane’s VOR Receiver should read either: 360 FROM or 180 TO, regardless of the aircraft’s location in relation to the VOR.
How the check is done:
1. Tune and Identify the VOT.
2. Twist the OBS Knob to center the CDI Needle.
3. Check for proper TO/FROM Indication.
4. The radial selected must be within:
5. +/- 4 degrees of 360 or 180.
Ground Check
With a VOR Ground Check:
• The Pilot must park the airplane in the designated ground spot.
• The Pilot must tune and identify the correct VOR.
• The Pilot must use the ground check sign to know:
• Which radial he/she should be on.
• Whether he/she should have a TO or a FROM Indication.
How the check is done:
1. Park aircraft in designated check spot.
2. Tune and Identify the Correct VOR.
3. Twist the OBS Knob to center the CDI Needle.
4. Check for proper TO/FROM Indication.
5. The radial selected must be within:
6. +/- 4 degrees of Designated Radial.
Airborne Check
With an Airborne VOR check:
• The Pilot must position the airplane over the designated location.
• The Pilot must tune and identify the correct VOR.
• The Pilot must use the information in the Chart Supplement to know:
• Which radial he/she should be on.
• Whether he/she should have a TO or a FROM Indication.
How the check is done:
1. Position aircraft over designated check spot.
2. Tune and Identify the Correct VOR.
3. Twist the OBS Knob to center the CDI Needle.
4. Check for proper TO/FROM Indication.
5. The radial selected must be within:
6. +/- 6 degrees of Designated Radial.
Dual VOR Check
With a Dual VOR check, the airplane must be equipped with 2 VOR Receivers.
How the check is done:
1. The pilot tunes both VOR Receivers to the same VOR.
2. The pilot centers both CDI Needles.
3. Check for proper TO/FROM Indications.
4. With both CDI Needles Centered:
5. The Selected Radials should be within 4 degrees of each other.
VOR Check Summary:
• VOT = +/- 4
• Ground Check = +/- 4
• Airborne Check = +/- 6
• Dual Check = within 4 degrees of each other
Author - Nate Hodell
CFI/CFII/MEI/ATP - Creator of wifiCFI - Owner of Axiom Aviation Flight School.
This information is included in the Navigation Aids: VOR Lessons on wifiCFI. Sign up today to watch videos, listen to podcasts, take lesson quizzes, join live webinars, print lesson quicktakes, and more by clicking this link >
where aviation comes to study
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How to Convert A Hexadecimal Number to Decimal in Excel
Sometimes we use hexadecimal numbers to mark products in daily life, and we want to convert these hexadecimal numbers to decimal numbers in some situations. We can convert number between two number types by convert tool online, actually we can also convert numbers by function in excel as well. In excel, =HEX2DEC(number) can help you to convert hexadecimal number to decimal properly, and on the other side, you can use =DEX2HEX(number) to convert decimal to hexadecimal number.
1. Convert Hex Number to Decimal in Excel
As we mentioned above, we can use HEX2DEC function to convert numbers conveniently. Just prepare a table with two columns, one column is used for recording HEX numbers, the second column is used for saving the converted decimal numbers.
Convert A Hexadecimal Number to Decimal 1
Step1: in B1 enter the formula:
=HEX2DEC(A2)
Convert A Hexadecimal Number to Decimal 2
Step2: Click Enter to get returned value. So 21163 in B2 is the mapping decimal number for 52AB.
Convert A Hexadecimal Number to Decimal 3
Step3: Drag the fill handle down to fill the following cells.
Convert A Hexadecimal Number to Decimal 4
Verify that all hexadecimal numbers are converted to decimal numbers correctly. You can also double check the result by convert tool online to make sure the result is correct.
Note:
Sometimes hexadecimal numbers are displayed like 0x52AB, user can remove 0x before 52AB and then use HEX2DEC function to convert number.
2. Convert Decimal to Hex Number in Excel
Prepare another table, the first column is Decimal, the second column is Hex Number.
Convert A Hexadecimal Number to Decimal 5
Step1: in B10 enter the formula:
=HEX2DEC(A2)
Convert A Hexadecimal Number to Decimal 6
Step2: Click Enter to get returned value. So 4D2 in B10 is the mapping hex number for 1234.
Convert A Hexadecimal Number to Decimal 7
Step3: Drag the fill handle down to fill the following cells.
Convert A Hexadecimal Number to Decimal 8
Note:
There are some other functions to convert numbers between different types. See below screenshot.
Convert A Hexadecimal Number to Decimal 9
Convert A Hexadecimal Number to Decimal 10
Convert A Hexadecimal Number to Decimal 11
3. Video: Converting Hex Numbers to Decimal and Decimal to Hex
In this video, we’ll explore two essential skills: converting Hexadecimal numbers to Decimal and Decimal numbers to Hex in Excel.
4. SAMPLE FIlES
Below are sample files in Microsoft Excel that you can download for reference if you wish.
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Home | | Chemistry | General properties of Lanthanides
Chapter: 11th 12th std standard Class Organic Inorganic Physical Chemistry Higher secondary school College Notes
General properties of Lanthanides
General properties of Lanthanides
The Lanthanide series include fifteen elements i.e. lanthanum (57 La) to lutetium (71 Lu). Lanthanum and Lutetium have no partly filled 4f- subshell but have electrons in 5d-subshell.
The position of f block elements in the periodic table, is explained above.
The elements in which the extra electron enters ( n- 2 )f orbitals are called f- block elements. These elements are also called as inner transition elements because they form a transition series within the transition elements. The f-block elements are also known as rare earth elements. These are divided into two series.
i) The Lanthanide series (4f-block elements)
ii) The Actinide series (5f- block elements )
The Lanthanide Series
The Lanthanide series include fifteen elements i.e. lanthanum (57 La) to lutetium (71 Lu). Lanthanum and Lutetium have no partly filled 4f- subshell but have electrons in 5d-subshell. Thus these elements should not be included in this series. However, all these elements closely resemble lanthanum and hence are considered together.
General properties of Lanthanides
1. Electronic configuration
The electronic configuration of Lanthanides are listed in the table . The fourteen electrons are filled in Ce to Lu with configuration [54 Xe ]4f1-14 5d1 6s2
2. Oxidation states
The common oxidation state exhibited by all the lanthanides is +3 (Ln3+) in aqueous solutions and in their solid compounds. Some elements exhibit +2 and +4 states as uncommon oxidation states.
La - +3
Ce - +3, +4, +2
Pr - +3, +4
Nd - +3, +4, +2
3. Radii of tripositive lanthanide ions
The size of M3+ ions decreases as we move through the lanthanides from lanthanum to lutetium. This steady decrease in ionic radii of M3+ cations in the lanthanide series is called Lanthanide contraction.
Cause of lanthanide contraction
The lanthanide contraction is due to the imperfect shielding of one 4f electron by another in the same sub shell. As we move along the lanthanide series, the nuclear charge and the number of 4f electrons increase by one unit at each step. However, due to imperfect shielding, the effective nuclear charge increases causing a contraction in electron cloud of 4f-subshell.
Consequences of lanthanide contraction
Basicity of ions
i) Due to lanthanide contraction, the size of Ln3+ ions decreases regularly with increase in atomic number. According to Fajan's rule, decrease in size of Ln3+ ions increase the covalent character and decreases the basic character between Ln3+ and OH- ion in Ln(OH)3. Since the order of size of Ln3+ ions
are
La3+> Ce3+ ............... >Lu3+
ii) There is regular decrease in their ionic radii.
iii) Regular decrease in their tendency to act as reducing agent, with increase in atomic number.
iv) Due to lanthanide contraction, second and third rows of d-block transistion elements are quite close in properties.
v) Due to lanthanide contraction, these elements occur together in natural minerals and are difficult to separate.
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
11th 12th std standard Class Organic Inorganic Physical Chemistry Higher secondary school College Notes : General properties of Lanthanides |
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Patents
1. Advanced Patent Search
Publication numberUS4053739 A
Publication typeGrant
Application numberUS 05/713,470
Publication dateOct 11, 1977
Filing dateAug 11, 1976
Priority dateAug 11, 1976
Also published asCA1097407A1, DE2735204A1, DE2735204C2
Publication number05713470, 713470, US 4053739 A, US 4053739A, US-A-4053739, US4053739 A, US4053739A
InventorsRobert Lynn Miller, Robert Neal Weisshappel
Original AssigneeMotorola, Inc.
Export CitationBiBTeX, EndNote, RefMan
External Links: USPTO, USPTO Assignment, Espacenet
Dual modulus programmable counter
US 4053739 A
Abstract
The inventive counter is operable to divide an input signal by the sum of two binary numbers, A and B. Each number is stored in memory. These numbers are alternately preset into a binary counter which also receives the input signal. A logic gate monitors the counter output and changes state when the number previously preset in the counter equals the accumulated count. The gate state transition is used to preset the counter with the alternate stored number. Thus, the process continues whereby the output from the logic gate represents the input signal divided by the sum of A and B.
Images(1)
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Claims(5)
We claim:
1. A multiple modulus counter for dividing a signal having a frequency f by a divisor N = M1 + M2 + . . . + Mx, where N, M1, M2, . . . , Mx are selected numbers, comprising:
counter means including an input for receiving the signal to be divided, an output for producing a signal representative of the count of signals received at the input, and means to input a preset count state;
a plurality of Mx preset means, each actuable to preset one of the numbers M1 . . . Mx into the counter means;
control means responsive to the count state at the counter output to sequentially actuate a successive one of the preset means in response to the counter counting to the count preset into the counter by the preceding preset means, the control means producing an output waveform having transitions corresponding to the actuation of predetermined preset means,
whereby the control means output waveform is of a frequency f/N.
2. A dual modulus counter for dividing a signal having a frequency f by a divisor N = A + B, where N, A and B are selected numbers, comprising:
counter means including an input for receiving the signal to be divided, an output for producing a signal representative of the count of signals received at the input, and means to input a preset count state;
first preset means actuable to preset the count A in the counter means;
second preset means actuable to preset the count B in the counter means; and
control means responsive to the count state at the counter output to sequentially actuate the second and first preset means in response to the counter counting the numbers A and B, respectively, the control means producing an output waveform having transitions at the times of actuating the first and second preset means,
whereby the control means output waveform is of a frequency f/N.
3. A frequency synthesizer comprising:
a reference signal source for generating a reference signal of frequency f;
a phase comparator for producing at its output an error signal representative of the phase difference of signals received at its input;
means for coupling the reference signal source to the first phase comparator input;
a signal controlled oscillator for producing an oscillator signal of predetermined frequency at its output responsive to a received control signal;
means for processing the phase comparator error signal and producing a control signal in response thereto;
means for coupling the produced control signal to the signal controlled oscillator;
prescaler means actuable to frequency divide the oscillator signal by one of two predetermined divisors P, P';
a dual modulus divider for frequency dividing the output from the prescaler by alternate stored divisors A and B, where A and B are selected numbers, the dual modulus divisor including means to actuate the prescaler means from its P divisor to its P' divisor upon transition from the A divisor to the B divisor and from its P' divisor to its P divisor upon transition from the B divisor to the A divisor; and
means for coupling the output from the dual modulus divider to the comparator second input,
whereby the oscillator signal tends to assume the frequency f/(AP + BP').
4. The frequency synthesizer of claim 3 wherein P' = P + 1.
5. The frequency divisor of claim 3 wherein the dual modulus divider comprises:
counter means including an input for receiving the prescaler output signal, an output for producing a signal representative of the count of signals received at the input, and means to input a preset count state;
first preset means actuable to preset the count A in the counter means;
second preset means actuable to preset the count B in the counter means; and
control means responsive to the count state at the counter output to sequentially actuate the second and first preset means in response to the counter counting the numbers A and B, respectively, the control means producing an output waveform having transitions at the times of actuating the first and second preset means.
Description
BACKGROUND OF THE INVENTION
The present invention pertains to the electronic signal processing art and, in particular, to a programmable frequency counter.
Programmable frequency counters have been well known in the electronic processing art, particularly in the frequency synthesizer field. Frequency synthesizers commonly employ standard phase lock loop circuitry wherein a reference frequency oscillator signal may be divided by a selected one of a plurality of divisors thus providing an output signal of desired frequency. Previous techniques employed in digital frequency synthesizers have used, in the feedback portion of a conventional phase lock loop, a variable prescaler, and first and second counters. The first counter has been programmable and is used to divide the output of the variable prescaler by a fixed number (N). The second counter, often referred to as a swallow counter, has been used to switch the variable prescaler to a new divisor, or modulus, which new modulus is present during the counting of "N". As is discussed at page 10-3 of the Motorola "McMOS HANDBOOK", printed 1974 by Motorola, Inc., the total divisor NT of the feedback loop is given by:
NT = (P + 1)A + P(N - A)
where, the variable modulus prescaler operates between two divisors P and P+1, the swallow counter has a fixed divisor A, and the programmable divider has the divisor N.
While the above described frequency synthesizer provided the desired function, it requires a large number of parts and thus is expensive to manufacture. It is desirable, therefore, to provide the frequency synthesizer function using fewer parts.
SUMMARY OF THE INVENTION
It is an object of this invention, therefore, to provide an improved dual modulus programmable counter which is particularly suited for application in frequency synthesizers.
It is a particular object of the invention to provide the above dual modulus programmable counter which employs a minimum of components and, therefore, results in a minimum cost.
Briefly, according to the invention, a multiple modulus divider divides a signal having a frequency f by a divisor N = N1 + M2 + . . . + Mx, where N, M1, M2, . . . , Mx are selected numbers. The improved counter comprises a counter means which includes an input for receiving the signal to be divided, an output for producing a signal representative of the count of signals received at the input, and means to input a preset count state. Also included are a plurality of Mx preset means, each of which is actuable to preset one of the numbers M1 . . . Mx into the counter means. A control means responds to the count state at the counter output to sequentially actuate successive ones of the preset means in response to the counter counting to the count preset into the counter by the preceeding preset means. The control means produces an output waveform having transitions corresponding to the actuation of the predetermined preset means whereby the control means output waveform is of a frequency f/N.
The improved dual modulus programmable counter may be used in combination with further components to comprise a frequency synthesizer. In particular, additional frequency synthesizer components comprise a reference signal source for generating a reference signal frequency f. This signal is coupled, via appropriate means, to the first input of a phase comparator which compares this signal to the signal received at its second input, and produces an error signal representative of the phase difference therebetween at its output. The phase comparator error signal is processed for application to the control signal of a signal controlled oscillator which, in turn, responds by producing an oscillator signal of predetermined frequency. The output from the signal controlled oscillator couples to a prescaler which is actuable to frequency divide the oscillator signal by one of two predetermined divisors P, P'. The aforementioned dual modulus divider frequency divides the output from the prescaler by alternate stored divisors A and B, where A and B are selected numbers. The divisor includes means to actuate the prescaler means from its P divisor to its P' divisor upon transition from the A divisor to the B divisor, and from its P' divisor to its P divisor upon transition from the B divisor to the A divisor. The output from the divider is coupled to the comparator second input whereby the oscillator signal tends to assume the frequency f/(AP' + BP).
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic diagram illustrating the inventive dual modulus counter; and
FIG. 2 is a schematic diagram illustrating a frequency synthesizer which employs the inventive counter.
DESCRIPTION OF THE PREFERRED EMBODIMENT OF THE INVENTION
Referring to FIG. 1, a signal of frequency f, which is to be divided by a divisor N, is applied at the input 12 of a standard binary counter 10. The binary counter 10, operating in the well known manner, produces a signal at its output terminal 14 in response to a predetermined count of the input signal f. The binary counter 10 also has preset count input terminals 16, 18. A binary number coupled to one of the preset count inputs 16, 18 will activate the counter 10 to the binary number. In the present preferred embodiment of the invention, binary counter 10 is of the count-down type which means that a count state preset at the input terminals 16 or 18 will be decremated one count for each received input pulse at input 12. The binary counter 10 responds to counting down to a zero count state by changing its output logic state at output terminal 14.
A change in the output state at output 14 of binary counter 10 activates the "C" input 22 of a conventional control flip-flop 24. Flip-flop 24 has a first "Q" output 26 and a second "Q" output 28. The control flip-flop 24 responds to transition state changes at its input 22 to alternately activate the Q output 26 high and low, with the Q output 28 correspondingly low and high.
The Q output 28 of the flip-flop 24 couples to the input terminals 32, 42 of a pair of preset storage registers 30, 40, respectively. Each register 30, 40 is programmed to contain a preset number. In this case preset register 30 contains the number A and preset register 40 contains the number B. Upon suitable activation at their inputs 32, 42 each register 30, 40 applies the number stored therein to the preset input terminals 16, 18 of the binary counter 10, activating the count in the same to the appropriate number A, B. Each number A, B corresponds to a modulus with which the input signal f will be divided. In this preferred embodiment of the invention a dual modulus system is provided. Thus, there are two preset registers 30, 40 each containing the number A, B respectively. In a generalized system, any one of a number of divisors of modulus M1 + M2 + . . . + Mx might be used, in which case there would be a preset register for each, each containing the appropriate number M1, M2, . . . Mx. For purposes of clarity the following discussion deals primarily with a dual modulus counter. Nonetheless, it should be understood that anyone of ordinary skill in the art could practice the invention by constructing a counter having more than two moduli.
Operation of the dual modulus programmable counter of FIG. 1 may be understood as follows.
Assume initially that the Q output 28 of the flip-flop 24 has activated preset register 30 to place the count A into the binary counter 10. Thus, each successive count of the input signal f reduces the counter by one whereby, finally, the counter reaches a count of zero. At this time the counter output 14 makes a transition thereby activating the control input 22 of the flip-flop 24. At this point the Q output 26 and Q output 28 of flip-flop 24 make a transition to the opposite logic state. This transition causes the second preset register 40 to input the count B into the binary counter 10. Now successive input counts at input 12 of binary counter 10 due to the input signal f reduce the count state of the counter 10 until it again reaches zero, at which point an output transition at output 14 once again activates the control input 22 of the flip-flop 24, thus activating preset register 30 to again input the count A into the binary counter 10.
Henceforth, the cycle repeats and the Q output 26 of the flip-flop 24 assumes a waveform having a frequency f/N, where N = A + B. Thus, with a minimum of components at input signal f is divided by two moduli A, B, thereby dividing the input signal f by the sum of the two moduli, N. As is discussed with reference to FIG. 2, the fact that the control flip-flop 24 produces an output transition after the A count period renders the instant dual modulus programmable counter extremely useful in frequency synthesizer applications.
FIG. 2 illustrates the preferred embodiment of a frequency synthesizer which employs the novel dual modulus programmable counter. There a standard phase lock loop chain includes a reference oscillator 100 which produces a reference signal of frequency f. The signal f is fed to the first input 112 of a phase detector 110. Phase detector 110 has a second input 114 and an output 116. Acting in the conventional manner, the phase detector 110 produces an error signal at its output 116, which error signal is representative of the phase difference between signals received at the input terminals 112, 114.
In the conventional manner, the output error signal at output terminal 116 is low pass filtered through a low pass filter circuit 118 and applied to the control input 122 of a voltage controlled oscillator 120. The voltage controlled oscillator 120 produces an oscillator signal of predetermined frequency at its output 124 responsive to a control signal received at its control input 122. This oscillator output signal is the output signal fout of the frequency synthesizer.
The output terminal 124 of the voltage controlled oscillator 120 also feeds to the input terminal 132 of a variable modulus prescaler 130. The variable modulus prescaler 130 responds to a signal at its divisor input 134 to divide signals received at its input terminal 132 by either one of two moduli P, or P' reproducing the output frequency divided signal at its output terminal 136. In the preferred embodiment of the invention, P' = P + 1, however it should be understood that the selection of the P' modulus is one of individual designer's choice. The frequency divided output 136 of the variable modulus prescaler 130 is applied to the input terminal 142 of the dual modulus programmable counter 150. The dual modulus programmable counter 150 is seen to be identical to the preferred embodiment thereof illustrated in FIG. 1. For example, input terminal 142 is the input of a binary counter 140 corresponding to the binary counter 10 of FIG. 1. Binary counter 140 has an output 144 which feeds to the control input 152 of a control flip-flop 154. The control flip-flop 154 has a Q output 156 and a Q output 158. The Q output 158 actuates the inputs 162, 172 of the preset storage registers 160, 170 respectively. As before, each preset register 160, 170 contains preset numbers A, B, respectively, which, upon actuation via the input terminals 162, 172 feed their corresponding number into the binary counter 140 via the preset input terminals 146, 148.
The Q output 156 of the flip-flop 154 feeds to the modulus control terminal 134 of the variable modulus prescaler 130. A transition in logic state at input 134 causes the variable modulus prescaler 130 to alternate between the P and P+1 divisors. Finally, the Q output 158 of the control flip-flop 154 feeds to the second input 114 of the phase comparator 110.
Operation of the frequency synthesizer of FIG. 2 is understood as follows.
The reference oscillator 100 feeds a signal of frequency f to the first input 112 of the phase detector 110. Phase detector 110, in turn, produces an error signal at its output 116 which, when low pass filtered via the filter 118, controls the voltage controlled oscillator 120. The oscillator output signal from the voltage controlled oscillator 120 is frequency divided by the variable modulus prescaler 130. Assuming that the variable modulus prescaler 130 is activated to its P modulus, the variable modulus prescaler 130 will produce an output transition at its output terminal 136 when it has counted P counts in the oscillator signal. At this time the first count is received by the binary counter 140 at its input 142. Stored within the binary counter 140 initially is the binary number A. Thus, this binary preset count is decremented by one count. This process continues until the variable modulus prescaler 130 counts to the number P, A times. After the binary counter 140 has counted down from its preset input A, it produces an output at output terminal 144 which in turn is applied to the control input 152 of the control flip-flop 144. This transition at the control input 152 causes the Q output 156 and Q output 158 to flip to their opposite states. Thus, the Q output 156 activates the variable modulus prescaler 130 to begin dividing by its second modulus P+1. Also, the Q output 158 causes the number B stored in register 170 to be fed into the binary counter 40. Now, the binary counter 140 does not change its output state at its output terminal 144 until the variable modulus prescaler has counted P+1 counts a total of B times.
Thereafter, the cycle repeats whereby the waveform at the Q output 158 of the flip-flop 154 is of a frequency fout /Nt, where Nt = A(P) + B(P+1). Now, in the conventional manner, the waveform fout /Nt is phase compared with the reference oscillator 100 signal f, whereby the two tend to phase lock producing the output signal fout = f/NT.
Thus, the dual modulus programmable counter 150 replaces the variable counter and the swallow counter of the prior art when used in a frequency synthesizer which provides an output signal which is the frequency division of a reference signal. Since the inventive dual modulus programmable counter does not require both a programmable counter, and a swallow counter, as has been known in the prior art, a significant reduction in parts count, and thus cost, has been achieved.
While a preferred embodiment of the invention has been described in detail, it should be understood that many modifications and variations thereto are possible, all of which fall within the true spirit and scope of the invention.
Patent Citations
Cited PatentFiling datePublication dateApplicantTitle
US3353104 *Jun 14, 1965Nov 14, 1967Ltv Electrosystems IncFrequency synthesizer using fractional division by digital techniques within a phase-locked loop
US3594551 *Nov 29, 1966Jul 20, 1971Electronic CommunicationsHigh speed digital counter
US3605025 *Jun 30, 1969Sep 14, 1971Sperry Rand CorpFractional output frequency-dividing apparatus
US3714589 *Dec 1, 1971Jan 30, 1973Lewis RDigitally controlled phase shifter
US3959737 *Nov 18, 1974May 25, 1976Engelmann Microwave Co.Frequency synthesizer having fractional frequency divider in phase-locked loop
US3982199 *Jan 6, 1975Sep 21, 1976The Bendix CorporationDigital frequency synthesizer
Referenced by
Citing PatentFiling datePublication dateApplicantTitle
US4184068 *Nov 14, 1977Jan 15, 1980Harris CorporationFull binary programmed frequency divider
US4231104 *Apr 26, 1978Oct 28, 1980Teradyne, Inc.Generating timing signals
US4241408 *Apr 4, 1979Dec 23, 1980Norlin Industries, Inc.High resolution fractional divider
US4316151 *Feb 13, 1980Feb 16, 1982Motorola, Inc.Phase locked loop frequency synthesizer using multiple dual modulus prescalers
US4325031 *Feb 13, 1980Apr 13, 1982Motorola, Inc.Divider with dual modulus prescaler for phase locked loop frequency synthesizer
US4327623 *Mar 31, 1980May 4, 1982Nippon Gakki Seizo Kabushiki KaishaReference frequency signal generator for tuning apparatus
US4330751 *Dec 3, 1979May 18, 1982Norlin Industries, Inc.Programmable frequency and duty cycle tone signal generator
US4357527 *Jan 25, 1979Nov 2, 1982Tokyo Shibaura Denki Kabushiki KaishaProgrammable divider
US4390960 *Nov 21, 1980Jun 28, 1983Hitachi, Ltd.Frequency divider
US4468797 *Feb 3, 1982Aug 28, 1984Oki Electric Industry Co., Ltd.Swallow counters
US4559613 *Jun 29, 1982Dec 17, 1985The United States Of America As Represented By The Secretary Of The Air ForceDigital frequency synthesizer circuit
US4574385 *Feb 16, 1984Mar 4, 1986Rockwell International CorporationClock divider circuit incorporating a J-K flip-flop as the count logic decoding means in the feedback loop
US4651334 *Dec 24, 1984Mar 17, 1987Hitachi, Ltd.Variable-ratio frequency divider
US4658406 *Aug 12, 1985Apr 14, 1987Andreas PappasDigital frequency divider or synthesizer and applications thereof
US4891825 *Feb 9, 1988Jan 2, 1990Motorola, Inc.Fully synchronized programmable counter with a near 50% duty cycle output signal
US5065415 *Feb 21, 1990Nov 12, 1991Nihon Musen Kabushiki KaishaProgrammable frequency divider
US5066927 *Sep 6, 1990Nov 19, 1991Ericsson Ge Mobile Communication Holding, Inc.Dual modulus counter for use in a phase locked loop
US5195111 *Aug 13, 1991Mar 16, 1993Nihon Musen Kabushiki KaishaProgrammable frequency dividing apparatus
US5202906 *Dec 23, 1987Apr 13, 1993Nippon Telegraph And Telephone CompanyFrequency divider which has a variable length first cycle by changing a division ratio after the first cycle and a frequency synthesizer using same
US5235531 *Dec 13, 1991Aug 10, 1993Siemens AktiengesellschaftMethod and arrangement for dividing the frequency of an alternating voltage with a non-whole-numbered division factor
US5495505 *Dec 20, 1990Feb 27, 1996Motorola, Inc.Increased frequency resolution in a synthesizer
US5781459 *Apr 16, 1996Jul 14, 1998Bienz; Richard AlanMethod and system for rational frequency synthesis using a numerically controlled oscillator
US5842006 *Sep 6, 1995Nov 24, 1998National Instruments CorporationCounter circuit with multiple registers for seamless signal switching
US6035182 *Jan 20, 1998Mar 7, 2000Motorola, Inc.Single counter dual modulus frequency division apparatus
US6072404 *Apr 29, 1997Jun 6, 2000Eaton CorporationUniversal garage door opener
US6725245May 3, 2002Apr 20, 2004P.C. Peripherals, IncHigh speed programmable counter architecture
USRE32605 *Jun 28, 1985Feb 16, 1988Hitachi, Ltd.Frequency divider
WO1981002371A1 *Jan 5, 1981Aug 20, 1981Motorola IncAn improved frequency synthesizer using multiple dual modulus prescalers
WO1981002372A1 *Jan 5, 1981Aug 20, 1981Motorola IncImproved divider with dual modulus prescaler
WO1982003477A1 *Mar 30, 1982Oct 14, 1982Inc MotorolaFrequency synthesized transceiver
Classifications
U.S. Classification708/103, 377/52, 331/25, 331/1.00A, 331/16, 377/47
International ClassificationG06F7/68, H03K23/66, H03L7/193, H03L7/18
Cooperative ClassificationH03K23/665, H03L7/193, H03L7/18, H03K23/667, G06F7/68
European ClassificationH03K23/66P, H03K23/66S, G06F7/68, H03L7/18, H03L7/193
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Knowledge BaseYou're questions answered.
Is whey protein keto friendly?
Whey protein can be a suitable addition to a ketogenic diet, depending on the specific product and how it fits into your daily macronutrient goals. The ketogenic diet emphasizes high fat, moderate protein, and very low carbohydrate intake.
Keto-Friendliness of Whey Protein
• Carbohydrate Content: To maintain ketosis, it's crucial to limit carbs. Some whey protein powders, especially those that are isolates, are very low in carbohydrates, typically containing less than 1 gram per serving, making them an excellent choice for keto dieters1.
• Fat Content: While whey protein does not naturally contain much fat, some keto-specific protein powders might include added fats from sources like MCTs (medium-chain triglycerides) to align more closely with keto macronutrient ratios2.
• Protein Levels: Moderate protein consumption is vital on a keto diet to prevent muscle loss without knocking you out of ketosis. Whey protein is effective because it provides high-quality protein that can help meet these needs without exceeding them3.
Choosing the Right Whey Protein for Keto
• Check the Label: Look for whey protein isolates rather than concentrates as they typically contain fewer carbohydrates.
• Avoid Added Sugars: Ensure the whey protein powder does not contain added sugars or high-carb fillers, which can disrupt ketosis.
• Consider Your Daily Macros: Incorporate whey protein into your overall daily macronutrient goals. It's important to balance your intake of fats, proteins, and carbs to stay within ketogenic guidelines.
Overall, whey protein can be part of a ketogenic diet when chosen carefully and consumed as part of a well-planned keto eating strategy. As always, monitor your body's response and adjust your diet accordingly to maintain ketosis and achieve your dietary goals.
References:
1. Volek, J. S., & Phinney, S. D. (2012). The Art and Science of Low Carbohydrate Performance. Atria Books. Information on how dietary proteins influence ketosis and athletic performance.
2. Stubbs, B. J., & Cox, P. J. (2015). Metabolic effects of exogenous ketone supplementation – An alternative or adjuvant to the ketogenic diet as a cancer therapy? Journal of Nutrition & Metabolism, 12, 35.
3. Paoli, A., Rubini, A., Volek, J. S., & Grimaldi, K. A. (2013). Beyond weight loss: a review of the therapeutic uses of very-low-carbohydrate (ketogenic) diets. European Journal of Clinical Nutrition, 67(8), 789.
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ATOMIC STRUCTURE
< Early Atomic models Atomic structure: TOC Hydrogen atomic spectrum >
NATURE OF LIGHT & QUANTUM THEORY
The early theories describing the atomic structure are based on classical physics. However these theories could not explain the behavior of atom completely. The modern view of atomic structure is based on quantum theory introduced by Max Planck.
Before learning the quantum theory, it is necessary to understand the nature of light.
LIGHT
Light is considered as an electromagnetic radiation. It consists of two components i.e., the electric component and the magnetic component which oscillate perpendicular to each other as well as to the direction of path of radiation.
electromagnetic radiation representation
The electromagnetic radiations are produced by the vibrations of a charged particle. The properties of light can be explained by considering it as either wave or particle as follows (dual nature).
WAVE NATURE OF LIGHT
According to the wave theory proposed by Christiaan Huygens, light is considered to be emitted as a series of waves in all directions. The following properties can be defined for light by considering the wave nature.
Wavelength (λ): The distance between two successive similar points on a wave is called as wavelength. It is denoted by λ.
Units: cm, Angstroms (Ao), nano meters (nm), milli microns (mµ) etc.,
Note:
1 Ao = 10-8 cm.
1 nm= 10-9m = 10-7cm
Frequency (ν): The number of vibrations done by a particle in unit time is called frequency. It is denoted by 'ν'.
Units: cycles per second = Hertz = sec-1
Velocity (c): Velocity is defined as the distance covered by the wave in unit time. It is denoted by 'c'.
Velocity of light = c = 3.0 x 108 m.sec-1 = 3.0 x 1010 cm.sec-1
Note: For all types of electromagnetic radiations, the velocity is a constant value. The relation between velocity (c), wavelength (λ) and frequency (ν) can be given by following equation.
velocity = frequency x wavelength
c = νλ
Wave number (): The number of waves spread in a length of one centimeter is called wave number. It is denoted by . It is the reciprocal of wavelength, λ.
units: cm-1, m-1
Amplitude: The distance from the midline to the peak or the trough is called amplitude of the wave. It is usually denoted by 'A' (a variable). Amplitude is a measure of the intensity or brightness of light radiation.
PARTICLE NATURE OF LIGHT
Though most of the properties of light can be understood by considering it as a wave, some of the properties of light can only be explained by using particle (corpuscular) nature of it. Newton considered light to possess particle nature. In the year 1900, in order to explain black body radiations, Max Planck proposed Quantum theory by considering light to possess particle nature.
PLANCK'S QUANTUM THEORY
Black body: The object which absorbs and emits the radiation of energy completely is called a black body. Practically it is not possible to construct a perfect black body. But a hollow metallic sphere coated inside with platinum black with a small aperture in its wall can act as a near black body. When the black body is heated to high temperatures, it emits radiations of different wavelengths.
The following curves are obtained when the intensity of radiations are plotted against the wavelengths, at different temperatures.
Following are the conclusions that can be drawn from above graphs.
1) At a given temperature, the intensity of radiation increases with wavelength and reaches a maximum value and then starts decreasing.
2) With increase in temperature, the wavelength of maximum intensity (λmax) shifts towards lower wavelengths. According to classical physics, energy should be emitted continuously and the intensity should increase with increase in temperature. The curves should be as shown by dotted line.
In order to explain above experimental observations Max Planck proposed the following theory.
Quantum theory:
1) Energy is emitted due to vibrations of charged particles in the black body.
2) The radiation of energy is emitted or absorbed discontinuously in the form of small discrete energy packets called quanta
3) Each quantum is associated with definite amount of energy which is given by the equation E=hν.
Where
h = planck's constant = 6.625 x 10-34 J sec = 6.625 x10-27 erg sec
ν= frequency of radiation
4) The total energy of radiation is quantized i.e., the total energy is an integral multiple of hν. It can only have the values of 1 hν or 2 hν or 3 hν. It cannot be the fractional multiple of hν.
5) Energy is emitted and absorbed in the form of quanta but propagated in the form of waves.
EINSTEIN'S GENERALIZATION OF QUANTUM THEORY
Einstein generalized the quantum theory by applying it to all types of electromagnetic radiations. He explained photoelectric effect using this theory.
Photoelectric Effect: The ejection of electrons from the surface of a metal, when the metal is exposed to light of certain minimum frequency, is called photoelectric effect
The frequency of light should be equal or greater than a certain minimum value characteristic of the metal. This is called threshold frequency, νo
The photoelectric effect cannot be explained by considering the light as wave. Einstein explained photoelectric effect by applying quantum theory as follows:
1. All electromagnetic radiations consists of small discrete energy packets called photons. These photons are associated with definite amount of energy given by the equation E=hν.
2. Energy is emitted, absorbed as well as propagated in the form of photons only.
3. The electron is ejected from the metal, only when a photon of sufficient energy strikes the electron. When a photon strikes the electron, some part of the energy of photon is used to free the electron from the attractive forces in the metal atom and the remaining part is converted into kinetic energy.
hν = W + K.E
Where
W = energy required to overcome the attractions
K.E = kinetic energy of the electron
Since the frequency corresponding to the minimum energy required to overcome the attraction is called threshold frequency, νo, the above equation can be written as:
hν = hνo + K.E
or
K.E = hνo- hν = h(νo- ν)
< Early Atomic models Atomic structure: TOC Hydrogen atomic spectrum >
Author: Aditya vardhan Vutturi
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Satellites Orbiting Earth
How a Satellite Works
Satellites are very complex machines that require precise mathematical calculations in order for them to function. The satellite has tracking systems and very sophisticated computer systems on board. Accuracy in orbit and speed are required for the satellite to keep from crashing back down to Earth. There are several different types of orbits that the satellite can take. Some orbits are stationary and some are elliptical.”Satellite Orbit”
Low Earth Orbit
A satellite is in “Low Earth Orbit” when it circles in an elliptical orbit close to Earth. Satellites in low orbit are just hundreds of miles away. These satellites travel at high speeds preventing gravity from pulling them back to Earth. Low Orbit Satellites travel approximately 17,000 miles per hour and circle the Earth in an hour and a half.
Polar Orbit
This is how a satellite travels in a polar orbit.
This is how a satellite travels in a polar orbit. These orbits eventually pass the entire surface of the Earth.
Polar Orbiting Satellites circle the planet in a north-south direction as Earth spins beneath it in an east-west direction. Polar Orbits enable satellites to scan the entire surface of the Earth. Like pealing an orange peal in a circular motion from top to bottom. Remote sensing satellites, weather satellites, and government satellites are almost always in polar orbit because of the coverage. Polar orbits cover the Earth’s surface thoroughly. The polar obit occupied by a satellite has a constant location in which it passes over. ALL POLAR ORBITING SATELLITES INTERSECT The North Pole at their same point. While one Polar orbit satellite is over America, another Polar Satellite is passing over the North Pole. So the North Pole has a constant flow of UHF and higher microwaves hitting it. The illustration shows that the common passing point for Polar Orbiting Satellites is over the North Pole.
A polar orbiting satellite will pass over the Earths equator at a different longitude on each of its orbits; however, Polar Orbiting satellites pass over the North Pole every time. Polar orbits are often used for earth mapping, earth observation, weather satellites, and reconnaissance satellites. This orbit has a disadvantage. No one spot of the Earth’s surface can be sensed continuously from a satellite in a polar orbit.
This is from U.S. Army Information Systems Engineering Command.
“In order to fulfill the military need for protected communication service, especially low probability of intercept/detection (LPI/LPD), to units operating north of 65 degree northern latitude, the space communications architecture includes the polar satellite system capability. An acceptable approach to achieving this goal is to fly a low capacity EHF system in a highly elliptical orbit, either as a hosted payload or as a “free-flyer,” to provide service during a transition period, nominally 1997-2010. A single, hosted EHF payload is already planned. Providing this service 24 hours-a-day requires a two satellite constellation at high earth orbit (HEO). Beyond 2010, the LPI/LPD polar service could continue to be provided by a high elliptical orbit HEO EHF payload, or by the future UHF systems.” (quote from www.fas.org)
THERE IS A CONSTANT 24 HOUR EHF AND HIGHER MICROWAVE TRANSMISSION PASSING OVER THE NORTH POLE!
“Geo Synchronous” Orbit
This is how a satellite travels in a Equitorial orbit
This is how a satellite travels in a “Geo Synchronous” orbit. Equatorial orbits are also called “Geostationary”. These satellites follow the rotation of the Earth.
A satellite in a “Geo Synchronous” orbit hovers over one spot and follows the Earths spin along the equator. Go to this link for more information on “Geo synchronous Orbits”. Earth takes 24 hours to spin on its axis. In the illustration you can see that an “Geo Synchronous” Orbit follows the equator and never covers the North or South Poles. The footprints of “Geo Synchronous” orbiting satellites do not cover the polar regions, so communication satellites in “Geo Synchronous” orbits in cannot be accessed in the northern and southern polar regions.
Because the “Geo Synchronous” satellite does not move from the area that it covers, these satellites are used for telecommunications, gps trackers, television broadcasting, government, and internet. Because they are stationary, their orbits are much farther from the Earth than the Polar orbiting satellites. If a stationary satellite is too close to the Earth, it will crash back down at a faster rate. They say there are about 300 “Geo Synchronous” satellites in orbit right now. Of course, these are the satellites that the public is allowed to know about, that are not governmentally classified.
Satellite Anatomy
This is the Atatomy of a Satellite.
This is the Anatomy of a Satellite.
A satellite is made up of several instruments that work together to operate the satellite during its mission. This illustration to the left demonstrates the parts of a satellite.
The command and data system controls all of the satellite functions. This is a very complex computer system that communicates all of the satellite flight operations, where the satellite points, and any other mathematical operations.
The Pointing control directs the satellite in order for the satellite to keep a steady flight path. This system is a complex sensor instrument that keeps the satellite pointing in the same direction. The satellite uses a propulsion system called “momentum wheels” that adjusts the position of the satellite into its proper place. Scientific observation satellites have more precise propulsion systems than do communications satellites.
The Communications system has a transmitter, a receiver, and various antennas to transmit data to the Earth . On Earth, Ground control sends instructions and data to the satellite’s computer through the Antenna. Pictures, data, television, radio, and many other data is sent by the satellite back to practically everyone on Earth.
The Power system needed power and operate the satellite is an efficient solar panel array that obtains energy from the Sun’s rays. Solar arrays make electricity from the sunlight and store the electricity in rechargeable batteries.
The Payload is what a satellite needs to perform its job. A weather satellite would have a payload that consist of an Image sensor, digital camera, telescope, and other thermal and weather sensing devices.
The Thermal Control is the protection required to prevent damage to the satellite’s instrumentation and components in. Satellite are exposed to extreme temperature changes. Temperatures range from 120 degrees below zero to 180 degrees above zero. Heat distribution units and thermal blankets to protect the electronics and components from temperature damage.
Satellite Footprints
A single satellite footprint
Here you can see one footprint covers an enormous area.
Geostationary satellites have a very broad view of Earth. The footprint of one Echo Starbroadcast satellite covers almost all of North America. They stay over the Earth at same the same location so we always know where they are. Direct contact with the satellite can be made because Equatorial Satellites are fixed.
Many communications satellites travel in Equatorial orbits, including those that relay TV signals into our homes; However, the size of the footprint of one satellite covers the entire Northern America.
The multi path effect that occurs when satellite transmissions are obstructed by topographical entities also provides insight on microwave global warming. Microwaves are being bombarded upon our planet. Our planet absorbs and obstructs the waves from space. Microwaves penetrate through all of our atmosphere and bounce and echo off of the Earth. Imagine the footprint overlaps that are being produced by the thousands of satellites in orbit right now?
coverage 8 pic
Here you can see the footprint overlapping the that satellites make. Each satellite covers an enormous area.
The closer the satellite is to something the more power will be exerted on the object. The farther the waves have to go the less power they will have. Because the atmosphere is so much closer to the satellite, there is a stronger beam of energy going through the clouds and atmosphere. This stronger power causes a higher rate of warming in the atmosphere than it does on the surface of the Earth.
The illustration to the right shows how eight satellites microwave an enormous part of our Earth. When the radio signals reflect off of surrounding terrain; buildings, canyon walls, hard ground multi path issues occur due to multiple waves doubling over themselves. These delayed signals can cause poor signals. Ultimately, the water, ice, and Earth are absorbing and reflecting microwaves in many different directions. Microwaves passing through Earths atmospheres are causing radio frequency heating at the molecular level.
System spectral efficiency
“In wireless networks, the system spectral efficiency is a measure of the quantity of users or services that can be simultaneously supported by a limited radio frequency bandwidth in a defined geographic area.” The capacity of a wireless network can be measured by calculating the maximum simultaneous phone calls over 1 MHz frequency spectrum. This is measured in Erlangs//MHz/cell, Erlangs/MHz/sector, Erlangs/MHz/site, or Erlangs/MHz/km measurements. Modern day cell phones take advantage of this type of transmission. These cell phones transmit a microwave transmission that is twice the frequency of a microwave oven in your home.
This is a misconception of how microwave frequencies travel.
This is a misconception of how microwave frequencies travel.
An example of a spectral efficiency can be found in the satellite RADARSAT-1. In 1995 RADARSAT-1, an Earth observation satellite from Canada, was launched in an orbit above the Earth. RADRASAT-1 provides images of the Earth, scientific and commercial, used in agriculture, geology, hydrology, arctic surveillance, oceanography, cartography, ice and ocean monitoring, forestry, detecting ocean oil slicks, and many other applications. This satellite uses continuous high microwave transmissions. A Synthetic Aperture Radar (SAR) system is a type of sensor that images the Earth at a single microwave frequency of 5.3 GHz. SAR systems transmit microwaves towards the surface of the Earthy and record the reflections from the surface. This satellite can image the Earth during any time and in any atmospheric condition.
This is how microwave frequencies travel
This is how microwave frequencies actually travel.
A Common misconception about microwave transmissions is that the transmission is directly beaming straight into the receiving antennae. (See misconception illustration) This however, is not true. Transmissions are spread into the air in a spherical direction. The waves travel in every direction until they find a receiver or some dielectric material to pass into.
When a microwave transmission is sent to a receiving satellite dish the transmission is sent in a spherical direction. (See how microwaves travel illustration) The signal passes through all parts of that sphere until it finds a connection. All microwaves, not received by an antennae, pass through the dielectric material in the earth. Dielectric material is primarily water and ice.
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The Celestial Sphere
Humans perceive in Euclidean space -> straight lines and planes. But, when distances are not visible (i.e. very large) than the apparent shape that the mind draws is a sphere -> thus, we use a spherical coordinate system for mapping the sky with the additional advantage that we can project Earth reference points (i.e. North Pole, South Pole, equator) onto the sky. Note: the sky is not really a sphere!
From the Earth’s surface we envision a hemisphere and mark the compass points on the horizon. The circle that passes through the south point, north point and the point directly over head (zenith) is called the meridian.
This system allows one to indicate any position in the sky by two reference points, the time from the meridian and the angle from the horizon. Of course, since the Earth rotates, your coordinates will change after a few minutes.
The horizontal coordinate system (commonly referred to as the alt-az system) is the simplest coordinate system as it is based on the observer’s horizon. The celestial hemisphere viewed by an observer on the Earth is shown in the figure below. The great circle through the zenith Z and the north celestial pole P cuts the horizon NESYW at the north point (N) and the south point (S). The great circle WZE at right angles to the great circle NPZS cuts the horizon at the west point (W) and the east point (E). The arcs ZN, ZW, ZY, etc, are known as verticals.
The two numbers which specify the position of a star, X, in this system are the azimuth, A, and the altitude, a. The altitude of X is the angle measured along the vertical circle through X from the horizon at Y to X. It is measured in degrees. An often-used alternative to altitude is the zenith distance, z, of X, indicated by ZX. Clearly, z = 90 – a. Azimuth may be defined in a number of ways. For the purposes of this course, azimuth will be defined as the angle between the vertical through the north point and the vertical through the star at X, measured eastwards from the north point along the horizon from 0 to 360°. This definition applies to observers in both the northern and the southern hemispheres.
It is often useful to know how high a star is above the horizon and in what direction it can be found – this is the main advantage of the alt-az system. The main disadvantage of the alt-az system is that it is a local coordinate system – i.e. two observers at different points on the Earth’s surface will measure different altitudes and azimuths for the same star at the same time. In addition, an observer will find that the star’s alt-az coordinates changes with time as the celestial sphere appears to rotate.
Celestial Sphere:
To determine the positions of stars and planets on the sky in an absolute sense, we project the Earth’s spherical surface onto the sky, called the celestial sphere.
The celestial sphere has a north and south celestial pole as well as a celestial equator which are projected reference points to the same positions on the Earth surface. Right Ascension and Declination serve as an absolute coordinate system fixed on the sky, rather than a relative system like the zenith/horizon system. Right Ascension is the equivalent of longitude, only measured in hours, minutes and seconds (since the Earth rotates in the same units). Declination is the equivalent of latitude measured in degrees from the celestial equator (0 to 90). Any point of the celestial (i.e. the position of a star or planet) can be referenced with a unique Right Ascension and Declination.
The celestial sphere has a north and south celestial pole as well as a celestial equator which are projected from reference points from the Earth surface. Since the Earth turns on its axis once every 24 hours, the stars trace arcs through the sky parallel to the celestial equator. The appearance of this motion will vary depending on where you are located on the Earth’s surface.
Note that the daily rotation of the Earth causes each star and planet to make a daily circular path around the north celestial pole referred to as the diurnal motion.
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Chiropractic and Bedwetting
Aug 18, 2022
Chiropractic
Welcome to McKinnon Marie, your trusted source for alternative and natural medicine. In this article, we will explore how chiropractic care can help with bedwetting issues, providing you with a comprehensive understanding of the topic.
Understanding Bedwetting
Bedwetting, medically known as nocturnal enuresis, is a common issue, especially in children. It refers to involuntary urination during sleep. While it can be distressing for both children and parents, it is important to remember that bedwetting is typically a developmental phase that most children outgrow with time.
How Chiropractic Care Can Help
Chiropractic care offers a holistic approach to addressing bedwetting concerns. By focusing on the nervous system and overall spinal health, chiropractors can identify and correct any potential underlying issues contributing to the problem.
The Spine and Nervous System Connection
The spine plays a crucial role in the proper functioning of the nervous system. Misalignments or subluxations in the spine can disrupt communication between the brain and other parts of the body, potentially affecting bladder control. Chiropractic adjustments aim to realign the spine, allowing for optimal nerve function and restoring balance to the body.
Reducing Interference and Restoring Balance
Chiropractors use gentle, non-invasive techniques to address spinal misalignments. These adjustments promote proper nerve flow, enhancing the function of the bladder and reducing bedwetting episodes. By restoring balance to the body, chiropractic care provides a natural and drug-free solution for individuals struggling with bedwetting.
The Benefits of Chiropractic Care for Bedwetting
Choosing chiropractic care for bedwetting offers several advantages:
• Non-Invasive: Chiropractic adjustments are gentle and non-invasive, making them a safe option for children and adults alike.
• Addressing Underlying Causes: Chiropractors focus on identifying and resolving the root cause of bedwetting, rather than just treating the symptoms.
• Drug-Free Solution: Chiropractic care provides a natural alternative to medication, reducing the need for pharmaceutical intervention.
• Improving Overall Well-being: Through spinal adjustments, chiropractic care promotes overall health and well-being, supporting the body’s ability to function optimally.
• Complementary Approach: Chiropractic care can be used alongside other treatments or therapies, enhancing their effectiveness.
Consult with Our Experts at McKinnon Marie
At McKinnon Marie, we take a patient-centered approach to alternative and natural medicine. Our experienced chiropractors specialize in addressing bedwetting concerns and providing comprehensive care.
If you or your child are experiencing bedwetting issues, we invite you to schedule a consultation with our team. Our chiropractors will assess your specific situation, develop a personalized treatment plan, and guide you on a journey towards improved well-being.
Trust McKinnon Marie for exceptional alternative and natural medicine solutions. Contact us today to learn more about chiropractic care for bedwetting.
Jayson Jeffries
Great info on bedwetting!
Nov 8, 2023
Mike Dempsey
I never knew chiropractic care could be related to bedwetting. It's fascinating to learn about these potential connections.
Aug 4, 2023
Steven Gerhardt
I've always been curious about alternative medicine. This article provides valuable insight into the potential benefits of chiropractic care for bedwetting.
Jul 4, 2023
Tim Hopper
I appreciate the detailed explanation of how chiropractic care can provide relief for bedwetting. It's an eye-opening read.
Apr 7, 2023
Jason Threat
As a parent, I'm eager to learn about non-invasive solutions for bedwetting. This article offers valuable information.
Mar 20, 2023
Jeff Emmot
The potential link between chiropractic care and bedwetting is intriguing. I'm eager to learn more about this connection.
Feb 10, 2023
Mecca Robbins
Interesting read. I appreciate the comprehensive explanation of how chiropractic care can potentially help with bedwetting.
Jan 23, 2023
Xavier Luna
Thanks for shedding light on this issue. It's important to explore alternative treatments for bedwetting.
Jan 22, 2023
Kim Guinn
I've heard about the effectiveness of chiropractic care for various issues, so it's great to see it explored in the context of bedwetting.
Dec 4, 2022
Al Venzon
It's great to see alternative medicine being explored for common issues like bedwetting. Thanks for bringing attention to this topic.
Oct 21, 2022
Jamie Lowe
This is a fascinating topic! I've never considered the connection between chiropractic care and bedwetting before.
Oct 13, 2022
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If you're familiar with the fitness scene, you may have heard of high-intensity metcon, which promises a faster metabolism, weight loss, and muscle growth packaged into a short but efficient workout. These workouts differ from traditional, more familiar forms of exercise, like cardio and weightlifting, as they pack many techniques into one intense, condensed workout.
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The term "metcon" stands for "metabolic conditioning", a type of training that has gained popularity in the last few years, primarily due to CrossFit-style workouts. These workouts tax the body to burn fat, build muscle, and increase endurance.
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Types of Metcon Workouts
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Metcon and High-Intensity Interval Training (HIIT) are often mentioned together since they both use a circuit method. However, metcon alternates both high-intensity and lower-intensity exercises to tax and condition the body's metabolism, while HIIT workouts use timed intervals of high-intensity exercise only.
However, standard HIIT intervals can be combined with metcon circuits to facilitate extra conditioning.
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Metcon vs. Weightlifting
Overall, metcon workouts are more intense and burn more calories than traditional weightlifting but are more taxing. Weightlifting focuses on completing a set number of lift repetitions at a particular weight, resting between moves. Though metcon often incorporates weightlifting moves, its emphasis on quick movement between exercises and including lower-intensity exercises to eliminate rest separates it from weightlifting.
You can add metcon circuits at the end of a weightlifting workout to add cardiovascular and metabolic conditioning to a workout that is otherwise purely focused on strength.
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Experienced gym-goers and weightlifters are prime candidates for metcon workouts, as they'll recognize familiar moves within the fast-paced circuit. Incorporating metcon circuits into your regimen can help you build more muscle or lose more weight than standard exercise routines alone if you're already familiar with the gym or have your own equipment.
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Who Shouldn't Try Metcon
Metcon workouts aren't for beginners. If you're new to exercise, metcon isn't a great place to start. Start slow to build your exercise tolerance and familiarity with the exercises — specifically the correct form. Consider beginning with high-intensity interval training before transitioning into metcon.
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This site offers information designed for educational purposes only. You should not rely on any information on this site as a substitute for professional medical advice, diagnosis, treatment, or as a substitute for, professional counseling care, advice, diagnosis, or treatment. If you have any concerns or questions about your health, you should always consult with a physician or other healthcare professional.
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use strict; use Win32::OLE qw(in); use HTML::Template; { package Wrapper::Notes::Template; use strict; use File::Spec; sub new { my ($class,$document, $attachmentdir) = @_; my $self = { document => $document, attachments => $attachmentdir }; bless $self, $class; $self; }; sub document { $_[0]->{document} }; sub param { my ($self,@args) = @_; if (scalar @args) { my $result; if ($_[1] eq 'Attachments') { my $result = []; my $body = $self->document->GetFirstItem('Body'); my @attachments = grep { warn join ":",$_->{Name}, $_->{Type},$_->{Text}; $_->{Type} == 4 } (@{$self->document->Items()}); mkdir $self->{attachments}; for my $attname (@attachments) { my $url = File::Spec->catfile($self->{attachments},$attname); $url = File::Spec->rel2abs($url); #warn "Extracting $attname to $url"; my $f = $self->document->getAttachment($attname); if ($f) { $f->extractFile($url); push @$result, { name => $attname, url => $url }; }; }; return $result; } elsif ($_[1] eq 'EmbeddedObjects') { my $result = []; my $body = $self->document->GetFirstItem('Body'); my $attachments = $body->EmbeddedObjects; if ($attachments) { mkdir $self->{attachments}; for my $att (Win32::OLE::in $attachments) { warn $att->{Type}; my $url = File::Spec->catfile($self->{attachments},$att->{Name}); $url = File::Spec->rel2abs($url); $att->extractFile($url); push @$result, { name => $att->{Name}, url => $url }; }; }; return $result; } else { $result = $self->document->{$_[1]}; }; if (ref $result) { return [ map { "value" => $_ }, @$result ]; } else { $result; }; } else { return (map { $_->Name } (Win32::OLE::in ($self->document->Items()))), "Attachments", "EmbeddedObjects"; }; }; }; my ($server,$database) = ('server','mail/corion.nsf'); my $Notes = Win32::OLE->new('Notes.NotesSession') or die "Cannot start Lotus Notes Session object.\n"; my ($Version) = ($Notes->{NotesVersion} =~ /\s*(.*\S)\s*$/); print "The current user is $Notes->{UserName}.\n"; print "Running Notes \"$Version\" on \"$Notes->{Platform}\".\n"; my $Database = $Notes->GetDatabase($server, $database); my $AllDocuments = $Database->AllDocuments; my $Count = $AllDocuments->Count; print "There are $Count documents in the database.\n"; my $Index = 4419; while (++$Index <= $Count) { my $Document = $AllDocuments->GetNthDocument($Index); my $wrapper = Wrapper::Notes::Template->new($Document,sprintf "email/mail.%05g",$Index); my $template = HTML::Template->new( filename => 'lotus-email.tmpl', die_on_bad_params => 0, loop_context_vars => 1, associate => [ $wrapper ], case_sensitive => 1, ); my $outfile = sprintf "email/mail.%05g.html", $Index; open MAIL, ">", $outfile or die "Couldn't create '$outfile' : $!\n"; $template->output( print_to => *MAIL ); close MAIL; last unless $Index <= 4420; # magic number! }
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Accelerator Physics
Second Edition
I
Accelerator Physic
Second E d i t i o n S. Y. Lee
Department of Physics, Indiana University
\jjjp World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • BANGALOf
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
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ACCELERATOR PHYSICS (Second Edition) Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-256-182-X 981-256-200-1 (pbk)
To the memory of my parents
Preface
Since the appearance of the first edition in 1999, this book has been used as a textbook or reference for graduate-level "Accelerator Physics" courses. I have benefited from questions, criticism and suggestions from colleagues and students. As a response to these suggestions, the revised edition is intended to provide easier learning explanations and illustrations. Accelerator Physics studies the interaction between the charged particles and electromagnetic field. The applications of accelerators include all branches of sciences and technologies, medical treatment, and industrial processing. Accelerator scientists invent many innovative technologies to produce beams with qualities required for each application. This textbook is intended for graduate students who have completed their graduate core-courses including classical mechanics, electrodynamics, quantum mechanics, and statistical mechanics. I have tried to emphasize the fundamental physics behind each innovative idea with least amount of mathematical complication. The textbook may also be used by undergraduate seniors who have completed courses on classical mechanics and electromagnetism. For beginners in accelerator physics, one begins with Sees. 2.1-2.4 in Chapter 2, and follows by Sees. 3.1-3.2 for the basic betatron and synchrotron motion. The study continues onto Sees. 2.5, 2.8, and 3.7 for chromatic aberration and collective beam instabilities. After these basic topics, the rf technology and basic physics of linac are covered in Sees. 3.5, 3.6, 3.8 in Chapter 3. The basic accelerator physics course ends with physics of electron storage rings in Chapter 4. I have chosen the Frenet-Serret coordinate-system of (x, s, z) for the transverse radially-outward, longitudinally-forward, and vertical unit base-vectors with the righthand rule: z = x x I. I have also chosen positive-charge to derive the equations of betatron motion for all sections of the Chapter 2, except a discussion of ±-signs in Eq. (2.29). The sign of some terms in Hill's equation should be reversed if you solve the equation of motion for electrons in accelerators. The convention of the rf-phase differs in linac and synchrotron communities by 4>\mac = ^synchrotron — (TT/2) • To be consistent with the synchrotron motion in Chapter 3, I have chosen the rf-phase convention of the synchrotron community to describe the synchrotron equation of motion for linac in Sec. 3.8. In this revised edition, I include two special topics: free electron laser (FEL) vii
viii
PREFACE
and beam-beam interaction in Chapter 5. In 2000, several self-amplified spontaneous emission (SASE) FEL experiments have been successfully demonstrated. Many light source laboratories are proposing the fourth generation light source using high gain FEL based on the concept of SASE and high-gain harmonic generation (HGHG). Similarly, the success of high luminosity B-factories indicates that beam-beam interaction remains very important to the basic accelerator physics. These activities justify the addition of two introductory topics to the accelerator physics text. Finally, the homework is designed to solve a particular problem by providing step-by-step procedures to minimize frustrations. The answer is usually listed at the end of each homework problem so that the result can be used in practical design of accelerator systems. I would appreciate very much to receive comments and criticism to this revised edition. S.Y. Lee Bloomington, Indiana, U.S.A. November, 2004
Preface to the first edition
The development of high energy accelerators began in 1911 when Rutherford discovered the atomic nuclei inside the atom. Since then, high voltage DC and rf accelerators have been developed, high-field magnets with excellent field quality have been achieved, transverse and longitudinal beam focusing principles have been discovered, high power rf sources have been invented, high vacuum technology has been improved, high brightness (polarized/unpolarized) electron/ion sources have been attained, and beam dynamics and beam manipulation schemes such as beam injection, accumulation, slow and fast extraction, beam damping and beam cooling, instability feedback, etc. have been advanced. The impacts of the accelerator development are evidenced by many ground-breaking discoveries in particle and nuclear physics, atomic and molecular physics, condensed-matter physics, biomedical physics, medicine, biology, and industrial processing. Accelerator physics and technology is an evolving branch of science. As the technology progresses, research in the physics of beams propels advancement in accelerator performance. The advancement in type II superconducting material led to the development of high-field magnets. The invention of the collider concept initiated research and development in single and multi-particle beam dynamics. Accelerator development has been impressive. High energy was measured in MeV's in the 1930's, GeV's in the 1950's, and multi-TeV's in the 1990's. In the coming decades, the center of mass energy will reach 10-100 TeV. High intensity was 109 particles per pulse in the 1950's. Now, the AGS has achieved 6 x 1013 protons per pulse. We are looking for 1014 protons per bunch for many applications. The brilliance of synchrotron radiation was about 1012 [photons/s mm2 mrad2 0.1% (AA/A)] from the first-generation light sources in the 1970's. Now, it reaches 1021, and efforts are being made to reach a brilliance of 1029 - 1034 in many FEL research projects. This textbook deals with basic accelerator physics. It is based on my lecture notes for the accelerator physics graduate course at Indiana University and two courses in the U.S. Particle Accelerator School. It has been used as preparatory course material for graduate accelerator physics students doing thesis research at Indiana University. The book has four chapters. The first describes historical accelerator development. The second deals with transverse betatron motion. The third chapter concerns synchrotron motion and provides an introduction to linear accelerators. The fourth deals with synchrotron radiation phenomena and the basic design principles
ix
x
PREFACE TO THE FIRST EDITION
of low-emittance electron storage rings. Since this is a textbook on basic accelerator physics, topics such as nonlinear beam dynamics, collective beam instabilities, etc., are mentioned only briefly, in Chapters 2 and 3. Attention is paid to deriving the action-angle variables of the phase space coordinates because the transformation is basic and the concept is important in understanding the phenomena of collective instability and nonlinear beam dynamics. In the design of synchrotrons, the dispersion function plays an important role in particle stability, beam performance, and beam transport. An extensive section on the dispersion function is provided in Chapter 2. This function is also important in the design of low-emittance electron storage ring lattices. The SI units are used throughout this book. I have also chosen the engineer's convention of j = —i for the imaginary number. The exercises in each section are designed to have the student apply a specific technique in solving an accelerator physics problem. By following the steps provided in the homework, each exercise can be easily solved. The field of accelerator physics and technology is multi-disciplinary. Many related subjects are not extensively discussed in this book: linear accelerators, induction linacs, high brightness beams, collective instabilities, nonlinear dynamics, beam cooling physics and technology, linear collider physics, free-electron lasers, electron and ion sources, neutron spallation sources, muon colliders, high intensity beams, vacuum technology, superconductivity, magnet technology, instrumentation, etc. Nevertheless, the book should provide the understanding of basic accelerator physics that is indispensable in accelerator physics and technology research. S.Y. Lee Bloomington, Indiana, U.S.A. January, 1998
2 Electrostatic Accelerators I. 1 Transfer Matrix and Stability of Betatron Motion II.3 Floquet Transformation xi vii ix 1 4 5 6 6 9 17 18 19 19 20 22 23 23 24 24 24 35 36 37 39 41 41 42 47 47 51 52 .2 Courant-Snyder Parametrization II.4 Radio-Frequency (RF) Accelerators I.1 High Energy and Nuclear Physics III.2 Solid-State and Condensed-Matter Physics III.1 Natural Accelerators I.1 Acceleration Cavities II.Contents Preface Preface to the first edition 1 Introduction I Historical Developments I.3 Other Applications Exercise 2 Transverse Motion I Hamiltonian for Particle Motion in Accelerators I.3 Other Important Components III Accelerator Applications III.3 Induction Accelerators I.4 Particle Motion in Dipole and Quadrupole Magnets Exercise II Linear Betatron Motion II.3 Equation of Betatron Motion I.2 Accelerator Magnets II.5 Colliders and Storage Rings I.2 Magnetic Field in Frenet-Serret Coordinate System I.1 Hamiltonian in Frenet-Serret Coordinate System I.6 Synchrotron Radiation Storage Rings II Layout and Components of Accelerators II.
9 Mechanisms of emittance dilution and diffusion Exercise Off-Momentum Orbit IV. Action. Flexible momentum compaction (FMC) lattices C.7 Experimental Measurements of Dispersion Function IV.8 Beam Injection and Extraction III. 7T jump schemes B.4 Lattice Design Strategy Exercise Linear Coupling VI.4 Dispersion Suppression and Dispersion Matching IV.3 Experimental Measurement of Linear Coupling CONTENTS 57 60 65 66 67 73 85 85 91 92 101 105 108 110 115 117 121 129 129 133 136 139 141 143 145 146 146 149 155 156 157 161 172 173 178 178 183 184 186 186 189 193 III IV V VI . FMC in double-bend (DB) lattices IV.3 Application of Dipole Field Error III.5 Courant-Snyder Invariant and Emittance II.1 Closed-Orbit Distortion due to Dipole Field Errors III.1 Dispersion Function IV.9 Minimum {H) Modules Exercise Chromatic Aberration V.7 Transverse Spectra III.I Chromaticity Measurement and Correction V.2 %-Function.4 Action-Angle Variable and Floquet Transformation II. and Integral Representation IV.6 Transport Notation IV.2 Effects of an isolated Linear Coupling Resonance VI.2 Extended Matrix Method for the Closed Orbit III.2 Nonlinear Effects of Chromatic Sextupoles V. Other similar FMC modules D.6 Stability of Betatron Motion: A FODO Cell Example II.5 Achromat Transport Systems IV.3 Chromatic Aberration and Correction V.6 Application of quadrupole field error III.5 Basic Beam Observation of Transverse Motion III.3 Momentum Compaction Factor IV.8 Transition Energy Manipulation A.4 Quadrupole Field (Gradient) Errors III.8 Effect of Space-Charge Force on Betatron Motion Exercise Effect of Linear Magnet Imperfections III.1 The Linear Coupling Hamiltonian VI.xii II.7 Symplectic Condition II.
5 Summary of Synchrotron Equations of Motion Exercise II Adiabatic Synchrotron Motion II.2 RF Phase Modulation and Parametric Resonances III.1 Normalized Phase-Space Coordinates III.4 Frequency Spread and Landau Damping Exercise IX Synchro-Betatron Hamiltonian Exercise 3 Synchrotron Motion I Longitudinal Equation of Motion I.4 Small-Amplitude Synchrotron Motion at the UFP II.5 RF Voltage Modulation III.3 Measurements of Synchrotron Phase Modulation III.6 Measurement of RF Voltage Modulation Exercise xiii 196 197 197 202 202 209 211 212 213 216 216 220 221 225 228 232 237 239 240 244 245 246 247 248 249 251 252 253 255 258 259 261 263 268 268 271 277 280 288 295 297 .2 Bucket Area II. 1 Nonlinear Resonances Driven by Sextupoles VII.3 Small-Amplitude Oscillations and Bunch Area II.1 Fixed Points II.1 Impedance VIII.1 The Synchrotron Hamiltonian I.4 Betatron Tunes and Nonlinear Resonances Exercise VIII Collective Instabilities and Landau Damping VIII.5 Linear Coupling Using Transfer Matrix Formalism Exercise VII Nonlinear Resonances VII.CONTENTS VI.4 Some Practical Examples I.4 Linear Coupling Correction with Skew Quadrupoles VI.6 Experimental Tracking of Synchrotron Motion Exercise III RF Phase and Voltage Modulations III.2 The Synchrotron Mapping Equation I.3 Effect of Wakefield on Transverse Wave VIII.2 Higher-Order Resonances VII.2 Transverse Wave Modes VIII.4 Effects of Dipole Field Modulation III.5 Synchrotron Motion for Large-Amplitude Particles II.3 Evolution of Synchrotron Phase-Space Ellipse I.3 Nonlinear Detuning from Sextupoles VII.
Shunt impedance C.3 Particle Acceleration by EM Waves A. HOMs 301 302 305 308 309 312 315 317 318 320 322 326 326 327 334 340 343 343 345 353 356 359 362 363 367 369 373 381 383 383 387 387 388 388 389 390 391 392 395 396 399 401 . Exercise VII Longitudinal Collective Instabilities VII.1 Historical Milestones VIII.2 Nonlinear Synchrotron Motion at 7 « 7T IV.4 Beam Loading Compensation and Robinson Instability .2 Fundamental Properties of Accelerating Structures A.3 Beam Loading VI. Alvarez structure E. . G.5 The QI Dynamical Systems Exercise V Beam Manipulation in Synchrotron Phase Space V. traveling wave.2 Collective Microwave Instability in Coasting Beams VII. .1 Pillbox Cavity VI. and coupled cavity linacs .4 Synchrotron Motion with Nonlinear Phase Slip Factor IV.xiv IV CONTENTS Nonadiabatic and Nonlinear Synchrotron Motion IV.2 Capture and Acceleration of Proton and Ion Beams V. Standing wave. EM waves in a cylindrical wave guide B.2 Low Frequency Coaxial Cavities VI.7 The Barrier RF Bucket Exercise VI Fundamentals of RF Systems VI. Loaded wave guide chain and the space harmonics F.1 Longitudinal Spectra VII.3 Longitudinal Impedance VII. .I RF Frequency Requirements V. 3 Beam Manipulation Near Transition Energy IV. Transit time factor B.6 Double rf Systems V.. 1 Linear Synchrotron Motion Near Transition Energy IV. Phase velocity and group velocity C. TM modes in a cylindrical pillbox cavity D.4 Microwave Single Bunch Instability Exercise VIII Introduction to Linear Accelerators VIII.5 Beam Stacking and Phase Displacement Acceleration V.3 Bunch Compression and Rotation V.4 Debunching V. The quality factor Q VIII.
1 Non-relativistic Reduction I.2 Interaction of the Radiation Field with the Beam I. 1 Damping of Synchrotron Motion II.7 Radiation Integrals II.4 Quantum Fluctuation Exercise II Radiation Damping and Excitation II.8 Beam Lifetime Exercise III Emittance in Electron Storage Rings III.2 The Coherent Beam-Beam Effects II. FODO cell lattice B.3 Frequency and Angular Distribution I. Three-bend achromat III. Minimizing emittance in a combined function DBA E.1 The beam-beam force II. Minimum (H)-function lattice D.4 Longitudinal Particle Dynamics in a Linac VIII.2 Radiation Field for Particles at Relativistic Velocities I.CONTENTS VIII.6 Vertical Beam Width II.2 Insertion Devices III.1 Emittance of Synchrotron Radiation Lattices A.3 Experiments on High Gain FEL Generation Exercise II Beam-Beam Interaction II.4 Radiation Excitation and Equilibrium Energy Spread II.3 Damping Rate Adjustment II. Double-bend achromat (Chasman-Green lattice) C.5 Radial Bunch Width and Distribution Function II.5 Transverse Beam Dynamics in a Linac Exercise 4 Physics of Electron Storage Rings I Fields of a Moving Charged Particle I.3 Nonlinear Beam-Beam Effects xv 402 407 410 417 422 424 424 427 433 435 437 438 441 445 448 453 455 456 456 462 466 467 467 469 473 475 476 478 486 489 497 498 500 506 509 510 513 517 519 521 .1 Small Signal Regime I.2 Damping of Betatron Motion II.3 Beam Physics of High Brightness Storage Rings Exercise 5 Special Topics in Beam Physics I Free Electron Laser (FEL) I.
2 Langevin Equation of Motion II.1 Cauchy Integral Formula III.2 Fixed Points I.4 Fokker-Planck Equation B Numerical Methods and Physical Constants I Fourier Transform I.7 Vector Operation V Maxwell's equations V.xvi CONTENTS II.4 Some Simple Fourier Transforms II Model Independent Analysis III Model Independent Analysis II.3 Digital Filtering I.3 Poisson Bracket I.6 Gauss' and Stokes' theorems IV. . 522 II.2 Dispersion Relation IV Useful Handy Formulas IV.2 Cylindrical waveguides V.5 Floquet Theorem II Stochastic Beam Dynamics II.3 Voltage Standing Wave Ratio 533 533 533 534 534 535 536 537 537 538 539 541 543 543 544 544 545 546 546 547 548 549 549 549 550 551 551 551 551 552 552 553 553 553 554 554 556 .2 The Hankel transform IV.I Lorentz Transformation of EM fields V.1 Canonical Transformations I.3 Accelerator Modeling III Cauchy Theorem and the Dispersion Relation III. 1 Generating functions for the Bessel functions IV. 2 Independent Component Analysis II.4 Liouville Theorem I.4 Experimental Observations and Numerical Simulations .2 Discrete Fourier Transform I. .B e a m Interaction in Linear Colliders 525 Exercise 527 A Basics of Classical Mechanics I Hamiltonian Dynamics I.3 The complex error function IV. .3 Stochastic Integration Methods II.5 B e a m .l Central Limit Theorem II.5 Cylindrical Coordinates IV.1 Nyquist Sampling Theorem I.4 A multipole expansion formula IV.
CONTENTS VI Physical Properties and Constants xvii 557 561 563 571 Bibliography Index Symbols and Notations .
D. I have benefited greatly from the collaboration with Drs. and Y. H. Yan and Prof. who made many useful suggestions to this revised edition. V. M. Kang. Your comments and corrections will be highly appreciated. Cousineau. Beltran. Li. A. Fung. M. I would like to thank S. Huang. Jeon. During the course of this work. The responsibility for all errors lies with me. K. C.Y. Guo. Bai. I owe special thanks to Prof.Acknowledgments I would like to thank students and colleagues. xviii .M. X. Ranjbar. Huang. Wang. Breitzmann. particularly D. Y. Y. Zhang. Chao. S. W. X. K. I owe special thanks to Margaret Dienes for editing the first edition of this book. Jau-Jiun Hsiao for making critical suggestions to the new chapter in this revised edition. David Caussyn. Ng. I would like to thank Angela Bellavance for pointing out mis-prints during a USPAS program in 2001. Ellison. Riabko. A. who helped me polish the lecture notes into a book form.
etc. These include electromagnetism. The counting rate in a detector is given by £u. In recent years. and Robert Schrieffer in 1957. waste treatment. Fine meshed superconducting wires are usually used in high-field magnets. plasma physics. High energy was measured in MeV's in the 1930's. As physicists probe deeper into the inner structure of matter. in biological and medical research with synchrotron light sources. where a is the cross-section of a reaction process. Understanding of the microscopic basis of superconductivity was achieved by John Bardeen. solid-state properties of materials. and quantum physics. and is about 1014 ppp in the 1990's. Advances in technology have allowed remarkable increases in energy and luminosity2 for fundamental physics research.G. The beam intensity was about 109 particles per pulse (ppp) in the 1950's. 1 . and is measured in TeV's in the 1990's. Leon Cooper. The race to build modern particle accelerators began in 1911 when Rutherford discovered the nucleus by scattering a-particles off Aluminum foil. Accelerators have also been used for radiotherapy. accelerators have found many applications: they are used in nuclear and particle physics research. The commonly used dimension is cm"2 s" 1 . etc. atomic physics. Since 1970. superconductivity. The physics and technology of accelerators and storage rings involves many branches of science.1 nonlinear mechanics. in industrial applications such as ion implantation and lithography. in material science and medical research with spallation neutron sources. and indeed new energy frontiers usually lead to new physics discoveries. High temperature superconductor was discovered by K. high energy and high luminosity colliders have become basic tools in nuclear and particle physics research. 2 The luminosity C is denned as the probability of particle encountering rate per unit area in a collision process (see Exercise 1. when men built bows and arrows for hunting. Mueller and J. Bednorz in 1986. A major application of particle accelerators is experimental nuclear and particle physics research. spin dynamics. high energy provides new territory for potential discoveries. Superconducting thin films deposited on the cavity surface are used for superconducting cavities.Chapter 1 Introduction The first accelerator dates back to prehistoric-historic times. The evolution of 1 Superconductivity was discovered by Heike Kamerlingh Onnes in 1911.A. food sterilization. The Meissner effect was discovered in 1933.7).
. high-brilliance photon beams from high-brightness electron beams in storage rings3 have been extensively used in biomedical and condensed-matter physics research.- . .6). 1.fi^&$rf~~~'**' n"*""u""! ••»•"«" — .1%Aw/w)) is plotted _ $• as a function photon beam {"„«««. | . . • « . innovative ideas provide substantial jump in beam enPTP"V el &y- _ y .J°t-=Tvr""°° -^ § photons/(mm 2 mrad 2 s •^ — ~ 1 ^ ^ 2 : • | (0. is plotted as a function of time._\ 5 and average Photon » * " " " * * " . .-.5 Brilliance denned as In 2 ^ ^ .„. i .-. 1980 2000 year ° Rectifier generator In recent years. 1Q6 J c v i T ^ 1940 1960 i . /ri™»)HT' „»"'" .S y S e a t e d by high . 102 108 _ Br. ..=.2 shows the peak photon brilliance (number of photons/(mm2 mrad2 s (0. .2 CHAPTER! INTRODUCTION accelerator development can be summarized by the Livingston chart shown in Fig. . I.. .tr^l^. Note * n a ' . . i . .S quality electron beams in • storage rings and m hnacs.° I | | ^jf jrfp -^--^rfi o 'CPQ_—&-^P Electron lin a c » J P j ^ T ^ H«*ron Synchrotron (strong focusing) TTlP -1-Ile HflsTlpH U d 4 I l e U llTIP 1 1 I l e m l b S s | < 109 _1 S . > l o 2 1 I " I " " I ^' " I " " 1 U I lo18 " = £ B I ' I \ IQ15 _ g* S lol2 l i | | I 1 1 | o 2 « ! * g I I" " I" " I' ' l ^ ' I l I TOW «»->*«~' ""^r * S K W M ..-*"V« _ .„!<. . .^fLe—^=_Ji c'°1°"°". o /—\ / ^ / »SLSXl^p^ ^ \ x " " I . — - pjg u r e U .. i . / »siin ^ ^ „ .io24 • .i<J 10~ 2 I I I " I Z j A 10" 1 10° 101 photon energy (keV) 102 1 0 " 2 10" 1 10° 101 photon energy (keV) Besides being used for fundamental material science research. .1 10 c.1.^ygtf—® g Proton lln. which is beam energy doubling in every two years.c . Figure 1." ctor '° cu " d . xhe Livingston Chart: The equivalent fixed target proton beam energy versus time in years. 2020 2040 drawn to guide the trend.loie _ s"ctLo1u." ' ~~~~. _ 10 28 / ^ ^ ^ \ t i g u r e 1. .„ — ^ \ • ..m.1% Aw/w)) as a function of photon beam energy from storage rings and linacs.2: The peak . KE = s/(2m p c 2 ) (see Exercise 1.„. i . high-intensity neu3 The brightness of a beam is denned as the beam's intensity divided by its phase-space volume. 'I'0** \ — lO 1311 TESU LCLS ' — Undulator (8-8 GeV) v '• - 8 . Cu _tJ" K c K • ? e n e l . • ' s ' ' 10 2 0 - — ^ l . where the equivalent fixed target proton energy.
Frontiers in accelerator physics and technology research Accelerator physics is a branch of applied science. and b and t quarks. wakefield control. wakefields.+/J. This led to the discovery of W and Z bosons. high-field superconducting magnets and the stability of high-brightness beams are important issues. and higher luminosity leads to higher precision in experimental results. a high-intensity proton source can be used to drive secondary beams such as kaons.ACCELERATOR PHYSICS 3 tron sources driven by powerful proton beam sources may provide energy amplification for future global energy needs.~ collider studies are also of current interest. China. dedicated meson factories such as the $-factory at Frascati National Laboratory in Italy and the B-factories at SLAC and Cornell in the U. . high acceleration gradients. Some of these topics in beam physics are as follows. the CERN Linear Collider (CLIC). Stochastic cooling has been successfully applied to accumulate anti-protons. and condensed-matter physics 4 See e. synchrotron light sources with high-brightness electron beams are used in medical. Innovations in technology give rise to new frontiers in beam physics research. and muons. Some proposed e+e~ colliders are the Next Linear Collider (NLC). biological. For lepton colliders. Taking advantage of radiation cooling. and high power rf sources are important. 1997). Ionization cooling is needed for muon beams in fi+fJ.~ colliders.g. • High-brightness beams: Beam-cooling techniques have been extensively used in attaining high-brightness hadron beams. • High energy: For high energy hadron accelerators such as the Tevatron at Fermilab. Since higher energy leads to new discoveries.S. With high-intensity y u beams. the National Spallation Neutron Source Design Report (Oak Ridge. Current research topics include high rf power sources. the Large Hadron Collider (LHC) at CERN.4 Furthermore. • High luminosity: To provide a detailed understanding of CP violation and other fundamental symmetry principles of interactions. the Japan Linear Collider (JLC). High intensity heavy-ion beams have also been actively pursued for inertial fusion evaluation. and at KEK in Japan were built in the 1990's. and the Tau-Charm factory is being contemplated in Beijing. which employs superconducting rf cavities. pions. Since the neutron flux from spallation neutron sources is proportional to the proton beam power. and the contemplated Very Large Hadron Collider (VLHC). /J. high acceleration gradient structures. etc.. physics and technology for high-intensity low-loss proton sources are important. and the TeV Superconducting Linear Accelerator (TESLA). the frontiers of accelerator physics research are classified into the frontiers of high energy and high brightness. Electron cooling and laser cooling have been applied to many low energy storage rings used in atomic and nuclear physics research.
high power rf sources. space-charge effects. Since the magnetic force is perpendicular to both v and B. Petti and A. novel acceleration techniques. Lennox. food sterilization.3) e. This book deals only with the fundamental aspects of accelerator physics. sterilization of medical tools. ARNS 44. the charged particle will move on a circular arc.33564 x p [GeV/c/u]. P. . First. high gradient accelerating structures. Recent research topics in accelerator physics include beam cooling. etc. beam-beam interactions. nonlinear beam dynamics. A high power tunable free-electron laser would be useful for chemical and technical applications. beam manipulation techniques. reliability. when the magnetic flux density is perpendicular to v. etc. beam instrumentation development.g. INTRODUCTION research. etc.1) The charge particle can only gain or lose its energy by its interaction with the electric field E. • Accelerator applications: The medical use of accelerators for radiation treatment.L. The momentum rigidity of the charged particle is Bp [T-m] = ? = ^ x 3. 155 (1994). and ease in operation. In particular. magnet technology. beam-cooling physics and technologies. I Historical Developments A charged particle with charge q and velocity v in the electromagnetic fields (E.. rf physics and technology. etc. (1. ion sources. etc.5 isotope production.4 CHAPTER 1. nonlinear beam dynamics. the bending radius is where m and p = mv are the mass and momentum of the particle. q Z 5 See (1. collective beam instability. material testing. electron-beam welding. requires safety. Higher beam power density with minimum beam loss can optimize safety in industrial applications such as ion implantation. B) is exerted by the Lorentz's force F: F = q(E + vxB).. It serves as an introduction to more advanced topics such as collective beam instabilities.J. Sub-picosecond photon beams would be important to time-resolved experiments. the technical achievements in accelerator physics of past decades will be described. Accelerator technology research areas include superconducting materials.
etc. to the surprise of many physicists. Barnett et al. S. This discovery created an era of search for high-voltage sources for particle acceleration that can produce high-intensity high-energy particles for the study of nuclear transmutation.H. 1997. stable or radioactive ions. 7See J. Rutherford employed a particles escaping the Coulomb barrier of Ra and Th nuclei to investigate the inner structure of atoms. heavy elements have been measured with energies up to 3 x 1020 eV. Interest in the relativistic heavy ion collider (RHIC) was amplified by the cosmic ray emulsion experiments. high vacuum components for attaining excellent beam lifetime. 323 (1983) and R. 1998). Sci. 19. Simpson. Feb. p. p. undulators and wigglers to produce high brilliance photon beam. 31. Rutherford also used a particles to induce the first artificial nuclear reaction. continuous (CW. . Pions were discovered in 1947 in emulsion experiments. and others. the existence of a positively charged nucleus with a diameter less than 10"11 cm.6 He demonstrated. Nuclei range from n and H to Ni. They are designed to accelerate electrons (leptons) or hadrons. Neddermeyer. Anderson. and A and Ze are the atomic mass number and charge of the particle. Cosmic rays Cosmic rays arise from galactic source accelerators. An event with energy 3 x 1020 eV had been recorded in 1991 by the Fly's Eye atmospheric-fluorescence detector in Utah (see Physics Today. a + 14N — 17O > + H. 6The kinetic energy of a particles that tunnel through the Coulomb barrier to escape the nuclear force is typically about 6 MeV. Accelerators are classified as follows. (Particle Data Group) D54. Jan.I. I. Ann. and the revolution of quantum mechanics in the early 20th century. In 1919. Accelerators are composed of ion sources. Phys. in no specific chronological order.I Natural Accelerators Radioactive accelerators In 1911. Nucl. cavity and magnet components that can generate and maintain electromagnetic fields for beam acceleration and manipulation. DC or coasting beam) or bunched and pulsed.A. 33. electrostatic or radio frequency. Rev.7 Muons were discovered in cosmic-ray emulsion experiments in 1936 by CD. devices to detect beam motion. Accelerators can be classified as linear or circular. targets for producing secondary beams. This led to the introduction of Bohr's atomic model. Rev. 1 (1996). HISTORICAL DEVELOPMENTS 5 where Bp is measured in Tesla-meter. and the momentum is measured in GeV/c per amu.
J.G. it has the brand name Pelletron. the rectifier units are replaced by an electrostatic charging belt. which increases the peak acceleration voltage. Nucl. (1. ds*is the differential for the line integral that surrounds the surface area. Phys.6 CHAPTER 1. Cockcroft-Walton electrostatic accelerator In 1930. INTRODUCTION 1.4) J Js Here £ is the induced electric field. Van de Graaff. Van de Graaff developed the electrostatic charging accelerator. Prog. R. 1. 704 (1934). 1 (1946). and the high-voltage terminal and the acceleration tube are placed in a common tank with compressed gas for insulation.10 Today the voltage attained in tandem accelerators is about 25 MV. John Douglas Cockcroft and Ernst Thomas Sinton Walton developed a highvoltage source by using high-voltage rectifier units. they reached 400-kV terminal voltage to achieve the first man-made nuclear transmutation: p + Li — » 2 He.D. economical. Buechner. The cascade type of X-ray tube is called the Coolidge tubes. When the Van de Graaff accelerator is used for electron acceleration. Rep. J.W. Since then. 195 (1960). Cockcroft and Walton shared 1951 Nobel Prize in physics. 11. Cockcroft and E. dS is the differential for the 8 J. Soc.J.9 In the Van de Graaff accelerator. Inst. Van de Graaff. when the magnetic flux changes. Placement of the high-voltage terminal at the center of the tank and use of the chargeexchange process in the tandem accelerator can increase the beam energy for nuclei. In 1932. Proc.J. and reliable radio frequency quadrupole (RFQ) accelerators. 1 0 R. 619 (1932).3 Induction Accelerators According to Faraday's law of induction. Methods 8. 9 R. $ is the total magnetic flux. Walton. $= f B-dS. A136. More recently. . Trump. the induction electric field along a beam path is given by ie-ds = $. A137.2 Electrostatic Accelerators X-ray tubes William David Coolidge in 1926 achieved 900-keV electron beam energy by using three X-ray tubes in series. Cockcroft-Walton accelerators have been widely used in first-stage ion-beam acceleration. they are being replaced by more compact.T. 229 (1932). Van de Graaff and tandam accelerators In 1931.S. Roy.8 The maximum achievable voltage was limited to about 1 MV because of sparking in air. and W. A144.
in Proc.K.1: Induction linac projects and achievements Laboratory / (kA) E (MeV) Beam width Repetition (ns) rate (Hz) ETA II LLNL 3 70 50 I ETA III LLNL 2 6 50 2000 ATA | LLNL | 10 | 50 | 50 | 1000 Project B: Betatron Let p be the mean radius of the beam pipe in a basic magnet configuration of a betatron. 1591 (1986). Table 1. R. A: Induction linac The induction linac was invented by N. Hyder. IEEE Trans. Barletta. Neil.J. Lett.C. Lett.I HISTORICAL DEVELOPMENTS 7 surface integral. Christofilos and R. Zi (1.12 A linear induction accelerator (LIA) employs a ferrite core arranged in a cylindrically symmetric configuration to produce an inductive load to a voltage gap. Beal.S. IEEE Trans. J or £ =-Bwp. Nucl.A. 1988). 3149 (1985).K. et al. NS 16. and V.J. LINAC96 (1996). Accel.5) will use magnetic field as a synonym for magnetic flux density. Eds.B..E.g. R. Rose. (Plenum Press. Christofilos in the 50's for the acceleration of high-intensity beams. 71 (1980). R. 57. Prono.C. Miller. the electric field at the voltage gap along the beam axis is used to accelerate the beam. Review of new developments in the field of induction accelerators. The induced electric field can be used for beam acceleration. 11. 3144 (1985). 13 See e. D.J. Part. Rev. NS32. 294 (1958) and references therein. and A. J. the induced electric field along the beam axis is given. Nucl. IEEE Trans.13 Table 1. Miller. A. Guenther. G. Sci. Caporaso.. NS32. If the total magnetic flux enclosed by the beam circumference is ramped up by a time-dependent magnetic flux density. G. N.W. 54. NATO ASI on High Brightness Transport in Linear Induction Accelerators. according to Faraday's law of induction. W. Phys.H.1 lists the achievements of some LIA projects. Nucl. Sci.B. Briggs. e. Sci. by <b £ • ds = 2np£ = np2Bm. Phys. M. Rev. and B is the "magnetic field"11 enclosed by the contour C. 2588 (1985). Hester. Each LIA module can be viewed as a low-Q 1:1 pulse transformer. in Proc.F. A properly pulsed stack of LIA modules can be used to accelerate high-intensity short-pulse beams with a gradient of about 1 MeV/m and a power efficiency of about 50%. Caporaso. 12 See 11 We .g. Simon Yu. When an external current source is discharged through the circuit.
8
CHAPTER 1. INTRODUCTION
Here £ is the induced electric field, and Bm is the average magnetic flux density inside the circumference of the beam radius. Thefinalparticle momentum can be obtained by integrating Newton's law, p = e£, i.e. p=-eBmp = eBgp, or Bg = -Bm. (1.6)
The betatron principle that the guide field Bg is equal to 1/2 of the average field Bm, was first stated by R. Wiederoe in 1928. u Figure 1.3 is a schematic drawing of a betatron, where particles circulate in the vacuum chamber with a guide field Bg, and the average flux density enclosed by the orbiting particle is £?av.
Figure 1.3: Schematic drawing of a betatron. The guidefieldfor beam particles is jBg, and the average flux density enclosed by the orbiting path is B av . It took many years to understand the stability of transverse motion. This problem was solved in 1941 by D. Kerst and R. Serber.15 When the magnetic field is shaped according to B, = £ > „ ( * ) " , (1.7)
where R is the reference orbit radius, r is the beam radius, and n is the index of focusing given by (see Exercise 1.14)
R fdBA
. .
Let x = r — R and z be small radial and vertical displacements from a reference orbit, then the equations of motion become — +cj2nz = 0, -—+L)2{l-n)x = 0. (1.9)
14In 1922, Joseph Slepian patented the principle of applying induction electric field for electron beam acceleration in the U.S. patent 1645304. 15D. Kerst and R. Serber, Phys. Rev. 60, 53 (1941). See also Exercise 1.14. Since then, the transverse particle motion in all types of accelerators has been called betatron motion.
I. HISTORICAL DEVELOPMENTS
9
Thus the motion is stable if 0 < n < 1. The resulting frequencies of harmonic oscillations are fx = foy/1 — n and fz = foy/n, where / 0 = W/2TT is the revolution frequency. In 1940 D. Kerst was the first to operate a betatron to achieve 2.3 MeV. In 1949 he constructed a 315-MeV betatron16 at the University of Chicago with the parameters p = 1.22 m, B% = 9.2 kG, Einj = 80 - 135 keV, / inj = 13 A. The magnet weighed about 275 tons and the repetition rate was about 6 Hz. The limitations of the betatron principle are (1) synchrotron radiation loss (see Chapter 4) and (2) the transverse beam size limit due to the intrinsic weak-focusing force.
1.4
Radio-Frequency (RF) Accelerators
Since the high-voltage source can induce arcs and corona discharges, it is difficult to attain very high voltage in a single acceleration gap. It would be more economical to make the charged particles pass through the acceleration gap many times. This concept leads to many different rf accelerators,17 which can be classified as linear (RFQ, linac) and cyclic (cyclotron, microtron, and synchrotron). Accelerators using an rf field for particle acceleration are described in the following subsections.
<u
o
Wideroe Linac
i
I —
I
-
i
i i
RF source S~\
r^
g->czi
c—i c—i i — i — i
i -+
Figure 1.4: Schematic drawing of the Wiederoe rf LINAC structure. Wideroe used a 1-MHz, 25-kV oscillator to make 50-kV potassium ions.
A. LINAC
In 1925 G. Ising pointed out that particle acceleration can be achieved by using an alternating radio-frequency field. In 1928 R. Wiederoe reported the first working rf accelerator, using a 1-MHz, 25-kV oscillator to produce 50-kV potassium ions (see Fig. 1.4). In 1931 D.H. Sloan and E.O. Lawrence built a linear accelerator using a 10-MHz, 45 kV oscillator to produce 1.26 MV Hg+ ion.18 An important milestone
Kerst et. a!., Phys. Rev. T8, 297 (1950). rf sources are classified into VHF, UHF, microwave, and millimeter waves bands. The microwave bands are classified as follows: L band, 1.12-1.7 GHz; S band, 2.6-3.95 GHz; C band, 3.95-5.85 GHz; X band, 8.2-12.4 GHz; K band, 18.0-26.5 GHz; millimeter wave band, 30-300 GHz. See also Exercise 1.2. 18 D.H. Sloan and E.O. Lawrence, Phys. Rev. 38, 2021 (1931).
17 The 16 D.W.
10
CHAPTER 1.
INTRODUCTION
in rf acceleration is the discovery of the phase-focusing principle by E. M. McMillan and V. Veksler in 1945 (see Ref. [17] and Chap. 2, Sec. IV.3). Since the length of drift tubes is proportional to J3X/2, it would save space by employing higher frequency rf sources. However, the problem associated with a high frequency structure is that it radiates rf energy at a rate of P = UriCVl (1.10)
where w,f is the rf frequency, C is the gap capacitance, and VT{ is the rf voltage. The rf radiation power loss increases with the rf frequency. To eliminate rf power loss, the drift tube can be placed in a cavity so that the electromagnetic energy is stored in the form of a magnetic field (inductive load). At the same time, the resonant frequency of the cavity can be tuned to coincide with that of the accelerating field. In 1948 Louis Alvarez and W.K.H. Panofsky constructed the first 32-MV drifttube linac (DTL or Alvarez linac) for protons.19 Operational drift-tube linacs for protons are the 200-MeV linacs at BNL and Fermilab, and the 50-MeV linacs at KEK and CERN. In the 1970's Los Alamos constructed the first side-coupled cavity linac (CCL), reaching 800 MeV. Fermilab upgraded part of its linac with the CCL to reach 400 MeV kinetic energy in 1995. The coupled cavity drift tube linac (CCDTL) that combines CCL and DTL has been shown to be efficient in accelerating high intensity low energy proton beams. After World War II, rf technology had advanced far enough to make magnetron and klystron20 amplifiers that could provide rf power of about 1 MW at 3 GHz (S band). Today, the highest energy linac has achieved 50-GeV electron energy operating at S band (around 2.856 GHz) at SLAC, and has achieved an acceleration gradient of about 20 MV/m, fed by klystrons with a peak power of 40 MW in a l-/xs pulse length. To achieve 100 MV/m, about 25 times the rf power would be needed. The next linear collider (NLC), proposed by SLAC and KEK, at a center-of-mass energy of 500 GeV to 2 TeV beam energy, calls for X band with an acceleration gradient of 50 MV/m or more. The required klystron peak power is about 50 MW in a pulse duration is about 1.5 us. The peak power is further enhanced by pulse compression schemes.
Alvarez, Phys. Rev. 70, 799 (1946). klystron, invented by Varian brothers in 1937, is a narrow-band high-gain rf amplifier. The operation of a high power klystron is as follows. A beam of electrons is drawn by the induced voltage across the cathode and anode by a modulator. The electrons are accelerated to about 400 kV with a current of about 500 A. As the beam enters the input cavity, a small amount of rf power (< 1 kW) is applied to modulate the beam. The subsequent gain cavities resonantly excite and induce micro-bunching of the electron beam. The subsequent drift region and penultimate cavity are designed to produce highly bunched electrons. The rf energy is then extracted at the output cavity, which is designed to decelerate the beam. The rf power is then transported by rf waveguides. The wasted electrons are collected at a water-cooled collector. If the efficiency were 50%, a klystron with the above parameters would produce 100 MW of rf power. See also E.L. Ginzton, "The $100 idea", IEEE Spectrum, 12, 30 (1975).
20The 19L.
I HISTORICAL DEVELOPMENTS
11
Superconducting cavities have also become popular in recent years. At the Continuous Electron Beam Accelerator Facility (CEBAF) at the Thomas Jefferson National Accelerator Laboratory in Virginia, about 160 m of superconducting cavity was installed for attaining a beam energy up to 4 GeV in 5 paths using 338 five-kW CW klystrons. During the LEP-II upgrade more than 300 m of superconducting rf cavity was installed for attaining an almost 100-GeV beam energy. Many accelerator laboratories, such as Cornell and Fermilab in the U.S. and DESY in Germany, are collaborating in the effort to achieve a high-gradient superconducting cavity for a linear collider design called the TeV Superconducting Linear Accelerator (TESLA). Normally, a superconducting cavity operates at about 5-10 MV/m. After extensive cavity wall conditioning, single-cell cavities have reached beyond 25 MV/m.21 B: RFQ In 1970,1.M. Kapchinskij and V.A. Teplyakov invented a low energy radio-frequency quadrupole (RFQ) accelerator - a new type of low energy accelerator. Applying an rf electric field to the four-vane quadrupole-like longitudinally modulated structure, a longitudinal rf electric field for particle acceleration and a transverse quadrupole field for focusing can be generated simultaneously. Thus the RFQs are especially useful for accelerating high-current low-energy beams. Since then many laboratories, particularly Los Alamos National Laboratory (LANL), Lawrence Berkeley National Laboratory (LBNL), and CERN, have perfected the design and construction of RFQ's, which are replacing Cockcroft-Walton accelerators as injectors to linac and cyclic accelerators. C: Cyclotron
The synchrotron frequency for a non-relativistic particle in a constant magnetic field is nearly independent of the particle velocity, i.e.,
wsyn =
e-Bo
7m
« wcyc = , * m
eB0
(1-11)
where 7 « 1 for non-relativistic particles, Bo is the magnetic field, and m is the particle mass. In 1929 E.O. Lawrence combined the idea of a constant revolution frequency and Ising's idea of the rf accelerator (see Sec. I.4A of Wiederoe linac), he invented the cyclotron.22 Historical remarks in E.O. Lawrence's Nobel lecture are
reach 30 MV/m and beyond. 22 E.O. Lawrence and N.E. Edlefsen, Science, 72, 376 (1930). See e.g. E.M. McMillan, Early Days in the Lawrence Laboratory (1931-1940), in New directions in physics, eds. N. Metropolis, D.M. Kerr, Gian-Carlo Rota, (Academic Press, Inc., New York, 1987). The cyclotron was coined by Malcolm Henderson, popularized by newspaper reporters; see M.S. Livingston, Particle Accelerators: A Brief History, (Harvard, 1969).
21See
e.g., J. Garber, Proc. PAC95, p. 1478 (IEEE, New York 1996). Single-cell cavities routinely
12
reproduced below:
CHAPTER 1.
INTRODUCTION
One evening early in 1929 as I was glancing over current periodicals in the University library, I came across an article in a German electrical engineering journal by Wideroe on the multiple acceleration of positive ions. . . . This new idea immediately impressed me as the real answer which I had been looking for to the technical problem of accelerating positive ions, . . . Again a little analysis of the problem showed that a uniform magnetic field had just the right properties - that the angular velocity of the ion circulating in the field would be independent of their energy so that they would circulate back and forth between suitable hollow electrodes in resonance with an oscillating electric field of a certain frequency which has come to be known as the cyclotron frequency. Now this occasion affords me a felicitous opportunity in some measure to correct an error and an injustice. For at that time I did not carefully read Wiederoe's article and note that he had gotten the idea of multiple acceleration of ions from one of your distinguished colleagues, Professor G. Ising, who in 1924 published this important principle. It was several years had passed that I became aware of Professor Ising's prime contribution. I should like to take this opportunity to pay tribute to his work for he surely is the father of the developments of the methods of multiple acceleration.
If two D plates (dees) in a constant magnetic field are connected to an rf electric voltage generator, particles can be accelerated by repeated passage through the rf gap, provided that the rf frequency is an integer multiple of the cyclotron frequency, a/rf = huj0. On January 2, 1931 M.S. Livingston demonstrated the cyclotron principle by accelerating protons to 80 keV in a 4.5-inch cyclotron, where the rf potential applied across the the accelerating gap was only 1000 V. In 1932 Lawrence's 11inch cyclotron reached 1.25-MeV proton kinetic energy that was used to split atoms, just a few months after this was accomplished by the Cockcroft-Walton electrostatic accelerator. Since then, many cyclotrons were designed and built in Universities.23 Figure 1.5 shows a schematic drawing of a classical cyclotron. The momentum p and kinetic energy T of the extracted particle are p = rwyPc and T = mc2(7 - 1) = p2/[(j + l)m]. Using Eq. (1.3), we obtain the kinetic energy per amu as
A-(7+l)muUJ
=K
{A) '
(L12)
where BoRo = Bp is the magnetic rigidity, Z and A are the charge and atomic mass numbers of the particle, mu is the atomic mass unit, and K is called the K-value or bending limit of a cyclotron. In the non-relativistic limit, the /{"-value is equal to the proton kinetic energy T in MeV, e.g. K200 cyclotron can deliver protons with 200 MeV kinetic energy.
23 M.S. Livingston, J. Appl. Phys, 15, 2 (1944); 15, 128 (1944); W.B. Mann, The Cyclotron, (Wiley, 1953); M.E. Rose, Phys. Rev., 53, 392 (1938); R.R. Wilson, Phys. Rev., 53, 408 (1938); Am. J. Phys., 11, 781 (1940); B.L. Cohen, Rev. Sci. Instr., 25, 562 (1954).
I. HISTORICAL
DEVELOPMENTS
13
F j ;I I • • i \ \ \ '. ', i \ (~)rf n i l i .' C i i ; i! D |"^ \\ \ \ \ \ \ ion source ,' ; ; ; I \ \ \ \ \ \ ^, .•'','•!'/
Figure 1.5: Schematic drawing of a classical cyclotron. Note that the radial distance between adjacent revolutions becomes smaller as the turn number increases [see Eq. (1.13)].
septum
The iron saturates at a field of about 1.8 T (depending slightly on the quality of iron and magnet design). The total volume of iron-core is proportional to the cubic power of the beam rigidity Bp. Thus the weight of iron-core increases rapidly with its K-value: Weight of iron = W ~ K15 ~ (Bp)3, where Bp is the beam rigidity. Typically, the magnet for a K-100 cyclotron weighs about 160 tons. The weight problem can be alleviated by using superconducting cyclotrons.24 The design of beam extraction systems in cyclotrons is challenging. Let VQ be the energy gain per revolution. The kinetic energy at ./V revolutions is K^ = eiVVo = e2B2r2/2m, where e is the charge, m is the mass, B is the magnetic field, and r is the beam radius at the ./V-th revolution. The radius r of the beam at the iV-th revolution becomes
r =I
(^) "V*.
(U3)
i.e. the orbiting radius increases with the square root of the revolution number N. The beam orbit separation in successive revolutions may becomes small, and thus the septum thickness becomes a challenging design problem. Two key difficulties associated with classical cyclotrons are the orbit stability and the relativistic mass effect. The orbit stability problem was partially solved in 1945 by D. Kerst and R. Serber (see Exercise 1.14). The maximum kinetic energy was limited by the kinetic mass effect. Because the relativistic mass effect can destroy particle synchronism [see Eq. (1.11)], the upper limit of proton kinetic energy attainable in a cyclotron is about 12 MeV (See Exercise 1.4.).25 Two ideas proposed to solve the dilemma are the isochronous cyclotron and the synchrocyclotron.
24 See 25 H.
H. Blosser, in Proc. 9th Int. Conf. on Cyclotrons and Applications, p. 147 (1985). Bethe and M. Rose, Phys. Rev. 52, 1254 (1937).
14
CHAPTER 1.
INTRODUCTION
Isochronous cyclotron
In 1938 R.H. Thomas pointed out that, by using an azimuthal varying field, the orbit stability can be retained while maintaining the isochronism. The isochronous cyclotron is also called the azimuthal varying field (AVF) cyclotron. From the cyclotron principle, we observe that
where Eo = me2 and ui is the angular revolution frequency. Thus, to maintain isochronism with constant w, the B field must be shaped according to
Bz = ^
e
= ^E{p) = ^\l-mY/2.
ec* ec2 [ V c / J
(1.15)
When the magnetic flux density is shaped according to Eq. (1.15), the focusing index becomes n < 0, and the vertical orbit is unstable. Orbit stability can be restored by shaping the magnetic pole-face. In 1938 L.H. Thomas introduced pole plates with hills and valleys in an isochronous cyclotron to achieve vertical orbit stability.26 Such isochronous cyclotrons are also called AVF cyclotrons. The success of sector-focused cyclotron led by J.R. Richardson et al. led to the proliferation of the separate sector cyclotron, or ring cyclotron in the 1960's.27 It gives stronger "edge" focusing for attaining vertical orbit stability. Ring cyclotrons are composed of three, four, or many sectors. Many universities and laboratories built ring cyclotrons in the 1960's.
Synchrocyclotron
Alternatively, synchronization between cyclotron frequency and rf frequency can be achieved by using rf frequency modulation (FM). FM cyclotrons can reach 1-GeV proton kinetic energy.28 The synchrocyclotron uses the same magnet geometry as the weak-focusing cyclotrons. Synchronism between the particle and the rf accelerating voltage is achieved by ramping the rf frequency. Because the rf field is cycled, i.e. the rf frequency synchronizes with the revolution frequency as the energy is varied, synchrocyclotrons generate pulsed beam bunches. Thus the average intensity is low. The synchrocyclotron is limited by the rf frequency detuning range, the strength of the magnet flux density, etc. Currently two synchrocyclotrons are in operation, at CERN and at LBL.
Thomas, Phys. Rev. 54, 580 (1938). Willax, Proc. Int. Cyclotron Conj. 386 (1963). 28 For a review, see R. Richardson, Proc. 10th Int. Conj. on Cyclotrons and Their Applications, IEEE CH-1996-3, p. 617 (1984).
27 H.A. 26 L.H.
I. HISTORICAL DEVELOPMENTS D: Microtron
15
As accelerating rf cavities are expensive, it would be economical to use the rf structure repetitively: microtrons, originally proposed by V. Veksler in 1944, are designed to do this. Repetitive use requires synchronization between the orbiting and the rf periods. For example, if the energy gain per turn is exactly equal to the rest mass of the electron, the cyclotron frequency at the n — 1 passage is given by <4.-i = — , (1-16) nm0 i.e., the orbit period is an integral multiple of the fundamental cyclotron period. Thus, if the rf frequency tuTf is an integral multiple of the fundamental cyclotron frequency, the particle acceleration will be synchronized. Such a scheme or its variation was invented by V. Veksler in 1945. The synchronization concept can be generalized to include many variations of magnet layout, e.g. the race track microtron (RTM), the bicyclotron, and the hexatron. The resonance condition for the RTM with electrons traveling at the speed of light is given by AW nArf = 2 T T — , (1.17) ecB where AE is the energy gain per passage through the rf cavity, B is the bending dipole field, Arf is the rf wavelength, and n is an integer. This resonance condition simply states that the increase in path length is an integral multiple of the rf wavelength. Some operational microtrons are the three-stage MAMI microtron at Mainz, Germany,29 and the 175-MeV microtron at Moscow State University. Several commercial models have been designed and built by Scanditron. The weight of the microtron also increases with the cubic power of beam energy. E: Synchrotrons, weak and strong focusing After E.M. McMillan and V. Veksler discovered the phase focusing principle of the rf acceleration field in 1945, a natural evolution of the cyclotron was to confine the particle orbit in a well-defined path while tuning the rf system and magnetic field to synchronize particle revolution frequency.30 The first weak-focusing proton synchrotron, with focusing index 0 < n < 1, was the 3-GeV Cosmotron in 1952 at BNL.
e.g., H. Herminghaus, in Proc. 1992 EPAC, p. 247 (Edition Frontieres, 1992). Goward and D.E. Barnes converted a betatron at Telecommunication Research Laboratory into a synchrotron in August 1946. A few months earlier, J.R. Richardson, K. MacKenzie, B. Peters, F. Schmidt, and B. Wright had converted the fixed frequency 37-inch cyclotron at Berkeley to a synchro-cyclotron for a proof of synchrotron principle. A research team at General Electric Co. at Schenectady built a 70 MeV electron synchrotron to observe synchrotron radiation in October 1946. See also E.J.N. Wilson, 50 years of synchrotrons, Proc. of the EPAC96 (1996).
30Prank 29see
16
CHAPTER 1.
INTRODUCTION
A 6-GeV Bevatron constructed at LBNL in 1954, led to the discovery of antiprotons in 1955. An important breakthrough in the design of synchrotron came in 1952 with the discovery of the strong-focusing or the alternating-gradient (AG) focusing principle by E.D. Courant, H.S. Snyder and M.S. Livingston.31 Immediately, J. Blewett invented the electric quadrupole and applied the alternating-gradient-focusing concept to linac32 solving difficult beam focusing problems in early day rf linacs. Here is Some
Recollection on the Early History of Strong Focusing in the publication BNL 51377 (1980) by E.D. Courant: Came the summer of 1952. We have succeeded in building the Cosmotron, the world's first accelerator above one billion volts. We heard that a group of European countries were contemplating a new high-energy physics lab with a Cosmotron-like accelerator (only bigger) as its centerpiece, and that some physicists would come to visit us to learn more about the Cosmotron. . . . Stan (Livingston) suggested one particular improvement: In the Cosmotron, the magnets all faced outward. This made it easy to get negative secondary beam from a target in the machine, but much harder to get positive ones. Why not have some magnets face inward so that positive secondaries could have a clear path to experimental apparatus inside the ring? . . . I did the calculation and found to my surprise that the focusing would be strengthened simultaneously for both vertical and horizontal motion. ... Soon we tried to make the gradients stronger and saw that there was no theoretical limit - provided the alterations were made more frequent as the gradient went up. Thus it seemed that aperture could be made as small as one or two inches against 8 x 24 inches in the Cosmotron, 12 x 48 in the Bevatron, and even bigger energy machines as we then imagined them. With these slimmer magnets, it seemed one could now afford to string them out over a much bigger circles, and thus go to 30 or even 100 billion volts.
The first strong-focusing 1.2 GeV electron accelerator was built by R. Wilson at Cornell University. Two strong-focusing or alternating-gradient (AG) proton synchrotrons, the 28-GeV CERN PS (CPS) and the 33-GeV BNL AGS, were completed in 1959 and 1960 respectively. The early strong-focusing accelerators used combined-function magnets, i.e., the pole-tips of dipoles were shaped to attain a strong quadrupole field. For example, the bending radius and quadrupole field gradients of AGS magnets are respectively p = 85.4 m, and Bx = (dB/dx) = ±4.75 T/m at B — 1.15 T. This corresponds to a focusing index of n = ±352. The strengths of a string of alternating focusing and defocussing lenses were adjusted to produce net strong focusing effects in both planes. The strong focusing idea was patented by a U.S. engineer, N.C. Christofilos,33
Courant, H.S. Snyder and M.S. Livingston, Phys. Rev. 88, 1188 (1952). Blewett, Phys. Rev. 88, 1197 (1952). 33 N.C. Christofilos, Focusing system for ions and electrons, U.S. Patent No. 2736799 (issued 1956). Reprinted in The Development of High Energy Accelerators, M.S. Livingston, ed. (Dover, New York, 1966).
3 2 J. 31 E.D.
I. HISTORICAL DEVELOPMENTS
17
living in Athens, Greece. Since then, the strong-focusing (AG) principle and a cascade of AG synchrotrons, proposed by M. Sands,34 has become a standard design concept of high energy accelerators. Since the saturation properties of quadrupole and dipole fields in a combined function magnet are different, there is advantage in machine tuning with separate quadrupole and dipole magnets. The Fermilab Main Ring was the first separate function accelerator.35 Most present-day accelerators are separate-function machines. For conventional magnets, the maximum dipole field strength is about 1.5 T and the maximum field gradient is approximately I/a [T/m] (see Exercise 1.12), where o is the aperture of the quadrupole in meters. For superconducting magnets, the maximum field and field gradient depends on superconducting coil geometry, superconducting coil material, and magnet aperture.
1.5
Colliders and Storage Rings
The total center-of-mass energy obtainable by having an energized particle smash onto a stationary particle is limited by the kinematic transformation (see Exercise 1.6). To boost the available center-of-mass energy, two beams are accelerated to high energy and made to collide at interaction points.36 Since the lifetime of a particle beam depends on the vacuum pressure in the beam pipe, stability of the power supply, intrabeam Coulomb scattering, Touschek scattering, quantum fluctuations, collective instabilities, nonlinear resonances, etc., accelerator physics issues have to be evaluated in the design, construction, and operation of colliders. Beam manipulation techniques such as beam stacking, bunch rotation, stochastic beam cooling, invented by S. Van de Meer,37 electron beam cooling, invented by Budker in 1966,38 etc., are essential in making the collider a reality. The first proton-proton collider was the intersecting storage rings (ISR) at CERN completed in 1969. ISR was the test bed for physics ideas such as stochastic beam cooling, high vacuum, collective instabilities, beam stacking, phase displacement acceleration, nonlinear beam-beam force, etc. It reached 57 A of single beam current at 30 GeV. It stopped operation in 1981. The first electron storage ring (200 MeV) was built by B. Touschek et al. in 1960
34M. 35The
a cascade of accelerators including proton linac, rapid cycling booster synchrotron, and a separate function Main Ring. 36 A.M. Sessler, The Development of Colliders, LBNL-40116, (1997). The collider concept was patented by R. Wiederoe in 1943. The first collider concept based on "storage rings" was proposed by G.K. O'Neill in Phys. Rev. Lett. 102, 1418 (1956). 3 7 S. Van de Meer, Stochastic Damping of Betatron Oscillations in the ISR, CERN internal report CERN/ISR-PO/72-31 (1972). 38 See e.g., H. Poth, Phys. Rep. 196, 135 (1990) and references therein.
Sands, A proton synchrotron for 300 GeV, MURA Report 465 (1959). Fermi National Accelerator Laboratory was established in 1967. The design team adopted
18
CHAPTER 1. INTRODUCTION
in Rome. It had only one beam line and an internal target to produce positrons, and it was necessary to flip the entire ring by 180° to fill both beams. Since the Laboratoire de l'Accelerateur Lineaire (LAL) in Orsay had a linac, the storage ring was transported to Orsay in 1961 to become the first e+e~ collider. The StanfordPrinceton electron-electron storage ring was proposed in 1956 but completed only in 1966. The e~e" collider moved from Moscow to Novosibirsk in 1962 began its beam collision in 1965. Since the 1960's, many e+e~ colliders have been built. Experience in the operation of high energy colliders has led to an understanding of beam dynamics problems such as beam-beam interactions, nonlinear resonances, collective (coherent) beam instability, wakefield and impedance, intrabeam scattering, etc. Some e+e~ colliders now in operation are CESR at Cornell, SLC and PEP at SLAC, PETRA and DORIS at DESY, VEPP's at Novosibirsk, TRISTAN at KEK, and LEP at CERN. The drive to reach higher energy provided the incentive for the high power klystron. The power compression method SLED (SLAC Energy Development), originated by P. Wilson, D. Farkas, H. Hogg, et al., paved the way to the SLAC Linear Collider (SLC). High energy lepton colliders such as NLC, JLC, and CLIC are expanding linear accelerator technology. On the luminosity frontier, the ^-factory at Frascati and B-factories such as PEP-II at SLAC and TRISTAN-II at KEK aim to reach 1033"34 crn"2 s""1. Proton-antiproton colliders include the Tevatron at Fermilab and SppS at CERN. The discovery of type-II superconductors39 led to the successful development of superconducting magnets, which have been successfully used in the Tevatron to attain 2-TeV cm. energy, and in HERA to attain 820-GeV proton beam energy. At present, the CERN LHC (14-TeV cm. energy) and the BNL RHIC (200-GeV/u heavy ion cm. energy) are under construction. The (40-TeV) SSC proton collider in Texas was canceled in October 1993. Physicists are contemplating a very large hadron collider with about 60-100 TeV beam energy.
1.6
Synchrotron Radiation Storage Rings
Since the discovery of synchrotron light from a then high energy (80-MeV) electron synchrotron in 1947, the synchrotron light source has become an indispensable tool in basic atomic and molecular physics, condensed-matter physics, material science, biological, chemical, and medical research, and material processing. Worldwide, about 70 light sources are in operation or being designed or built. Specially designed high-brightness synchrotron radiation storage rings are classified into generations. Those in the first generation operate in the parasitic mode
39Type II superconductors allow partial magnetic flux penetration into the superconducting material so that they have two critical fields BC\(T) and BC2(T) in the phase transition, where T is the temperature. The high critical field makes them useful for technical applications. Most type II superconductors are compounds or alloys of niobium; commonly used alloys are NbTi and NbaSn.
II. S.. Eds. 1 Acceleration Cavities The electric fields used for beam acceleration are of two types: the DC acceleration column and the rf cavity. booster synchrotrons. new acceleration schemes such as inverse free-electron laser acceleration. plasma wakefield acceleration. the advanced light source (ALS) at Lawrence Berkeley National Laboratory. and colliders. Vo /) is the effective peak accelerating voltage..40 The DC acceleration column is usually used in low energy accelerators such as the Cockcroft-Walton. the beam pulse is usually prebunched and chopped into appropriate sizes. etc. Some basic accelerator components are described in the following subsections. Before injection into various types of accelerators. laser . Third-generation light sources produce high-brilliance photon beams from insertion devices using dedicated high-brightness electron beams. (1996) and reference therein.. chopper. There are research efforts toward fourth generation light sources based on free electron laser from a long undulator.18) where AV = VQ sin(wrft + <> is the effective gap voltage. For a particle with charge e. These include the advanced photon source (APS) at Argonne National Laboratory. 40In recent years. (1. where charged ions are extracted by a high-voltage source to form a beam.g. storage rings. Figure 1. etc. No. the Japan synchrotron radiation facility (JSRF). 398. wrf is the rf frequency. and 4> is the phase angle. II.6 is a schematic drawing of a small accelerator complex at the Indiana University Cyclotron Facility. Chattopadhyay. pre-accelerators such as the high-voltage source or RFQ. The rf acceleration cavity provides a longitudinal electric field at an rf frequency that ranges from a few hundred kHz to 10-30 GHz. AIP Conf. et al. LAYOUT AND COMPONENTS OF ACCELERATORS 19 from existing high energy e+e~ colliders. Low frequency rf cavities are usually used to accelerate hadron beams. Particle beams are produced from ion sources. drift-tube linac (DTL). the European synchrotron radiation facility (ESRF). the energy gain/loss per passage through a cavity gap is AE = eAV. The beams can be injected into a chain of synchrotrons to reach high energy. have been proposed for high-gradient accelerators. See e. Van de Graaff. II Layout and Components of Accelerators A high energy accelerator complex is composed of ion sources. and high frequency rf cavities to accelerate electron beams. The second generation comprises dedicated low-emittance light sources. Advanced Accelerator Concepts. etc. buncher/debuncher. The beam can be accelerated by a DC accelerator or RFQ to attain the proper velocity needed for a drift-tube linac. Proc.
the Cooler Injector Synchrotron (CIS) at the Indiana University Cyclotron Facility.2 Accelerator Magnets Accelerator magnets requires stringent field uniformity condition in order to minimize un-controllable beam orbit distortion and beam loss. The circumference is 17. The superconducting magnets employ superconducting coils to produce high field magnets. and a transfer line are shown to illustrate the basic structure of an accelerator system.36 m. . the CIS synchrotron with 4 dipoles. Accelerator magnets are also classified into conventional iron magnets and superconducting magnets. Accelerator magnets are classified into field type of dipole magnets for beam orbit control.6: A small accelerator. DTL. The synchronization is achieved by matching the rf frequency with particle velocity. INTRODUCTION Figure 1.20 CHAPTER 1. The source. sextupole and higher-order multipole magnets for the control of chromatic and geometric aberrations. chopper. debuncher. Acceleration of the bunch of charged particles to high energies requires synchronization and phase focusing. and the phase focusing is achieved by choosing a proper phase angle between the rf wave and the beam bunch. The conventional magnets employ iron or silicon-steel with OFHC copper conductors. quadrupole magnets for beam size control. II. RPQ.
the superconducting coils are arranged to simulate the cosine-theta like distribution.8 T. and the total integrated dipole field is I Bdl = 2-rrpo/e = 2nBp.9 = / Bdl = -— Bdl. The iron plate can be C-shaped for a C-dipole (see Exercise 1.II.10 and the left plot of Fig. and Bp = po/e is the momentum rigidity of the beam. superconducting coils can be used. A gap between the iron yoke is used to shape dipole field. 1. that requires DC magnetic field. Figure 1. the maximum attainable field for iron magnet is about 1. To attain a higher dipole field. the pole shape is designed to attain uniform field in the gap. (1. The rectangular blocks shown in the left plot are oxygen free high conductivity (OFHC) copper coils. courtesy of R. the bending angle 6 is given by . These magnets are called superconducting magnets. The total bending angle for a circular accelerator is 2TT. Solid block of high permeability soft-iron can also be used for magnets in the transport line or cyclotrons. Gupta at LBNL). courtesy of G. .20) The conventional dipole magnets are made of laminated silicon-steel plates for the return magnetic flux for minimizing eddy current loss and hysteresis loss.7 T magnet flux. (1.7). For superconducting magnets. Berg at IUCF) and an SSC superconducting dipole magnet (right.7: The cross-sections of a C-shaped conventional dipole magnet (left. or H-shaped for H-dipole. LAYOUT AND COMPONENTS OF ACCELERATORS 21 Dipoles Dipole magnets are used to guide charged particle beams along a desired orbit.19) P o Jsi Bp Jsi where po is the momentum of the beam. Since iron saturates at about 1. From the Lorentz force law. For conventional magnets.
vacuum ports 4 1 B. II.7 shows the cross-section of the highfieldSSC dipole magnets.. * * « « • . e. INTRODUCTION Superconducting magnets that use iron to enhance the attainable magnetic field is also called superferric magnets. Quadrupoles A stack of laminated iron plates with a hyperbolical profile can be used to produce quadrupole magnet (see Exercise 1.22) Lil-. s.3 Other Important Components Other important components in accelerators are ion sources. azimuthal. 1. Electron Cyclotron Resonance Ion Sources and ECR Plasma (Inst. Bristol. the Lorentz force for a particle with charge e and velocity v along the J direction is given by F = evBi§ x {zx + xz) = —evB\ZZ + evB\XX. beam current and beam loss. New York. For a charged particle passing through the center of a quadrupole. R. 24) in a quadrupole. the magnetic field and the Lorentz force are zero. beam dump. where s — vt is the longitudinal distance along the s direction.. (1. Pub. Wolf. and x. of Phys. Geller. 1995).. The right plot of Fig. emittance meters. and z are the unit vectors in the horizontal.g. Bp ox d?z n we obtain d2x n (1 .22 CHAPTER 1. For high field magnets. z) from the center. 77m. where the magnetic field of an ideal quadrupole is given by B = B1(zx + xz). . ed.41 monitors for beam position. Handbook of Ion Sources (CRC Press. At a displacement (a. blocks of superconducting coils are used to simulate the cosine-theta current distribution (see Exercise 1. Defining the focusing index as „ =* .9).21) where B\ = dBz/dx evaluated at the center of the quadrupole. The equations of motion become 1d^_eBL (1. 5-12 T. and vertical directions. 1996).12).^LZ vl at1 jmv (123) vl at1 jmv Thus a focusing quadrupole in the horizontal plane is also a defocussing quadrupole in the vertical plane and vice versa.
For high energy experiments. and other nonlinear magnets for nonlinear stopband correction. The Tevatron at Fermilab facilitated the discovery of the top quark in 1995. power supplies. Computer control software retrieves beam signals. For synchrotron radiation applications in electron storage rings. Observation of a parton-like structure inside a proton provided proof of the existence of elementary constituents known as quarks. The LHC (7-TeV on 7-TeV proton-proton collider) at CERN will lead high energy physics research at the beginning of the 21st century. high energy accelerators are needed. wigglers and undulators are used to enhance the photon beam quality. beam stacking accumulation.1 fm) studies of the electromagnetic properties of nuclei. beam orbit and stopband correctors. Ill III. f2. Tevatron (1-TeV on 1-TeV proton-antiproton collider). The CEBAF 4-6-GeV continuous electron beams allow high resolution (0. sophisticated particle detectors are the essential sources of discovery. The timing and operation of all accelerator components (including experimental devices) are controlled by computers. sextupoles. such as the B-factories at SLAC and KEK and the ^-factory at DA$NE. High luminosity colliders. stochastic beam cooling. . etc. LEP (50-100-GeV on 50-100-GeV e+e~ collider) led the way in high energy physics in the 1990's. octupoles. The RHIC (100-GeV/u on 100-GeV/u heavy ion collider) will provide important information on the phase transitions of quark-gluon plasma. W*1 • • •. High energy accelerators have provided essential tools in the discovery of p. skew quadrupoles.l Accelerator Applications High Energy and Nuclear Physics To probe into the inner structure of the fundamental constituents of particles. etc. High energy colliders such as HERA (30-GeV electrons and 820-GeV protons). will provide dedicated experiments for understanding the symmetry of the fundamental interactions. Historical advancement in particle and nuclear physics has always been linked to advancement in accelerators.III. orbit bumps. Z°. etc. and septum. ACCELERATOR APPLICATIONS 23 and pumps. Radioactive beams may provide nuclear reactions that will lead to understanding of the nucleo-synthesis of elements in the early universe. SLC (50-GeV on 50-GeV e+e" collider). kickers. The advance in computer hardware and software provides advanced beam manipulation schemes such as slow beam extraction. J/^. The IUCF cyclotron has provided the opportunity to understand the giant Ml resonances in nuclei. and sets proper operational conditions for accelerator components.
proton and heavyion beams have become popular in cancer radiation therapy because these beam particles deposit most of their energy near the end of their path. Beam lithography is used in industrial processing. Beams have been used in radiation sterilization. Bert M. isotope production for radionuclide therapy. (a) Estimate the magnetic rigidity of proton beams at the IUCF Cooler Ring (kinetic energy 500 MeV). Since the discovery of X-ray in 1895.24 CHAPTER 1. and strategic equipment. p. Patent 2787564). INTRODUCTION III. and radiation treatments. etc. In particular. p is the beam momentum. Phys. . diagnosis. biology. By controlling the beam energy. Shockley in 1954 (U. Nath. and insulators. Radiation has been used in the manufacture of polymers. 43see e. invented by W. and neutron back-scattering have provided important tools for solid-state and condensed-matter physics research. Neutron sources have been important sources for research aimed at understanding the properties of metals. and biochemistry. and Ze is the charge of the particle.43 etc.42 synchrotron radiation sources.— . RHIC (momentum 250 GeV/c). Show that the magnet rigidity Bp is related to the particle momentum p by R \rp i _ P _ \ 3. what is the total length of dipole needed for each of these accelerators? 42The ion implantation.3357 p [GeV/c] for singly charged particles j G e V / .2 Solid-State and Condensed-Matter Physics Ion implantation. radiation has been used in medical imaging. solid-state physics. Coursey and R.g. c ] for particles with charge Ze ' &P Limj . ships. 3 Other Applications Electron beams can be used to preserve and sterilize agricultural products. 25. Particle beams have been used to detect defects and metal fatigue of airplanes. Radiation can be used to terminate unwanted tumor growth with electron. III. or ion beams. radiation hardening for material processing. Today. semiconductors. Free-electron lasers with short pulses and high brightness in a wide spectrum of frequency ranges have been used extensively in medical physics. proton. (b) If the maximum magnetic flux density for a conventional dipole is 1. Exercise 1: Basics 1. most of the beam energy can be deposited in the cancerous tumor with little damage to surrounding healthy cells.| 3^57 p where B is the magnetic flux density. p is the bending radius. has become an indispensable tool in the semiconductor industry. April 2000.7 Tesla.S. Tevatron (momentum 1 TeV/c) and SSC (momentum 20 TeV/c).
.l)uio.6 mm. The total power radiated by an accelerated charged particle is given by Larmor's formula: 1 2e2t>2 _ 1 2e2 dp^ dp. and 7 is the relativistic energy factor. Modern Microwave Technology (Prentice Hall.F. what is the equivalent field gradient? 4. Assuming that you can build a capacitor with a minimum capacitance of C = 1 pF. For an order of magnitude estimation. 5. if a sinusoidal voltage Vrt — t^costt/pt is applied to the dees. (a) Estimate the space-charge force for the SSC low energy booster at injection with kinetic energy 800 MeV and NQ = 1010 particles per bunch. If this force is exerted by a quadrupole.HIC). and the beam size is 3 /jm.. the maximum attainable kinetic energy is \/2eVmc2/n. show that a test charged particle traveling along at the same velocity as the beam.C. f e 2 JV o—n. experiences a repulsive space-charge force.r o 6 2 2r . (b) What happens if the test charged particle travels in the opposite direction in the head-on collision process? Estimate the space-charge force for the e+e~ colliding beam at SLC. the rms bunch length is as = 0. where the beam parameters are E = 47 GeV and NB = 2 x 1010. The resonance frequency of a LC circuit is / r = l/2-Ky/LC. what is the total length of dipoles needed in each accelerator? 2. (a) Let ip be the rf phase of the particle. 5 T (Tevatron) and 6./w. Use the following steps. NJ. 1987).5 mm thick will yield an inductance of about 3 x 10~7 H. Englewood Cliffs.6 T (SSC). ~ 47re0 3c3 ~~ 4TT€0 3m 2 c 3 ( dr ' dr ' "See V. Integrate this equation and show that the maximum kinetic energy attainable is \j2eVmc'1. where WQ = eB/m is the cyclotron frequency. with rms bunch length as = 2 m and beam diameter 4 mm. where e and m are the charge and mass of the particle. r - a r> a ( 2^607" 7 where 7 = l / \ / l . Veley./ 3 2 and e is the charge of the beam particle. Consider a uniform cylindrical beam with N particles per unit length in a beam of radius o. Here the number of particles per unit length is N = NB/(\/2nas).EXERCISE 1 25 (c) If superconducting magnets are used with magnetic fields B = 3. a 5-cm-radius single loop with wire 0. Show that the equation of motion in a uniform acceleration approximation is dip/dt = (7" 1 . (b) Denning a variable q = acos^>. where a = 2u>oeV/irmc2. the synchronous frequency is cj = eB/ym = UJQ/J. show that the equation of motion becomes {q/\/a2 — q2)dq = (27 — 2)UIQ dj. in the uniform acceleration approximation.5 T (R. dy2/dt — acostp. In a cyclotron. v. what value of inductance L is needed to attain 3 GHz resonance frequency? What is your conclusion from this exercise? Can you use a conventional LC circuit for microwave tuning? 44 3. to prove that.
dp _ 1 dE where w = j3c/p. INTRODUCTION where dr = dt/y is the proper time and pM is the four-momentum. and p is the bending radius. p. the gradient of the accelerating cavities will increase by a factor of 10. Classical Electrodynamics. The ratio of radiation power loss to power supply from an external accelerating source is = P _ 1 2e2 dE _ 2 re dE dE/dt ~ 47reo3m2c3«.78 x l(T 18 [m/(GeV) 3 ] for protons. . and p is the radius of curvature in the dipole. where the circulating beam currents are respectively 3 mA and 70 mA. 1 7.85 x l(r 5 [m/(GeV) 3 ] for electrons.26 CHAPTER 1. I is the total beam current. Assuming that electrons gain energy from 1 GeV to 47 GeV in 3 km at SLC. The radiated power becomes where ro is the classical radius of the particle. 45See J. what is the ratio of power loss to power supply? In the Next Linear Collider (NLC). The radiative energy loss per revolution of an isomagnetic storage ring becomes i. Calculate the energy dissipation per revolution for electrons at energy E = 50 GeV and 100 GeV in LEP. where p = 3096. Find the energy loss per turn for protons in SSC. Jackson. 2nd ed.(rf^) ~ S m c 2 ^ ' 1 where re = 2. and = 7 4nr0 3(mc 2 ) 3 = f 8. p changes direction while the change in energy per revolution is small..6 Tesla at 20 TeV. The power radiated 1 2e2 dp 2 = 1 2e2 dE 2 47re o 3m 2 c 3 Mr 47re0 3m 2 c 3 1 dx ' ' where dE/dx is the rate of energy change per unit distance. i. iii. ii. where the magnetic field is 6.e. Uo = ^CyE^/p. the circumference is 87120 m.853 m. the motion is along a straight path.45 (a) In a linear accelerator. What will be the ratio of radiation power loss to power supply? What is your conclusion from this exercise? (b) In a circular accelerator.175 m and the circumference is 26658. m is the mass.D.82 x 10~15 m is the classical radius of the electron. and the bending radius is 10187 m. Find the synchrotron radiation power loss per unit length in LEP and SSC. 468 (1975). Show that the power radiated per unit length in dipoles for a beam is where Ug is the energy loss per revolution.
(a) In fixed target experiments. is a measure of the probability (rate) of particle encounters per unit area in a collision process. What is the advantage of stretching the beam pulse length to 1 s in this experiment? (b) When two beams collide head-on. where si = s + fict. z. the luminosity is given by C = (dNB/dt)retarget. where the beam repetition rate is 0. „ 6+(62 + [l. s2 = s — fict. the luminosity is reduced by a factor exp{—b2/4a2}. and p\ and pi are the normalized distribution functions for these two bunches. L [cm~ 2 s~ 1 ]. f is the encountering frequency. z. . Consider a fixed target experiment. The luminosity. Show that the total energy for a head-on collision of two beams at an energy of 7 cm mc 2 each is equivalent to a fixed target collision at the laboratory energy of 7771c2 with 7 = 27c2m . + 1 [e_s]2)m+^a+{a2+s2)y2) 6 +(b2 + s2)V2\ where I is the length of the solenoid. beam particle per pulse is 1013. and the target thickness is 4 mg/cm 2 Au foil. the beam pulse length is 150 ns. Ni and JV2 are the numbers of particles. where dNB/dt is the number of beam particles per second on target. and ntarget is the target thickness measuring the number of atoms per cm2.4 Hz. show that the luminosity for two bunches with identical distribution profiles is £=fN1N1 4naxaz Show that when two beams are offset by a horizontal distance 6. 8.1. Using a Gaussian bunch distribution. Here NB is the number of particles per pulse (bunch) and / is the pulse repetition rate. where (dNB/dt) = NBf. J is the current density. b are the inner and outer cylindrical radii respectively.a2. the luminosity is £ = 2fNiN2[ pi(z. and CTS are respectively the horizontal and vertical rms bunch widths and the rms bunch length. 7. and s is the distance from one end of the solenoid. Thus the total counting rate of a physics event is it = (TphysA where <Tphys is the cross-section of a physics process. Find the instantaneous and average luminosities of the fixed target experiment. a. si)p2{x.EXERCISE 1 27 6.s ] 2 )i/ 2 . 1 p ( w ) = f x2 z2 s2 \ (2*)^axazas ^ { " ^ f " &f " 2^f) ' where crx. Show that the magnetic field on the axis of a circular cylindrical winding of uniform cross-section is Hs) = — {(t-°n»a+ia2 o^ ML. s2)dxdzdsd{fict). The average luminosity is given by (C) = (dNB/dt)ntATget.
J. where I\ is the total dipole current and (r. where y+ and y_ are the complex coordinates y = x + jz at an infinitesimal distance from the current sheet. INTRODUCTION (a) For an ideal solenoid.A. From elementary physics. 2568 (1966). R. where S is the cross-section area of the solenoid.28 CHAPTER 1. Apply this theorem to show that the cosine-theta current distribution on a circular cylinder gives rise to a pure dipole field inside the cylinder. the current / is positive if it points out of paper. show that the magnetic field becomes46 where n is the number of turns per unit length.z) = I*1 where j is an imaginary number. Note that the total energy stored in the magnet is given by the magnetic energy. and yo = xo + JZQ is the position of the current filament. (a) Show that the 2D magnetic field at location y = x + jz for a long straight wire is Bz(x.a). show that the magnetic field inside the current sheet is Bz = -ti0Ii/4a. 4689 (1967).47 (b) If the current per unit area of an infinitely long circular current sheet is X(r. Phys. High-field superconducting dipoles are normally made of current blocks that simulate the cosinetheta distribution.z)+jBx(x.B(y_) = jMdl/dy).6 -> a. . (b) For an ideal solenoid. and dl/dy is the current per unit length. This is the cosine-theta current distribution for a dipole. Bx = 0. the field at a distance r from a long straight wire carrying current / is B = ^oI/2-nr along a direction tangential to a circle with radius r around the wire. (c) The Beth current sheet theorem states that the magnetic fields in the immediate neighborhood of a two-dimensional current sheet are B(y+) . <j>) are the cylindrical coordinates with x = r cos <j> and z = r sin <j>. show that the inductance is L = non2(S = non2 x volume of the solenoid. <j>) = ( / i / 2 o ) cos <f> 5{r . Beth. 38. 9. 46Set 47See s = 1/2. and / is the current in each turn. 37. Appl.
Show that the dipole field of a window-frame dipole with two sheets of parallel plates having infinite permeability is given by B = fioNI/g = HQUI. show that the magnetic potential is <&m = —Kxz. where g is the gap between two parallel plates. Use the following identities: iry _ 4y ^ t a n h _ _ _ ^ _______ 1 Try _ 2 c o t h ___ 4y ^—^ + _ ^ _ _ _ _ . where N is the number of turns. Following Maxwell's equation.oN2(. g is the gap between two iron plates. Show that the inductance is L = /j. where NI is the number of ampere-turns per pole. The total power dissipation is P = [NI)2R. The achievable gradient is B\ = Bv0\e tip/a.Y. 151 (1991). where R = pi/A is the resistance. The inductance in an ideal quadrupole is _ 8M0iV2l 2 a2 (Xc o4 8noN2e 12x? j R i a2 2 X" where xc is the distance of the conductor from the center of the quadrupole.EXERCISE 1 29 10. A as the cross-sectional area of the conductor. and n = N/g is the number of turns per unit gap length. A300. Show that the magnetic field at the coordinate y = x + jz. Meth. Nucl.w/g = nan2 x volume of the dipole. Lee. between two sheets of parallel plates with infinite permeability is 48 Bz + jBx = Of tanh E i l l l + coth to!) 1. 1 . I is the current in each turn. In reality. with B = —V$ m . due to a thin current wire located at coordinates yo = XQ + jzo. The pole-tip field is jBpoie tip = Ka. where I and w are the length and width of the dipole. V x B = 0 in the current-free region. where wc is the width of the pole. Thus the pole shapes of quadrupoles are hyperbolic curves with xz = o 2 /2.Show that the gradient field is Bx = 2naNI/a2. and a is the half-aperture of the quadrupole. Inst. 12. The current flows in the xxz direction. and the magnetic field can be derived from a magnetic potential. For a quadrupole field with Bz = Kx. 48S. 11. Bx = Kz.9 Tesla. The equipotential curve is xz = constant. <&m. and p is the resistivity of the coil. To avoid the magnetic field saturation in iron. x2 should be replaced by x\ + xcwc. the pole-tip field in a quadrupole is normally designed to be less than 0.
INTRODUCTION 13. where fields are all transverse with phase velocity up = u/k. where Z = \fJTjl is the intrinsic impedance of the medium. the inductance per unit length is where the integral is carried out between two conductors. where / = Xvp is the current per unit length.z)e-^ks-ut\ H(f. the capacitance per unit length is C = X/V.30 CHAPTER 1. and df= dxx + dzz. For a transverse guided field propagating in the +s direction.~ . we assume E{r. show that lH-dr = X/eZ = Xvp.Z) = -V_L#E. because of the transverse nature of the electromagnetic field. t) = Hx(x. (a) Show that the frequency ui and the wave number k of the electromagnetic wave satisfy the dispersion relation UJ — fc/^/e/J. and surrounded by isotropic and homogeneous medium with permittivity e and permeability [i. z and s form the basis of an orthonormal coordinate system. where the external charge and current are zero. Show that there is a general relation: C L = /ie = 1/«|. z)e-^ks-ut\ B± = tfj_. where C and L are the capacitance and the inductance per unit length.t) = Ex{x. the electric field can be represented by £(Z. where V = cj>i — <j>2 is the potential difference between two conductors. Consider a pair of conductors with cross-sections independent of the azimuthal coordinate s. where <f> is the electric potential. By definition. VxE = on . . (b) Show that. (c) Similarly. Let x. and Vj_ is the transverse gradient with respect to the transverse coordinates. Using Ampere's law.Z). Maxwell's equations are V-(eE) = 0. The characteristic impedance of the transmission line is given by Rc = y/L/C = V/I. and A is the charge per unit length on conductors. Show that the transverse electromagnetic fields satisfy the static electromagnetic field equation. and the transverse plane wave obeys the relation H = ^ sxEj_.
. Kerst and R. For charged ion beams.5 RG174/U 0. (a) Assume that the vertical component of the magnetic flux density is B . there are thermionic sources.. show that the equations of motion become £ + wg(l-n)£ = 0. Rev. where UJQ = v/R = eBo/jm is the angular velocity of the orbiting particle. laser-driven electron sources.) > where n is the field index. In the cylindrical coordinate system.=fl0(l)-«Bb(l-B4^ + . etc. the transverse oscillations of charged particles in linear or circular accelerators are generally called betatron oscillations. 15. 60.z are respectively the radially outward and vertically upward directions. Ion sources are indispensable to all applications in accelerators.152 98. C + wg< = 0.307 93. Br. r ^ = Type Diameter Capacitance Inductance [cm] [pF/m] LuH/m] RG58/U 0. Show that the radial magnetic field with BT = 0 at z = 0 is nB0 (dBz\ V dr ) T = R R (b) Using i = (r-R)/R and ( = z/R. Show that the stability of betatron motion requires 0 < n < 1.73 I 96..8 | | Rc Delay time [Q] [ns/m] 50 50 50 | 14. Derive the transverse equations of motion for electrons in a betatron49 by the following procedures. 6 is the azimuthal angle. the equation of motion for electrons is at at where f.4 RG218/U I 1.EXERCISE 1 31 (d) Show that the capacitance and the inductance per unit length of a coaxial cable with inner and outer radii r\ and r-i are 2ne T P 1 r2 n—/—v> L = 7rln~ • In(r2/ri) 2vr r\ Fill out the following table for some commonly used coaxial cables. For electron beams. there are many different configurations for generating plasma 49See D.Bz are the radial and vertical components of the magnetic flux density. and 9 = v/r is the angular velocity. Phys. 53 (1941). Because of this seminal work. Serber. rf gun sources.
. INTRODUCTION sources for beam extraction.50 Charged ion beams are usually drawn from a spacecharge ion source at zero initial velocity.00861 16. 188 (1988). The Paraxial Ray Equation: In the free space. and Ar+ ion sources are given by the following table. Phys. N + . AIP Conf. The flow of charged ions is assumed to be laminar. and is the perveance of the ion source. proton. Rev. (j> is the azimuthal coordinate. Proc. Let s be the distance coordinate between the parallel plates with a = 0 at the emitter. s) for the cylindrical coordinates in paraxial geometry. Proc. Here the microperveance is defined as 1 fj. where r is the radial distance from the axis of symmetry. the condition of maximum space-charge shielding is equivalent to V = 0 and dV/ds = 0 at s = 0. See also A. Large Ion Beams (Wiley. the electric potential obeys the Laplace equation V2V = 0. and v is the velocity of the ion. 492 (1911). and s is the longitudinal coordinate. show that the Poisson equation becomes ds2 e0 \2e) where J = pv is the current density. Child.P = 1 x 10" 6 A/V 3 / 2 .0545 | 0. where Vo is the extraction voltage at the anode.32 CHAPTER 1.D.0272 | 0.51 (c) Show that the space-charge perveance parameters for electron. No. Symp. Rev. New York.P) 2. 1988). deuteron. AIP Conf. e and m are the charge and mass of the ion. 210 (1990). No. 51 C. 32. I D+ I He+ I N+ I A+ 1 e I p X (fJ.334 0. Forrester. In the space-charge dominated limit. 32. 0. Langmuir. The relation of the current to the extraction voltage is called Child's law.0385 ~0. Proc. The Poisson equation becomes d?V _ p where V is the electric potential. Phys. Production and Neutralization of Negative Ions and Beams. Int. and eo is the permittivity. The maximum beam current occurs when the electric field becomes zero at the emitter.g. He + . (b) For a space-charge dominated beam. p is the ion density in the parallel plate.0146 [ 0. Show that the maximum current is J = X^o3/V. the electric field between the anode and the cathode is maximally shielded by the beam charge. we expand the 50 See e. Assume a simplified geometry of two infinite parallel plates so that the the motion of ions is one-dimensional. and s = o at the anode.T. I. Using the basis vectors (f. 450 (1913). on Electron Beam Ion Sources. (a) In the non-relativistic limit with laminar flow.
1/(3) T/(5) where VJj correspond to nth-derivative of Vb with respect to s. . This result is the basis of beam position monitor design.K. known as the paraxial ray equation. Pierce. 1943). Introduction to Electron Optics. Show that the equation of motion for the radial coordinate. s) and the electric field E = Err + Ess are VW V(r. The multipole expansion can be obtained by using the identity cosn<? + jsinn9 = e^nB. (Oxford. Electron Optics and the Electron Microscope. Consider a line charge inside an infinitely long circular conducting cylinder with radius b. where a is the distance from the center of the cylinder. a sin <f>). Show that the induced surface charge density on the cylinder is 53 (b 6 \ a( A 2nb b2 + a2- b2 . Theory and Design of Electron Beams. where the overdot represents the time derivative. F. Radio Engineers' Handbook. csin<£). 1945). . The paraxial ray equation can be used to analyze the beam envelope in electrostatic accelerators. (Van Nostrand. 53 Let the image charge be located at c = (ccos<£.4>) '<Pvl) = -5S[1 + 2 SU) A f °° /a\n cosn <*-"4 1 where <^w is the angular coordinate of the cylindrical wall surface. The induced surface charge density is a = eoEr.52 17.R.EXERCISE 1 33 position vector as R = rr + ss.c? 2ba cos(<£ w . . and (j> is the phase angle with respect to the x axis. Zworykin et al. 1950). 52 V. The line-charge density per unit length is A. Let VQ(S) be the electric potential on the axis of symmetry. The electric field is E = —V$. (Wiley. becomes Vr" + \v'r> + -V"r = 0. Cosslett. (McGraw-Hill. we obtain c = b2/a and Ai = —A. and the coordinates of the line charge are a = (a cos <j>. J. The equation of motion for a non-relativistic particle in the electric field is mR = eE.s)=V0(s)-^rr*-^rri y(4) + ---. Terman. Show that the electric potential V(r. 1949).. where V replaces Vo for simplicity and the prime is the derivative with respective to s. then the electric potential for infinite line charges at f is $(r) = In \r — a -I In r — cl. Using the condition EQ = 0 on the conducting wall surface in the cylindrical coordinate. V.E.
.
Lattice design has to take these field errors into account. control and feedback-correction of collective beam instabilities. A closed orbit in a linac is the orbit with zero betatron oscillation amplitude. In the first-order approximation. In actual accelerators. the resulting closed orbits will also depend on the particle momentum. beam distribution. The dispersion function. we derive the particle Hamiltonian in the Frenet-Serret coordinate system. magnetic field errors are unavoidable. 1A closed orbit in a synchrotron is defined as a particle trajectory that closes on itself after a complete revolution. This defines a closed orbit. Betatron motion around the closed orbit is determined by an arrangement of quadrupoles. where p0 is the momentum of a reference particle. high-intensity and high-brightness beams require measurement and correction of linear and nonlinear resonances. It is now used for transverse motion in all types of accelerators. the method discussed in this chapter can also be applied to a linac or a transport line. therefore the closed orbit and the betatron motion will be perturbed. since the bending angle of a dipole depends on the particle momentum. where the betatron motion is equivalent to an initial value problem. Serber on the transverse particle motion in a betatron. Particle motion with a small deviation from the closed orbit will oscillate around the closed orbit.Chapter 2 Transverse Motion In an accelerator. the deviation of the closed orbit is proportional to the fractional off-momentum deviation (p — Po)/Po. 2In principle. We discuss the Floquet transformation to action-angle variables. defined as the derivative of the closed orbit with respect to the fractional off-momentum variable. we examine the properties of linear betatron motion. and the chromatic aberration of the betatron motion play a major role in the accelerator's performance. In particular. bending magnets are needed to provide complete revolution of the particle beam. Various aspects of transverse particle motion in synchrotrons will be discussed in this chapter. In synchrotrons. Kerst and R. II. 35 . Furthermore. The terminology of betatron motion is derived from the seminal work of D. I.2 In Sec. called the accelerator lattice. In Sec. and measurement. the transverse particle motion is divided into a closed orbit1 and a small-amplitude betatron motion around the closed orbit.
(2. The energy and momentum of the particle can be expressed as E = ymc2 = mc2dt/dT and p = rwyv' = mdf/dr. Section IV deals with the off-momentum closed orbit and its implications for longitudinal synchrotron motion. and (x. we study the effects of linear magnetic imperfections (dipole and quadrupole field errors) and their application in beam manipulation. In Sec. (2. and properties of the envelope function. B = V x A. v = df/dt is the velocity. and p is the mechanical momentum. Section VIII introduces the basic concept of transverse collective instabilities and Landau damping.V $ .36 CHAPTER 2.2) The canonical momentum P = dL/dv = p + eA. (2.1) at where p = ymv is the relativistic kinetic momentum. (2. . • • • pairs are conjugate phase-space coordinates. and 7 = l / \ / l — v2/c2 is the relativistic Lorentz factor.3) and Hamilton's equations of motion are dH X~WX' Px~ p_JJI dx' e ' {2A) where the overdot is the derivative with respect to time t. Ill. and Section VI describes linear betatron coupling.dA/dt. ^=F = e(E + vxB). Section IX lays out a general framework for the synchrotronbetatron coupling Hamiltonian.1) can be derived from Lagrange's equation d_ (dL\ _d£_() where the Lagrangian is L = —mc2Jl — v2/c2 — e$+ev-A. and also with the lattice design strategies for variable 7T and minimum dispersion action. m is the mass. The electric and magnetic fields are related to the vector potential A and the scalar potential $ by E = . Px). Section V describes the chromatic aberration and its correction. e is the charge. we examine the effects of low-order nonlinear resonances. The Hamiltonian for particle motion is given by H = P-v-L = c[m2c2 + {P. Thus Eq. where r is the proper time with dt/dr = 7. I Hamiltonian for Particle Motion in Accelerators The motion of a charged particle in electromagnetic field E and B is governed by the Lorentz force. VII. In Sec. TRANSVERSE MOTION beam emittance.eA)2}1'2 + e$.
9) 3 Using Eq. Z'(S) = -T(S)X{S) . ?o(s) is the reference orbit.8) P(s) where the prime denotes differentiation with respect to s. 2.6) where p(s) defines the radius of curvature. HAMILTONIAN FOR PARTICLE MOTION IN ACCELERATORS 37 Figure 2. (2. where s is the arc length measured along the closed orbit from a reference initial point. The particle trajectory around the reference orbit can be expressed as r(s) = fo(s) + xx(s) + zz(s) . x.I Hamiltonian in Frenet-Serret Coordinate System Let fo(s) be the reference orbit (see Fig. and T(S) is the torsion of the curve. . Here x and z are betatron coordinates. I. we find a centr i p etai = \d2r0/dt2\ = {ds/dt)2\(d/ds)(df0/ds)\ = v21(ds/ds)|. where v = ds/dt is the tangential velocity.5) The unit vector perpendicular to the tangent vector and on the tangential plane is3 x{s) = ~p(s)^ . (2. s and z form the orthonormal basis for the right-handed Frenet-Serret curvilinear coordinate system with x'{s) = -priis) + T{S)Z{S). s and i form the basis of the curvilinear coordinate system.7) The vectors x. where r(s) = 0.5). we discuss only plane geometry.I. (2.1). The magnitude of the bending radius is p = ^2/acentripetal = \ds/ds\.1: Curvilinear coordinate system for particle motion in synchrotrons. (2. Any point in the phase space can be expressed by r = TQ + xx + zz. The tangent unit vector to the closed orbit is given by i(s) = d*&. The unit vector orthogonal to the tangential plane is given by z(s) = x{s) x s{s) . For simplicity. (2. (2.
s.[fo(s) + xx(s) + zz{s)\ . (2. z) are (see Appendix A) ps = -d-~ = {l+xlp)P-s. — ps as the new Hamiltonian. z). dH x = —. >Px ~ ^ ' z dps_ . _dp1 ~ a 'Pz ~ 13~- . The conjugate momenta for the coordinates (x. dH .10) where P is the momentum in the Cartesian coordinate system. A x m2c2 + ^ + lyjp + (p.ps = --^-. . and the conjugate phase-space coordinates given by x.pz. s.px.11). A s = (l + x/p) A .e. . (2.z . The transformation reduces the degrees of freedom from three to two. 1/. 1 dp^ ~ KTi x _ ~ dp. we find .13) Note that As and ps are not simply the projections of vectors A and P in the s direction.1OJ dH at opx ox opz oz This is Hamilton's equations of motion with s as the independent variable. Hamilton's equation becomes .17) ~ atr' When the scalar and vector potentials 4> and A are independent of time. But the price to pay is that the new Hamiltonian depends on the new variable s.s .eAzfj =A-x. opz oz .z. _ ap. dH z=—.px = -—-. s. In the new coordinate system. ops os . z.^ 1 = P. ox Px = -?h oz = p.-H. (2.eAxf + (p.14) The next step is to use s as the independent variable instead of time t [16]. dH .X. i. Using the relation dH = (dH/dpx)dpx + {dH/dps)dps — 0 or ds s \apx) \dpsj dpx where the prime denotes differentiation with respect to s. Because of the repetitive nature of the accelerator. dH . (2. .38 CHAPTER 2.—. Az =A . z) = -P.p2 = . .12) (2. dH s= ^—. a _ ap. us Px = . t. opx ox .AX and Az are obtained by substituting the vector A in Eq. TRANSVERSE MOTION To express the equation of motion in terms of the reference orbit coordinate system (x. x.11) The new Hamiltonian becomes { (n - PA \2 ~\ 1/2 where AS. (2. the new Hamiltonian —ps is also time independent. (2. we perform a canonical transformation by using the generating function F3(P.
The total energy and momentum of the particle are E = H — e<j) and p = JE2/c2 — m2c2. t.eAzf) . -H). The periodic nature of the new Hamiltonian can be fruitfully exploited in the analysis of linear and nonlinear betatron motion.A2) hs dz _ ~ 1 dAs hs dz' °z~ _ 1 d{hsA2) hs dx 1 dAs ~ hsdx' KZ-1 (2. In accelerator applications.eAs. (2. 1.I. \dA. we can assume Ax = Az = 0. (x. 8A3 dx dz J s + 1 [dfoAa) 9AJ . for an accelerator with transverse magnetic fields.20) V$ = —X + ——S + —Z. z.eA°> where the phase-space coordinates are (x.z)x + Bz{x. furthermore. HAMILTONIAN FOR PARTICLE MOTION IN ACCELERATORS 39 the dependence of the new Hamiltonian on s is periodic. we expand the Hamiltonian up to second order in px and pz H « -p (l + ^ + i ± ^ [(px . hz = 1. The new Hamiltonian H = -ps is then given by k=.pz. we consider only the case with zero electric potential with $ = 0.19) In the Frenet-Serret coordinate system. s. /I2 = A • 5.eA>)2]V2 .{l+f) [ ( g T^ ) 2 -mV -{p* ~eA*)2 -{pz .21) > . z)..z)z. the scale factor becomes hx = l. We have hs = 1 + .px.2 Magnetic Field in Frenet-Serret Coordinate System (2. z' fts hs [~dx~~~~ds~\ ds hs ds where ^ = A • x.eAxf + (Pz . The two-dimensional magnetic field can be expressed as B = Bx(x. Since the transverse momenta px and pz are much smaller than the total momentum. ox hs os dz V. where _ 1 d(h.z. and A3 = A.A = ! \d(hsA1) | 8A2 | d(hsA3)l hs dx [ ds /is [dx dx ds dz \x+[dz dz dz J' 9$ 1 d$ 3$ vy v A v 1 [ ^ 3 _ d(hsA2)] .
an are called 2(n + l)th multipole coefficients with dipole 6o.. n> = 1 cm.Normally the normalization constant Bo is chosen as the main dipole field strength such that bo — I. The resulting magnetic flux density is given by4 oo Bz + jBx = Bo £ ( b n + jan){x + jz)n 71=0 (2.S. dipole roll do. 4The multipole expansion of the magneticfieldis usually rescaled to obtain Bz + jBx = BQ n=O ^(bn+JanX^^r. 62.BZ (see Exercise 2.1. skew sextupole 02. in our coordinate system. .3). sextupole 62. quadrupole 6i. Thus we have Bobo = —[Bp]/p. Using Maxwell's equation V x B = 0.24) where j is the imaginary number. skew quadrupole ai. The resulting bn and an coefficients are dimensionless. For straight geometry with hs = 1. In Europe.6) oz hs oz ox hs ox and As can be obtained through power series expansion General solutions of BX.40 CHAPTER 2. (2-27) where.x=z=o n ~ Bon\ dx" B = l = 0 ' ( b) where bn. and for RHIC and Tevatron. 5 Note that the multipole convention used in Europe differs from that in the U. we have d i d A d idAs_ o~I—5 <"5~T—3~ u\i. The complex 2D magnetic field representation in Bz + jBx is called the Beth representation (see Exercise 1. r\> = 2. the — and + signs are used for particles with positive and negative charges respectively. p is the bending radius. etc. where Bp is the momentum rigidity of the beam. physicists use 61.E ( ^ +io») (^ + ^ ) n .5 The effective multipole field on the beams becomes 1 -I 00 — (Bz + jBx) = T .10). TRANSVERSE MOTION with As = hsA2.02 for quadrupole and skew quadrupole.25) with bn .54 cm. and As can be expanded in power series as As = Sosft g *££{* + jzr 1 ] . ai for dipole and dipole roll.2. Vb where r^ is a reference radius. For high energy accelerators such as the SSC and LHC. and Bz = ^ and Bx = ~"^ 1 . (2. we obtain V\AS = 0..Bon! 3x. 5ft[.] represents the real part. and bo = 1. etc.
29) by changing the time variable to the coordinate of orbital distance s._dH_ opz . Bz = -Bo + -—^x = ^Bo + BlX.4 Particle Motion in Dipole and Quadrupole Magnets We consider a on-momentum particle with p = Po. expand the magnetic field up to first order in x and z. (2.29) can be derived through Newton's law of acceleration (see Exercise 2. i.3 Equation of Betatron Motion Disregarding the effect of synchrotron motion (see Sec.] represents the imaginary part of the expression.1. B = . ox Bx = —^z = BlZ. the betatron equations of motion become Pl ..e.e. and e is the charge of a particle.V $ m (see Exercise 2. 1.. i. B P \\ P) (2-29) I Bpp y p) where we neglect higher-order terms. (2. ox (2. 1. the magnetic field can also be derived from a scalar magnetic potential $ m . Alternatively. _ dH az With the transverse magnetic fields of Eq. p is the momentum of the particle.__dH_ ox . Hamilton's equations of betatron motion are given by . Po is the momentum of a reference particle. HAMILTONIAN FOR PARTICLE MOTION IN ACCELERATORS 41 Since V x B = 0 in the current free region.22). the upper and lower signs correspond to the positive and negative charged particle respectively._dH_ opx .3).I. The sign convention is chosen such that Bp is positive. Eq.2). The scalar magnetic potential is *m = -Bo3 [ £bj~^YL(^+ J * H Ln=0 J (2-28) where 3[. Bp = po/e is the magnetic rigidity. The equations of motion are given by ~ m ~± B p ' Z~^ TU 7 B p ' which can be transformed into Eq. (2.e.30) . IX).1. x" = f/v2s. i.
we have Kx = —Kz. and makes an angle 9/2 with the pole-faces in the rectangular dipole.31) and (2. where 1/p = 0.g.2: Schematic drawing of the particle trajectory in a sector dipole and in a rectangular dipole.32) are given below. A weak focusing accelerator requires 0 < n(s) < 1. • The focusing functions Kx. z" + Kz(s)z = 0. For a strong-focusing accelerator. Eq. Some observations about the linear betatron equations (2. to be discussed in next section. (2. Thus Eqs. This means that a horizontally focusing quadrupole is also a vertically defocussing quadrupole and vice versa. Such a dipole is called a sector dipole with perpendicular entrance and exit angles to the edge of the dipole field (see Fig.2. The solution of Hill's equation satisfies the Floquet theorem. where the entrance and exit angles of particle trajectories are not perpendicular to the dipole edge. The betatron equation of motion.2a).42 CHAPTER 2. there is an edge focusing/defocussing effect (see Exercise 2. • A horizontal bending dipole has a focusing function Kx = 1/p2. \ / (b) rectangular dipole • In a quadrupole. Kz are periodic functions of the longitudinal coordinate s.2). Kx = l/p2^Kl{s). and Kz = 0. (2. For non-sector type dipoles. becomes x" + Kx(s)x = 0. and the quadrupole gradient function B\ = dBz/dx is evaluated at the closed orbit. Kz = ±/fi(s). n(s) ss ±350 for the AGS. Note that the particle orbit is perpendicular to the pole-faces of the sector dipole magnet.31) and (2. \ / \ / \ 1 (a) sector dipole e/2/K / \ \ \ e' / JKj/Z / \ Figure 2. \ / \ . e.32) where Ki(s) = Bi(s)/Bp is the effective focusing function. (2.31) (2. TRANSVERSE MOTION where Bo/Bp = 1/p signifies the dipole field in defining a closed orbit.29). . and the upper and lower signs correspond respectively to the positive and negative charged particles. 2. \n\ 3> 1. The focusing index is given by n(s) = p2 Ki(s).32) are Hill's equations with periodic boundary conditions.
z) be local polar coordinates inside a dipole. Let (x.s. Expressing the scalar potential in power series of particle coordinates.z). show that '-»+•*-%. 1 + z/p & ' Derive Eq. where 6 = s/p is the angle associated with the reference orbit. *=E V.S. The particle coordinate is r — (p + x)x + zz.29) through the following geometric argument. . Thus the electric field and magnetic field can be expanded by scalar potentials with B = . and show that P2 - Bp{l+ p> ' Z ~ W1 + p> ' where the prime is the derivative with respect to s. transverse magnetic fields are o = ?_^£ 1 + x/p dz ' o z 1 dA. 2. The momentum of the particle is p = ymr. and p is the radius of curvature.(2.. where d9 = vsdt/(p + x). ds = pd9. 3.z). i. (a) Using Eq. with B = Bxx + Bzz. where both scalar potentials satisfy the Laplace equation & with V 2 $ . ' % where Bp = ymvs/e is the momentum rigidity and vs is the longitudinal velocity. In the curvilinear coordinates (x. (2. dp/dt = ymr.e. and the overdot corresponds to the derivative with respect to time t.3*. s. (c) Transform the time coordinate to the longitudinal distance s with ds — pd9. Derive Eq. where p is the bending radius. we then have _ 2 -.V $ m . show that f = xx + (p + x)9s + zz. (2. where j is constant in the static magnetic field.19). Inside the vacuum chamber of an accelerator. Similarly. f=[x-(p + x)82]x + [2x9 + (p + x)9]s + zz. In the Prenet-Serret coordinate system (x.1 1. where $ stands for either $ m or < e = 0. (b) Using dp/dt = ev x B. (2.1 43 Exercise 2.29) from the Hamiltonian of Eq. . E = . we have V x B = 0 and V x E = 0.V $ e . 92$ + a? + TTTx^rnrx &) =0xizi 1 5 .8). v *= TTW[1+hx]^] * 1 d .A ^jv .EXERCISE 2.. 1 <9$N where h = 1/p.
3tj+2.l)h2Aid . show that the magnetic potential.i(3i .3i(i - l)h2Ai_2:j+2 . etc. this serves as a general method for deriving the magnetic field map.19) for the particle motion in the solenoid is H = -p+ ^-[(p» . Assuming yloo = 0. the potential for a quadrupole is given by the ^4n term and the skew quadrupole arises from the A20 term.eAxf + {pzeAz)2}. TRANSVERSE show that Aij satisfies the following iteration relation: Aij+2 = -A'lj . show that the vector potential can be expressed as A=-[lrbo(s)-±rHo'(S) + --j 4>. Bz = z-£b2k+i(x2 + z2)k.l)2h3Ai-u + ih'A'^j . and (j> = (—zx + xz)/r.44 CHAPTER 2. 6A word of caution: the magnetic potential obtained here can not be used as the potential in the Hamiltonian of Eq. (b) The Hamiltonian of Eq. is6 $ = -Booz+±A2o{x2-z2) + Anxz+l-A3o{x3-3xz2) + \B'oW +\A2lx2z + UB'0'0 . In particular. up to the fourth order with i + j< < 4. where r = xx + zz.xz3) \A[IXZ\ 4. (2.ihA'Uj -3ihAi-ld+2 -i{i . r = \jx2 + z2. However.(3i + MOTION l)hAi+1J . (2. . Show that the vector potential is In a cylindrical coordinate system. and AQI — —Boo i n a rectangular coordinate system with h = h' = 0. where the prime is the derivative with respect to s.6x2z2 + z4) +±A2'0(-3x2z2 + z") + \A31(x*z .18).i(i .Aig = 0. The field components in the current-free region of an axial symmetric solenoid are 00 OO OO Bx = xJ2b2k+l(x2 k=0 + z2)k. k=0 Bs = Y2hk(x2 + z2)k. k=0 (a) Show that the coefficients are 2(feTi)*"' W^)h'2k+l' b2k+1 = b2k+2 = where the prime is the derivative with respect to s.Ai+2j .A21)z3 + ^-A40(x4 .l)(t - 2)h3Ai-.
2gx' . included in the g' terms. (c) Up to third order. .7 For normal multipoles with mid-plane symmetry with Bz{z) = Bx(-z). The linearized equation can be solved analytically. provides both horizontal and vertical focusing. Note also that the effects of the ends of a solenoid. b. i.^-x(x2 + z2). p.1 Show that the lineaxized equation of motion is (see also Exercise 2. Bx = zJ2 ^z2\ i.k=0 OO OO Bs = z £ «#****. have been included to obtain this Hill's equation in the rotating frame. Letting y = x + jz. show that the equation of motion is x" + 2gz'+g'z z"-2gx'-g'x = ^z'(x2 + z2) + ^-z(x2 + z \ = -^-x'(x2 + z2) .k=0 i.6. CERN 85-19.k=O OO where a. Transforming the coordinates into the rotating frame with y = ye-je{s)^ where g = fs rf Jo show that the system is decoupled. and the decoupled equation of motion becomes y" + g2y = 0.EXERCISE 2. 45 where g = ebo/2p = eB^/2p is the strength of the solenoid. 25 (1985). the most general form of expansion is Bz = £ b^z2". in the rotating frame. Bs(z) = -Bs{-z). Steffen. Thus the solenoidal field. c can be determined from Maxwell's equations: V x B = 0 and V • B = 0. Bx(z) = -Bx(-z). independent of the direction of the solenoidal field. Show that Maxwell's equations give the following relations: ai'k = WTibi+1*' Ci'k + p Ci .g'x = 0. Consider the transverse magnetic field in the Prenet-Serret coordinate system. Z o 5.1 >* = 2 T T 1 6 ^ ' 7 See K.2) x" + 2gz' + g'z = 0. show that the coupled equation of motion becomes y"-j2gy'-jg'y = 0. z" .
0\. where p —> oo.o 3 + — ^ p } 1} + ~p(B2'° + 1 7 + —] Bs 1 D + 2[Bl'° " ( ~ o '''' = BjiOz + ( J B i .n " * ' = Bloz + 2B2.^ ^ + ^{x+iz)z2 l + ---. B z ( z = 0 ) = B o f i + B l f l x + B 2 f i x 2 + B 3 f l x 3 + •••. the end field has an octupole-like magnetic multipole field. Thus for a finite length quadrupole with B[ 0 ^ 0.46 CHAPTER 2. R'" .0 ~ ^~^~' *> OR 0.e.^ ^)a. Show that in a pure multipole magnet.oxz + 3Bs. Assuming that we can measure the Bz at the mid-plane as a function of x. where S^o are functions of s..o + Blfix + B2iOx2-(B2fi -{SB3yo H Bx . 2B2. where j is the complex number. the magnetic field can be expanded as Bz+jBx = YJBn»{x n=0 +3 z T .lrr.n . s.-°i. .0 +^ .ox3 (B'°<°Vnn-r2J- 1/n (-D2.^ > ) ^ + ( ^ .ox2z--{3B3. i. 1 1" —2~) + 21. o .o + -g h0. show that the field map is Bz = Bo. B + ^)z2 B + B3. TRANSVERSE MOTION where the prime is the derivative with respect to s.2.
we study. we can apply the Floquet theorem (see Appendix A. The nominal betatron tunes are ux = 6. Sec. the Courant-Snyder invariant and emittance. (2. where four combined function magnets are arranged to form a basic focusing-defocussing periodic (FODO) cell.7 and vz = 6.Bi{s)/Bp. (2. betatron tune. The total length of a repetitive cell is 19. LINEAR BETATRON MOTION 47 II Linear Betatron Motion Particle motion around a closed orbit is called betatron motion.34) 8The focusing functions are Kx = l/p2 + B1(s)/Bp and Kz(s) = -B1{s)/Bp for negative charged particles. For example. BF BF BD BD ] • • • • s=0 BE s=L Figure 2.0208186 m~2. Let y. Since the amplitude of betatron motion is small. y' represent either horizontal or vertical phase-space coordinates. and L is the length of a periodic structure in an accelerator.3: A schematic drawing of the Fermilab booster lattice. .33) where Kx = 1/p2 . the linearized betatron equation of motion governed by Hill's equation x" + Kx{s)x = 0. 1 Transfer Matrix and Stability of Betatron Motion Because accelerator components usually have uniform or nearly uniform magnetic fields.889612 m and focusing function Kp = 0. The accelerator is made of 24 such FODO cells. Fig. A small trim focusing quadrupole is used to change the betatron tune.0244817 m~2 and Ku = -0.3 shows a schematic drawing of the Fermilab booster lattice. and thus the superperiod of the machine is 24. and the envelope equation. (2.II.7588448 m. z" + Kz{s)z = 0. Floquet transformation. 2. In this section we study linear betatron motion. Kz(s) = B^/Bp. It consists of four combined-function magnets of length 2. then Eq.33) becomes y" + Ky(s)y = 0.8. II. 8 The focusing functions are periodic with Kx<z(s + L) = KXyZ(s). 1.5) to facilitate the design of an accelerator lattice. the focusing functions KXtZ(s) are piecewise constant. Exploiting the periodic nature. in this section. and B^s) = dBz/dx evaluated at the closed orbit.
35) ! acoshiyf^Ks + b). (2.e. we neglect the subscript y hereafter. K> 0.— -fesinVKe\ .2/2 of Hill's equation. The transfer matrix for a constant focusing function K is [I cos^KE . we can express the solution of Eq.34) as y(a) = M{s\so)y{so). we obtain detM = 1. For any two linearly independent solutions 2/1. Since K is finite. . The solutions of Hill's equation with constant Ky are9 a cos{y/Ks + b). the transfer matrix for a > quadrupole reduces to i J > ^defocussing = ( 1/y i )' (2.48 CHAPTER 2. s) = yiy'2 . Letting be the betatron state-vector. K = 0.)-(*>) IW (2. as + b.38) Since the Wronskian obeys W(s) — [det M]W(so). f = \imt^ol/(Ke). M(s\sQ) = < ( 1 i ) / cosh J\K\£ v { VvWnhy^ •^focusing — [ _-i I f -j=sinh J\K\£\ v 1*1 Jl_ cosh^f^ J K < 0: defocussing quad. convention for the transfer matrix of a thin-lens quadrupole is M q u a d =(_ 1 1 / / JJ. (2. (2. where / > 0 for a focusing quadrupole and / < 0 for a defocussing quadrupole. In this case. where £ = s — SQ. K < 0.y[y2.39) where / is the focal length given by10 10The 9To simplify our notation. the Wronskian is independent of time. *(.37) where M(S\SQ) is the betatron transfer matrix. dW W(yi. TRANSVERSE MOTION with the periodic condition Ky(s + L) = Ky{s). i. ff = 0: drift space „ n t . -£• = 0. y and j / must be continuous.y2. and the integration constants a and 6 are determined by the initial values of y0 and y'o.— .— K > 0: focusing quad. In thin-lens approximation with £ — 0.
(2. (2.43) where C and S" are the derivatives of C and 5 with respect to s. (2. « '*«')• 49 (240) where 6 = £/p is the orbiting angle and p is the bending radius. This means that the effect of a dipole with a small bending angle is equivalent to that of a drift space.*>) . e. M{SM = ( ^ j J£.45) where {yo. S(s0. LINEAR BETATRON MOTION Similarly. we obtain detM(s|s 0 ) = W(C. s0) = 1.44).so)y'o. The transfer matrix for any intervals made up of subintervals is just the product of the transfer matrices of the subintervals. s0) = 0. (2. C"(s0.g. using the Wronskian of Eq.s) = 1. the solution of a second-order differential equation can be expressed as y(s) = C(s.so)yo + S(s. s0) = 1. so) = 0. In the small-angle approximation.so)yo + S'{s. y'(s) = C'(s.y'o) and {y. (2-41) where £ is the length of the dipole. and. M(s 2 |s 0 ) = M{s2\Sl)M(Sl\so). S'{s0.33) can be expressed in terms of the transfer matrix as11 yW = MWeoMo).y') are the particle phase-space coordinates at the entrance and exit of accelerator elements. . the transfer matrix for a pure sector dipole with Kx = 1/p2 is « < > = (->* .So) an< i S(S.44) Thus the solution of Eq. The 2x2 matrix M depends only on the function K(s) between s and s0. u The transfer matrix for the uncoupled betatron motion can be expressed as /x\ \ x' \ fMx(s2]Sl) 0 /x\ \ I x'\ \z'/1 I * \z'/2 V ° M2(s2\Sl)J \ z ' where the M's are the 2x2 transfer matrices. (2. (2.so)y'o.S. particle motion can be tracked through accelerator elements.42) Using these matrices. Combining all segments.II. and yo and y'o are the initial phase-space coordinates at SQ. The solutions C(s. the transfer matrix becomes Ms(S>s0)=(J I ) .SQ) are respectively called the cosine-like and sine-like solutions with boundary conditions C(s0.
e. we obtain M{s2 + L\Sl) = M(s2)M(s2\Sl) = M(s2\Sl)M(Sl). A2 = e-J'*. M(s) = M(s + L\s) = Mn---M2Mu where the Mi's are the transfer matrices of the constituent elements. Using the periodicity condition. A2 be the eigenvalues and V\.3 has 24 superperiods. the eigenvalues are the reciprocals of each other. (y°)=av1+bv2.50) (2. (2.e. Thus a necessary condition for orbit stability is to have a real betatron phase advance <£. The transfer matrix for passing through P superperiods is M(s + PL\s) = [M(s)] P .48) Thus M(s 2 ) and M(si) are related by a similarity transformation: M(s 2 ) = M(s2\s1)M{s1)[M{83\s1)]-1.46) where V\ and v2 are the eigenvectors associated with eigenvalues Ai and X2 respectively. and $ is complex if Trace (M) > 2. Let Trace(M) = 2cos($). 2. (2. and the LEP at CERN has 8. (2. Since M has a unit determinant. The transfer matrix M of one repetitive period composed of n elements is a periodic function of s with a period L. The eigenvalue satisfies the equation A2 . Ai = 1/A2. This implies that the transfer matrix of a periodic section has identical eigenvalues.52) . We find that $ is real if Trace(M) < 2. > Expressing the initial condition of beam coordinates (yoi2/o) a s a l i n e a r superposition of the eigenvectors.47) (2. the Fermilab booster shown in Fig. The number of identical modules that form a complete accelerator is called the superperiod P. i. we find that the particle coordinate after the mth revolution becomes (Vj) = Mm (Vj) = aX^+bX^v. The eigenvalues are Ai=e J *. For example. the AGS at BNL has 12. v2 be the corresponding eigenvectors of the matrix M. Let Ai.49) where 3 is the betatron phase advance of a periodic cell. i.50 CHAPTER 2.Trace(M)A + 1 = 0. (2. The necessary and sufficient condition for stable orbital motion is that all matrix elements of the matrix [M(s)]m remain bounded as m increases. Let L be the length of a module with K(s + L) = K{s).. or |Trace(M)| < 2. TRANSVERSE MOTION An accelerator is usually constructed with repetitive modules.e. and Ai + \2 = Trace(M). (2. and for passing through m revolutions becomes [M(s)] mP .51) The stability of particle motion requires that A^" and \™ not grow with m. i.
the values of the Courant-Snyder parameters 02. 7i V 7i/ afa = a i . 2 Courant-Snyder Parametrization The most general form for matrix M with unit modulus can be parametrized as . .57 ) We note that 7 is constant in a drift space._ / cos $ + a sin $ M= .54) The ambiguity in the sign of sin<& can be resolved by requiring /? to be a positive definite number if |Trace(M)| < 2.56) V7/2 V Ml -2M 21 M 22 Ml ATA where My are the matrix elements of M(s2\si). and by requiring Im(sin$)>0 if |Trace(M)| > 2. LINEAR BETATRON MOTION 51 II. This ambiguity will be resolved when the matrix is tracked along the accelerator elements.7 i « = -(«-s*)//5*.72 at S2 are related to Qi. M" 1 = I cos $ . ./?i.12 $ is the phase advance. /3 and 7 are Courant-Snyder parameters. 1. . = Icos$ +Jsin$. (2. .„. + (i^)!.7i at S\ by (P\ a = ( Mf. we obtain the De Moivere's theorem: Mk = (I cos $ + J sin $)* = I cosfc"J>+ J sin fc$.48).T .55) Using the similarity transformation of Eq. (2. and s* = Q1/71 is the location for an extremum of the betatron amplitude function with a(s*) = 0.J sin $. -MnMai -2MnM12 MUM22 + M12M21 M\2 -M12M22 \ \\a (0\ .53) where a. and 7 parameters have nothing to do with the relativistic Lorentz factor.^2. The definition of the phase factor $ is still ambiguous up to an integral multiple of 2TT. f). (2. . Using the property of matrix J.JJ. \ —7 sin < P cos < — a sin $ J J > /?sin$ \ T . and 3 = (- - ) ' with Trace(J) = °> J 2 = .. (2. T . 12The a. 72 = 7i = l/£*. The evolution of the betatron amplitude function in a drift space is &= i +1l ( s _-y =/s . . The evolution of betatron functions is shown in the following examples.I or ^7 = 1 + a2. I is the unit matrix. (2.„. P* ( 2 .
> = ^2. where the focusing function K(s) satisfies K(s) = K(s+L). y*(s) = o w f s j e " * ' .w'2 — w'(s2). and w and ip are the amplitude and phase functions. Let s2 — S\ = L be the length of a periodic beam line.w2w'1 sin ip ' 2 wiw2smip Mcos'iA \ smtp .60) (2. Eq.61) Here. (2. Any solution of Eq.^ = ipfa) -ip(si). (2. and the prime is the derivative with respect to s. w[ — w'(si). Since K{s) is real. Eq. we set the periodic boundary conditions to the amplitude and phase functions: u>i = w2 = w.61) is the betatron phase equation.34) is a linear superposition of the linearly independent solutions y and y*. (2.4r = 0. Sec 1. (2. (2. 7 2 = 7 l + 2 a 1 / / + /3 1 // 2 . 4. and we have chosen a normalization for the amplitude and phase functions. Eq.34) can be solved by using the Floquet transformation: y(s) = s w ( s ) ^ w . Thus a thin-lens quadrupole gives rise to an angular kick to the betatron amplitude function without changing its magnitude.5). (2. a2 = <*i + A / / . the evolution of betatron function is given by ft = ft. Passing through a thin-lens quadrupole.( .3 Floquet Transformation Since the focusing function K(s) is a periodic function. TRANSVERSE MOTION 2. Thus the mapping matrix M(s2\si) can be obtained easily as / V^ cos ip .58) where / is the focal length of the quadrupole.52 CHAPTER 2. Using the Floquet theorem (see Appendix A. (2.59) where a is a constant.60) is called the betatron envelope equation. .w2 = w{s2). w[ = w'2 = w'. tp(si + L) — tp(si) = $. II. the amplitude and phase functions satisfy w" + Kw .1 ^ c o s f i + wiwismip/ where w\ = w(si).
67). which will be referred to as the betatron amplitude function.l [ l + (|)2j= 0 ) or a' = Kf3-±(l + a2).53).67) reduces to the CourantSnyder parametrization of Eq. ^2 are values of betatron amplitude functions at si and s2 respectively. we obtain (2. The betatron phase advance is -Cm i^» + ^ . and ft) <*2.62) to Eq. .53). from Eq.60). where s2 = S\ + L with Kfa) = K(si). 71. and we have defined the betatron amplitude matrix B(s) and its inverse as We note. Applying Floquet theorem to a repetitive period. (2. LINEAR BETATRON MOTION 53 Equating the matrix M of a complete period in Eq. (265) Here the amplitude function j3(s) is also the local wave number of betatron oscillations. (2. or (/?) = L/$. (2. we obtain ft = ft>.63) (2. (2. «i.II. we have $ = L/{P). (2. a = ~ww' = -P/2. ax = a 2 . we obtain w2 = 0. ^ = •0(s2) — V'(si). The Courant-Snyder parameter a is related to the slope of the betatron amplitude function. The betatron wavelength is A^ = 2TT(/?}. and the transfer matrix of Eq.64) Thus the amplitude of the betatron motion is proportional to the square root of the Courant-Snyder parameter /3(s). (2.66) The transfer matrix from S\ to s2 in any beam transport line becomes ( y^(cos ip + ai sin ip) \/P\h sin i> \ ( v% 0 \ / cos^ sinV\/^7 ° ^ = B(s2)( C0S^ ^^B" 1 ^). Substituting /? = w2 back into Eq. (2. that the linear betatron motion becomes coordinate rotation after the normalization of the phase-space coordinates with the B" 1 matrix. (2-67) where A. In the smooth approximation.
and its derivative is periodic.4) is made of a pair of focusing and defocussing quadrupoles with or without dipoles in between. is called Floquet transformation. FODO CELL «F/S | | B (2J1) (2. 2. w-kfm 0 +^ =0 .e. {^QF 0 QD 0 ^QF}. fy are constants to be determined from initial conditions.34) becomes y(s) = aJfWJcoa[il>y{8)+ty] with %{s) = f -r-f^. or Qy. The phase change per revolution is P$. This is a pseudoharmonic oscillation with varying amplitude /J*/2(s). and QF and QD indicate the focusing and defocussing quadrupoles. The betatron tune vy. Example 1: FODO cell in thin-lens approximation A FODO cell (Fig. where /o is the revolution frequency.72) The phase function <j>y increases by 2n in one revolution. JO Py(S) (2. I . defined as the number of betatron oscillations in one revolution. from y(s) to the amplitude and phase functions /3s(s) and <j>y(s). . 1 B Figure 2. where the transfer ma« F / 2 trix for the dipoles (B) can be apEH proximated by drift spaces. The local betatron wavelength is A = 2irPv{s). We define new variables r/ a n d </>y:13 ""f.4: A schematic plot of a FODO cell.54 Betatron tune CHAPTER 2. is •fc-^-s/ WY (269) The betatron oscillation frequency is vyfo. i. Thus t h e linear b e t a t r o n motion is in fact a simple harmonic motion.70) where a. (2. Hill's equation can be transformed to «D I . TRANSVERSE MOTION We consider an accelerator of circumference C = PL with P identical superperiods. The general solution of Eq. 13 This transformation.
we have 2Lt (1 . 15The transfer matrices of dipoles are represented by those of drift spaces. and (3F and aF are values of the betatron amplitude functions at the center of the focusing quadrupole.sin($/2)) sin^ ' ° PD = Q D = (2J5) at the center of the defocussing quadrupole. Insertions (or straight sections) are usually used for physics experiments. FODO cells are usually repetitively used for beam transport in arcs and transport lines. The transfer matrix for vertical motion can be obtained by reversing focusing and defocussing elements. We can also use the transfer matrix of Eq. and Arid. (2. The accelerator lattice is usually divided into arcs and insertions. and the corresponding CourantSnyder parameters are values of the betatron amplitude functions at that position.2.74) sin$ The parameter $ is the phase advance per cell. rf cavities.. is15 .^ | or sinf = ^- (2-73) 2 M l + sin(*/2)) 0.U !)(!?)(! !)(!?)(-*!) = / l-$ 2L1(l + ^ ) \ V-^(l-|?) 1-$ ) where / is the magnitude of the focal lengths for the focusing and defocussing quadrupoles. (2. point = — ^ (2 . injection and extraction systems. The above procedure can be performed at any position of the FODO cell. point = ± c o s ( $ / 2 ) (2™) at the midpoint between the QF and the QD. The betatron tune for a machine with N FODO cells is v = N$/2TT. in the thin-lens approximation. Arcs are curved sections that transport beams for a complete revolution.8). Because of the repetitive nature of FODO cells. "mid. and L\ is the drift length between quadrupoles.14 The transfer matrix for the horizontal betatron motion. (2.II LINEAR BETATRON MOTION 55 where O represents either a dipole or a drift space.Sin2 . etc. the transfer matrix can be identified with the Courant-Snyder parametrization of Eq. For example.53) to obtain cos$=^Trace(M) = l .J . where we neglect the effect of 1/p2 focusing and edge focusing.67) to find the betatron amplitude functions at other locations (see Exercise 2. .
i. The solid and dashed lines in the upper plot of Fig. The AGS lattice can be well approximated by 60 FODO cells with a phase advance of 52. Px 1 + sin $/2 . which made of 20 combined-function magnets. The phase advance of a doublet cell.5.e. The phase advance of each FODO cell is about 52. Example 2: Doublet cells The values of the horizontal and vertical betatron functions in FODO cells alternate in magnitude.5 show the betatron amplitude functions /3x(s) and fiz(s) for the AGS. The lower plot shows schematically the placement of combined-function magnets. The middle plot shows the dispersion function D(s).sin $/2 . we consider a doublet beam line. to be discussed in Sec. 1 . The AGS lattice has 12 superperiods. (2. IV.726 m for a complete circumference of 807. and a half-cell length of L\ = 6. Note that the superperiod can be well approximated by five regular FODO cells. I~^J ' jinn -— ft ~ l .8. and Ll and L2 are the lengths . Some examples of paraxial beam transport beam lines are the doublet. each composed of 20 combined-function dipoles. is sinf = ^ . The middle plot shows the dispersion function Dx. In some applications. shown schematically in the bottom plot of Fig. 2. in thin-lens approximation. the triplet.8°.8° for a betatron tune of 8. The beam size variation increases with the phase advance of the FODO cell. The upper plot shows fix (solid line) and @z (dashed line). a paraxial beam transport system provides a simpler geometrical beam matching solution. 2.5: The betatron amplitude functions for one superperiod of the AGS lattice. / is the focal length of the quadrupoles.77) where we have assumed equal focusing strength for the focusing and the defocussing quadrupoles. 2.s i n $ / 2 ' l + sin$/2' at the focusing and defocussing quadrupoles respectively.56 CHAPTER 2. TRANSVERSE MOTION Figure 2.12 m. In the following example. shown schematically in Fig. and the solenoidal transport systems.6.
Thus we obtain y' = -^(tanV-f). ' quadrupoles can be filled with Ll dipoles. the horizontal and vertical betatron amplitude functions are nearly C identical along the transport line.6. (2.79) If Li < ^2. of t h e drift spaces shown in Fig. y" + K{s)y = 0. (2.2. II. we find the new Hamiltonian *=*+lK- (-4 28 ) . LINEAR BETATRON MOTION DOUBLET CELLS Q D Q Q f | | <® W T~| Q] i 1 L2 57 Figure 2.82) 3= ~ f r = tp se°2 * = h[y2 + {N+ayn (2'83) Applying the canonical transformation and using Eq. y') are conjugate phase-space coordinates. provided that P satisfies Eq. Thus the doublet can be considered as an example of the paraxial transport system. (2. can be derived from a pseudo-Hamiltonian where (y.81) where ip is the phase factor.6: A schematic plot of a doublet transport line. (2.12).^ $ . (2. .66).y ) . where < > Q two quadrupoles a r e separated K F T~] by a distance L\. and the conjugate action variable is (2.II. 2. where y' = dFi/dy is verified easily.80) The Hill equation.4 Action-Angle Variable and Floquet Transformation H=l-y'2+1-K(s)y2. We observe that Eq. and the P] ^ long drift space Li between two ' . (2. Other paraxial transport systems are triplets and solenoidal focusing channels (see Exercise 2.13) ~ ~ sin* / U t ( 2 7 8 ) Anin = 7 .66).2. This suggests a generating function F ^ V) = [y'dy = -|g(tanV> .70) is a solution of Hill's equation. T h e maximum and minimum values of t h e betatron amplitude function are (see Exercise 2.
58
CHAPTER 2. TRANSVERSE
MOTION
Hamilton's equation gives if)' = dH/dJ = l//3(s), which recovers Eq. (2.61). Since the new Hamiltonian is independent of the phase coordinate ip, the action J is invariant, T--%-*• ds dip Using Eq. (2.83), we obtain / y = \l2f3J cos-0, /2T y'=-J-— [smip + acosi>], <285>
(2.86)
where a = —/3'/2- Now it is easy to verify that the action J is16
J=^[
Z7T ./torus
dy'dy=±<fy'dy.
Z7T J
(2.87)
The phase space area enclosed by the invariant torus is equal to 2n J. Figure 2.7: The horizontal and vertical betatron ellipses for a particle with actions Jx = Jz = 0.57T mm-mrad at the end of the first dipole (left plots) and the end of the fourth dipole of the AGS lattice (see Fig. 2.5). The scale for the ordinate x or z is in mm, and that for the coordinate x' or z' is in mrad. For the left plots, the betatron amplitude functions are Px = 17.0 m, ax = 2.02, A = 14.7 m, and az = -1.84. For the right plots they are /3X = 21.7 m, ax = -0.33, fiz = 10.9 m, and az = 0.29. Figure 2.7 shows the phase-space ellipses (x, x') and (z, z') for a particle with actions Jx = Jz = 0.5TT mm-mrad at the ends of the first and the fourth dipoles of the AGS lattice (see Fig. 2.5). Such a phase-space ellipse is also called the Poincare map, where the particle phase-space coordinates are plotted in each revolution. The consecutive phase-space points can be obtained by multiplying the transfer matrices,
16The
Jacobian of the transformation from (y,y') to {J,ip) is equal to 1.
II. LINEAR BETATRON MOTION
59
where M x and M z are the transfer matrices of one complete revolution. The Poincare map of betatron motion at a fixed azimuth s is also called the Poincare surface of section. If the betatron tune is not a rational number, the consecutive phase-space points of the particle trajectory will trace out the entire ellipse. The areas enclosed by the horizontal and vertical ellipses are equal to 2-irJx and 2-KJZ respectively. As the particle travels in the accelerator, the shape of the phase-space ellipse may vary but the area enclosed by the ellipse is invariant. A. Normalized phase space coordinates We define the normalized conjugate phase-space coordinate Vy as
Vy = py' + ay = -JzpJs\ni>.
(2.89)
A particle trajectory in the normalized phase-space coordinates (y, — Vy) is a circle with radius \/2/3J. The shape of the normalized phase-space ellipse is independent of the location s. In terms of the betatron amplitude matrix of Eq. (2.68), the normalized phase space coordinates are expressed as
B. Using the orbital angle 9 as the independent variable The Hamiltonian H of Eq. (2.84) depends on the independent variable s. Because /3(s) is not a constant, the phase advance is modulated along the accelerator orbital trajectory. Sometimes it is useful to obtain a global Fourier expansion of particle motion by using the generating function
F2(r/,,J)=(^-^j
+ u9JJ
(2.91)
to compensate the modulated phase-advance function. Here 6 — s/R is the orbiting angle of the reference orbit. The new conjugate coordinates (T/S, J) are
$ = if>- f ' ^ + v6, Jo p J = J, (2.92)
and the new Hamiltonian becomes H = vJ/R. Changing the time coordinate from s to 9, the new Hamiltonian is re-scaled and becomes H = RH = vJ. (2.93)
60
CHAPTER 2. TRANSVERSE
MOTION
The transformation from betatron phase-space coordinates to action-angle variables is y= fifij cos (i> + x{s)-vO), Vv = py' + ay= - ^ S J s i n ( ^ + x(s) - vff), (2.94) (2.95)
where x = /oS ds/0 and (V>, J) are conjugate phase-space coordinates. The transformation is useful in expressing a general betatron Hamiltonian in action-angle variables for obtaining a global Fourier expansion in the nonlinear resonance analysis. Hereafter, the notation (•ip, J) will be simplified to (ip, J).
II. 5
Courant-Snyder Invariant and Emittance
Py' + ay= -a/31/2(s) sin (i/0(s) + 6). (2.96)
Using the general solution y(s) of Eq. (2.70), we obtain
The Courant-Snyder invariant defined by
C(y, y') = ^[v2 + H + Py')2] = iv2 + 2 W + Py'2
(2.97)
is equal to twice the action, which is independent of s. The trajectory of particle motion with initial condition (yo, y'o) follows an ellipse described by C(y, y') = e. The phase space enclosed by (y, y') of Eq. (2.97) is equal to 7re (see Fig. 2.8).
Figure 2.8: The Courant-Snyder invariant ellipse. The area enclosed by the ellipse is equal to 7re, where e is twice the betatron action; a,/3 and 7 are betatron amplitude functions. The maximum amplitude of betatron motion is i/Pe, and the maximum divergence (angle) of the betatron motion is y/ye.
A. The emittance of a beam A beam is usually composed of particles distributed in the phase space. Depending on the initial beam preparation, we approximate a realistic beam distribution function
II. LINEAR BETATRON MOTION
61
by some simple analytic formula. Neglecting dissipation and diffusion processes, each particle in the distribution function has its invariant Courant-Snyder ellipse. Given a normalized distribution function p(y,y') with / p(y,y')dydy' = 1, the moments of the beam distribution are
(y) = I Vp(y, y')dydy',
(y1) = J y'p(y, y')dydy', <# = /(</'- {y')?p{y, yVydy',
(2.98) (2.99) ( 2 - 10 °)
°l = f(y - (v))2p(y, vVydy1,
°m/ = J(y - (y)W - (y'))p(y> y')dydv' = r w ,
where ay and ayi are the rms beam widths, oyyi is the correlation, and r is the correlation coefficient. The rms beam emittance is then defined as €rms = \jo2yO2y, _
a2yyl
= OyOy, Vl ~ r2.
(2.101)
If the accelerator is composed of linear elements such as dipoles and quadrupoles, the emittance defined in Eq. (2.101) is invariant. The rms emittance is equal to the phase-space area enclosed by the Courant-Snyder ellipse of the rms particle (see Exercise 2.2.14). Although incorrect, the term "emittance" is often loosely used as twice the action variable of betatron oscillations. The betatron oscillations of "a particle" with an "emittance" e is y(s) = JJe cos [v^){s) + 8}. (2.102) Figure 2.8 shows a Courant-Snyder invariant ellipse for a given emittance of a beam. For a beam with rms emittance 7re,17 the rms beam width is y/3(s)e, and the beam rms divergence y' is ^Jry(s)e. Since 7 = (1 + a2)/j3, the transverse beam divergence is smaller at a location with a large /3(s) value, i.e. all particles travel in parallel paths. In accelerator design, a proper f3(s) value is therefore important for achieving many desirable properties. B. The cr-matrix The a-matnx of a beam distribution is defined as
"ill
ZXt
:f) = «y-W»-M)').
(2,03)
where y is the betatron state-vector of Eq. (2.36), y+ — {y,y') is the transpose of y, and (y) is the first moment. The rms emittance denned by Eq. (2.101) is the
17The accelerator scientists commonly use 7r-mm-mrad for the unit of emittance. However, the factor 7 is often omitted. In beam width calculation, we get cry = ^KCy/ly/ir. The synchrotron T light source community also uses nano-meter (nm) as the unit for emittance. In fact, the factor n is implied and omitted in the literature.
62
CHAPTER 2. TRANSVERSE
MOTION
determinant of the cr-matrix, i.e. erms = \/det<7 (see also Exercise 2.2.14). Using the transfer matrix of Eq. (2.37), we obtain a(s2) = M(s2\s1)a{s1)M(s2\siy. (2.104)
It is easy to verify that y^(J~1y is invariant under linear betatron motion, thus the invariant beam distribution is p(y,y') = piv^-'y)C. Emittance measurement The emittance can be obtained by measuring the cr-matrix. The beam profile of protons and ions is usually measured by using wire scanners or ionization profile monitors. Synchrotron light monitors are commonly used in electron storage rings. More recently, laser light has been used to measure electron beam size in the submicron range. Using the rms beam size and Courant-Snyder parameters, we can deduce the emittance of the beam. Two methods commonly used to measure the rms emittance are discussed below. C l . Quadrupole tuning method Using Eq. (2.104), we find the rms beam radius R2 at a drift-distance L downstream of a quadrupole:18 an(s2) = R22 = an(s1)(l + ^ ^ - - L g ] V cm(si) / +^-L2, o-ii(si) (2.106) (2.105)
where g = Bi£q/Bp is the effective quadrupole field strength, L is the distance between the quadrupole and the beam profile monitor, and <7y(si)'s are elements of the a matrix at the entrance of the quadrupole with ej?ms = (Ji\O22 — a\2, and crn(s2) is the 11-element of the cr-matrix at the profile monitor location s2 (see Exercise 2.2.14(d) for an equation with thick quadrupole lens). The R\ data measured with varying quadrupole strength g can be used to fit a parabola. The rms emittance erms can be obtained from the fitted parameters. This method is commonly used at the end of a transport line, where a fluorescence screen or a wire detector (harp) is used to measure the rms beam size.
18If the transfer matrix between the tuning quadrupole at si and the profile monitor at S2 is m, we obtain cru = an [mu + (cumu/iTu) - "Ji2ff]2 + (7ni2erms)/crii> where <Jn is the 11-element of the cr-matrix at s2 , oy's are elements of the cr-matrix at si, m y 's are elements of the transfer matrix, and g = BiEq/Bp is the effective focusing strength of the quadrupole.
II. LINEAR BETATRON MOTION C2. Moving screen method
63
Using a movable fluorescence screen, the beam size at three spots can be used to determine the emittance. Employing the transfer matrix of drift space, the rms beam radii at the second and third positions are [ i?2 = an + 2L1<7i2 + L-^a^, \ R2 = an + 2{U + L2)a12 + (L, + L2)2a22, /r, i n 7 \ {Z'W'>
where an = R\, on and 022 are elements of the a matrix at the first screen location, and L\ and L2 are respectively drift distances between screens 1 and 2 and between screens 2 and 3. The solution an and <722 of Eq. (2.107) can be used to obtain the rms beam emittance: erms = yan<?22 ~ a\iIf screen 2 is located at the waist, i.e. dR\jdLx = 0, then the emittance can be determined from rms beam size measurements of screens 1 and 2 alone. The resulting emittance is (2.108) e2 = (R\Rl - Rl) /L\. This method is commonly used to measure the electron emittance in a transfer line. D. The Gaussian distribution function The equilibrium beam distribution in the linearized betatron phase space may be any function of the invariant action. However, the Gaussian distribution function (2.109) P{y,v') =-^exp(——-(<T 222/2 -2(Ti22/2/'+ crU2/'2)) \ z det a / is commonly used to evaluate the beam properties. Expressing the normalized Gaussian distribution in the normalized phase space, we obtain p(y^y) = ^e-(y2+^2"2y, (2.110)
where (y2) = (p2) = a* = f3yerms with an rms emittance erms. Transforming (y, Vv) into the action-angle variables (J, ip) with y = ^2/3yJcos^, Vy = -sj2^Jsmi), (2.111)
where the Jacobian of the transformation is
the distribution function becomes p{J) = —e-J'*-,
'rms
p(e) = -^—e-^™,
£€rms
(2.113)
64
CHAPTER 2. TRANSVERSE
MOTION
Table 2.1: Percentage of particles in the confined phase-space volume e/erms 12 14 I 6 I 8 Percentage in ID [%] 63 86 95 98 Percentage in 2D [%] 40 74 90 96
where e = 2J. The percentage of particles contained within e = ne rms is 1 - e""/2, shown in Table 2.1. The maximum phase-space area that particles can survive in an accelerator is called the admittance, or the dynamical aperture. The admittance is determined by the vacuum chamber size, the kicker aperture, and nonlinear magnetic fields. To achieve good performance of an accelerator, the emittance should be kept much smaller than the admittance. Note that some publications assume 95% emittance, i.e. the phase-space area contains 95% of the beam particles, eg5% « 6e rms for a Gaussian distribution. For superconducting accelerators, a dynamical aperture of 6<r or more is normally assumed for magnet quench protection. For electron storage rings, quantum fluctuations due to synchrotron radiation are important; the machine acceptance usually requires about 10cr for good quantum lifetime. Accelerator scientists in Europe use e = 4erm3 to define the beam emittance. This convention arises from the KV distribution, where the rms beam emittance is equal to 1/4 of the total emittance [see Eq. (2.131)].
E. Adiabatic damping and the normalized emittance
The Courant-Snyder invariant of Eq. (2.97), derived from the phase-space coordinate y, y', is not invariant when the energy is changed. To obtain the Liouville invariant phase-space area, we should use the conjugate phase-space coordinates {y,py) of the Hamiltonian in Eq. (2.18). Since py = py' = mcP'yy', where m is the particle's mass, p is its momentum, and Pj is the Lorentz relativistic factor, the normalized emittance defined by en = $fe (2.114) is invariant. Thus the beam emittance decreases with increasing beam momentum, i.e. e = €n//?7- This is called adiabatic damping. The adiabatic phase-space damping of the beam can be visualized as follows. Because the transverse velocity of a particle does not change during acceleration, the transverse angle y' = py/p becomes smaller as the momentum of the particle increases, and the beam emittance e — en//?7 becomes smaller. It is worth pointing out that the beam emittance in electron storage rings increases with energy as j 2 because of the quantum fluctuation to be discussed in Chap. 4. The
II. LINEAR BETATRON MOTION
65
corresponding normalized emittance is proportional to 7 3 , where 7 is the relativistic Lorentz factor. On the other hand, the beam emittance in electron linac will be adiabatically damped at high energies.
II.6
Stability of Betatron Motion: A FODO Cell Example
In this section, we illustrate the stability of betatron motion using a FODO cell example. We consider a FODO cell with quadrupole focal length /1 and - / 2 , where the ± signs designates the focusing and defocussing quadrupoles respectively. The transfer matrix of {§QFi O QD2 O |QF X } is _ _ / ( 1
1+
OWl
LA/1
(A / I
Lt\(
1 \
0\
^-7^-2&
2L l ( l + ^ )
+
\ h
h
hh
+
2/f + i!ih>
h
h
V2 I
where L\ is the drift length between quadrupoles. Identifying the transfer matrix with the Courant-Snyder parametrization, we obtain
co8«, = l + £ - £ - - £ y , C0S$z = 1 _ ^ + ^ l _ J L .
(2-116) (2 . 117 )
The stability condition, Eq. (2.52), of the betatron motion is equivalent to the following conditions: |1+2X2-2X1-2X1X2| <1 and |1 - 2X2 + 2XX - 2X^X2\ < 1, (2.118)
where Xx = Z-i/2/i and X2 = Li/2/ 2 . The solution of Eq. (2.118) is shown in Fig. 2.9, which is usually called the necktie diagram. The lower and the upper boundaries of the shaded area correspond to $ x , z = 0 or -K respectively. Since the stable region is limited by X1]2 < 1, the focal length should be larger than one-fourth of the full cell length. The stability condition of the above FODO cell example seems to suggest that the phase advances $x and $ z of a repetitive module should be less than TT.19 However, this is not a necessary condition. The phase advances of a complex repetitive latticemodule can be larger than TT. For example, the phase advance of a flexible momentum compaction (FMC) module is about 3TT/2 (see Sec. IV.8 and Exercise 2.4.17) and the phase advance of a minimum emittance double-bend achromat module is about 2.4?r
19The phase advance $ x of a double-bend achromat is larger than 7r (see Sec. IV.5). Thus a simple FODO cell working as double-bend achromat is unstable.
66
CHAPTER 2. TRANSVERSE
MOTION
(see Sec. III.l; Chap. 4). In general, the stability of betatron motion is described by ICOS^JI < 1 and |cos$ 2 | < 1 for any type of accelerator lattice or repetitive transport line.
Figure 2.9: Stability diagram of a FODO cell lattice. The lower and upper boundaries correspond to <&I]Z = 0 or 180° respectively.
II.7
Symplectic Condition
_ J. In general, the transfer matrix of a
The 2x2 transfer matrix M with detM = 1 satisfies MJM = J, where M is the transpose of the matrix M, and J = I Hamiltonian flow of n degrees of freedom satisfies MJM = J, where M is the transpose of the matrix M, and
J =
(2.119)
(-I
o)'
With
j2 =
"7'
J = ~J<
J~l = -J
(2-120)
with / as the n x n unit matrix. A 2n x 2n matrix, M, is said to be symplectic if it satisfies Eq. (2.119). The matrices / and J are symplectic. If the matrix M is symplectic, then M~l is also symplectic and detM = 1 . If M and ./V are symplectic, then MN is also symplectic. Since the set of symplectic matrices satisfies the properties that (1) the unit matrix I is symplectic, (2) if M is symplectic then M~x is symplectic, and (3) if M and iV are symplectic, then MN is also symplectic, the set of symplectic matrices form a group denoted by Sp(2n). The properties of real symplectic matrices are described below.
II. LINEAR BETATRON MOTION
67
• The eigenvalues of symplectic matrix M must be real or must occur in complex conjugate pairs, i.e. A and A*. The eigenvalues of a real matrix M or the roots of the characteristic polynomial P(X) = \M — \I\ = 0 have real coefficients. • Since \M\ = 1, zero can not be an eigenvalue of a symplectic matrix. • If A is an eigenvalue of a real symplectic matrix M, then I/A must also be an eigenvalue. They should occur at the same multiplicity. Thus eigenvalues of a symplectic matrix are pairs of reciprocal numbers. For a symplectic matrix, we have K~\M or P(A) = X2nP{\) If we define Q(X) = X~nP{X), then Q(X) = Q(\). (2.123)
A
- XI) K = M~x -XI=
-XM~X{M - A"1/)
(2.121)
(2.122)
II.8
Effect of Space-Charge Force on Betatron Motion
The betatron amplitude function w — Jpy of the Floquet transformation satisfies Eq. (2.60). Defining the envelope radius of a beam as
Ry = yffay,
where ty is the emittance, the envelope equation becomes ^' + ^ - - ^ = 0,
(2.124)
(2.125)
where the prime corresponds to the derivative with respect to s. Based on the Floquet theorem, if Ky is a periodic function of s, i.e. Ky(s) = Ky(s + L), where L is the length of a repetitive period, the solution of the envelope equation can be imposed with a periodic condition, Ry(s) = Ry(s + L). The periodic envelope solution, aside from a multiplicative constant, is equal to the betatron amplitude function. The envelope function of an emittance dominated beam is equal to \lpyty. What happens to the beam envelope when the space-charge force dominates the beam dynamics? Here we discuss some effects of the space-charge force on betatron motion.
68
CHAPTER 2. TRANSVERSE MOTION
A. The Kapchinskij-Vladimirskij Distribution It is known that the Coulomb mean-field from an arbitrary beam distribution is likely to be nonlinear. In 1959, Kapchinskij and Vladimirskij (KV) discovered an ellipsoid beam distribution that leads to a perfect linear space-charge force within the beam radius. This distribution function is called the KV distribution.20 Particles, in the KV distribution, are uniformly distributed on a constant total emittance surface of the 4-dimensional phase space, i.e.
**> V~ * ?-) = J^5 G? (*2 + ^ ) + h 0 2 + *?) - l) . •
(2-126)
where N is the number of particles per unit length, e is the particle's charge, a and b are envelope radii of the beam, x and z are the transverse phase-space coordinates, and Vx — R'x, and Vz = R'z are the corresponding normalized conjugate phase-space coordinates. Some properties of the KV distribution are as follows. 1. With the phase-space coordinates transformed into action-angle variables, the KV distribution function becomes rfJ,,7,) = ^ ( ^
+
^ - l )
(2.127)
Thus beam particles are uniformly distributed along an action line
3j-+Jy
f-x
€z
=\
^
(2.128)
where ex and ez are the horizontal and vertical emittances. The envelope radii are
a = y[KZ, b = JpJ;.
(2.129)
2. Integrating the conjugate momenta, the distribution function becomes
^-sH'-S-if)
3. The rms emittances of the KV beam are _(x^_e _<f!)_!i
(2i3o)
where the 9(^) function is equal to 1 if f > 0, and 0 if £ < 0. In fact, the KV particles are uniformly distributed in any two-dimensional projection of the four-dimensional phase space.
(2m)
Thus the rms envelope radii are equal to half of the beam radii in the KV beam.
20I.M. Kapchinskij and V.V. Vladimirskij, Proc. Int. Conf. on High Energy Accelerators, p. 274 (CERN, Geneva, 1959).
(2. = Q' ('3) 214 ( 2 . w2 v x (2-138) (2. beam particles can be viewed as a charge distribution in an infinite long wire with a line-charge density given by Eq. Including the mean-magnetic-field. A noteworthy feature of the KV distribution function is that the resulting mean-field inside the beam envelope radii is linear! If the external focusing force is also linear.z) = ^-(-~^—-x+—^-rz). Neglecting the longitudinal variations. and N is the number of particles per unit length.130). and Kx is the "normalized" space-charge perveance parameter given by (2. LINEAR BETATRON MOTION B.139) . Thus Hill's equations of motion become H*.133) ' b(a + b) J v where 7 is the relativistic energy factor. z) is i?/ A Ne ffj-j^n x'2 z'2(x-x')x + (z-z')z + E{x>z) = 2^abJJdxdz 2ne0 \a(a + b) @{l-^-¥\x-xiy {z-z>y v .+ A = 0. (2. Performing Floquet transformation of the linear KV-Hill equation x = wxe?*' and z = wzeji>'.137) we obtain < + (K* ~ " T ^ M ) W.135 > where the prime is a derivative with respect to the longitudinal coordinate s.136) where r 0 = e2/4-n:eomc2 is the classical radius of the particle.II. the force on the particle at (x. b(a + b) ) where eo is the vacuum permittivity. the KV distribution is a self-consistent distribution function. The electric field at the spatial point (x. z) is F(x. <+ \ a(a + b)J w% (K> ~ 7T^) Wz + —3=Q. v ' 27reo72 \a(a + b) (2. \ b(a + b)J w* <= 4w2 i>l = —. The Coulomb mean-field due to all beam particles 69 The next task is to calculate the effect of the average space-charge force.W -^) I=0' z" + {K^-wfvyjz *« = ^ f .
the space-charge parameter. i. The usefulness of the KV equation has been further extended to arbitrary ellipsoid distribution functions provided that the envelope functions a and b are equal to twice the rms envelope radii. and the emittances ex and ez are equal to four times the rms emittances.140) and (2. (2. b) = ~(Kxa2 + Kzb2) .2KSC ln(a + b) + ^ +^ .J. 7. (2. (2. (2. Sacherer. To understand the physics of the mismatched envelope. and R. Lee and R.143) we can derive the KV equations (2. it is advantageous to extend the envelope equation to Hamiltonian dynamics as discussed below. For space-charge dominated beams. Nncl. we obtain the KV envelope equations. or simply the KV equations: a!' + Kxa-^--% a + b a6 OK c2 OK f2 = Q.K.70 CHAPTER 2. b{s)=b{s + L).142). IEEE Trans. 1101 (1971).139) by ^ e j . (2. Part. J. (2. (2. The matched beam envelope solution can be obtained by a proper closed orbit condition of Eq. ibid.145) 21 P. Lapostolle. For beams with an initial mismatched envelope. E. and the beam emittance. Pb = b'. Kx(s) = Kx(s + L). Gluckstern.142) A numerical integrator or differential equation solvers can be used to find the envelope function of the space-charge dominated beams. 5. P. the KV equation can be solved by imposing the periodic boundary (closed orbit) condition (Floquet theorem) a(s) = a(s + L). Accel. Sci. (2.L. 1105 (1971). (2.D. the envelope solution can vary widely depending on the external focusing function.141) from the envelope Hamiltonian: ffenv = \(pl+pl + Kxa2 + Kzb2) .140) b" + Kzb-^-% = Q.M. the envelope equation can be solved by using the initial value problem to find the behavior of the mismatched beams. NS-18. 61 (1973).138) by JTX and Eq. TRANSVERSE MOTION Multiplying Eq.e. Lapostolle.P.M. F. and identifying a = wxJTx and b = wzy/e^. C. . 83 (1976). Lawson.141) Solving the KV envelope equation is equivalent to finding the betatron amplitude function in the presence of the space-charge force. ibid.144) With the envelope potential defined as Vem(a.21 If the external force is periodic. Cooper. Hamiltonian formalism of the envelope equation Introducing the pseudo-envelope momenta as Pa = a'.2KSC ln(o + 6) + ^ + ^ .
i. .150) (9 Tin _ -ft^tot _ .C Kenv / .^sclna+ A = \V\ + Venv(a). we can obtain the tune of the envelope oscillation. where _ K^L (2. V 1 0 K e nv / \o . we consider the effect of space charge on the envelope function.146) where am and bm are the matched envelope radii.e. For example. a2m=(^)[K + V^l\.( a m . (2.^ . With a = b in Eq. Now. the matched envelope solution is amo = \JtxL/2-K = ^expx. where the focusing function is Kx = (2TT/L) 2 .15) independent of the envelope-oscillation amplitude. bm) = .II. if we start from the condition with envelope momenta pa = pb = 0. (2. LINEAR BETATRON MOTION 71 the matched beam envelope can be easily understood as the equilibrium solution of the envelope Hamiltonian.( a m . bm) = 0. L .148) When the space-charge force is negligible.e. the envelope Hamiltonian is #env = \vl + \ ( f ) 2 « 2 . The matched envelope radius is obtained from the solution of dVem/da = 0. The second-order derivative at the matched radius becomes The tune of the mismatched envelope oscillation is twice the tune of betatron motion (see also Exercise 2. The envelope oscillations of a mismatched beam can be determined by the perturbation around the matched solution env = 2~d^T^ ~ ^ + 2 ~ 9 6 ^ ( " m ) + ' ' ' • ( ^ Using the second-order derivatives. and the betatron amplitude function is equal to L/2-K. D. the matched envelope radii are located at the minimum potential energy location. i. .^ . (2.140). \9 • rn -t A>-J\ Here L is the betatron wavelength. An example of a uniform focusing paraxial system First we consider a beam in a uniform paraxial focusing system.2.
Phys. Lee and A.C.exL) = 1. Rev. See Eq. 1609 (1995). There is a large envelope detuning from 2/z to \/2/i. Rev. Chen and R.150) for the matched envelope radius. When the space-charge perveance parameter is zero. Phys. and L tot and $tot are the total length and total phase advance of a transport system. is the betatron phase advance. the envelope tune approaches 22The Laslett (linear) space-charge tune shift is related to the space-charge perveance parameter by £sc = Ai/sc = KscLtot/4iWex = KV./2ir for the unbetatron tune) as a function of the maximum amplitude of the envelope amplitude. where /j. E 51. Lett. A nonlinear envelope resonance can be excited when perturbation exists and a resonance condition is satisfied. and obtain which is the phase advance per unit length of small amplitude envelope oscillation in the presence of the Coulomb potential. 2195 (1994). C.4199 in this example.72 CHAPTER 2. See also Ref. Rev.150) indicates that the betatron amplitude function increases by a factor K + %/«. the phase advance of > the small-amplitude envelope oscillations can maximally be depressed to 2\/2 TT/L. [4] for an exploration of the space-charge dynamics.23 Figure 2.2 + 1 due to the space-charge force. Phys. TRANSVERSE MOTION is the effective space-charge parameter. At a large envelope amplitude. 72. where v is the tune. the phase advance of the envelope oscillation is twice of that of the betatron oscillation. n = 2.28175. Riabko.2817 radian (or v = fj. Davidson. E49. Riabko et al. 3529 (1995). 5679 (1994). The matched radius is RQ — am^/2Tr/(fj.10: The phase advance of the envelope oscillations divided by the original betatron phase advance for a high space charge beam with Ksc = 10. . Phys.22 Equation (2. The ordinate R is the normalized maximum envelope radius of the beam. Figure Ksc = 10 perturbed oscillation 2. Next we evaluate the second-order derivative of the potential at the matched radius. A. (2.Y. Rev. 23S. as K — oo. and when the space-charge force is large.10 shows the envelope tune of a space charge dominated beam with and a phase advance of /x = 2.. E 51.
in the thin-lens approximation. of Michigan Press. When the space charge parameter K is large. Show that the mapping matrix M for a short quadrupole of length i. 2. / > 0. i. 1971). (2. is -(-}:) where / = l i m ^ o ^ ^ ) " 1 ! is the focal length of a quadrupole. The focusing function K(s) for most accelerator magnets can be assumed to be piecewise constant. When a particle enters a dipole at an angle S with respect to the normal edge of a dipole (see drawing below).s\. (2. Show that the transfer matrices for the horizontal and vertical betatron motion due to the edge focusing are 24Using edge focusing.5 GeV. See L. The ZGS was made of 8 dipoles with a circumference of 172 m attaining the energy of 12. ( K(s)=K<0. Greenbaum. For a focusing quad. Exercise 2. 18. the betatron tune can be depressed to zero.e. 1963. _ \J\K\smhyf\K\s V^F Zcosh^\K\s with s = S2 . . This phenomenon is usually referred to as edge focusing.2 1. (2.150) into Eq. M(S2\Sl)=(l J)./sc = fsc = KV. there is a quadrupole effect.EXERCISE 2. Effect of space charge force on particle motion The single particle betatron phase advance per unit length is obtained by substituting Eq. Ann Arbor. Show that K(s) = 0. ) cos VKs . Near the matched envelope radius (or small amplitude envelope oscillations). where v = $tot/(27r) is the tune of the accelerator. the envelope tune approaches y/2 times the unperturbed betatron tune. Its first proton beam was commissioned on Sept. the zero-gradient synchrotron (ZGS) was designed and constructed in the 1960's at Argonne National Laboratory.K). / < 0. $* = -J-(VK2 Lt 2TT + 1 .135). M( S2 | Sl )= cos \fRs -VKsmvKs -4T sin \/TCs \ ) ).24 We use the convention that 5 > 0 if the particle trajectory is closer to the center of the bending radius.2 73 twice the unperturbed betatron tune.153) When the space charge parameter n is small. and for a defocussing quad. the incoherent space-charge (Laslett) tune shift is equal to A. A Special Interest (Univ.
dBz/dx = 0. where 9 is the bending angle. calculate the mapping matrix for the basic period and discuss the stability condition. T h u s t h e A*z = I tan* i M* = tani5 / ^ \ v ^ .2b). p is the bending radius.74 CHAPTER 2."i / zontal defocussing and vertical focusing. (a) Show that the horizontal and vertical transfer matrices are (p/Vl — n) sin(-\/l — n s/p) \ yr _ ( cos(\/l — n s/p) x \ —(%/! ~ n/p) sin(\/l — n s/p) cos(\/l — n s/p) ) ' M _ ( cos(Vn s/p) (p/Vn) sin(^/n s/p) \ Z V —(y/n/p) sin(Vl — n s/p) cos(-v/n s/p) ) ' (b) Show that the betatron tunes are vx = (1 —n)1//2 and vz = n1/2. 2. The particle orbit enters and exits a sector dipole magnet perpendicular to the dipole edges. Note that a sector magnet gives rise to horizontal focusing. The entrance and exit edge angles of a rectangular dipole are 5\ = 0/2 and 82 = 6/2.1. Kz{s) = n/p2 = constant and Kx = (1 — n)/p2.0) dx x=o' ( where we have chosen the coordinate system shown in Fig. and the stability condition is 0 < n < 1. is the length of the dipole.' / \ / ' ^ \ 7 ^ ^ \ / \ ~ - 3. with Kx = Kz = 0. 4.^ ^^~ ^ \ "~~\^ V j \^ \ / ^__ . The focusing index n is ' p(s) dBz(s. Find the horizontal and vertical transfer matrices for a rectangular dipole (Fig.e. and (.x. TRANSVERSE MOTION \~J~ 1 / V—— lJ where 8 is the entrance or the exit angle edge effect with S > 0 gives rise to hori. Solve the following problems by using the uniform focusing approximation with constant n. where p is the radius of the accelerator.. are introduced into the accelerator lattice adjacent to each combined-function dipole. fl i\ where 0 is the bending angle. i. For a weak-focusing accelerator. .. Assuming that the gradient function of the dipole is zero. 6. The path length for a particle orbit in an accelerator is C = j ^[l + {x/p)f + xl2 + z^ds. 2..0) Bz{s. /cos0 psin0\ .Q. (c) If N equally spaced straight sections. 5. show that the transfer matrix is . / of the particle with respect to the normal direction of t h e dipole edge.
In a strong-focusing synchrotron. and QD is a defocussing quadrupole.25 (d) Find the phase advance $ that minimizes the betatron amplitude function at the focusing quadrupole location. I /9 = acosh2^\K\s + bsmh. where there are no quadrupoles.As and s? = s0 + As. Show that the average orbit length of a particle executing betatron oscillations is longer by C \R? ' Thus the orbit length depends quadratically on the betatron amplitude. show that the betatron amplitude function is 25Find /?'s at Si = So . (a) Express a. /?o and 70 at the beginning of the element. In the smooth approximation. Using Eq. OO represents either a drift space or bending dipoles of length L\. show that p'" + 4/3'K + 2/3K' = 0.EXERCISE 2. Solve this equation for a drift space and a quadrupole respectively. Using the thin-lens approximation. focusing quadrupole < fi = a cos 2y/l(s + b sm2^/Ks + c. a at the quadrupoles and at the center of the drift space as a function of L\ and $. . defocussing quadrupole. vz and z are the average radius. and the betatron oscillations can be expressed as where R. the vertical betatron tune. (2. and calculate the derivative from these numbers. (a) Find the mapping matrix and the phase advance of the FODO cell and discuss the stability condition.2^/\K\s + c. A FODO cell is composed of QF 0 0 QD 0 0 . the art (or science) of magnet arrangement is called lattice design. The length of a FODO cell is L = 1L\.2 75 Show that the average orbit length of the particle with a vertical betatron action Jz is longer by A C _ 1 A + a2 where az and j3z are betatron amplitude functions. where QF is a focusing quadrupole. and c in terms of parameters c*o. 7. and the vertical betatron amplitude respectively. and Xz is an arbitrary betatron phase angle of the particle. the betatron amplitude function is approximated by (f3z) = R/vz. 6. and show that the solution of this equation must be one of the following forms: drift space ( 0 = a + bs + es 2 .66). The basic building blocks of a lattice are usually FODO cells. (b) Find the parameters /3. 8. (b) In a drift space. (c) Verify that /3' = — la numerically at the center of the drift space.
the injection or extraction kickers are located at a high /9 locations with a 90° phase advance. TRANSVERSE MOTION where P* is the betatron function at the symmetry point s = s* with P' = 0. (2. . 11.72 at s2 are related to ai.67) becomes M(s2\si) = I^MBj" 1 .e. Show also that 7 = (1 + Q2)/P is equal to 1/P*. Mf2 / \71/ where My are the matrix elements of M(s2\si).76 CHAPTER 2. Use the transfer matrix M(s2\si) of Eq. Use these equations to verify 9. i. i. the displacement at a downstream location is Ax2 = dy/Pifhsmij). (2. your solution to part (a). sin^N . and if> = ip(s2) — ip(si) is the betatron phase advance between s\ and S2.80) into Eq. when a particle is kicked at si by an angle 9. We consider a FOFO focusing channel where the focusing elements are separated by a distance L. 10.1. where B 2 and Bi are the betatron amplitude matrices at s = s2 and s\ respectively.67) to show that. 12.4). 7 is constant in a drift space. where fii and p2 are values of betatron functions at si and s2 respectively. Transforming the betatron phase-space coordinates onto the normalized coordinates with Y= or TPV' v^TP{ay+fi^ show that the betatron transfer matrix in normalized coordinates becomes Cri 1 M(s2 \si)N= I cosV".48).93). .A>7i at s\ by fp2\ I M\x -2MnM12 M11M22 + M12M21 -2M 2 1 M 2 2 M\2 WAX \ a2 = -M11M21 V72/ V M22! -M12M22 I I a i .e. p2. (c) Using the similarity transformation Eq. Show that the Floquet transformation of Eq. (2. The focusing channel can be considered as a focusing-focusing (FOFO) channel. (2. Use the thin-lens approximation to evaluate beam transport properties of a periodic FOFO channel. . Often a solenoidal field has been used to provide both the horizontal and the vertical beam focusing for the production of secondary beams from a target (see Exercise 2.94) transforms the Hamiltonian of Eq. Show that the transfer matrix of Eq. (2. the betatron transfer matrix becomes coordinate rotation with rotation angle equal to the betatron phase advance. show that the Courant-Snyder parameters a2. (2. To minimize the kicker magnet strength 9.The quantity ^PiP2sim/> is usually called the kicker arm.
xi = j xp(x. Nucl. These quadrupole doublets can be used to maintain round beam configuration during beam transport.(x1)) = roxax. Lapostolle.6). The doublet pairs are repeated at intervals L2 2> L\ for beam transport (Fig. (c) Show that the minimum betatron amplitude function is |8* = ^/ii(4/2-LiLa)/4L 2 .x')dxdx' = l.2 (a) Show that the phase advance of a FOFO cell is Sin 2=2V7' 77 $ 1 [I where / is the focal length given by f~l = g2i = ®2/£. 30 (1991). The first and second moments of beam distribution are 1 r I r. (a) Show that the betatron phase advance in a doublet cell is •0 = i>x. (d) Sketch the betatron amplitude functions and compare your results with that of the FODO cell transport line. IEEE Trans. 2. Sci. Oxx' = jf £ ( z i .. . g = Bu/2Bp is the effective solenoid strength.x')dxdx'. B\\ is the solenoid field. describe the properties of betatron motion in a doublet transport line. where / is the focal length of the quadrupoles. Buon. (x1) = -fiY.x')dxdx'. CERN 91-04. The statistical definition of beam emittance is applicable to all phase space coordinates. ^ = JfY.x'). °l = jf £ t e . The doublet configuration consists of a focusing and defocussing quadrupole pair separated by a small distance L\ as a beam focusing unit. .x'i = J x'p(x.x') be the distribution function with fp{x.{x))(x'i . (b) Show that the maximum and minimum values of the betatron amplitude function are Anax = i/sin $.<*'»2.z = arcsin (yfLiLz/Zf] . Statistical definition of beam emittance:26 We consider a statistical distribution of N non-interacting particles in phase space (x. 14.(*»2> 4 = }f £(*. Let p{x. /3min = / sin $. 26See P. 13. 1101 (1971). NS-18. and @ = g£ is the solenoid rotation angle. £ is the length of the solenoid.EXERCISE 2. and J. (b) Show that the maximum betatron amplitude function is approximately /?max = (£l +L2 + LiL2/f)/smif). Using the thin-lens approximation with equal focal length for the focusing and defocussing quadrupoles.
Ly/KsinVKlq + ga f (ai) f-j= sinV#l q . show that the total phase-space area is A = nab = 47rerms. particles are distributed in the Courant-Snyder ellipse: I(x. Use this result to show that x ^ r ^ x of Eq. (b) Show that the rms emittance defined above is invariant under a coordinate rotation X = a: cos 0 +a. Use the coordinate rotation to show that 9 9 trms — ai Q — at.r2.\_ (7X2 + 2axx' + ftx12). i. and show that the correlation coefficient R = aYV. Show that ax and ax. and r is the correlation coefficient. to ensure that the phase-space area of such an ellipse is Tre. (a) Assuming that particles are uniformly distributed in an ellipse x2/a2 + xl2/b2 = l..106) becomes \ axx' ax' / 2 \ axx' ax' / 1 a2x(s2) = ox(Sl) (cos VK£q . according {?'. is zero if we choose X tan 2 0 = . For a thick quadrupole lens.'sin 0.I r ^ r [-4= sin \fiCL + L cos -JKI} . The rms emittance is defined as £rms = PxVx' Vl ./ara the rotation angle to be AA ' A . reach extrema at this rotation angle.78 CHAPTER 2. in the linear betatron motion.* % . (2. where M+ is the transpose of the matrix M.e. X' = -x sin 0 + x' cos 0. TRANSVERSE MOTION Here ax and ax> are rms beam widths. The transport equation for the amatrix can be used to measure the cr-matrix elements and derive the rms beam emittance. ft. y) =MM«)(f*. (c) In accelerators. +L cos VKia) )2 + . 7 1 ' — /75—5 a 0 or Show that x}<7~lx = erms VPi ( 13 -a\ ( a2x axx. (d) Show that the a matrix is transformed.105) is invariant under betatron motion and thus an invariant beam distribution function is a function of xla~lx. x') = jx'2 + 2axx' + fix2. y) ^(-si-i)*. where a. (2. 7 are betatron amplitude functions. The factor 4 has often been used in the definition of the full emittance. e = 4e rms . show that Eq.
where the particle betatron coordinate obeys Hill's equation y" + Ky(s)y = 0. i. E51.Y. where R is the average radius. and L is the length of the drift space between the quadrupole and the profile monitor. we have (Ky) = (2nQy/C)2 obtained from Floquet transformation to Hill's equation. Rev.dH.' « * e* >" <*><&•»• For a linear Hamiltonian.e. Rev. Riabko et al. (a) Show that the envelope equation of motion is Y" + Ky(s)Y-^ = 0. Show that the betatron motion is Le 2-KS . Riabko. 3529 (1995). 27See S. where L is the wavelength of the betatron oscillations. . i. Show that the rms emittance is conserved. Consider a beam of noninteracting particles in an accelerator with focusing function Ky(s).EXERCISE 2. Y) are conjugate envelope phase-space coordinates with P = Y'.K. 28Using the smooth approximation. Let Y be the envelope radius of the beam with emittance e.dH... The "potential energy" of the envelope Hamiltonian is V — 1g AK rY 2 +4. Ym = \jLej2n. _ dx dx' _ da ' ds Show that de2 n dH dx „ .e. E51. Phys. Lee and A. where K(s) is the focusing function. . where (P. dH. . and Qy is the betatron tune.28 The equivalent betatron amplitude function is Py = L/2TT. (e) Particle motion in synchrotrons obeys Hamiltonian dynamics with . . where C is the circumference. 1609 (1995). (b) Show that the envelope equation can be derived from the envelope Hamiltonian27 tfenv = \P2 + \Ky{s)Y2 + ~ . Y(s) = ^/3(s)e. .. .2 79 where K = B\/Bp and tq and the focusing function and the length of the quadrupole. Phys. What would your conclusion be if the Hamiltonian were nonlinear? 15. Ky(s) = (2n/L)2. . The matched beam radius is given by dVenv/dY = 0.dHxs •dJ = -2o^x &F>" ^ X a* » + 2 " . The corresponding average betatron wavelength is C/Qy.) = R/Qy.2Y3' t venv V In a smooth focusing approximation. we have dH/dx = Kx. . . A. and the average betatron amplitude function is (/?j.
The Courant-Snyder phase-space ellipse of a synchrotron is yy2 + 2ayy' + j3y'2 = e. 16. Show that the normalized envelope R satisfies the equation: Using (R. . where a. in fact. The envelope Hamiltonian is. 29The easiest way to estimate the emittance growth is to transform the injection ellipse into the normalized coordinates of the ring optics. Show that the exact solution of the envelope equation is R2 = y/l +a2 +acos{2i/(f> + x). (c) Let us make Floquet transformation to the envelope equation in part (a) with B. where a is the envelope mismatch amplitude. The deviation of the injection ellipse from a circle in the normalized phase space corresponds to the emittance growth. and show that the injection ellipse becomes (P where U+ (axp-fila)2\ M ) 2 arf-aP! + 0 ft YP + T ~e' 2 _ Y = Tpy' P = j?to + M.-v R + — . linear. the resulting oscillation will be sinusoidal. and v is the betatron tune. If the injection optics is mis-matched with 71J/2 + 2ot\yy' + /3\y'2 = e.29 (a) Transform the injection ellipse into the normalized coordinates of the ring lattice. Thus the envelope of a mis-injected beam bunch will oscillate at twice the betatron oscillation frequency (the quadrupole mode).J l •_ 1 f ds where /? is the betatron amplitude function. TRANSVERSE MOTION and the solution of the envelope equation is y2 = /^Ii7 +Acos(2? F +x) ' where the parameters A and x Sire determined by the initial beam conditions.80 CHAPTER 2. Note: if the square of the rms beam width is plotted as a function of revolution turns. fi and 7 are the Courant—Snyder parameters. PR = dR/d(f>) as the conjugate phase space coordinates. we obtain the envelope Hamiltonian as H = \P^ + Venv(R). where the envelope potential is Venv . find the emittance growth factor.
the lattice betatron functions are usually designed to an appropriate (3* z value with symmetry condition: axz = 0 .0 m and ax = 2. The luminosity.s2)dxdzdsd(fict). <*s are respectively the rms beam sizes in x.Pet and s2 = s — Pet. F-=\Xam-y/Xlm-l) / . show that the luminosity. K bvoxo*z where R is the reduction factor.14) *mm = I (7i/3 + 017 ~ 2aia) = —!— {/3a2x.15). oz = VPz^z. \l/2 . in a short bunch condition with as <S (3XZ.z.00 mm and 6 = 1. iS C = 2fNlN2 f I pi(x. nonlinear betatron detuning arises from space-charge forces. etc.si)p2(x. the phase-space area of the mis-injected beam will decohere and grow. Because the betatron tune depends on the betatron amplitude. where s\ = s 4. where a = 5. 17. + -ya2x + 2aaxx. or at a symmetry point in a storage ring. nonlinear magnetic fields. (a) Assuming Gaussian bunch distribution with P[X'Z' i S> ~ ^- 1 (2*)V2axoz*s 6XP / \ *2 *2 s2 I 2^ " 2^ " 2^/ ' where ax = VPx^x.02. .2.z. What happens to the beam if the beam is injected into a perfect linear machine where there is no betatron tune spread? Show that the tune of the envelope oscillations is twice the betatron tune (see Exercise 2. and ax = \/P%ex and a* = v //3je 2 are rms beam size at the IP.2.).2 81 (b) Transform the ellipse to the upright orientation.2. Find the final beam emittance after nonlinear decoherence. Show that the emittance growth factor is Fi=(xmm+y/x^iy (d) Let the betatron amplitude function at the injection point be fix = 17. chromaticities.EXERCISE 2. Note that the rms quantities ax. The injection ellipse of a beam with emittance 5TT mm-mrad is given by x2fa2 + x'2/b2 = 1.00 mrad. and show that the major and minor axes of the ellipse are F+={Xmm + JX2im-l) / . At an interaction point (IP) of a collider. C. z. measuring the probability of particle encounters in a head on collision of two beams. The resulting betatron amplitude functions in the straight section become f)XtZ = (3* z + s2/0x>z (see Exercise 2. s directions. where the mismatch factor Xmm is (see Exercise 2.ax> and axx> can be measured from the injected beam. (c) In general.8). \l/2 .
17. Bennewitz and W. 2 18. 38. Haberli. W. with 100 us duration and 5 Hz repetition rate. (c) For a round beam with A = Ax = Az. Paul. show that the luminosity reduction factor for two identical Gaussian distributions is R(A H{Ax.82 CHAPTER 2.47047^)3 ' R(A) = ^AeA where the latter approximate identity is valid up to about A < 2. A ^ y/ZA(l+ 0. 139. Most colliders operate at a condition Ax. Ann. Focusing of atomic beams: 30 There are now two types of polarized ion sources: the atomic-beam polarized ion source (ABS). Plot R(A) as a function of A and show > that the actual luminosity is C = R(A)C0 = {NlN2 v^V 2 erfc(A) 47re^<rs for a given as. ABS has produced polarized H~ ions with about 75% polarization at a peak current of 150 fik. 159 (1951). Naturwiss. which is a quadrupole or a sextupole. show that (see Section 7.G.2836. In a short bunch approximation with Ax ^> 1 and Az 3> 1.. Paul.Az) ^]- 2 ^J f A l + {C2/Ai)){1 ^ + {C2/Ai]) where AXtZ = /?* z/crs is a measure of the betatron amplitude variation at the interaction point. The luminosity reduction due to finite bunch-length is called the hour-glass effect. 32H. [26]) C = R(AZ)CO = ™ 4^Vpxex€zpz V7r * VlA0(f). 373 (1967). i.AZ) ss 1. Nucl. 31The 30See . Ax > 1. where ej_ = ex = ez. show that the reduction factor becomes R=2Azre^£=A^e4A1 where KQ is the modified Bessel function.z « 1. in Ref. Asymptotically. we have R{A) —• 1 for A —)• oo. Sri.1. and the optically pumped polarized ion source (OPPIS) producing mainly hydrogen and deuterium ions. [25]) D/^ i-A A' C .. Similarly. Priedburg. Does the luminosity decrease at A < 1? (d) For a flat beam with [5% > as. Rev.e.31 The principle of the ABS is to form atomic beams in a discharge tube called a dissociator. H.07703A2) erfc(A) » ( 1 + 0. As the beam travels through the beam tube. TRANSVERSE MOTION (b) Because of finite bunch-length as. and W. Z. Calculate the reduction factor as a function of Az and show that the luminosity is (use 3. This exercise illustrates e.364.32 The non-uniform magnetic field preferentially selects one spin state (Stern-Gerlach effect).5. the spin states of the atoms are selected in a separation magnet.g.4 + 0. we obtain R(AX.3 of Ref.3. Phys. 489 (1954). Plot £ as a function of A. OPPIS has been able to produce a polarized H~ ion source up to 400 /iA with 80% polarization at a normalized emittance near 1 7r mm-mrad.
F. and the polarized ions are drawn by the electric field to form a polarized ion beam. (a) Show that the sextupole field focuses the spin state of the atomic beam with lower magnetic dipole energy. it defocuses the spin state with higher magnetic dipole energy. Let ft = gfj. What is the focal length? 33 B.EXERCISE 2. A paraxial focusing system (lithium lens): A strong paraxial focusing system can greatly increase the yield of the secondary beams. the polarized ions are formed by the bombardment of electron beams. where g is the Lande ^-factor. The selected atoms. ro = 10 mm. which have a preferential one-spin state. ro is the radius of the Li conductor. w no = 4TT X 10~7 Tm/A is the permeability. (c) If the temperature of the dissociator is 60 K. The Li lens was first used at Novosibirsk for focusing the e + e" beams. et al.a « \iB for the hydrogen-like atom. The electron spin is quantized with respect to the magnetic field.e.788 x 10~ 5 eV/T is the Bohr magneton. The magnetic flux density is where / is the current.B J be the magnetic moment of the atomic beam. i. and /j.33 A cylindrical lithium rod carrying a uniform current pulse can create a large magnetic field. where in the high-field regime the nuclear spin can be flipped by rf field. the lithium lens or a strong solenoid has been used. /j. The atoms not contained in the beam pipe will be pumped away. 19. Here the the magnetic field is slowly changed to align all atomic polarization into the uniform field ionizer region. and JJ. Bayanov.B = 5. Thus the force acting on the hydrogen atom is F = V(jl-B) = ±^aV\B\ for two quantized spin 1/2 states of the hydrogen atom. It is worth pointing out that there is no preferred direction of the spin projection inside the sextupole. (a) Find the focusing function for the 8-GeV kinetic energy antiprotons if / = 500 kA. Methods. in other words. r is the distance from the center of the rod. will pass through the transition region. the electron spin is quantized along the B direction. Nucl. Inst. and / is the angular momentum of the atom. 190.2 83 the focusing effect due to a sextupole field. and the length is 15 cm. what is the velocity spread of the atomic beam? Discuss the effect of velocity spread of the atomic beam. (b) When a quadrupole is used to replace the sextupole magnet. The magnetic energy of the atomic beam in the magnetic field B is W = -p-B. It became the essential tool for anti-proton collection at Fermilab. To this end. show that the effective force on the atom is a dipole field. 9 (1981). .
OOdegDIP)" ! CIS =1/5 of Cooler circumference =86.E2=EANG.3 x A1!3 fm.364m/4 Ll:= 2. The atomic weight is 6.TAPE STOP . f o r edge a n g l e ANG := TWOPI/4 00 : DRIFT.L=L1."CIS BOOSTER (1/5 Cooler).DELTAP=0.0. Low energy synchrotrons often rely on the bending radius Kx = 1/p2 for horizontal focusing and edge angles in dipoles for vertical focusing.SUP. choose the length of the Li lens to be less than 10% of the nuclear reaction length.341 ! c e l l length 17. e e LCELL:=4. SUP: LINE=(BD. What is the effects of changing the edge angle and dipole length? Discuss the stability limit of the lattice.82m / 5 =17. E1=EANG.*TW0PI/360 ! u s e r a d .84 CHAPTER 2.00) ! a superperiod USE.013 x 105 N/m).941 g. CTPIA = vr(rp + R\).27324 EANG:=12. ANGLE=ANG.#S/E TWISS. JRA = 1. and A is the atomic mass number. Find the lattice property of the low energy synchrotron described by the following input data file (MAD). where rp = 0. TRANSVERSE MOTION (b) The total nuclear reaction cross-section between the antiprotons and the Li nucleus is given by the geometric cross-section. TITLE. show that the nuclear reaction length is about 1 m. 20. (c) Find the magnetic pressure P = B2/2fio that acts to compress the Li cylinder in units of atmospheric pressure (1 atm = 1. and the density is 0.SUPER=4 PRINT.8 fm.0 ! dipole length L2:=LCELL-L1 ! s t r a i g h t section length RH0:=1. To minimize the beam loss.5 g/cm 3 .L=L2 BD : SBEND.364m ! I t accelerates protons from 7 M V to 200 M V in 1-5 Hz. i.e. K2=0.
and feed-down from higher-order multipoles. Let v _ ( W A > v _ (Vo\ y+~{y'o) y-~\y'o-o)' be the phase-space coordinates of the closed orbit just before and just after the kick element located at s0. can be represented by oo ABZ +jABx = Bo £ > „ +jan) (x+jz)n. III. The effect of linear betatron coupling due to the skew quadrupole term. This section addresses the linear betatron perturbations resulting from the dipole and quadrupole field errors. dipole field errors may arise from errors in dipole length or power supply. where ABdt is the integrated dipole field error and Bp = po/e is the momentum rigidity of the beam. etc. VI. A.^ .. (2.154) where the perturbing fields ABZ and ABX. The perturbed closed orbit and Green's function First. (2. dipole roll giving rise to a horizontal dipole field. EFFECT OF LINEAR MAGNET IMPERFECTIONS 85 III Effect of Linear Magnet Imperfections In the presence of magnetic field errors.25). Bo is the main dipole field.)• <2155) . z" + Kz(s)z = . we will show that linear magnet imperfections have two major effects: (1) closed-orbit distortion due to dipole field error.The closed-orbit condition becomes M (5)-(«-. ai. and illustrates possible beam manipulation by using the perturbing fields.l Closed-Orbit Distortion due to Dipole Field Errors Up to now. and (2) betatron amplitude function distortion due to quadrupole field error. Based on our study of the betatron motion in Sec. we have assumed perfect dipole magnets with an ideal reference closed orbit that passes through the center of all quadrupoles. Hill's equations are x" + Kx(s)x = ^ . hi is the sextupole field error. we consider a single thin dipole field error at a location s = So with a kick-angle 8 = ABdt/Bp in an otherwise ideal accelerator. b0 and bx are respectively the dipole and quadrupole field errors. The a's are skew magnetic field errors. II. similar to Eq. and the solenoid will be discussed in Sec. In reality. n=0 Here j is an imaginary number.III. a closed orbit not centered in the quadrupoles.
82. the .^(s o )|) 2 sin TTU (2. the values of the betatron amplitude Figure 2. the closed orbit is dominated by the fifth error harmonic. (2.s0)9(s0).11 shows the closed-orbit perturbation in the AGS booster due to a dipole field error of 6. Po functions at kick dipole location s 0 . Since the betatron tune of the AGS booster is 4.158) (2. The right plot shows the same closed orbit as a function of the longitudinal distance.11: The left plot shows schematically the closed-orbit error of the AGS booster resulting from a horizontal kicker with kick-angle 9 = 6.\iP(s) . where G(s. The orbit response arising from a dipole field error is given by the product of the Green function and the kick angle. Since the betatron tune of 4.e. TRANSVERSE MOTION where M is the one-turn transfer matrix of Eq.82 is close to the integer 5.a ° c o s ™ ) ' are ( 2 . (sin7ri/ .COS7ny' y'°= 2iirT^. s0) = V S ) / ? ( S o ) cos( W .156 ) where v is the betatron tune and ao.86 CHAPTER 2.67). we obtain yco(s)=G(s. The closed orbit at other location s in the accelerator can be obtained from the propagation of betatron oscillations.159) is the Green function of Hill's equation. (2. The right plot of Fig. The resulting closed orbit at SQ is on a yo = 2^.82 mr.53) for an ideal accelerator. 2. The left plot is a schematic drawing of the resulting closed orbit around an ideal orbit. Using Eq. i.82 mr at the location marked by a straight vertical line.
12: Left. a schematic plot of the closed-orbit perturbation due to an error dipole kick when the betatron tune is an integer. B. if the betatron tune is a half-integer.III. y') in the presence of a dipole error when the betatron tune is an integer. 2. Here Ay' = 8.12). Since Hill's equation with dipole field errors. 2. the closed orbit can be obtained by a linear superposition of dipole kicks. A9(t) = (AB(t)/Bp)dt. AB(t) .159) shows that the closed orbit becomes infinite when the condition SHITTY = 0 is encountered. showing 5 complete oscillations in Fig. dipole field errors are distributed around the accelerator. JP(s) rs+c . the angular kicks of two consecutive revolutions cancel each other (see right plot of Fig. Right. The orbit kicks in every turn due to a dipole error coherently add up. The left plot of Fig. Figure 2. 2. However. Distributed dipole field error In reality. is in the same direction in each revolution. i. Since the angular kick Ay' = 9. where 6 is the dipole kick angle. the closed orbit becomes very sensitive to dipole field error.e. Thus the betatron tunes should also avoid half integers. Equation (2. the closed orbit does not exist. In other words. a schematic plot of the particle trajectory resulting from a dipole kick when the betatron tune is a halfinteger. making the closed orbit unstable. For the closed orbit. This is why the betatron tunes are designed to avoid an integer value.11. if the betatron tune is near an integer. is linear. we will show later that the quadrupole field error will produce betatron amplitude instability at a half integer tune. On the other hand. it is better to choose a betatron tune closer to a half-integer. here the angular kicks from two consecutive orbital revolutions cancel each other. EFFECT OF LINEAR MAGNET IMPERFECTIONS 87 closed-orbit perturbation is dominated by the n = 5 harmonic.12 shows schematically the evolution of a phase-space trajectory (y. where 9 is the kick angle of the error dipole.
"P k=-oo (2.165) which has simple poles at all integer harmonics.159).160) shows that the closed orbit may not exist at all if the betatron tune is an integer. Bp (2. and ip(s) = v<j>{s). The resulting closed orbit is usually dominated by a few harmonics near [u]. Js t) ^ | ^ d t . The orbit response of the inhomogeneous Hill equation is yco(s) = fS+CG(s.160) is the closed-orbit solution of the inhomogeneous Hill equation where A S = ABZ for horizontal motion and AB — —ABX for vertical motion. (2. the closed-orbit displacement yco(s) becomes Vcois) = y/W) £ -P^e**.160) is a periodic function of 2n. The integer stopband integrals Since square bracketed term in the integrand of Eq. (2. The simple pole structure in Eq.88 CHAPTER 2. we expand it in a Fourier series: f(<l>)=f?l\4>)^®= £ fke^. (2. (2. C. TRANSVERSE MOTION where <j>(s) = (l/i/) /05 dt//3(t). In Fourier harmonics.163) where the Fourier amplitude /& is the integer stopband integral given by with /_* = f%. It is easy to verify that Eq. which is an integer nearest to the betatron tune. (2. In a single stopband approximation. The presence of sin -KV in the denominator of Eq.162) where the Green function is given by Eq. the closed orbit can be approximated by y/ffOQH/Ml COs([t/]0 + x) 2 [y .165) indicates that the closed orbit is most sensitive to the error harmonics closest to the betatron tune. (2.[„]) Vco{S) " .
Closed-orbit correction Closed-orbit correction is an important task in accelerator commissioning. the effective angular kick is e = l£Ay = T> (2167) where B\ = dBz/dx is the quadrupole gradient. N. is not known a priori. E. EFFECT OF LINEAR MAGNET IMPERFECTIONS D. For example. the perturbing field error AB/Bp. and the rms angular kick angle. (2. and / is the focal length. Now we consider the dipole field error generated by quadrupole misalignment. . a statistical argument is usually used to estimate the rms closed orbit. the harmonic correction scheme. The remaining closed orbit can generally be corrected by the stopband correction scheme. Substituting Eq. If the closed orbit is large.1 mm will result in a rms closed-orbit distortion of 2 mm. The coefficient in curly brackets is called a sensitivity factor for quadrupole misalignment.169 ) where 0t is the angular kick of the zth corrector. and 0ims are respectively the average ^-function.III.166) where /?av. or the x2-minimization method. any major known source of dipole error should be corrected. we obtain Wms « ( 9 ^ / " TV^Q} A2/"ns. the stopband near k = [v] is / = 2^E^e-'W<. Statistical estimation of closed-orbit errors 89 In practice. (2. The sensitivity factor increases with the size of an accelerator.rms * ^-J^ 2v2| SHITTY VNOm*.166). ?/co.167) into Eq. due mainly to random construction errors in the dipole magnets and misalignment errors in the quadrupoles. M ( 2 . one can adjust the real and imaginary parts independently. (2. the number of dipoles with field errors. (2.168) |k2v/2/av|sin7rz/| v J where Nq is the number of quadrupoles and / a v is the average focal length. When quadrupole magnets are misaligned by a distance Ay. First. With a few dipole correctors. Placing these correctors at high-/? locations with a phase advance between correctors of [v\<t>i « n/2. the beam lifetime and dynamical aperture can be severely reduced. if the sensitivity factor is 20. an rms quadrupole misalignment of 0. During the design stage of an accelerator.
manipulation with an internal target. of Nc correctors. (2.. and higher order terms associated with betatron motion are neglected. Possible schemes of local orbit bumps are the "four-bump method" discussed in Sec. VPi (i = 1 . . The BPM resolution for proton storage rings is about 10 to 100 /im.170) where Co is the orbit length of the unperturbed orbit. Effects of dipole field error on orbit length The path length of a circulating particle is C = j y/(l + x/pf + x'2 + z'2 ds « Co + j . N C ) . F.. 34The BPM resolution depends on the stability of the machine and on the number of bits and the effective width of the pickup electrode (PUE). Let yi<co and A* be the closed-orbit deviation and BPM resolution of the ith BPM. Since a dipole field error gives rise to a closed-orbit distortion. .d s + ---. extraction.3.34 The aim is to minimize x ~k N-> hi. the resolution is reduced by a factor of 16.62.I 2 A? by varying 61.. = — w ( a k c o s kcj>i + bk s i n kfa). In many beam manipulation applications such as injection.For example... For example. etc. Because the closed orbit is not sensitive to errors in harmonics far from the betatron tune. III. the fcth stopband can be corrected by adjusting the ak and bk coefficients. these harmonics can hardly be changed by closed-orbit correction schemes. Let Nm be the number of BPMs and Nc the number of correctors. where ft and <f>i are the betatron amplitude function and the betatron phase at the ith kicker location. the circumference of the closed orbit may be changed as well. if Nc dipole correctors are powered with 9..3 and the "three-bump method" (see Exercise 2. the BPM resolution for the data acquisition system with a 12-bit ADC and a 40-mm effective width PUE is about 10 fim.90 CHAPTER 2. If an 8-bit ADC is used. TRANSVERSE MOTION The harmonic closed-orbit correction method uses distributed dipole correctors powered with a few harmonics nearest the betatron tune to minimize a set of stopband integrals fk.4). These orbit correction schemes minimize only the errors in harmonics nearest the betatron tune. Another orbit correction method is the ^-minimization procedure. A few harmonics can be superimposed to eliminate all dangerous stopbands.. local closed-orbit bumps are often used.
etc. tidal action. EFFECT OF LINEAR MAGNET IMPERFECTIONS 91 We consider the closed-orbit change due to a single dipole kick at s = s0 with kick angle 90. Thus the change in orbit length due to a dipole field error is equal to the dispersion function times the orbital kick angle. traffic and mechanical vibration. However.III. J p (2.174) 36The FFT spectra of a transverse phase-space coordinate display rotational harmonics at integer multiples of the revolution frequency and the betatron lines next to the rotation harmonics. This method can be used to measure the betatron tune. The modulation frequency from ground vibration is typically less than 10 Hz. particle motion in an accelerator will not be affected by a small-amplitude modulation provided that the modulation frequencies do not induce betatron or synchrotron resonances. III.|V. an rf dipole field operating at a betatron sideband35 can kick the beam out of the vacuum chamber. . the power supply ripple can produce modulation frequency at some harmonics of 50 or 60 Hz. the change in the total path length becomes AC = <£ D(s)^%^-ds.(a) . the dipole field errors are generated by power supply ripple. These betatron frequency lines are called the betatron sidebands.7 for details. Bp (2. Using Eq. this is called rf knock-out. For example.162). the equation of motion for a dipole field error in a constant-focusing quadrupole is x" + Kx = ^ .. and thus the circumference is modulated at some modulation frequencies. particle motion will be strongly perturbed. .171) where D(8o) = J f^lfAds P = 2 sin TT^J.^(so)|)<fa (2. IV). When dipole field errors are distributed in a ring. we find the change in circumference as AC = C-C0 = 80 <f G*(s>s°) ds = D(s0) 0o. if the modulation frequency is equal to the betatron or synchrotron frequency. (2. Normally. For example. See Sec. III.173) Bp J In many cases. J yliS0) / ^ p ^ cos (Try.154) for the closed orbit of the betatron oscillation can be solved by the extended 3x3 transfer matrix method. ground vibration. and the frequency generated by mechanical vibrations is usually of the order of kHz.2 Extended Matrix Method for the Closed Orbit The inhomogeneous differential equation (2.172) is the value of the dispersion function at s0 (see Sec. (2.
(2. . Proc.1 / / 1 Ayjf \ .155) is equivalent to /j/b\ \y'o = (I 0 0 \ /Afn 0 1 9\\M21 Mi2 0 \ (yo\ M22 0 ] \ y ' o \.178) \ i/ Vo o i ) \ o o i)\ \ ) where the M's are matrix elements of the 2x2 one-turn transfer matrix for an ideal machine. TRANSVERSE MOTION The betatron phase-space coordinates before and after the focusing quadrupole can be obtained from the extended transfer matrix by /x\ \x'\ W2 / cosVKe jg sin VKI %ff?(l-cosVKe)\ =\-y/K sin y/Kl cosVKt £%sany/Ke V 0 0 ^ 1 AW: /x\ U ' > (2175) where I = s2 — s\. (2. Examples are the local-orbit bump. 36S. Let Aj/q be the quadrupole misalignment. The resulting extended transfer matrix in the thin-lens approximation is Mquad = / 1 0 0 . In thin lens approximation. etc.175) becomes M(s 2 | S l ) = /I 0 0\ -1// 1 d .J. Similarly. For example. (2.. 1991). rf knock-out. the closed-orbit equation (2.176) V o oiy where 6 = ABZ£/Bp and / = 1/JW are respectively the dipole kick angle and the focal length of the perturbing element.177) V 0 0 1 ) The 3x3 extended transfer matrix can be used to obtain the closed orbit of betatron motion.36 III.3 Application of Dipole Field Error Sometime. Lee. one-turn kicker for fast extraction. and 9 is the dipole kick angle. Piscataway. the 3x3 extended transfer matrix can be used to analyze the sensitivity of the closed orbit to quadrupole misalignment by multiplying the extended matrices along the transport line. (2.92 CHAPTER 2. Eq.. IEEE PAC Con}. 1639. S.Y. (IEEE. N. we create imperfections in an otherwise perfect accelerator for beam manipulation. The dipole field error can also arise from quadrupole misalignments. Tepikian. p.
the two-bump method has been used at favorable phase-advance locations in accelerators. or i/9i#i cosliru-ipn] + \fW162 c o s ^ i / . EFFECT OF LINEAR MAGNET IMPERFECTIONS A. For example. Using four bumps. we obtain Vco(s) = ^ f ^ .4). and to avoid the limiting-aperture in the accelerator. we obtain / \ / M s = -{\[K. Figure 2. Orbit bumps 93 To facilitate injection. to Sj.^ ] + \fW%®% coslnv—ipte] + \fW$n COSTTJ/ — 0. V^^isinfTr^-^i] + i / S ^ s i n ^ .4) has also been used for local orbit bumps.III.^ ] + v^^sinTrv = 0.0\ sinipn + \JJ182 sin^42)/ sin^43. the middle bump dipole has negligible field strength.179) where 9t = (ABAs)i/Bp and (ABAs). The three-bump method (see Exercise 2. Expressing 63 and 64 in terms of 6\ and 62. B.3. avoiding unwanted collisions in colliders.E JK 0i cosfri/ .^ | ) . Although the slope of the bumped particle orbit can not be controlled in the three-bump method. y'co(si) = 0. we discuss the four-bump method facilitated by four thin dipoles with kick angles Q{ (i = 1. are the kick-angle and the integrated dipole field strength of the i-th kicker. Occasionally. or special-purpose beam manipulation.37 the orbit of the beam can be bumped to a desired transverse position at specified locations.. where xpji = ipj—ifii is the phase advance from s. extraction.i / ^ ] + V ^ ^ s i n ^ .2. etc. (2. The conditions that the closed orbit is zero outside these four dipoles are VCO{SA) = 0. . Using Eq.\i> .160). this method is usually used for the local orbit correction because of its simplicity. internal target area. In this example. a fast kicker magnet is usually powered in about 20-100 ns rise and fall times in order to bump beam bunches into the ex37Other examples are orbit bump at the aperture restricted area.3. Fast kick for beam extraction To extract a beam bunch from the accelerator. to avoid unwanted collisions. the counter-circulating e+ and e~ beams. I I I I \ (0 1 s n ^ *-**J) [ sjPtfi = {yjPxOiSYMpn + y'&^sin^M/sinvW The orbit displacement inside the region of the orbit bump can be obtained by applying the transfer matrix to the initial coordinates. we can adjust the orbit displacement and the orbit angle to facilitate ease of injection and extraction. Since the two outer bumps happen to be nearly 180° apart in the betatron phase advance.13 shows an example of a local orbit bump using three dipoles. or the p and p beams in a collider can be made to avoid crossing each other in a common vacuum chamber with electrostatic separators. 2 sin 7 v ~[ T v (2.
flx(s) is the amplitude function at location s. B^ is the kicker dipole field. a kicker kicks the beam from the bumped orbit to the the extraction channel at x\.14 shows a schematic drawing of the cross-section of a Lambertson septum magnet. The magnetic kicker employs ferrite material to minimize eddy-current effects. The rise and fall times of the kickers range from 10 ns to 100's ns. where a septum is located. etc.38 With the transfer matrix of Eq. current sheet septum.181) where 6y = f B^ds/Bp is the kicker strength (angle). To achieve a minimum kicker angle. where the former will not affect the circulating beams. Depending on the application. In this example. where the uniform dipole field bends the beam into the extraction channel. the septum is located about 90° phase advance from the kicker. such as wire septum. . 38The kicker is an electric or magnetic device that provides an angular deflection to the charged particle beam in fast rise and fall times so that it can selectively deflect beam bunches. The septum is a device with an aperture divided into afield-freeregion and a uniformfield region.13: A simple orbit bump in the AGS booster lattice.67). (2. The iron in the Lambertson magnet is shaped to minimize the field leakage into the field-free region and the septum thickness. Figure 2. that is of the order of 4-10 mm depending on the required magnetic field strength. (2. The quantity in curly brackets in Eq. the kicker array for stochastic cooling. Since the first and third kickers are nearly 180° apart in the betatron phase advance. the local orbit bump is essentially accomplished with these outer two kickers. Similar constraints apply to the kicker in the transverse feedback system. there are 3 focusing and 2 defocussing quadrupoles between two outer bump dipoles. A beam is bumped from the center orbit xc to a bumped orbit x^. The symbol X marks the three dipole kicker locations. and the latter can direct the beam into an extraction or injection channels.94 CHAPTER 2. The electric kicker applies the traveling wave to a stripline type waveguide. TRANSVERSE MOTION Figure 2. At the time of fast extraction. and A<f>x(s) is the phase advance from s^ of the kicker to location s. Px{sy) is the betatron amplitude function evaluated at the kicker location. Lambertson septum.181) is called the kicker lever arm. and the values of the betatron amplitude function at the septum and kicker locations are also optimized to obtain the largest kicker lever arm. the transverse displacement of the beam is AzCo(s) = ypx{sk)/3x(s)sm(A<t>x(s))}ek. (2. etc. one can choose among different types of septum. traction channel.
/30 is the value of the betatron amplitude function at the rf dipole location. and x^.o o .8. C is the circumference.14: A schematic drawing of the central orbit xc. if the betatron tune is 8. and Wo is the orbital angular frequency.182) with y VPo Hill's equation becomes 1 r» ds v Jo P where vm = w m /w 0 is the modulation tune. Hill's equation is cPv °° -r^ + K{s)y = 6asinumt £ 6(s-nC). The arrows indicated a possible magnetic field direction for directing the kicked beams downward or upward. rf knock-out In the presence of a localized rf dipole. and the particular solution rjco is the coherent time dependent closed orbit. "S n=-oo (2. oo). bumped orbit x\>. large betatron . Xf. The solution of the inhomogeneous Hill's equation is r] = Acosi>(j) + Bsinis(j) + r}co.182) where 9a = AB£/Bp and uim are respectively the kick angle and the angular frequency of the rf dipole. and t = s//3c is the time coordinate. The periodic delta function reflects the fact that the beam particles encounter the kicker field only once per revolution.. The blocks marked with X are conductor-coils. and kicked orbit x^ in a Lambertson septum magnet. C. (2. Performing the Floquet transformation to Eq. ^ = J L 2*R[v>-\n'+Vm)*) °° v2B3^9 Sln(n + ^" (2"185) Note that the discrete nature of the localized kicker generates error harmonics n + vm for all n € ( . For example. (2. The ellipses marked beam ellipses with closed orbits xc.III.184) where A and B are the amplitude of betatron motion determined by the initial conditions. EFFECT OF LINEAR MAGNET IMPERFECTIONS 95 Figure 2. Effects of rf dipole field.
v(n + i/m) sin v<j>\ where the last approximate identity is obtained by expanding the term in the sum with n + vm sa u. shown in the upper trace.41 39It is worth pointing out that the coherent growth time of the betatron oscillation is inversely proportional to \vm . Now we examine the coherent betatron motion of the beam in the presence of an rf dipole at i/m « v (modulo 1) with initial condition y — y' = 0.D." The fractional betatron tune. ^2 r Sln(n + ^rn)0 . Rev.40 The rf dipole can be adiabatically turned on to induce coherent betatron oscillations for betatron tune measurement without causing serious emittance dilution. The rf dipole was on from 1024 to 1536 revolutions starting from the triggering time. 4051 (1994).186) indicates that the beam is driven coherently by the rf dipole. Lett. To measure this small effect in the environment of the existing power supply ripple.. The solution is y(S> = —9^R— Z7r-K n=-oo " ~ \n + ^W ^ 1A—(„. is equal to the knockout tune. Ph. is measured by averaging the phase advance from the Poincare map (see Sec. the two-kick method was used to measure the instantaneous betatron tune change at the moment of the second kick. Beyond the coherent time. 4673 (1998).1.8. .15 shows the measured betatron coordinate (lower curve) at a beam position monitor (BPM) after applying rf knock-out kicks to the beam in the IUCF Cooler Ring. This two-kick method can be used to provide a more accurate measurement of the dependence of the betatron tune on the betatron amplitude. This process is related to the Landau damping to be discussed in Sec.. the beam would have been driven out of the vacuum chamber. Ellison et a/. where data from two BPMs are used. At revolution number 2048. III. £ 5 0 .1. Rev. Phys. Equation (2. 41M. Thus the method is called the "rf knock out. the beam was imparted a transverse kick. Phys. that. Note the linear growth of the betatron amplitude during the rf dipole-on time. Rev. and the amplitude of betatron motion grows linearly with time.0. Phys..4. See M. VIII. E56. the beam motion is out of phase with the external force and leads to damping.39 Figure 2. Bai et al.96 CHAPTER 2.2. 40The power supply ripple at the IUCF cooler ring gives rise to a betatron tune modulation of the order of 2 x 10~3 at 60 Hz and its harmonics.u\ (mod 1). and retaining only the dominant term. and the fractional part of the betatron tune (upper curve). 80. TRANSVERSE MOTION oscillations can be generated by an rf dipole at any of the following modulation tunes: vm — 0. 6002 (1997).8. in this experiment.2.— The localized repetitive kicks generate sidebands around the revolution lines. Had the rf dipole stayed on longer. On the other hand. the dependence of the betatron tune on the betatron action is typically 10~4 per lir mm-mrad. Thesis. Indiana University (1999).5).
This method is usually referred to as the beam transfer function (BTF). Figure 2. the beam profile restored back to its original shape. VIII). the beam was imparted by a one-turn kicker. After another 512 revolutions. The rf dipole was turned on for 512 revolutions. Furthermore. . the beam profile became larger because the beam was executing coherent betatron oscillations. EFFECT OF LINEAR MAGNET IMPERFECTIONS 97 Figure 2. the measurement of the coherent betatron tune shift as a function of the beam current can be used to measure the real and imaginary parts of the transverse impedance (see Sec. the beam profile restored to its original shape.15: The lower curve shows the measured vertical betatron oscillations of the beam of one BPM at the IUCF cooler ring resulting from an rf dipole kicker at the betatron frequency. Note the linear growth of the betatron amplitude during the rf knockouton time. After the rf dipole was adiabatically turned off. where data from two BPMs are used.III. The beam profile appeared to be much larger during the time that the rf dipole was on because the profile was an integration of many coherent synchrotron oscillations. As the rf dipole is adiabatically turned off. Figure 2. and the profile was obtained from the integration of many coherent betatron oscillations. The upper curve shows the fractional part of the betatron tune obtained by counting the phase advance in the phase-space map.16: The beam profile measured from an ionization profile monitor (IPM) at the AGS during the adiabatic turn-on/off of an rf dipole. When the rf dipole was on.16 shows the vertical beam profile measured at the AGS during the adiabatic turn-on/off of an rf dipole. The induced coherent betatron motion can be used to overcome the intrinsic spin resonances during polarized beam acceleration.
Con}. i = 1. The ORM method minimizes the difference between the measured and model matrices Rexp a n d Rmodei. 128 (1994). • • •.. sextupole field strength.98 CHAPTER 2. is the rms error of ith measurements.. iVm) vs the dipole kick at Oj (j = 1. etc. where iVj. Proc.. sextupole misalignment. Nb). 1994 European Part. TRANSVERSE MOTION D. The measured response matrix needs calibration in the kicker angle and BPM gain. and the machine stability during the experimental measurement..42 We consider a set of small dipole perturbation given by Oj.Nh i = l. quadrupole roll. Accel. If the closed-orbit response to a small dipole field perturbation can be accurately measured. Nm.. Con}. i. j = l. AIP Conf. The outcome of response matrix modeling depends on the BPM resolution. Sj) of the actual machine.162) shows that the beam closed orbit in a synchrotron is equal to the propagation of the dipole field error through the Green's function that is an intrinsic property of the betatron motion. KeXP«- fj9i ' 42 See J. etc. Proc.. Here the number of index k is Nh x Nm.Nm. Proc. Experimentally. Safranek.. Lee.. Orbit Correction and Analysis in Circular Accelerators. where <7. The response matrix R. N^. we measure Ry (i = 1. . It is worth mentioning that iVb is not necessarily equal to Nm. defined as yi = R t j 9 j .. is the number of dipole kickers. the calibration of the BPM gain factor. 315. where JVm is the number of beam position monitors. The resulting response functions can be used to calibrate quadrupole strengths.. and the model response matrix can be calculated from MAD[19]. The full set of the measured response matrix R can be employed to model the dipole and quadrupole field errors. • • •. j = 1. The ORM method has been successfully used to model many electron storage rings..J. where y stands for either x or z.. Accel. J. quadrupole misalignments. 1027 (1997). The measured closed orbit from the dipole perturbation is yi. No. BPM gains. Safranek and M.. J. SYNCH[20]. Orbit response matrix for accelerator modeling Equation (2. 1995 IEEE Part.Let Wk = I R ™»M«..e.J. Safranek and M.187) is equal to Green's function Ry = Gy(si. Proc. Lee. dipole field integral. (2.B «P«I / 2 _ 18g ) be the difference between the closed-orbit data measured and those derived from a model. The orbit response matrix (ORM) method measures the closedorbit response induced by a known dipole field perturbation. the Green's function of Hill's equation can be modeled.. 2817 (1995). or C0MF0RT[21] programs.. the number of BPMs and kickers..
The singular value decomposition (SVD) algorithm decomposes the matrix W into w = ovVk dw m = U A v T (2192) where V T is a real orthonormal Np x Np matrix with V V T = V T V = 1. and A i s a m x n diagonal matrix. and gt is the gain factor of the ith BPM.190).e. the quadrupole strength and roll. and V is composed of orthonormal eigenvectors of * WTW. the BPM gain factor. A is a diagonal Np x Np matrix with elements An = \f\l > A22 = y/X^--. i. (2.(VA. Once the SVD of matrix W is obtained. and U = AVA" 1 is a {Nm • Nb) x Np matrix with U T U = I. 43The SVD decomposition ofamxn matrix W in Eq. Np is the number of parameters.1 = l/\fK and 0 for all remaining diagonal elements with i > r. The ORM accelerator modeling is to minimize the error of the vector W by minimizing the %-square (x2) defined as ^=5vnr£w2.e. etc. A2. The iterative procedure continues until I Aw m | or the change of x 2 are small. WTW = VA 2 V T . Some of these parameters are kicker angle calibration factor. In our application to accelerator physics. (2.1 ^) W(wn). Wk(wm + Awm) « Wk(wm) + ^ A w m = 0.> 0. The ORM modeling is to find a new set of ium-parameters such that ||W(«»m)|| = 0. = 0 (i > r) is equivalent setting Aw* = 0 for i > r.191) To evaluate Awm. EFFECT OF LINEAR MAGNET IMPERFECTIONS 99 where / . sextupole strength. This means that these dynamical parameters have no relevance to the measured data.190) First. . A^. (2-189) We consider sets of parameters w m 's that are relevant to accelerator model and orbit measurement. (2. Here.193) where A " 1 is a diagonal matrix with AJ"/ = l/\f\[.III. The SVD-method sets all eigenvalues A < Ac. = 0.192) can also be carried out in such a way that U and V are respectively orthonormal real mxm and nxn matrices with U T U = UU T = 1 and V T V = W T = 1. we invert matrix W = ^ t . (i > r). one finds Awm as Awm = . (2. The idea is to find a new set of parameters wm + Awm that satisfies Eq. (2. (i > r) to A. i. the dipole angle and dipole roll. • • • are eigenvalues of the matrix WTVV. which has the dimension of is (Nb • Nm) x Np. is the calibration factor of the j t h kicker. where Ac is called the tolerance level and r is called the rank of the matrix W. Setting all A. {Nb • Nm) » Np. we begin with parameters wm and evaluate W(iu m ). 43 Here Ai. • • •.
the dimension of the matrix W. (eq23green) at a calibrated vertical steerer angle. For example.17: Left. (2) BPM gain <?. the proton storage ring (PSR) at Los Alamos National Laboratory accumulates protons for 3000 turns and the beam bunch is extracted after accumulation for high-intensity short-pulse neutron production.. 2.g. (4) dipole angle calibration. digitized betatron oscillation data of one BPM are used to derive betatron amplitude. . 1481 (1997). Right. TRANSVERSE MOTION The response matrix modeling has been successfully implemented in many electron storage rings.17). "First Test of Orbit Response Matrix in Proton Storage Ring".. top plot shows the closed orbit data compared with Green's function of Eq. For high-power synchrotrons. Technote: PSR03-001 (2003).44 In accelerator modeling.. beam particles are injected. The bottom right plot shows a similar comparison after ORM modeling.100 CHAPTER 2. et al.45 The right plots of Fig. The betatron oscillations of each BPM can be used to obtain the betatron amplitude. 4 5 X. phase and tune. Huang et al. e. These steps are sometimes essential in attaining a reliable set of model parameters. (1) kicker angle calibration fj. p. etc. dipole and quadrupole power supplies. The closed orbit data can be obtained by averaging betatron oscillations in a single turn injection. Chu.17 shows an example of typical fit in ORM modeling. phase and tune. accelerated and extracted in a short time duration. BPM gain.M. Proceedings of PAC 1997. and closed orbit offset. and the closed orbit (see the left plot of Fig. (5) dipole roll. sextupole misalignment. It is advantageous to model accelerator parameters in sequences. These information can be used in the ORM analysis for accelerator modeling. quadrupole strength and roll. 44 C. where the BPM resolution is about l~10 /zm. where the BPM resolution is usually of the order of 100 fim. etc. can be large. 2. The inversion of a very large matrix may become time consuming. The method has been used to calibrate kicker angle. (3) quadrupole strength AKi. Analysis of the Orbit Response Matrix Measurement for PSR. The method is also applicable to proton synchrotrons. (Nm • Nb) x Np. Figure 2.
Rev. Rev. 1684 (1999).25). Vadim Sajaev. Why don't we choose a half-integer betatron tune? This section addresses the effect of quadrupole field errors that can arise from variation in the lengths of quadrupoles. Thesis. 82. III. from the horizontal closed-orbit deviation in sextupoles. II. C. What happens to the betatron motion if some quadrupole strengths deviate from their ideal design values? We found in Sec.4 Quadrupole Field (Gradient) Errors The betatron amplitude function discussed in Sec. Stanford University (1999).194) where K0(s) is the focusing function of the ideal machine discussed in Sec.3C. Betatron tune shift Including the gradient error. Ph. one uses an rf dipole pinger to excite coherent betatron oscillation and measures the response function with turn-by-turn BPM digitizing system (See Sec.X. II depends on the distribution of quadrupole strengths. PEP-II and Advanced Photon Source.X. EFFECT OF LINEAR MAGNET IMPERFECTIONS 101 The success of accelerator modeling depends critically on the orbit and tune stability. III. . Model Independent Analysis Using turn-by-turn BPM data excited by resonant pinger discussed in Sec. This method has been successfully applied to SLC linac. and Y. ABX = B2(xcoz0 + x0zg). and C. (2. Yan. E. (2. These errors correspond to the bi term in Eq.Y. Beams 6.D. where B 2 = d2Bz/dx2. C.co.X. This process is called feed-down. in Appendix B).46 For the application of MIA in a storage ring.l that the effect of dipole field error on the closed orbit would be minimized if the betatron tune was a half-integer. II. III. proper set of experimental data for attaining relevant parameters. Hill's equation for the perturbed betatron motion about a closed orbit is 1 l + [K0(s) + k(s)}y = 0. Phys.47 etc. the number of BPMs and orbit steerers.III. Wang. and a quadrupole field gradient B2a. The perturbed focusing function K(s) = K0(s)+k(s) satisfies 4 6 J. Wang. we obtain ABZ = -B 2 (xeo + 2zCoZ/3 + x% . Phys. 104001 (2003). Irwin. C. Wang.T. Lett. ST Accel. J.T. Yao. Irwin and Y.25). called Model Independent Analysis (MIA). (2. A. 151 (2000). one can also carry out response matrix analysis for accelerator modeling. Proceedings of EPAC 2000. and k(s) is a small perturbation. Yan. from errors in the quadrupole power supply. Thus an off-center horizontal orbit in a sextupole generates a dipole field ^B2xl0.z}). p. 47 Substituting x = xco + x@ and z = zp into the sextupole field of Eq.
196) The betatron tunes are particularly sensitive to gradient errors at high-/3 locations. where C is the circumference. (2. and negative for a defocussing quadrupole. Let Mo be the one-turn transfer matrix of the ideal machine. M 0 (s) = / c o s $ 0 + J s i n $ 0 . The one-turn transfer matrix at s2 is M(sj) = M{s2 +C|5i) m(«i) M(Sl\s2). and the betatron tune shift is Az^ = -^f3(si)k(sx)dsi. TRANSVERSE MOTION a weaker superperiod condition K(s + C) — K(s). A = P(si). z where A$ = $ — $ 0 . / is the 2x2 unit matrix. . The transfer matrix of this infinitesimal localized perturbing quadrupole error is The one-turn transfer matrix M(si) = Mo(si)m(si) becomes A/re \—( c o s ^ o + Qisin$ 0 -/?i^(si)rfsisin$ 0 \— ji sin$ 0 — [cos$o + aisin$ 0 ]^( s i)^ s i /?isin$ 0 \ cos $o — ai sin $ 0 / ' where ot\ = a(si).102 CHAPTER 2. and \-7(s) -a(s)J Here a(s). cos<3> — cos$o = —y8(si)fc(si)dsisin$0) z or A$ «-/3(s1)A. Thus the power supply for high-/3 quadrupoles should be properly regulated in high energy colliders. (2. where $o = 27r^o is the unperturbed betatron phase advance in one complete revolution. The phase advance of the perturbed machine can be obtained from the trace of M. and 7(s) are betatron amplitude functions of the unperturbed machine. we again consider an infinitesimal quadrupole kick at s\ of Eq.(si)rfs1. the tune shift is Au = ~ f p(Sl)k(Sl)dSl. /3(s). it is positive for a focusing quadrupole.e. v0 is the unperturbed betatron tune. For a distributed gradient error. Here the betatron tune shift depends on the product of the gradient error and the betatron amplitude function at the error quadrupole. i. i.e. We consider now the gradient perturbation with an infinitesimal length ds\ at Si.195). B. and 71 = 7(si). and high-brightness storage rings. Betatron amplitude function modulation (beta-beat) To evaluate the effect of the gradient error on the betatron amplitude function.
the gradient error function i/o/32k(s). which is a periodic function of s. (2.197) and integrating over the distributed gradient errors.& + &)].67).3.198) = -TT^f- where (j> = (l/^o) Jo ds/p.$ 0 Removing the subscript 2 in Eq./32 = /2(s2) fa = V)(si)/t'o> a n dfa= ^(^V^o are the values of the unperturbed betatron functions. wherefi2= P{S2) is the value of the perturbed betatron function and $ is the perturbed betatron phase advance. The half-integer stopband integrals In a manner similar to the closed-orbit analysis.10) C. (2.200) where JP = 7T'1>P 2TT J Hs) er^ ds (2.199) becomes WW "A f V K. (2. It is easy to verify that Ap/p satisfies (see Exercise 2.196) is equal to the zero harmonic of the stopband integral. (2. we obtain llr = -^r^ms^os^+^fa)]^ k(fa)p2(fa)cos[2v0(n + <f>-fa)}dfa. where ft = /?(si)./32 and A $ = $ .) -'2j^^-(p/2r (2-202) f 2 2 0 2) . (2. we obtain (A/?2) sin $ 0 = AM12 .fa)]. (2. Since M i 2 = J32 sin $.Ill EFFECT OF LINEAR MAGNET IMPERFECTIONS 103 Using Eq.e. We note that the tune shift of Eq. The solution of Eq.201) is the pth harmonic half-integer stopband integral. (2.^ M s i A f t c o s p i ^ T r . i.197) where A/32 = ft . Av = Jo/2. we find the change of the off-diagonal matrix element as A[M(s2)]i2 = -Msi/Sift s\n[vo(<j>i -fa)}sin[zvo(27r +fa./?2 cos $ 0 A $ = . can be expanded in a Fourier series vof32k(s)= £ p=—oo Jpe^.
199) is JW (2. Figure 2. The change of the slope y' is proportional to the displacement y. This will lead to an ever increasing betatron amplitude. This means that the betatron tune should differ from a half-integer by at least the stopband width. (2. . Thus an integer betatron tune is also a half-integer resonance. Thus the half-integer stopband gives rise to unstable betatron motion.18 shows the behavior of a quadrupole kick at a half-integer tune. (2. and p is the integer nearest 2^o.12. quadrupole kicks will resemble the left plot in Fig. 0 -p) > (2-203) where \ ls a phase angle.Since the quadrupole kick is proportional to the displacement y.195)] Ay = 0 and Ay' = -k(si)ydsl = -y/f. The right plot shows the effect of zero tune shift 7r-doublets. The "closed orbit" of the betatron amplitude function will cease to exist.104 CHAPTER 2. The left plot of Fig. 2. Therefore the betatron tune should not be a half-integer. as shown in the left plot of Fig. that produce a local perturbation to betatron motion. The right plot shows the effect of zero tune shift 7r-doublets. and beam loss may occur. this is called a half-integer resonance. and y is the displacement from the center of the closed orbit.18. The amplitude function becomes infinite when 2v0 approaches an integer. The evolution of phase-space coordinates resulting from a quadrupole kick is [using Eq. the coherent addition of the kick angle in each revolution gives rise to the unstable particle motion. which give rise only to local betatron perturbation. which changes sign in each consecutive revolution at a half-integer betatron tune. When the betatron tune is an integer. The leading term of Eq. The stopband width is defined by 5vp ~ \JP\ such that |A/3(s)//?(s)|max « 1 at v0 ss | ± \5vp. 2. TRANSVERSE MOTION This indicates that the betatron amplitude function is most sensitive to those error harmonics of P2k(s) nearest 2v0. where the quadrupole kicks are coherently additive. a quadrupole error can generate coherent additive phase-space kicks every revolution. the beam size will increase by at least a factor of \/2.18: Left. where / = l / ( M s i ) is the focal length of the error quadrupole. 2. schematic plot of a particle trajectory at a half-integer betatron tune resulting from a defocussing error quadrupole kick Ay' = —y/f. When the betatron tune is a half-integer. When the betatron tunes are inside the stopband.
Pellegrin. H. R. 48 48R.205) where /3ir2 are betatron amplitude functions at quadrupole pair locations. Boer. IEEE Trans.L. 136 (1991). M.18. the average focal length. 63 (1989). Employing such modules. IV. p. Further detailed discussions can be found in the literature. Statistical estimation of stopband integrals 105 Again. Proc. we obtain AAtfiALi + /32AK2AL2 = 0. CERN 84-15. Using the zero tune shift condition. Sci. Strehl. p. SLAC PUB-2522 (1980). and (AK/K)ims are respectively the average /3 value. i. (2.205) also produces a zero stopband integral at p = [2v]. the stopband integral can be estimated by statistical argument as 47TJav \ -K /rms where P&v. p. and Proc. 1933 (1985). NS32. 105. p. if we do not know a priori the gradient error. the number of quadrupoles. J[2v] — 0- Since the stopband integral J[2v] of a zero tune shift 7r-doublet is zero. CERN 89-05. 869 (1983). EFFECT OF LINEAR MAGNET IMPERFECTIONS D. in the design stage of an accelerator. on High Energy Accelerators. 385 (1984). 2. a zero tune shift quarter-wave quadrupole pair produces a maximum contribution to the half-integer stopband. the doublet has little effect on the global betatron perturbation shown in the right plot of Fig. Zero tune shift 7r-doublets can be used to change the dispersion function and the transition energy (to be discussed in Sec. Effect of a zero tune shift 7r-doublet quadrupole pair A zero tune shift 7r-doublet (or the zero tune shift half-wave doublet) is composed of two quadrupoles separated by 180° in betatron phase advance with zero tune shift. AIP Conf. 99 (1987). where the betatron perturbation due to the first quadrupole is canceled by the second quadrupole. p. CERN 91-04. In this section we discuss some basic beam diagnosis tools. CERN 87-10. . and the rms relative gradient error. R. (2. Jung. Shafer.e. and AKiALi is the integrated field strength of the iih quadrupole. J. On the other hand.III.fsv.5 Basic Beam Observation of Transverse Motion Measurements of beam properties are important in improving the performance of a synchrotron. F. Littauer. Koziol. we can correct half-integer betatron stopbands. E. J. Serio. Conf. III.8). 11th Int. Zero tune shift half-integer stopband correctors We find that the zero tune shift ?r-doublet produces a zero stopband width.Nq. The zero tune shift condition of Eq. P. Nucl. 459 (1980).
split electrodes and buttons.g. E = U+ + [/_ is the sum signal. the beam position is y * 2 WTTT w U+-U= 2 E' wA (2-206) where U+ and C/_ are either the current or the voltage signals from the right (up) and left (down) plates.106 CHAPTER 2. which is the self-inductance of the loop.19: A schematic drawing of electric beam position monitors. Similarly.206) may require nonlinear calibration. we can obtain the closed-orbit information from the DC component.g. (2. we can measure the betatron motion. The button BPMs are used mainly in electron storage rings. where the bunch length is small. Measurements of the normalized difference signal with proper calibration provide information about the beam transverse coordinates. e. sampling the position data at a slower rate. configurations. the induced image electric charges on the plates can be transmitted into a low impedance circuit. the relation Eq. A. The voltage is proportional to the rate of variation of the magnetic flux associated with the beam current linked to the loop. If we digitize beam centroid positions turn by turn. Figure 2. An electrostatic monitor is equivalent to a current generator. As the beam passes by. On the other hand. The BPM can have an electrostatic. Beam position monitor (BPM) Transverse beam position monitors (BPMs) or pickup electrodes (PUEs) are usually composed of two or four conductor plates or various button-like geometries. small secondary loop winding. the strip-line type has a larger transfer function. or the induced voltage can be measured on a high impedance port such as the capacitance between the electrode and the surrounding vacuum chamber. e.is called the difference signal or the A-signal. A = U+ . In general. where the image charge is detected by the shunt capacitance of the electrode to ground. . Depending on the geometry of the PUE. TRANSVERSE MOTION Figure 2.19 shows a sketch of some simple electric BPM geometries used mainly in proton synchrotrons. The split-can type BPM has the advantage of linear response. a magnetic loop monitor is equivalent to a voltage generator with a series inductor.U. and w/2 is the effective width of the PUE. or a magneto-static.
Dots in the left plot of Fig. B. The fractional part of the horizontal betatron tune is ux = 0. EFFECT OF LINEAR MAGNET IMPERFECTIONS 107 Figure 2.i) ellipse. . /3i and p2 are the values of betatron amplitude function at two BPMs. From the FFT spectrum. (2. (2. The invariant phase-space ellipse becomes x\+ o ( IK «/£• esc VV P 2 fci2- cot V>2i Zi \2 j =2AJ. we find that the horizontal and vertical tunes of this experiment were ux = 3.20: The measured betatron coordinates at two horizontal BPMs. where the solid line is obtained from Eq. Figure 2. where tp21 = "4>2 — ipi is the betatron phase advance between two BPMs. The solid line is drawn to guide the eye. The observed vertical betatron tune at vz = 0.208) where the area enclosed by the (z 2> zi) ellipse is 27iV/?i/32 |sin^2i|<A and J is the betatron action.21 shows the measured (x2. The phase-space trajectory can be optimally derived from the measured betatron coordinates at two locations with a phase advance of an odd multiple of 90°.III.758 and vz = 4.a.20 shows the data for the horizontal betatron oscillation of a beam after a transverse kick at the IUCF cooler ring. the betatron tune can be determined from the FFT of the transverse oscillations (see Appendix B).208) by fitting y/Si/ft and ip2i parameters . (2. Measurements of betatron tune and phase-space ellipse If the betatron oscillations from the BPM systems can be digitized turn by turn. and a\ = —P[/2 at the first BPM. 2. The top plot shows the digitized data at two BPM positions (xi and x2).002.683 respectively. we obtain . Using Eq.242 ± 0.62).317 may result from linear coupling or from a tilted horizontal kicker._csc^2i l~vmX2~ (coty>2i+Qi) A Xu (2-207) . The FFT spectrum of the BPM data (middle plot) shows the fractional part of the horizontal betatron tune. The lower plot shows the FFT spectrum of the Xi data. vs the revolution number. after the beam is imparted a magnetic kick. Here a total of 385 data points are used in the FFT calculation.
Because the horizontal tune in this example is 3. 2.£i. and 2ft J = 8 x 10"6 m2. IV. we can derive the betatron amplitude function by measuring the betatron tune as a function of the quadrupole strength. (2. the phase-space ellipse of (a:i. for the orientation. ihi = 80°.n+i vs ii i B . Other applications. Because the betatron tune is nearly 3. If ft is independently measured.20. The turn by turn digitized data require a high bandwidth digitizer and a large memory transient recorder.196).21: The left plot shows the phase-space ellipse (x2. 2. The right plot of Fig. will be discussed in Sec. The area enclosed by this nearly circular ellipse is 27rft| sin2irux\J.x{) of Fig. A.20). Two BPMs separated by about 90° in phase advance are useful for obtaining a nearly upright transverse phase-space ellipse. ft-Function measurement Using Eq. the action of the ellipse can be determined.6 Application of quadrupole field error By using the quadrupole field error. measured. the phase space is an upright circle.n+i) is nearly a circle. The area enclosed is 27rft| sm2ni/x\J. obtained by plotting betatron coordinates of successive revolutions of single digitized BPM data.21 shows ii. and ft ^ for the size of the ellipse.8. (2. TRANSVERSE MOTION Figure 2. such as ?r-doublets for dispersion function manipulation. If the betatron amplitude function ft is independently measured.n. The right plot shows a poor man's phasespace ellipse.108 CHAPTER 2.208) with y/falPi = 1-4.758. The area enclosed by the ellipse is 27iVft/?2 I sin •jfel | J. if available hardware is limited. However. the optical properties of the lattice can be altered. Examples of ^-function measurement and the betatron tune jump are described below. or manipulated. 2. III. the action of the betatron orbit can be obtained. the phase-space ellipse can be obtained by using digitized data of successive turns of a single BPM.75 (see Fig. The solid line shows the ellipse of Eq. The average betatron ampli- .
Tune jump The vector polarization of a polarized beam is defined as the percent of particles whose spins lie along a quantization axis. Figure 2. the fractional horizontal tune is seen to increase with the strength of the horizontal defocussing quadrupole.22 shows an example of the measured fractional part of the betatron tune vs the strength of a quadrupole at the IUCF cooler ring. The "average" betatron amplitude function at the quadrupole can be derived from the slopes of the betatron tunes.HI. B. the actual horizontal tune is below an integer. The slope of the betatron tune vs the quadrupole field variation is used to determine the betatron amplitude functions.79284739 for protons. Since the fractional part of the horizontal tune increases with the defocussing quadrupole strength. Because the fractional parts of betatron tunes are qx — 4 — vx and qz = 5 — uz. In this example. acceleration of a polarized beam may encounter spin depolarization resonances [22]. where iV± are the numbers of particles with their spin projection lying along and against the quantization axis.0011596522 for electrons.N-)/(N+ + AL). where the horizontal and vertical tunes are determined from the FFT spectrum of the betatron oscillations. where qx and qz are the fractional parts of the betatron tunes. For polarized beams in a planar accelerator. and 0.49 Thus G7 is called the spin tune. . Since the spin tune increases with the beam energy. the polarization of a proton beam is P = (N+ .g.z) at a quadrupole becomes where AKl is the change in the integrated quadrupole strength. For the IUCF cooler ring. EFFECT OF LINEAR MAGNET IMPERFECTIONS 109 Figure 2. According to the Thomas-BMT equation. e. the polarization vector precesses about the vertical axis at Gj turns per revolution. where the "imperfection resonance" 49G = 1.22: An example of betatron amplitude function measurements. and thus the horizontal betatron function is larger than the vertical one. tude function (px.2)/2 is the anomalous gfactor and 7 is the Lorentz relativistic factor. the quantization axis can be conveniently chosen to lie along the vertical direction that coincides with the vertical guide field. the slope of the horizontal betatron tune is larger than that of the vertical. we have i/x = 4 — qx and vz = 5 — qz. where G = (g .
The AGS had 96 closed-orbit correctors for imperfection resonance harmonics. 51 An rf dipole has recently been used to overcome these intrinsic spin resonances. Similarly. Because of the integer and half-integer stopbands. the magnitude of tune jump is limited to about Avz « 0. a 5% partial snake has recently been used to overcome all imperfection resonances.3.7 Transverse Spectra A. —00 (2. Lett. TRANSVERSE MOTION arises from the vertical closed-orbit error.g. Transverse spectra of a particle A circulating particle passes through the pickup electrode (PUE) at fixed time intervals To. Huang et al. where To = 2-KR/jic is the revolution period.210) 50 At AGS. M. When the G7 value reaches an intrinsic spin resonance.8. Phys. nonlinear resonances. 73. 80. Phys.50 The intrinsic resonance in low/medium energy synchrotrons can be overcome by the tune jump method. the bandwidth of PUEs is normally from 100's MHz to a few GHz. The amount of tune change is Al/* = h f ^ds> (-0) 2 29 where ABi is the quadrupole gradient of tune jump quadrupoles. The imperfection resonance can be corrected by vertical orbit correctors to achieve proper spin harmonic matching. Rev.51 III. See. Lett. etc. The AGS had 10 fast ferrite quadrupoles to produce a tune jump of about 0. Rev. should be carefully evaluated. .110 CHAPTER 2. and @c is the speed. the betatron tune is suddenly changed to avoid the resonance.3 in about 2. Placement of tune jump quadrupoles to minimize the stopband integral can reduce non-adiabatic perturbation to the betatron motion. 4673 (1998). With a large tune jump. beam dynamics issues such as non-adiabatic betatron amplitude function mismatch. A 20 G-m rf dipole was used to replace 10 ferrite quadrupoles with an integrated field strength of f Bids = 15 T.5 jus.g. the important half-integer stopbands are located at p = 17 and 18. This betatron tune jump can be achieved by using a set of ferrite quadrupoles with a very fast rise time. Bai et al. e. linear betatron coupling. 52 We assume that the bandwidth of PUEs is much larger than the revolution frequency.. R is the average radius. See.. Since the betatron tune of AGS is about 8. and the "intrinsic resonance" is produced by the vertical betatron motion.. e. non-adiabatic closed-orbit distortion. In fact. H. the non-adiabatic closed-orbit perturbation due to the misalignment of tune jump quadrupoles can also be analyzed. The current of the orbiting charged particle observed at the PUE52 is 00 /(t) = e£<J(t-nT 0 ). 2982 (1994).
Figure 2. The rf current is twice the DC current because negative frequency components are added to their corresponding positive frequency components. and S(t) is the Dirac 5-function. (d(t)) = (I(t))y0.212) d{t) = I(t)y{t) = I(t) [y0 + ycosuj?t}. (2. shown in the bottom plot. If we apply a transverse impulse (kick) to the beam bunch. we obtain I(t) = f £ e***' = ^ + 2 f £ cosnWoi. and observed in a spectrum analyzer (bottom plot).23 shows the periodic time domain current pulses.Because the negative frequency components are added to their corresponding positive frequency components. The top plot of Fig.211) where j is the complex number. where y0 is the offset due to the closed-orbit error or the BPM misalignment.e." Passing the signal into a spectrum analyzer for fast Fourier transform (FFT). we observe a series of power spectra at integral multiples of the revolution frequency nfo. The BPM measures the transverse coordinates of the centroid of the beam charge distribution (dipole moment). The DC current is e/T0. y is the amplitude of the betatron oscillation. J 0 n=-oo 1f> i0 n=i (2.23: A schematic drawing of current pulses in the time domain (upper plot). and W = f5c/R = 2ftfo is the angular frequency. The DC component of the dipole moment can be obtained by applying a low-pass filter to the measured BPM signal. Expressing the periodic delta function in Fourier series. 2. and up = Qyuio is the betatron angular frequency with betatron tune Qy. o Note that the periodic occurrence of current pulses is equivalent to equally spaced Fourier harmonics. and the rf current is 2e/To.III.213) . the rf current is twice the DC current. i. The middle plot shows the frequency spectra of the particles occurring at all "rotation harmonics. given by (2. in the frequency domain (middle plot). the beam will begin betatron oscillations about the closed orbit. EFFECT OF LINEAR MAGNET IMPERFECTIONS 111 where e is the charge of the particle.
n=—oo (2. the betatron tune can be measured by measuring the frequency of betatron sidebands. The Fourier transform of the rectangular current pulse becomes !(«>) = f" / 1 r°° l~K J-oo We-**dt sidebands are classified into fast wave.112 CHAPTER 2. the betatron oscillations obtained from the BPM signals contain sidebands around the revolution frequency lines. the distribution can be a cosine-like function. This topic is important to collective instabilities.215) where the density distribution is normalized according to / /•To/2 J-To/2 p(t . v ' where A is the bunch width in time. (-1) 228 . 10 otherwise. We discuss two simple examples as follows. and slow wave. 1. We ask what happens if the beam distribution has a finite time span with 00 I{t) = Nae Y. (2.e. 53The = Hr-^-\^J^-n^- [ALewosinwAl ^2. / = (n ± Qj. backward wave. Expanding the dipole moment in Fourier harmonics.53 B.nT0)dt = I. to be discussed briefly in Sec. the betatron oscillation can be measured. VIII. If the beam is confined by a barrier rf wave or a double rf system. Naturally.)/o with integer n. On the other hand. TRANSVERSE MOTION where the betatron oscillation can effectively be removed. by employing a band-pass filter. i. Fourier spectra of a single beam with finite time span We note that a periodic 5-function current pulse in time gives rise to equally spaced Fourier spectra at all revolution harmonics./2A if-A<*-nT0<A. If the beam is confined by a sinusoidal rf cavity to form a bunch. we obtain dp = ~ y T (e>(»-+Qv)»ot + ei(n-Q. p(t-nT0).216) There are many possible forms of beam distribution.)uot\ ( 2 2 1 4 ) In the frequency domain. the beam distribution can be approximated by the rectangular distribution: p(t-nT0) = {.
' 0 n=—oo .The frequency spectra of a long bunch. In the frequency domain. Since all particles in the bunch are assumed to have an identical revolution frequency. The form factor serves as the envelope of the revolution comb shown in Fig.«*)• (2-220) Figure 2." T ° ) 2 / 2 " ' . (2.219) The Fourier transform of the current pulse can be carried out easily to obtain / H = [^e. e. 2. (2. In general.e . the Fourier spectra can extend to about l/at.218) and (2. the 5-function pulses are replaced by pulses with finite frequency width. (2. the Fourier spectra of Eqs.III.( ' . Figure 2. EFFECT OF LINEAR MAGNET IMPERFECTIONS 113 2.nT0) = _ L .24: The form factors for the Fourier spectra of a Gaussian bunch and a rectangular bunch with rms bunch length at = 1 ns.221) n=-oo . The beam distribution for electrons in storage rings is usually described by a Gaussian distribution due to the quantum fluctuation p(t . If there is a revolution frequency spread.24 shows the envelope factor for Gaussian and rectangular beam distributions with an rms bunch length of 1 ns. will have coherent spectra limited by a few hundred MHz. The beam current observed at a PUE is I(t)=e £ °O 5{t-n^) 71 •'" =^ AT- OO £ <**"**». if the beam has a time width at. Note that the coherent signal of a rectangular bunch can extend beyond the Gaussian cut-off frequency. Fourier spectra of many particles and Schottky noise We consider N charged particles evenly distributed in the ring.g.220) are (^-function pulses bounded by the envelope factors. 1 m bunch length (3.23.w2 *< 2/2 ] £ *(« . the spectrum of the beam pulse is truncated by a form factor that depends on the time domain distribution function.3 ns). C.
(2. the first coherent Fourier harmonic is Bu0. However.g. N > 108.223) d(*) = E ^ T «*(<"/«*+ *) £ e ^ . the frequency spectrum is practically outside the bandwidth of PUEs. and the spectrum is simply invisible. e." Similarly. the average power of the dipole moment can be measured.114 CHAPTER 2. If the particles are randomly distributed.225) 5 4 The analysis of equally spaced short bunches in the ring has identical Fourier spectra. the coherent betatron sidebands of a nearly uniform distribution are beyond the bandwidth of PUEs. and the spacing of Fourier harmonics is also Nco0. the frequency spectra of the transverse dipole moment of N equally spaced particles give rise to a betatron sideband around the coherent orbital harmonics Nui0 ± uip. This means that the beam appears to have no rf signal.224) Normally. if there are B bunches in the ring. When N is a very large number. The beam that fills the accelerator is called a "DC beam. i. Similarly. It is important to realize that particles are not uniformly distributed in a circular accelerator.e. the average power of the dipole moment is P^ = ^[T\d2{t)\dt. . e.g. n=lt=l (2. the coherent betatron frequency becomes too high to be visible to PUEs. TRANSVERSE MOTION Note that the first Fourier harmonic is located at Nw0.54 If the number of particles is large.W ." or a "coasting beam. i=l 1* n=-oo (2.222) The beam signal arising from random phase in charged particle distribution is called the Schottky noise. This is called the Schottky noise signal. o i + A0i(i)).f " > . The power spectrum at each revolution harmonic from a low noise PUE is proportional to the number of particles. the dipole moment of the «th particle is oo di(t) = ej)i cos(u0it + Xi) E n——oo pit 1i S(t ~ nTi ~ *oi) °° = ^cos(cj0it + Xi) E e ^ ( . The longitudinal signal of N particles in a PUE is I(t) = eJ2 £ Sit-U-nT. N > 108. The dipole moment of the beam becomes n=-oo (2.) = 2ixe E i=\ n=-oo oo N Z>*eWt~ei) cc N n=-ooi=l « Are/o + 2 e / o E E c o s ( n a .
Beam injection and extraction Electrons generated from a thermionic gun or photocathode are accelerated by a high voltage gap to form a beam. The charge exchange injection involves H~ or H^ ions. this means that dipole moments of particles with frequencies within T~l m 10~2 Hz may interfere with one another.3.III. e. T is of the order of minutes. a duoplasmatron. 55 The chicane magnet may sometimes be replaced by punching a hole through the iron of a main dipole magnet provided that the saturation effect at high field is properly compensated.5S Since the injection orbit coincides with the closed orbit of the circulating protons without violating Liouville's theorem. and extracted by a voltage gap to form a beam.25a. The resulting Schottky power can be contaminated by particle-to-particle correlation. The Schottky signal can be used to monitor betatron and synchrotron tunes. as shown in Fig.226) The power spectrum resembles the single-particle frequency spectra located at nuo ± up.g. The injection procedure is as follows. frequency and phase space distributions. i. III. The beam is accelerated by a DC accelerator or an RFQ for injection into a linac (DTL). i. The closed orbit of the circulating beam is bumped onto the injection orbit of the H~ or Hj beam by a closed-orbit bump and a set of chicane magnets. Al. ions are produced from a source.12). the resulting phase-space area will be minimized.8 Beam Injection and Extraction A. except that we must take into account the effect of emittance blow-up through multiple Coulomb scattering due to the stripping foil (see Exercise 2.e. The strip or charge-exchange injection scheme There are many schemes for beam injection into a synchrotron. The beam is captured in a linac or a microtron and accelerated to a higher energy for injection into other machines. . pav = E «=i ^ 4Jt at u = n(uo) ± (w/j).e. EFFECT OF LINEAR MAGNET IMPERFECTIONS 115 where 2T is the sampling time. where a stripping foil with a thickness of a few /zg/cm2 to a few mg/cm2 is used to strip electrons. The injection efficiency for this injection scheme is high. It is the essential tool used for stochastic beam cooling. betatron sidebands around all rotation harmonics. (2. 2. the Schottky power is proportional to the number of particles. The medium energy beam is then injected into various stage of synchrotrons. For practical consideration. Measurements with varying sampling times can be used to minimize the effect of particle correlation. Since the phases u)itOi and xi a r e random and uncorrelated. etc. The injected beam can be painted in phase space by changing the closed orbit during the injection. Similarly.
116 CHAPTER 2. the closed orbit is bumped near the septum (dashed ellipse) so that the injected beam marked (1) is captured within the dynamical aperture. The efficiency may be enhanced by employing betatron resonances. TRANSVERSE MOTION Although the intensity of the H~ source is an order of magnitude lower than that of the H + source. Injection of the electron beam is similar to that of proton or ion beams except that the injected electrons damp to the center of the phase space because of the synchrotron radiation damping (see Chap. a higher capture efficiency and a simpler injection scenario more than compensate the loss in source intensity. The procedure is to bump the closed orbit of the circulating beam near the injection septum. and the injected beamlet will damp and merge with the circulating beam bunch. Figure 2. The injection efficiency is usually lower.7 eV binding energy. the particle distribution in betatron phase space can be optimized. cooling. Betatron phase-space painting. the closed-orbit of circulating beams is bumped (kicked) close to a septum magnet so that the injection beam bunch is within the dynamical aperture of the synchrotron. As the bumped orbit is moved during the injection time. Most modern booster synchrotrons and some cyclotrons employ a H~ source. the injected beam can avoid the septum in the succeeding revolutions marked (2) and (3).25b. 2. A2. At the time of injection pulse. radiation damping The injection of protons or heavy ions into a synchrotron needs careful phase-space manipulation. since the last electron in H~ has only about 0. The stable phase-space ellipse is shown as the dashed line in Fig. At the completion of the injection procedure. the bump is removed. The combination of phase-space painting . the phase space is painted and the injected beam is accumulated. However. This procedure is called phase-space painting. (b) The process of betatron phase-space accumulation. 4). If the betatron tune and the orbit bump amplitude are properly adjusted.25: (a) A schematic drawing of a chicane magnet that merges the H~ and H + orbits onto the stripper. Because of the betatron motion. During the injection. it can easily be stripped by a strong magnetic field at high energy. etc.
and of rf phase displacement acceleration. etc. and stacking accumulation of proton or polarized proton beams. The method will be discussed in Chap.9 Mechanisms of emittance dilution and diffusion A. Slow extraction employs nonlinear magnets to drive a small fraction of the beam particles onto a betatron resonance. residual gas scattering. III. The extracted beam can be delivered to experimental areas. intrabeam scattering. noise may arise from various sources such as power supply ripple. III. the value of the betatron amplitude function at the septum location. this is called stochastic slow extraction. A fast kicker is fired to take the beam into the extraction channel of a septum magnet (see Sec.3). Other injection methods A method that has been successfully applied at the ISR is momentum phase-space stacking. 3. etc. medical treatment. This method is also commonly used in low energy cooler rings for cooling. orbit bump is usually excited. Fast single turn extraction and box-car injection When a beam bunch is ready for extraction. B. The efficiency depends on the thin-septum thickness.Ill EFFECT OF LINEAR MAGNET IMPERFECTIONS 117 and damping accumulation can be used to provide high-brightness electron beams in storage rings. ground vibration.3). More recently. betatron phase advance between the nonlinear magnet and the septum location. Beam extraction Bl. Emittance diffusion resulting from random scattering processes In actual accelerators. . VII. called the box-car injection scheme. B2. A3. etc.1. efforts have been made to improve the uniformity of the extracted beam by stochastic excitation of the beam with noise. The slow extraction using the third order resonance will be discussed in Sec. beam particles can be slowly extracted by employing the half integer resonance (see Exercise 3. or be transferred into a another synchrotrons.3. Slow extraction Slow (beam) extraction by peeling-off high intensity beams can provide a higher duty factor56 for many applications such as high reaction rate nuclear and particle physics experiments. This requires an understanding of the momentum closed orbit or the dispersion function. This 56The duty factor is defined as the ratio of beam usage time to cycle time. Similarly. Large-amplitude particles moving along the separatrix are intercepted by a thin (wire) septum that takes the particles to another septum on the extraction channel.
particularly at high /3x-function locations. nonresonant and non-adiabatic ground vibration. The incoherent space-charge . and multiple Coulomb scattering from gas molecules. multiple scattering on the residual gas. Other effects are due to the angular kicks from synchrotron radiation. Al. VII). (2. the resulting change in the betatron action is A / = I(y. 3 and 4.8). lifetime effect due to nonlinear resonances (see Sec. and the beam is composed of particles with many different betatron phases. intrabeam Coulomb scattering. beam lifetime is further reduced by beam-beam effects. B.8). etc. i.3. intrabeam Coulomb scattering. injection and extraction kicker noises.. TRANSVERSE MOTION can induce emittance dilution and beam lifetime degradation. these will be addressed in Chaps. This effect can also be important in the strip-injection of the H" and Hj ion sources from the stripping foil. II. etc. Touschek effect (to be discussed in Chap.227) If the angular kicks are uncorrelated. The space-charge effect is characterized by an incoherent Laslett tune shift parameter £sc = Avsc (see Exercise 2.e. i. The emittance growth rate can be obtained from the well-known multiple Coulomb scattering. etc. Beam Lifetime The single beam lifetime is determined by nuclear scattering on residual gas in the beam pipe. (2. ion or electron trapping due to residual gas scattering. y' + O).4 II. diffusion processes caused by rf noise. Our understanding of betatron motion provides us with a tool to evaluate the effect of noise on emittance dilution.118 CHAPTER 2.12 gives an example of estimating the emittance growth rate. Exercise 2. Aerms = 2(A/) = {pe2) « (/3X)(92). photo desorption. Space charge effects The repulsive Coulomb mean-field field of a beam can generate defocussing force to reduce the effective external focusing. Multiple scattering from gas molecules inside the vacuum chamber can cause beam emittance dilution. If the betatron angle y' is instantaneously changed by an angular kick 9. y') = 0{ay + (3y') + ^B2. the increase in emittance due to the random scattering processes is obtained by averaging betatron oscillations and kick angles.2).228) Random angular kicks to the beam particles arise from dipole field errors. particle loss due to beam-beam collisions. In a collider. quantum fluctuation resulting from energy loss.3. The tune shift parameter for low energy linacs at the ion source can be large. the betatron tune can be detuned to a value nearly 0 (see Sec.e.I(y. beam aperture limitation.
where Hill's and the envelope equations are \y"+(k(s)-4r))y V p(s)/ I y" + k(s)y . II. and /3(s) is the betatron amplitude function. For synchrotrons.Ill EFFECT OF LINEAR MAGNET IMPERFECTIONS 119 tune shift for low energy synchrotron has a typical value of 0. e = 4erms is the KV beam emittance. and the over-dots are derivative with respect to the independent variable (time-coordinate) 4>. the space-charge terms in Hill's and envelope equations can be considered as a small perturbation unless a resonance condition is encountered.' 7 / 2 ^% =0 (y<Rb). For a KV beam. Making Floquet transformation with we transform Hill's and envelope equations into J +A .2 . where A is a ^-independent constant shift in the equilibrium radius and r is the independent term depending on the dynamics of the machine. We expand the envelope radius around the unperturbed closed orbit with R = 1 + r + A. Bl.^ (2. = 0. all particles are within the envelope radius.8.i l .6.136). Ksc is the space-charge perveance parameter. Rb = J/3(s)e is the KV beam envelope radius. The coherent envelope oscillations due to space-charge force We consider a simple KV model of ID paraxial system (see Sec. We try to illustrate possible mechanisms.0. = 0. (2. We expand the spacecharge factor: vP{a)KK = k / + g ^CQSM + A (2234) . Yet.230) Here y stands for either the particle's horizontal or vertical betatron coordinate. Rb + k(s)Rb . (2.232) (2. k(s) is the focusing function. which is about 10% or less of the betatron tunes. \y\<Rb(s) (2.^ *• y = o. defined in Eq.229) \y\> Rh(s) = 0. almost all low energy synchrotrons suffer space-charge induced emittance growth.233) R + SR-^-^ii^ where v is the betatron tune.
19. and it generates a perturbation term.232). vx = 3. obtained from a PIC simulation calculation for the Proton Storage Ring (PSR) at Los Alamos National Laboratory.234) into Hill's equation (2.£sc. Cousineau. The envelope radius.120 in Fourier series. The reduced envelope radius R shown in Fig. we obtain V+A - 2jRr- U + E In cos(n^ + xn) J T) = 0. n=l (2.4i/fsc)r « 2 ^ s c E In cos{n<j) + Xn). we obtain A = £sc/2i/ and OO r + {4v2 . after a closer inspection by substituting R = 1 + A + r into Eq.Rightfully. The parameter £sc is the Laslett (incoherent) linear space-charge tune shift parameter and £Sc<7n and Xn a r e the Fourier amplitude and phase of the n-th harmonic. (2.26 m circumference. the original betatron amplitude function (dashed line). It serves as a compressor to compress 1.26 clearly shows 4 oscillations in one circumference. fsc is called the linear space-charge tune shift parameter. (2.237) The space-charge force plays two roles in the envelope equation.238) may cause a large particle oscillation at n — 2{v — £sc) for the Mathieu instability. TRANSVERSE MOTION U~^t ZscQn = . Ph. 5 7 S. where the Fourier harmonic in the intrinsic betatron amplitude function serves as a harmonic perturbation to the envelope equation. . Indiana University.57 The PSR is a fixed energy synchrotron with 90. where CHAPTER 2.19 and vz = 2.233).19.234) into Eq. thesis. One speculates that a large envelope oscillation shown in Eq.239) The particle tune is vp « v . Since the vertical betatron tune of the PSR is about 2. (2. (2.16 ms (3214 turns in PSR) of proton pulse from the 800 MeV Linac into a high intensity proton pulse of about 180 ns. 2002. (2.D. (2. However.^ T ~ # =^t \MsWds' ^| sc (2'235) (2. It decreases the envelope tune from j / e n v = 2v to venv = 2v — £sc.236) cos{n<j> + Xn)d<t>.j . Substituting Eq.26 shows the space-charge perturbed vertical betatron amplitude function (solid line). Substituting Eq. or the perturbed betatron amplitude function. the dominant perturbing harmonic in the envelope equation is 4. and the normalized envelope radius R (dotted line). is resonantly excited by the harmonic n « vem with ^-n^l^^rs^+Xn)- (-3) 228 Figure 2. 2.239).
37 x 1013 particles) in the PSR at LANL. However. In reality. this difficult experiment has not been successfully demonstrated.e. we find that the resonance strength is actually zero. 6. The Fourier components of its betatron amplitude functions will be zero except harmonics 0. If particle motion inside the beam core is not affected by the envelope oscillation. its stopbands will only occur at the betatron tune values of 0. the harmonic content of the betatron amplitude function arises from the distribution of the focusing function. If the > stopband width fscgn is not zero. The ratio of these two betatron amplitude function. 18. 12. .26: The square root of the perturbed vertical betatron amplitude function (solid line).238). Detailed theoretical and numerical nalyses of 2D Hill's and envelope equations (including the effect of off-momentum particles) would be very valuable. (2. we illustrate a possible emittance dilution mechanism. We consider a stable beam orbiting in a synchrotron with space-charge tune shift parameter £sc at an equilibrium beam radius Rb{s). the envelope oscillation of a beam can not affect particle motion inside the envelope. shown as dotted line. No artificial quadrupole error is needed in generating the envelope stopband. (2.EXERCISE 2. 12.3 121 Figure 2. For example. is compared with the square root of the intrinsic betatron amplitude function (dashed line). 6. the envelope radius will be resonantly excited as shown in Eq. This causes a mismatch in the betatron phase-space ellipses for all particles inside and near the beam envelope radius and results in emittance dilution due to phase space mismatch. the envelope equation of Eq. We note that the envelope stopband can arise from the harmonic content of the intrinsic betatron amplitude function. 18. for a beam with high intensity (4. if an accelerator has a 12-superperiod symmetry in its focusing function.238) will not be resonantly excited if the envelope tune is chosen to be far away from these stopbands. • • •. Note that the average of R is slightly larger than 1. We speculate that the emittance dilution for space charge dominated beams can be minimized by correcting some half-integer harmonics of betatron amplitude functions. (2. When new particles are injected into the beam core.231). i. what is the mechanism for emittance dilution? In the following. is the reduced envelope radius R defined in Eq. the envelope tune is pushed toward an integer stopband venv —• n. the space-charge parameter £sc increases. Likewise. • • -.
3. i t where fk are integer stopband integrals. where NB is the number of particles per bunch and crs is the bunch length.160) is a solution of the above equation. we find that a particle at a distance r from a uniformly distributed paraxial beam bunch experiences a space-charge defocussing force ~ 2mc2Nr0 _ a 2 7 2 r' where 7 is the relativistic Lorentz factor.<l>x) = S{<j>-<!>{) is G(<j>. a is the radius of the beam bunch.B for vertical and radial betatron motion is given. (b) Show that the Green function G(4>. we should replace N by N = JVB/-V/27T<TS. (c) Expanding /33/2AB/Bp in Fourier series ^ = l j ^ wh A = . and f = xx + zz. (2. show that the equation of motion becomes Show that Eq. Use the Green function to verify the solution given by Eq.<j>i) in i ~ + u2\G{<t>. 2. show that the integer stopband width !?[„] is given by r ^ ] w 5v\f[v}\/V€r™s< w n e r e fy\ is the integer nearest the betatron tune.2) where the error field A. and limiting the closed-orbit deviation to less than 20% of the rms beam size. (a) Defining a new coordinate r\ = y/y/fi with <p = {1/v) /os ds//3 as the independent variable. TRANSVERSE MOTION Exercise 2. N is the number of particles per unit length. From Exercise 1. Particle motion in the presence of dipole errors is (Sect. In a Gaussian bunch.3 1. respectively. II.160).(j>i) = [COSI^(TT — \(f> — 0i|)]/2^sin7ri/. ro is the classical radius of the particle.^ . by AB = -ABX and A S = ABZ. .122 CHAPTER 2. show that the closed orbit arising from the dipole error is yco(s) = y/m £ k—-oo ^*»- (d) Using a single stopband approximation.L / ^ e . (2.
we have F G a u s s . 3. . the fraction of particles within la of a round Gaussian beam is 1 .161) can be derived from the 58This formula is derived on the basis of uniform beam distribution. show that Eq.e . where the beam distribution is not uniform. i. show that the space-charge force is equivalent to a defocussing quadrupole with strength R7.2 m.632. i. This is called the Laslett space-charge tune shift.EXERCISE 2. Thus the Laslett space-charge tune shift is also called the incoherent space-charge tune shift. For a round Gaussian distribution. a factor that depends on the distribution function should be used to estimate the space-charge tune shift [see part (c) of this Exercise]. Using r] = yJsfWy and p .4. Find the space-charge tune shift by using the data shown in the figure below. It is in fact a tune spread.3. = (1 . Draw a line in the figure for emittance vs the number of particles per bunch for a space-charge tune shift of 0. where {/3y) is the average betatron amplitude function and eN is the rms normalized emittance.*« 9 > where ATSC = 2Nro/(l32j3) is the normalized space-charge perveance parameter used frequently in the transport of space-charge-dominated beams in linacs.e" 1 ) « 0.e. 1. (b) The rms beam radius is a2 = (py)eN/p''y. Show that the betatron (Laslett) tune shift induced by the space-charge force is given by58 __ FBNBro ^ 2irRKsc "sc ~ 27reNj872 ~ 47re ' where F& = Jjp* is the bunching factor and e = eN //3*y is the emittance of the beam. and /3 and 7 are Lorentz's relativistic factors.\ \ f 2Nr° 9 00 ^ . The beam intensity in low energy synchrotrons is usually limited by the space-charge tune shift. (c) The Fermilab booster synchrotron is operating at 15 Hz with a circumference of 474. In estimating the space-charge tune shift for an actual accelerators. Particles with large betatron amplitudes have a small space-charge tune shift. = dr]/d(j> with <j> = {l/fy) /os ds//3y as conjugate phase-space coordinates. The rms bunch length at injection is about 1 m.1 .e. The Laslett tune shift is the betatron tune shift for particles with small betatron amplitudes. and <j> as time variable. the formula for the space-charge tune shift should be adjusted by a beam distribution form factor Fdjst. When you are applying this formula to an actual accelerator.3 123 (a) Using the result of Sec. (2.
A commonly used algorithm is based on the "three-bumps" method. it can be considered as a lattice made of 60 FODO cells with betatron tunes vz = 8.tf))* = ~f* {£P3/29(v')dv'] di>.(AH(J. and the orbit distortion is localized between the first and third steering magnets. Jo g(r.62. op (a) Letting r) = y/2J/u cos tp and dtp/d<j> = v. and 63 be the three bump angles. Let 61.124 Hamiltonian CHAPTER 2. 5.e. The AGS is composed of 12 superperiods with 5 nearly identical FODO cells per superperiod. Show that these angles must be related by 0 --6 / ^ ~ s i n ^ 3 1 Y /?2 Sin ^32 ' 6-6 / ^ ~ s i n ^ 2 1 V ^ 3 S m ^32 ' where ft is the ^-function at the ith steering magnet. where <Aff(J. 4. i.^))^ + [AH . + viTi2)+AH(T.')dr. Assuming as 3> o>. 2X<7? pis) = -jL-e^M.tf)). where three steering dipoles are used to adjust local-orbit distortion. show that the Hamiltonian becomes H = vJ+ (AHiJ. and P(r) = ^e-(*2+*W. a local orbit bump can be attained by two steering dipoles 6\ and 63 if and only if ^31 = nv.) = ^-. Show that the betatron tune is shifted by the perturbation Av — d(AH)^.). Obviously. where N& is the number of particles in a bunch. . evaluate the space-charge tune shift as a function of the amplitude r.12 m. (b) We consider a cylindrical Gaussian bunch distribution p(x.z) = NBp(r)p(s). The circumference is 807. AH = -S r^g(r.s./dJ.'. ijjji = ipj . where the phase advance is an integer multiple of IT.>].8 and vx = 8. The closed orbit can be locally corrected by using steering dipoles. Replacing r by yjifijjv cos ip. V27T<7S are respectively the transverse and longitudinal Gaussian distributions with rms width aT and as.fi is the phase advance from the ith to the j t h steering dipoles.6. TRANSVERSE MOTION H=±<p*. where TQ = e2/A-Ke^rru? is the classical radius of the particle. show that the Lorentz force for a particle at distance r from the center of the bunch is ^s)^{l-e-^)p{s).
How far from the closed orbit of the circulating beam should the septum be located? What effect. We assume that the values of the betatron amplitude functions are fix = /3Z = 10 m.1 T.25 in about 2. Where should the stripper be located with respect to the center of the circulating beam? What is the minimum width of the stripper? Sketch a possible injection system scenario including local orbit bumps. In the H~ or H^ strip injection process. What are the stopband integrals due to these tune jump quadrupoles? iii.025. During the beam accumulation process. . The injection beam arrives through the center of a septum while the circulating beam closed orbit is bumped near the septum position. and a vertical one to horizontal dipole field error. the betatron amplitude functions are Ar = Pz = 10 m. the orbit bump is reduced to avoid beam loss through the septum.s. This means that each quadrupole changes the betatron tune by -0.202) for the AGS.5 fj.EXERCISE 2. the emittances are ex = ez = 2.3 125 (a) Estimate the closed orbit sensitivity factor of Eq. the closed orbit is bumped onto the stripper location during the injection pulse. where xc0 is the closed orbit and xp is the betatron displacement. At the septum location. What are the favorable configurations for these quadrupoles from the beam dynamics point of view? iv. what is the minimum length of the kicker? What advantage. if any.5TT mmmrad for the injection beam. The phase advance between the septum and the kicker is 60°. /3Z = 8 m. i. a set of 10 ferrite quadrupoles located at high-/^ locations are powered to change the vertical tune by At>z = —0. (2. Multi-turn injection of heavy ion beams requires intricate phase-space painting techniques. The extraction septum is located 40 mm from the center of the closed orbit of the circulating beam. The septum (current sheet) thickness is 7 mm. if any. The injection beam and the circulating beam merge at the same phase-space point. and ex = ez = 40TT mm-mrad for the circulating beam. the betatron functions are f}x = 10 m. At extraction. (c) During the polarized beam acceleration at AGS.168). What is the kicker angle required for single-turn extraction? Assuming that the maximum magnetic flux density for a kicker is 0.5 n mm-mrad for the injection beam. 7. A ferrite one-turn kicker is located upstream with $x = 10 m and pz = 8 m. does an orbit bump provide? 9. Discuss a scenario for efficient single-bunch extraction. (b) Estimate the the rms half-integer stopband width of Eq. the 95% emittance of the beam is adiabatically damped to 5 7r mmmrad at Bp = 10 Tm. does the betatron tune have on the beam-accumulation efficiency? 8. Particle motion in the presence of closed-orbit error is X = Xco + Xp. and the thickness of the wire septum is 1 mm. (a) Show that an off-center horizontal closed orbit in quadrupoles gives rise to vertical dipole field error. Are there advantages to installing 12 quadrupoles? What are they? 6. What is the effect of these tune jump quadrupoles on the horizontal tune? ii. (2. We assume that the 95% emittances are 50 ir mm-mrad for the stored beam and 2.
Zgngier. and oto and ai are related to the derivatives of the betatron amplitude functions. a'o = KopQ . (a) Show that where Aif = Kx . LAL report 77-35. 75-90 (1987). where B? = &iB2/dx2\x_0z_0. . TRANSVERSE (b) The magnetic field of a nonlinear sextupole is MOTION ABZ + jABx = Bf-(x2-z2 + 2jxz). 1977.2 t t l . (c) In thin-lens approximation.126 CHAPTER 2.apPx -/Mi ' „ _ ffi . where il>o and ipi are the unperturbed and the perturbed betatron phase functions. Thus Pa and fix satisfy the Floquet equation: P'o = . This exercise provides an alternative derivation of Eq. Show that the effective quadrupole gradient is dBz/dx\eS = xC0B210.59 We define the betatron amplitude deviation functions A and B as _ aiffp .W. (jE^As/Bp) is the integrated sextupole strength. Montague. In the presence of gradient error. the betatron amplitude functions and the betatron tunes are modified. P[ = . the phase-space trajectory of A vs B is a circle.Ko. CERN 87-03. d^/ds = l//3i. and (Po) is the averaged value of betatron function in the quadrupole. Show that a horizontal closed-orbit error in a normal sextupole produces quadrupole field error. (d) In thin-lens approximation. (b) In a region with no gradient error.e. show that the change of A in a sextupole is A A ss Pogeg. a i = Kxpx .199). and xco is the closed-orbit deviation from the center of the sextupole. (2.7 o . # 0 / d s = 1/A>.A) vm' where /3o and fi\ are respectively the unperturbed and the perturbed betatron amplitude functions associated with the gradient functions KQ and K\. where geff = {B2As/Bp)xC0 is the effective quadrupole strength.2 a 0 .7 l . B. 69See also H. show that A2 + B2 = constant. show that the change of A at a quadrupole with gradient error is AA = j VWi AX ds » (P0)g. i. where g = + / Aif ds is the integrated gradient strength of the error quadrupole.
8. where P is the pressure. T is the temperature. .3). Show that the emittance growth rate is 1 -2e»~\ ^ ) TO> re _ 1rfe _ o . and Ag is the gram molecular weight of a gas. Because the emittance growth is proportional to the betatron function.641 x Kr6j9-Pg[ntorr]4g [g/cm 2 /s]. velocity. Po + Pi Show that this equation reduces to Eq.4 p 2 ^A*. The radiation length is 716. /3c and zp are momentum. where /3c is the velocity of the beam. l J> where (/?j_) is the average transverse betatron amplitude function in the accelerator.4 . n is the number of moles. and p is the momentum of the beam.4. . XQ is the radiation length. particularly at high-/? locations. z p is the charge of the projectile. 2l where Z and A are the atomic charge and the mass number of the medium. [nTorr]. Xog is the radiation length of the gas. 12. This effect can also be important in the strip-injection process. P g is the equivalent partial pressure of a gas at room temperature T = 293 °K.3 127 (e) If we define the average betatron phase function as where 1 f* ( 1 1\ .A. Show that the half-integer stopband integral Jp is approximately zero at p = [2v] for two quadrupole kickers separated by 180° in betatron phase advance with zero betatron tune shift. (a) Using the ideal gas law. and R = 8.314 [J (°K mol)" 1 ]. This exercise estimates the emittance dilution rate based on the multiple scattering formula (see the particle properties data) for the rms scattering angle fl2_2g2^J13. IV. V is the volume. show that the equivalent target thickness in [g/cm 2 /s] at room temperature is x = 1. 11. PV = nRT.4g edt ' PeN [irmmmrad] \pc[GeV]J XOg [g/cm2] .199) in the limit of small gradient error. better vacuum at high-/3x location is useful in minimizing the multiple scattering effects.EXERCISE 2. Multiple scattering from gas molecules inside the vacuum chamber can cause beam emittance dilution. and x is the target thickness. and charge number of the beam particles.6[MeV]zp\2 x 6 where p. 7 is the Lorentz relativistic factor. Such a zero tune shift it-doublet can be used to change 7 T with minimum effects on betatron motion (see Sec. ^ M A 1 + AJ*1 n rsa+C / i show that the function B satisfies 0 d(j> 2 +4 ^ = . (2. 1 5 7(fli) M / zp yP.
. the H~ passes through a thin foil of thickness tfon [/ig/cm2].foji tf0ii[^g/cm2] A Ac = 117. the efficiency of charge exchange is small. /3c is the velocity of the beam.60 60If the stripping foil is too thin. Show that the emittance growth per passage is IT?O #i.foii is the betatron amplitude function at the stripper location. p is the momentum of the injected beam. P2{pc[MeV})2 X 0 [g/cm 2 ] L " where /3j_.128 CHAPTER 2. and Xg is the radiation length. and the proton yield is little. A compromise between various processes is needed in the design of accelerator components. Estimate the emittance growth rate per passage through carbon foil with H~ beams at an injection energy of 7 MeV if /3j_ fOii = 2 m and tfoii = 4 [^g/cm2].8 —T7— ' L —57Trmmmrad . If the foil is too thick. the beam emittance will increase because of multiple Coulomb scattering. TRANSVERSE MOTION (b) During the H~ strip-injection process.
we can study the motion of off-momentum particles perturbatively. Its effect on betatron motion will be addressed in Sec V. the dispersion action." and a particle with momentum po is called a synchronous particle. IV.6 we introduce the standard transport notation. 61 The revolution frequency of a synchronous particle is defined as the revolution frequency of the beam. . IV. where we introduce the phase focusing principle of synchrotron motion.2. For a particle with momentum p.What happens to particles with momenta different from p0? Here we study the effect of off-momentum on the closed orbit.g. we will find that the off-momentum closed orbit is proportional to 5 in the first-order approximation. IV. and the dispersion function is defined as the derivative of the off-momentum closed orbit with respect to 5. including dipole field errors and quadrupole misalignment. |<5| < 10"4 for SSC. Ill. In Sec.7 we describe methods of dispersion measurements and correction. we discussed the closed orbit for a reference particle with momentum Po.e. a beam is made of particles with momenta distributed around the synchronous momentum po. IV. and < 2 • 10"2 for typical electron storage rings. In Sec.240) for 5 = 0 were discussed in Sec.4. the momentum deviation is Ap ~ p — po and the fractional momentum deviation is 5 = Ap/po. OFF-MOMENTUM ORBIT 129 IV Off-Momentum Orbit In Sec. 1 Dispersion Function Expanding Eq. Solutions of Eq.IV.61 However. In Sec. The frequency of the radio-frequency (rf) cavities has to be an integer multiple of the revolution frequency of the beam. By using closed-orbit correctors. and in Sec. (2. a synchronous particle synchronizes with the rf electric field applied to the beam.8 methods of transition energy manipulation. IV. i. in particular. < 3-10"2 for anti-proton accumulators. e. the integral representation. We will discuss the properties of the dispersion function. In Sec.5 we discuss the achromat transport system. The momentum compaction factor and transition energy are discussed in Sec. we can achieve an optimized closed orbit that essentially passes through the center of all accelerator components. we obtain / X+{P>(I 1-5 K(. II. and the %-function will be introduced in Sec. and in Sec. IV. (2. we examine the method of dispersion suppression in a beam line. Since 5 is small.The fractional momentum deviation 6 = Ap/p is typically small. < 5-10"3 for RHIC. IV.)\ {i 6 + + 6) + 6))x-p(i 5y (2 . Minimum (H) lattices are discussed in Sec. IV.3. This closed orbit is called the "golden orbit.240) where K(s) = B\/Bp is the quadrupole gradient function with Bi = dBz/dx evaluated at the closed orbit. IV. IV. The name "synchrotron" for circular accelerators is derived from the synchronism between the orbiting particles and the rf field.9.29) to first order in x/p. < 10"4 for the IUCF Cooler Ring.1.
(2.240) can be expressed as a linear superposition of the particular solution and the solution of the homogeneous equation:62 x = xp(s) + D{s)6. (2. D" + (Kx{s)+AKx)D = .245) where L is the length of a repetitive period. The local closed-orbit condition of Eq. However.241) In this section. the closed-orbit condition is imposed on the dispersion function63 D(s + L) = D(s). To the lowest order in 5. the inhomogeneous equation can easily be solved by the matrix method. we will neglect the chromatic perturbation term AKx(s). this local periodic closed-orbit condition facilitates accelerator lattice design. where D(s)S is the off-momentum closed orbit Aside from the chromatic perturbation AKX. The solution of a linear inhomogeneous dispersion equation is a sum of the particular solution and the solution of the homogeneous equation: ( ^ ) = "<*>O + (i). . where C is the circumference. (2.245) for repetitive cells is not a necessary condition. with Kx = ±-K(s).130 CHAPTER 2. «. Since Kx{s) and p(s) are usually piecewise constant for accelerator components. TRANSVERSE MOTION For an off-momentum particle with 5 ^ 0. the solution of the linearized inhomogeneous equation (2. the solution of the homogeneous equation Xp is the betatron motion around the off-momentum closed orbit. the dispersion function obeys the inhomogeneous equation D" + Kx(s)D = l/p. AKx = [-^ + K(s)}5 + O(S2). The solution of the inhomogeneous equation is called the dispersion function. the displacement x is x = xco(s) + xp(s) + D(s)6.244) Since Kx(s) and p(s) are periodic functions of s. (2. 63The closed-orbit condition for the dispersion function is strictly required only for one complete revolution D(s) = D(s + C) and D'(s) = D'(s + C).m 62Including the dipole field error. D\s + L) = D\s). where xp(s) and D(s) satisfy the equations x"p + (Kx(s) + AKx)xfi = 0.242) (2.+ O(S).243) (2. where i c o is the closed-orbit error discussed in Sec. III.
and d and dl are the particular solution. OFF-MOMENTUM ORBIT 131 where the 2 x 2 matrix M(s2\si) is the transfer matrix for the homogeneous equation. and B represents bending dipole(s). and 9 is the bending angle. d'). is represented by C = {^QF B QD B ^QF}.IV. Example 1: Dispersion function of a FODO cell in thin-lens approximation A FODO cell with dipole. Using thin-lens approximation. 2. as shown in Fig. The transfer matrix in Eq. where Kx = 1/p2. where QF and QD are focusing and defocussing quadrupoles. we obtain V ^k s i n ^ s ) k ^ ^sinh^S \ /O-y'l-fCil v J / lf^<0- The transfer matrix for a pure sector dipole. the transfer matrix of a sector dipole becomes M= \M\ 9 \. (2. Let d be shorthand notation for the twocomponent dispersion vector with d) = (d.249) 0 0 /I t 0 1 1 J where p is the bending radius. Vo o i / \o o i / \o o i / \o o i / V° ° !/ . we obtain M= (I -i 0 0\ (I L \L9\ 1 0 0 1 9 (I U 0 0\ (I L \L0\ 1 0 0 1 9 ( 1 -£ 0 0\ 1 0 . (2.4.246) can be expressed by the 3 x 3 matrix For a magnet with constant dipole field and field gradient. In the small angle approximation. VO 0 1 J where £ = p9 is the length of the dipole. is ( cos0 -{l/p)sin6 psin9 cosfl p(l—cos#)\ sinl? .
The dispersion function at the defocussing quadrupole location is £g(l-l8in(*/2)) sin 2 ($/2) ' UD~ D°-°- (2-253) The middle plot of Fig. which can be approximated by a lattice made of 60 FODO cells. The dispersion function at the focusing quadrupole location becomes The dispersion function at other locations in the accelerator can be obtained by the matrix propagation method. we obtain *-A o _. We list here some characteristic properties of the dispersion function of FODO cells. The closed-orbit condition of Eq. (2. (2. (2. and it is smaller with shorter cell length and smaller bending angle.245) becomes (D\ {") = [-&+& l~fi 2 ^ . • The dispersion function at the focusing quadrupole is larger than that at the defocussing quadrupole by a factor (2+sin(§/2))/(2-sin($/2)). which is about 2 at $ = 90°.247).£ ) J m S i n ( 1-$ 21(1+ i ) 2L0(1 + ^ ) \(D\ (225o) Note that D and D' in Eq.5 shows the dispersion function of the AGS lattice.250) are values of the dispersion function and its derivative at the focusing quadrupole location. the product of the cell length and the bending angle of a FODO cell. Using the Courant-Snyder parameterization for the 2x2 matrix. and / is the focal length of the quadrupoles. • The dispersion function is proportional to L6. the dispersion function is proportional to the inverse quadratic power of the phase advance. 2. • When the phase advance is small. 6 is the bending angle of a half cell.132 CHAPTER 2.f . Eq. TRANSVERSE MOTION where L is the half cell length.2£(l + sm(g/2)) where $ is the phase advance per cell. .
OFF-MOMENTUM ORBIT 133 Example 2: Dispersion function in terms of transfer matrix In general.M22) + Mi2M23 ~ 2-Mu-M22 _ MnM2i + (1 .a s i n $ ) D .258) Since the dispersion function satisfies the homogeneous betatron equation of motion in regions with no dipole (1/p = 0). For a FODO cell.cos $ . (2.Afn)M23 ^ " 2-Mu-M22 u _ M13(l .a sin $) " 2(1-cos $) ./3sin$ D'\ IV. (2. but in regions with dipoles. it is not invariant. Px (2. the transfer matrix of a periodic cell can be expressed as (Mu M = Mu M22 M13\ M23 V 0 M21 0 1 ) . ( cos $ + a sin $ —7sin$ 0 /?sin<I> (1 .53). p and 7 = (l+a2)/f3 are the Courant-Snyder parameters for the horizontal betatron motion at a periodic-cell location s. (2.245). a.2 The dispersion %-function is defined as n(D. we obtain _ Mi 3 (l . Mu. Action.c o s $ + asin$) + M23. 'HF < 7iD. where _ ^ S i n $ ( l + Isinf)2 2(1 +sin | ) sin4 f ' _L02sincfr(l-Isinf)2 D~ 2(1 . the 3x3 transfer matrix is cos$ — asin<3> 7 s i n $ D + (1 — cos$ + asin<I>) D' \.257) This representation of the transfer matrix is sometimes useful in studying the general properties of repetitive accelerator sections.254) where Mu. [2-2i3b) l^WJ where $ is the horizontal betatron phase advance of the periodic cell. the 'H-function is invariant.flsin$ " 2(1-cos $) ' _ -M137 sin $ + M23(l . Solving M13 and M23 as functions of D and D'.e.sin f) sin4 f ' l ' .cos$ .IV. and Integral Representation + 2axDD' + /3XD'2 = -j-[D 2 + (&£>' + axD)2\. and D and D' are the value of the dispersion function and its derivative at the same location. D') = jxD2 ^-Function. Using the closed-orbit condition of Eq. M2i and M22 are given by Eq. i. 0 1 ) (2. the dispersion %-function at the defocussing quadrupole is larger than that at the focusing quadrupole.
and 3 dispersivesections for injection.261) In a straight section. J d is not constant. the normalized dispersion coordinate Xd is nearly constant. etc. D ~ \fWx. 2.= £ > = y ^ c o s S d . In a region with dipoles. is identical to the betatron phase advance. Figure 2.134 CHAPTER 2.28 shows the normalized dispersion coordinates for the IUCF Cooler Ring. The change of the dispersion function across a thin dipole is AD = 0 and AD1 = 6. AXd = 0. TRANSVERSE MOTION and the dispersion H-function is proportional to the inverse cubic power of the phase advance. (2. the normalized dispersion phase-space coordinates for the double-bend achromat (DBA) lattice (see Sec.1 Example 1. IV. Now we define the normalized dispersion phase-space coordinates as { Xd = .5. rf cavities. aside from a constant. In contrast.262) where 0 is the bending angle of the dipole. Note that Xd is indeed nearly constant. Figure 2.. which is composed of 3 achromat straight-sections for electron cooling.A) shows different behavior.5) that is approximately made of 5 FODO cells. PA = JPxD1 + -^D where the dispersion action is given by = -p Jd = ^n{D.e. Jd is invariant and $ d . and Pd is small.27: Left: Normalized dispersion phase-space coordinates Xd and Pd are plotted in a superperiod of the AGS lattice.27 shows the normalized dispersion phase-space coordinates in one superperiod of AGS lattice (see Fig. and (Pd. Right: the coordinates are shown in Xd vs Pd. (2.D'). The scales for both Xd and Pd are m1/2. IV. 2. i. momentum stacking.260) Jd sin $ d . Figure 2. etc.34).e. Xd) propagate in a very small region of the dispersion phase-space (see also Fig. APd = y[px AD' = Jfx9 AJ d = (PXD' + axD)0. The machine is made of six 60°-bends . i. Y* (2. and the change in dispersion action is For FODO-cell lattice shown in Sec.
39. With the substitutions AB Ap 1 n . Note that the achromat sections are described by a single point at origin with Xd = Pd = 0. OFF-MOMENTUM ORBIT 135 Figure 2.Ap .38 and 2.28: Left: Normalized dispersion phasespace coordinates X& and Pd of the IUCF Cooler lattice are plotted. the minimization of (H) plays no important role in beam dynamics. The resulting dispersion phase-space coordinates are much larger than those of minimum emittance DBA lattices shown in Fig. 2.38. a minimum fiz inside dipole will provide a criterion for the magnet gap g. 2. .263) Po P where p is the bending radius. Thus the corresponding (3X will be large in dipole. For an ion storage ring. 2. Integral representation of the dispersion function The dispersion function can also be derived from the dipole field error resulting from the momentum deviation. the normalized dispersion coordinates increase in magnitude. it is preferable to design a machine with a minimum j3z inside dipoles.31. 2.i and Pd are m1/2. and ds is the infinitesimal length of the dipole. forming a 3 similar double-bend achromat modules. the normalized dispersion coordinates are located on invariant circles. Since the power required in the operation of a storage ring is proportional to g2.IV. the horizontal betatron phase-advance is also nearly n. It is worth pointing out that the lattice function and the dispersion phase-space coordinates of the IUCF Cooler Ring differ substantially from the low emittance DBA lattice to be shown in Figs. The scales for both X. The angular kick due to the off-momentum deviation is given by 9= 45*.28. Inside dipoles. Thus the dispersion phase $d advances nearly n in the dispersion matching section. (2. that are nearly half-circles as shown in Fig. In dispersion matching sections. Right: The coordinates are shown in Xd vs Pd at the end of each lattice elements. Instead. Since the dispersion phase-advance is equal to the horizontal betatron phase-advance in a straight section. where the dispersion function is shown in Fig.
where / is the revolution frequency.29). Because a high energy particle travels faster. Transition energy and the phase-slip factor 137 The importance of the momentum compaction factor will be fully realized when we discuss synchrotron motion in Chap. and v is the speed of the circulating particle. we discuss the phase stability of synchrotron motion discovered by McMillan and Veksler [17]. OFF-MOMENTUM ORBIT A. and 7Tmc2 or simply 7T is the transition energy. . To = l / / 0 is the revolution period of a synchronous particle. (2.^ = ^ . the converse is true. Phase stability of the bunched beam acceleration Let V(t) = Vo sin(hojot + <f>) be the gap voltage of the rf cavity (see Fig. The acceleration voltage at the rf gap and the acceleration rate for a synchronous particle are respectively given by Vs = Vo sin fa. In the meantime. Eo = foeVo sin fa. 7T « vx. This is the isochronous condition. <j> is an arbitrary phase angle.Ap . Without a longitudinal electric field. its speed compensates its longer path length in the accelerator. h is an integer called the harmonic number.^ . with 7 > j T .IV. where Vo is the amplitude. a higher momentum particle will have a revolution period shorter than that of the synchronous particle. 3. the fractional difference of the revolution periods between the off-momentum and on-momentum particles is AT AC AD . Particles with different momenta travel along different paths in an accelerator. At 7 = 7T the revolution period is independent of the particle momentum. Below the transition energy.270) . Since the revolution period is T = C/v. A / / / o = -1)5. 2. W = 2TT/O is the angular revolution frequency. and 6 = Ap/p 0 is the fractional momentum deviation. and /o is the revolution o frequency of a synchronous particle. or equivalently. where fa is the synchronous phase angle. For FODO cell lattices. the time slippage between a higher or lower energy particle and a synchronous particle is TQT)5 per revolution. l. which is the operating principle of AVF isochronous cyclotrons.269) Here jT = J\/ac is called the transition-7. so that a higher energy particle will arrive at a fixed location earlier than a synchronous particle. A synchronous particle is denned as an ideal particle that arrives at the rf cavity at a constant phase angle <j> = fa. with 7 < 7T and 77 < 0. where C is the circumference. All particles at different momenta travel rigidly around the accelerator with equal revolution frequencies. (2. The phase-slip factor r) is ^ = Q c . Above the transition energy. B.
P d ( s ) = 2 l i n ^ I. Since the change in path length due to betatron motion is proportional to the square of the betatron amplitude [see Eq. (2.4. Since D(s) is normally positive. (2.iri/x)dt p \lPx{t) ^ 265) IV. and the last approximate identity uses thin-lens approximation.2). Bx is the betatron amplitude function. vx is the betatron tune. (2. p S i n ( ^ ( t ) ~ ^ ( S ) " *"')dtMomentum Compaction Factor Since the synchronization of particle motion in a synchrotron depends critically on the total path length. $ is the phase advance of a FODO cell.160).266) which depends linearly on the dispersion function D(s).4ix(s) . The normalized dispersion functions can then be expressed as 1 xd{s) rs+c JPx{t) = 7T-/ 2 sin rcvx Js 1 fs+c cos{tpx{t) . the momentum compaction factor for a FODO lattice is given by Qc~ (£>F + DD)8 62 l_ 2L ~sin 2 ($/2) ~ ^ 2 > where L and 9 are the length and the bending angle of one half-cell. it is important to evaluate the effect of the off-momentum closed orbit on path length.3 . the dispersion function becomes jBJs) rs+c \/Px(t) where C is the circumference. The momentum compaction factor is then denned by where {D)i and 6j are the average dispersion function and the bending angle of the ith dipole. On the other hand. For example. the effect is small. TRANSVERSE MOTION in Eq.t) is the Green function of the horizontal Hill equation. ipx is the betatron phase function. . the deviation of the total path length for an off-momentum particle from that of the on-momentum closed orbit is given by AC = j -ds = U ^-ds] 6.136 CHAPTER 2. and Gx(s.170)]. and vx is the betatron tune (see Exercise 2. the total path length for a higher momentum particle is longer.
the acceleration rate is E = feVo sin 4>. The rate of change of the energy deviation is (see Chap. P £>o UQ where AE = E — EQ is the energy difference between the non-synchronous and the synchronous particles. f 0 < 4>s < TT/2 if 7 < 7T or rj < 0. 7 0 < <j>s < TT/2 for 7 < 0. where / is the revolution frequency. and the rf phase angles of a synchronous particle. TRANSVERSE MOTION where e is the charge.138 CHAPTER 2.e. It is usually called synchrotron motion. the phase focusing principle requires for 7 > 0. = ^ at Jo ^ *M**. 3.272) Equations (2. I) l(^) = ^ ( s i n ^ S i n ^' ±U .This is the equation of motion for a biased physical pendulum system. and n/2 < (j>s < -K 7 A non-synchronous particle will arrive at the rf cavity at a phase angle <f> with respect to the rf field. LBNL). where 4> can vary with time. (Courtesy of D. Li.29: Schematic drawing of an rf wave.272) form the basic synchrotron equation of motion for conjugate phase-space coordinates <f> and AE/OJQ. Figure 2. and the higher and lower energy particles. The differential equation for the small amplitude phase oscillation is ~ ^ - = ^ ^ ( s m 0 - W s ) * toPE* (^"^ (2 ' 273) Thus the phase stability condition is given by T?COS0S < 0. where 0 is the actual angular position of the particle in a synchrotron. At phase angle (f>.*) = -hfiu. \ TT/2 < <t>s < -K if 7 > 7T or rj > 0. Sec.271) and (2. Eo is the energy of the synchronous particle. is = hV^ = Po (2. the equation of motion for the rf phase angle <fr = —h9. . For a stable synchrotron motion. i. Similarly. and the overdot indicates the derivative with respect to time t.
Eq. u)syn — 0 as 7 — 7 T . If the arc is composed of modular cells.rms is the rms emittance. However. thus the transition energy problems can be eliminated. The synchrotron frequency of small-amplitude phase oscillations is given by I heV0\rj cos <j)s q». 64The curved transport line is usually called the arc. offers an attractive solution to these problems. Thus the beam size of a collider at the interaction point can be minimized by designing a zero dispersion straight section. Such a lattice is called an imaginary 7T lattice. a lower energy particle arrives later and gains more energy from the cavity. OFF-MOMENTUM ORBIT 139 Below the transition energy.29). when the beam is accelerated through the transition energy. microwave instability due to wakefields. A sudden change > > in the synchronous phase angle of the rf wave will not cause much beam dilution. IV. (2. the dispersion function can not be zero there. Thus the energy of the particle will becomes smaller than that of the synchronous particle. etc. Methods of achieving a negative compaction lattice will be addressed in Sec. Similarly. where the transition 7T is an imaginary number. with 0 < 0S < TT/2.8. This process gives rise to the phase stability of synchrotron motion. rf cavities. . such as FODO cells. and the straight section that connects arcs is usually called the insertion. On the other hand. and interaction regions for colliders. If the betatron and synchrotron motions are independent of each other. internal targets. In many applications.e.274) Particles are accelerated through the transition energy in many medium energy synchrotrons such as the AGS. the CERN PS. A zero dispersion function in the rf cavity region can also be important to minimize the effect of synchro-betatron coupling resonances. i. Particle motion in an imaginary 7T lattice is always below transition energy. etc.. needed for injection.245). An accelerator lattice with a negative momentum compaction factor. the Fermilab booster and main injector. i. 2. where ex.i{D)i9i < 0.4 Dispersion Suppression and Dispersion Matching Since bending dipoles are needed for beam transport in arc sections. the dispersion function should be properly matched in straight sections for optimal operation. Attaining an imaginary 7T lattice requires a negative horizontal dispersion in most dipoles. synchrotron motion around the transition energy region is very slow. and the KEK PS.ms + £>2(s) ((Ap/po)2>. Fortunately. beam loss and serious beam phase-space dilution can result from space-charge-induced mismatch. extraction. IV. The synchronous angle has to be shifted from (j>s ton — (j>s across the transition energy within 10 to 100 fis./?^ • (2 . insertion devices.=<*V 2. which simplifies lattice design.e. a higher energy particle arrives at the rf gap earlier and receives less energy from the rf cavity (see Fig.IV. the rms horizontal beam size is given by CT2(S) = /3i(s)e:t)1. the synchronous phase angle should be TT/2 < <f>s < n at 7 > j r . nonlinear synchrotron motion. Y. the dispersion function is usually constrained by the periodicity condition.64 We discuss here the general strategy for dispersion suppression.
65 Here M is the 2 x 2 transfer matrix of each cell.245). a strategy for dispersion function suppression can be derived.' f)- (?)-(••)(. Brown and R Servranckx. the dispersion function of the repetitive half achromat is D = d/2. Dispersion suppression Applying the first-order achromat theorem. [12].+ . D' = d'/2. If the dipole bending strength of the adjoining —I section is halved. We consider a curved (dipole) achromatic section such that Mn = I. The transfer matrix of n cells is R l = ^ ( M . and d is the dispersion vector. i. and / is a 2 x 2 unit matrix. (2. p. Let the 3x3 transfer matrix of a basic cell be where M is the 2 x 2 transfer matrix for betatron motion. Thus the achromat condition w = 0 can be attained if and only if Mn = I or d = 0. (d/2\ !/0\ «» = (-. The proof of this theorem is given as follows. the transfer matrix becomes _ and the dispersion function will be matched to zero value in the straight section. Eq. A unit matrix achromat works like a transparent transport section for any dispersion functions. + l)J^M» wy (2276) where w = (Mn — I)(M — I)'1^. .+ M .e.)• 65 See K.140 First-order achromat theorem CHAPTER 2. We note that one half of this achromatic section can generally be expressed as Using the closed-orbit condition. 121 in Ref. TRANSVERSE MOTION The first-order achromat theorem states that a lattice of n repetitive cells is achromatic to first order if and only if Mn = I or each cell is achromatic. An achromat section matches any zero dispersion function modules.
the beam closed orbit depends on particle momentum.5 Achromat Transport Systems If the dispersion function is not zero in a transport line. A. The double-bend achromat A double-bend achromat (DBA) or Chasman-Green lattice is a basic lattice cell frequently used in the design of low emittance synchrotron radiation storage rings. Is the dispersion function unique? A trivial corollary of the first-order achromat theorem is that a dispersion function of arbitrary value can be transported through a unit achromat transfer matrix.4 offers an example of an achromat. IV.iy>dy (2278) where M is the transfer matrix of the basic module with dispersion vector d. Is the dispersion function obtained unique? This question is easily answered by the closed-orbit condition Eq. IV. the dispersion function of an accelerator lattice is uniquely determined. The transport matrix of n identical modules is R"=(M. the fitting procedure is straightforward. Since the machine tune can not be an integer because of the integer stopbands. This is usually called the missing dipole dispersion suppressor (see Exercise 2. (2. With use of computer programs such as MAD and SYNCH. Using the closed-orbit condition. i. However. Eq. (M» -D(M . it is possible to design a transport system such that the beam positions do not depend on beam momentum at both ends of the transport line. OFF-MOMENTUM ORBIT 141 Thus we reach the conclusion that the matched dispersion function is equal to the dispersion function of the matched arc.e. a small modification in the quadrupole strengths is needed for dispersion suppression.245). A possible variant uses —I sections with full bending angles for dispersion suppression by varying the quadrupole strengths in the —I sections. The reduced bending strength scheme for dispersion suppression is usually expensive because of the wasted space in the cells. a 3 x 3 unit matrix.e. (2. Now we consider the case of an accelerator or transport line with many repetitive modules. In the case of unit transport. Such a beam transport system is called an achromat. any arbitrary value of dispersion function can be matched in the unit achromat.IV. we easily find that the dispersion function of the transport channel is uniquely determined by the basic module unless the transport matrix is a unit matrix.245) for the entire ring. i. A . When edge focusing is included. The achromat theorem of Sec. which however do not form a unit transfer matrix.4. Mn = I.3c).
4. where O and 0 0 can contain doublets or triplets for optical match. II. It is represented schematically by [00] B {0 QF 0 } B [00]. Lower plot: triplet DBA. the dispersion matching condition is (see Exercise 2. (2.28 °) Note that the focal length needed in the dispersion function matching condition is independent of the dipole bending angle in thin-lens approximation. The zero dispersion value at the entrance to the dipole is matched to a symmetric condition D'c = 0 at the center of the focusing quadrupole. Upper plot: standard DBA cell. 9 and L are the bending angle and length of the dipole. 2. The required focal length and the resulting dispersion function become f= \{Ll + \L)' D<=(Li + \L)e- ( 2 .30 shows a basic DBA cell. The top plot of Fig. In thin-lens approximation.280) renders a horizontal betatron phase advance $ x larger than n in the dispersion matching section (from the beginning of the dipole to the other end of the other dipole).13) [Dc\ 0 ( = 1 -1/(2/) 0 OWl Lx 0\ (I L L9/2\ 1 0 0 1 0 0 1 0 /0\ 0 .30: Schematic plots of DBA cells. TRANSVERSE MOTION DBA cell consists of two dipoles and a dispersion-matching section such that the dispersion function outside the DBA cell is zero. and it can easily be obtained from the geometric argument.6) indicates that betatron function matching section [00] can not . the betatron function depends on the magnet arrangement in the [00] section. where the quadrupole triplet is arranged to attain betatron and dispersion function match of the entire module. The dispersion function at the symmetry point is proportional to the product of the effective length of the DBA cell and the bending angle. and L\ is the distance from the end of the dipole to the center of the quadrupole. and possible other quadrupoles in the dispersion matching section.279) Vi ) \ o o i A o o i/\o o i M i / where / is the focal length of the quadrupole. The dispersion matching condition of Eq.142 CHAPTER 2. (2. Figure 2. where [00] is the zero dispersion straight section and {0 QF 0 } is the dispersion matching section. We consider a simple DBA cell with a single quadrupole in the middle. The stability condition of betatron motion (see Sec. Although this simple example shows that a single focusing quadrupole can attain dispersion matching.
R24 elements describe the linear .6 Transport Notation In many applications. OFF-MOMENTUM ORBIT 143 be made of a simple defocussing quadrupole. The design strategy is to use achromatic subsystems. or a triplet.3. An example of achromatic subsystem is the unit matrix module (see Sec. where a quadrupole triplet is located symmetrically inside two dipoles. Some properties of the triplet DBA storage ring can be found in Exercise 4. R23 Ru. Such DBA lattice modules have been widely applied in the design of electron storage rings. Achromatic modules can be optically matched with straight sections to form an accelerator lattice. medical radiation treatment. The achromatic transport system find applications in high energy and nuclear physics experiments.282) Note that the 2x2 diagonal matrices for the indices 1. (2. IV. j=i ( M = 1. and 5 = Ap/p0 is the fractional momentum deviation of a particle. 2.6. . Other achromat modules The beam transport system in a synchrotron or a storage ring requires proper dispersion function matching.4 are respectively the horizontal and vertical M matrices.16). The achromatic transport modules are also important in the transport beamlines (see Exercises 2. The transport of the state vector in linear approximation is given by Wiis2) = '£Rij{82\a1)Wj{81). The Ru.2. A quadrupole doublet. A simple DBA cell is the triplet DBA (lower plot of Fig. is usually used in the [00] section.281) \wj \ 6J PcAt where /3c is the speed of the particle.---. /3cAi is the path length difference with respect to the reference orbit. B. and 3.13 to 2. (2.6). the particle coordinates in an accelerator can be characterized by a state vector fWA W2 W5 ( x \ x' W= Z\ = I.JV.30). A unit matrix module can be made of FODO or other basic cells such that the total phase advance of the entire module is equal to an integer multiple of 2TT.4 on the first order achromat theorem).4. and other beam delivering systems.4. This compact lattice was used for the SOR ring in Tokyo. IV.
(2. and we obtain R l (2.C. D. The program TRANSPORT 6 6 has often been used to calculate the transport coefficients in transport lines. . particle transport through a thin sextupole is given by AT' _ S (r2 „. Here we used the convention that 5 > 0 corresponds to a focusing sextupole.R26 elements are the dispersion vector dot Eq. FNAL Report TM-1046 (1981). 72\ A. Carey et al. we get the momentum compaction factor as ac = R56. 233 — X ) J . the nonlinear dependence of the state vector can be expanded as Wi(s2) = J2RijWj(s1) + 3=1 3=1 k=l 1=1 J:J2TijkWj(s1)Wk(sl) 3=1k=l + E E E ^yH^(ai)W*(«i)Wi(«i) + • • • • For example.C. Without synchrotron motion. 66K. D. Brown. Fermilab-Pub-98-310 (1998). Rothacker. . The corresponding transport matrix elements are Ti2 1 1 Q U rp J ^ c rp Q ——7T.i l i — i>. D. The Ri6. In general.283) x z z x z5 x5 zS2 xS2 T . CERN 80-04 (1980). Tracing the transport in one complete revolution.C. Carey. we have R55 = Re6 = 1. TRANSVERSE MOTION betatron coupling. Iselin and F.144 CHAPTER 2.+ 1 11 - l •n-21 — —~c. All other elements of the R matrix are zero.275). •••. ^436 = — T J (^2166 — ~ 7 i ^4366 = + T - Similarly.' S where 5 = —B^tjBp is the integrated sextupole strength.L. Ch. Carey. -^43 = +"7i -1216 — + 7 . SLAC-R-530. particle transport through a thin quadrupole is given by .
The slope of this measurement is used to obtain the "measured" dispersion function.co is the closed orbit. The lower plot compares the measured dispersion function (rectangles) with that obtained from the MAD program (solid line). The resulting closed orbit will have the characteristic modulation frequency. the fast betatron oscillations are averaged to zero. On the other hand. 2.31 shows the closed orbit at a BPM location vs the rf frequency at the IUCF Cooler Ring. In the lower plot of Fig. Fitting the resulting closed orbit with the known modulation frequency. (2. Using Eq. A high dispersion straight section is used for momentum-stacking injection and zero dispersion straight sections are used for rf and electron cooling.284).e. we can determine the dispersion function more accurately.13). Figure 2. we can deduce the dispersion function at the BPM location.31: The upper plot shows the closed orbit at a BPM vs the rf frequency for the IUCF Cooler Ring. The upper plot of Fig. and also on the effects of power supply ripple. The dispersion function can be measured from the derivative of the closed orbit with respect to the off-momentum of the beam.4. OFF-MOMENTUM ORBIT 145 IV. To improve the accuracy of the dispersion function measurement. 2. if the BPM signals are sampled at a longer time scale. . r\ is the phase-slip factor. / 0 is the revolution frequency. dxco D dxco OQA\ (2 ' 284) = d[K^)=~rik^ where a.IV.67 The accuracy of the dispersion function measurement depends on the precision of the BPM system. i. and the momentum of the beam is varied by changing the rf frequency. 67Note here that the IUCF Cooler Ring lattice belongs generally to the class of double-bend achromats (see Exercise 2. The DC output provides the closed orbit of the beam.7 Experimental Measurements of Dispersion Function Digitized BPM turn by turn data can be used to measure the betatron motion.31 the "measured" dispersion functions of the IUCF Cooler Ring is compared with that obtained from the MAD program [19]. we can induce frequency modulation to the rf frequency shift.
and the design principle of the imaginary 7T lattices. This is called the "negative momentum compaction" or the "imaginary j T " lattice. The revolution period deviation AT for an off-momentum particle Ap = p — p0 is given by Eq. To avoid all the above unfavorable effects. p. For example. IV). Here we examine the strategy of 7T jump schemes pioneered by the CERN PS group. We discuss below the methods of ac manipulation. These problems can be avoided by an accelerator having a negative momentum compaction factor. Risselada. The jT jump schemes have been used successfully to ease beam dynamics problems associated with the transition energy crossing. Proc. which leads to unstable longitudinal motion resulting in serious beam loss. 1991. Since the frequency spread of the beam Aw = —T]u)(Ap/p0) vanishes at the transition energy. 161. 1974). (2. Hardt. DC. causing beam loss (see Chap. particle acceleration through the transition energy is unavoidable. 3. on High Energy Accelerators (USAEC. Furthermore. The accelerator becomes isochronous at the transition energy (7 = 7 T ).68 6 8 W. it is appealing to eliminate transition crossing. 7T jump schemes In many existing low to medium energy synchrotrons. Proc. CERN Accelerator School. VII). there is little or no Landau damping of microwave instability near transition (see Chap. Alternatively. CERN-91-04.146 CHAPTER 2. Con}. Sec. A. 3. All modern medium energy synchrotrons can be designed this way to avoid transition energy. the bunch area may grow because of collective instabilities. such as longitudinal microwave instability and nonlinear synchrotron motion. particles with different momenta may cross transition at different times. one can design an accelerator lattice such that the momentum compaction factor ac is negative.8 Transition Energy Manipulation Medium energy accelerators often encounter problems during transition energy crossing. 9th Int. See also T. There are many unfavorable effects on the particle motion near the transition energy. As a result. the momentum spread of a bunch around transition can become so large that it exceeds the available momentum aperture. and thus the beam never encounters transition energy. TRANSVERSE MOTION IV.268). the transition energy jump scheme. Washington. . Sec. Finding a suitable 7T jump scheme can provide beam acceleration through transition energy without much emittance dilution and beam loss. This has become a routine operation at the CERN Proton Synchrotron (PS). in these schemes some quadrupoles are pulsed so that the transition energy is lowered or raised in order to enhance the acceleration rate at the transition energy crossing.
and D* is the perturbed dispersion function at s = Si.so)—dso « £G s (s. s i)Ki. we obtain C0Aac = -*£ KiD'Di. is the dipole angle. If N 7 T -jump quadrupoles are used to change the momentum compaction factor. (2-289) Thus we usually employ zero tune shift quadrupole pairs for the 7T jump. t (2.IV. An important constraint is that the betatron tunes should be maintained constant during the 7T jump in order to avoid nonlinear betatron resonances. the closed orbit is Axco(s) = Js rs+C 147 Gx(s. assumed positive for a focusing quadrupole. /\B (2-285) where 9^ is the dipole angular kick in thin-lens approximation. (2. A(2s = 7 .fe KiDtD^J 5.287) where A is the unperturbed dispersion function. From Hill's equation. Thus the change of the orbit length for the off-momentum particle is given by AC « £ DA « . N (2. = --?-][X i tf i = 0. the angular kick resulting from the ith 7T jump quadrupole is given by 9t = -Ki \xC0{Sl) + D*5]. i.288) Note that the change in momentum compaction (called jT jump) depends on the unperturbed and perturbed dispersion functions at kick-quadrupole locations. the dispersion function is given by D(s0) = [C Jo Gxi"'"o)ds p(s) « £ Gx(sh 30)9. and GX(S. (2. AQ.I The effect of quadrupole field errors on the closed orbit Consider T quadrupoles for the 7T jump.264) where 0. Similarly..159). Equation (2. A. (2. We would like to evaluate the change of V orbit length for off-momentum particles due to the 7T jump quadrupoles.287) indicates that quadrupoles at nonzero dispersion locations can be used to adjust the momentum compaction factor.e. Note here that the kick angle 9t from quadrupoles at nonzero-dispersion function locations can be used to perturb the dispersion function and change the orbit length for off-momentum particles. .SQ) is the Green's function of Eq.£ / M ^ = °. OFF-MOMENTUM ORBIT In the presence of dipole field error.286) where Ki = —Bii/Bp is the strength of the zth 7T jump quadrupole. and we neglect higher-order terms in 5.
285): [D*{s) ..£ KiD\ + £ KiKjGx(si. ..)D. The resulting change in momentum compaction becomes N (2.295) (2.F)~XQ = (1+ F + F2 + F3 + .. (2.294) This result can be easily proved by using the zero tune shift condition: /3XtkKk + Ar.. TRANSVERSE MOTION A. s^KiD* 6. Using the 7r-doublets.292) C0Aac = -*£Ki(l ij + F + F2 + .) y I>iA. sjDiDj. and the change in the momentum compaction factor is given by C0Aac = .3 7 T jump using zero tune shift 7r-doublets When zero tune shift pairs of quadrupoles separated by IT in the betatron phase advance are used to produce a jT jump. the matrix F satisfies Fn = 0 for n > 2. cos{nvx .D{s)} 5 = . The perturbed dispersion function at these quadrupole locations can be solved to obtain D* = (1 .cos(7T!/a. i (2-291) where Fji = —Gx(sj.ipk\) = . (2-293) A.2 The perturbed dispersion function The change in the closed orbit resulting from the quadrupole kicks can be obtained by substituting Eq.fc+ijKWi = 0 and the -K phase advance condition: COS(7TZ/X \tpk .\ipi . (2.290) Thus the closed orbit solution is Di=Dj + '£FjiB'i. (2.148 CHAPTER 2. the perturbed dispersion function becomes DT^il + F^Dj. i ij (2.\ipi - ipk+1\).C O S ^ \ipk+i 1pj\).1pj\) = .286) into Eq.296) Here three points are worth mentioning: .Si)Ki. i (2.£ Gx(s.
and thus £ 4 KiDf <x J \ Kifii = 0 because of the zero tune shift condition. thus making a negative momentum compaction factor. 2829 (1991). Teng proposed an innovative scheme using negative dispersion at dipole locations. Part. The change in the momentum compaction factor contains a linear and a quadratic term in K.e. Fermilab (1988). 3. the 7r-doublet does not produce a large perturbation in the betatron amplitude function. On the other hand. Thus if all quadrupoles used for 7T jump are located in FODO cells. 915 (1989). In 1972. the amount of tune jump is second order in the quadrupole strength.C. IEEE Trans. ) ^ % 149 2.71 In this method a systematic closed-orbit stopband is created near the betatron tune to induce dispersionwave oscillations resulting in a high 7T or an imaginary 7 T . Sci. 72 L. Garren. 1955). AA = D* . a lattice having a very small or even negative momentum compaction factor can also be designed. Technical Memo. s . Vladimirskij and Tarasov70 introduced reverse bends in an accelerator lattice and succeeded in getting a negative orbit-length increase with momentum. 71 R.A = -<?. the resulting lattice is less tunable and the dispersion functions can be large. (2. p. Since the stopband integral of Eq. where the dispersion function can be matched by a straight section with a phase advance of TT to yield little or no contribution to positive orbit-length increment. Another method of designing an FMC lattice is called the harmonic approach.M. If the zeroth harmonic term dominates. Accel. Tarasov. . Courant. Proc. Teng. 70 V.D. Wienands. Thus the dynamical aperture may be reduced accordingly. Vladimirski and E. NS-32. G. 81 (1972). 1991 IEEE PAC. 69 This statement can be expressed mathematically as follows. i. the term linear in Ki vanishes because of the zero tune shift condition.69 The resulting change in the momentum compaction factor is a quadratic function of Ki. Theoretical Problems of the Ring Accelerators (USSR Academy of Sciences. we have D\ oc ft. T.IV. p. 2308 (1985). OFF-MOMENTUM ORBIT 1.72 This concept is the basis for flexible momentum compaction (FMC) lattices. Botman. Proc. 1989 IEEE PAC. and U. The change in the dispersion function is linear in K. 4. Guignard.K. Gupta and J. Nucl. However. Moscow.(*. 7T jump using quadrupoles in straight sections can be made linear in quadrupole strength.I. Collins. If the 7T jump quadrupole pairs are located in the arc. E. Flexible momentum compaction (FMC) lattices Alternatively.201) at p = [2vx] due to the tune jump quadrupole pair is zero because of the zero tune shift condition. where the unperturbed dispersion function is dominated by the zeroth harmonic in the Fourier decomposition. A.V. Beta Theory.A. which require parts of the lattice to have negative dispersion functions. B.
Y. TRANSVERSE MOTION Recently.74 F D CL O O EL QF/2 B Q D B Dispersion Matching Section Q r l QD2 F D CL O O EL B i i QD B QF/2 ni i i I I i 11 l i i r n I n i i i m i i l l i i n i l in I Ma Mb Mc Figure 2. Trbojevic.260) are handy. Rev. the normalized dispersion vector changes by AP = \fp\B and AX = 0. Tepikian. the dispersion function satisfies the homogeneous equation. Outside the dipole (p = oo). Proc. For attaining proper dispersion function matching.e. D. 3040 (1993). S. and S. Ng. The module forms the basic building blocks for a ring with a negative momentum compaction factor or an imaginary j T . P = Jfciy +-?£=D =-yfiTiSiniii. K.32: A schematic drawing of a basic module made of two FODO cells and an optical matching section. D. 1991 PAC. the maximum value of the dispersion function can be optimized to less than that of the FODO lattice. K. This dispersion phase-space plot can be helpful in the design of lattices and beam-transfer lines.d. Lee. S. i. 168 (1993). Proc. Therefore. The dispersion phase-space maps are carefully matched to attain a lattice with a pre-assigned 7T value. 1990 EPAC. and S. In the thin-element approximation. K. p.Y. Trbojevic. i. E48. the normalized dispersion coordinates Xi and Pi of Eq. 74 D. B. Lee.i and X^ lie on a circle P2 + X2 = 2JdThe phase angle ipd of the normalized coordinates is equal to the betatron phase advance. Ng. . 159 (1991).244) indicates that AD = 0 and AD' = 6 in passing through a thin dipole with bending angle 9. Fermilab Internal Report FN-595 (1992).Y. X = -L. and S. and 73 D. 1993 IEEE Part. p.Y. (2. at the same time. and the dispersion action J^ is invariant. P. Conf.D = y2JdCos</. in normalized Pd-X^ space. Holmes. It has also been used to lower the dispersion excursion during a fast 7T jump at RHIC. 1536 (1990).150 CHAPTER 2. Peggs. S. Trbojevic.I The basic module and design strategy A basic FMC module has two parts: (1) the FODO or DOFO cell.Y. Phys. Proc.e. Trbojevic. Lee. and D. Gerig. Ng. Accel. Finley. R. Eq. (2.Y. Trbojevic et al. The module can be made very compact without much unwanted empty space and. and D. where the negative dispersion function in dipoles provides a negative momentum compaction factor.73 re-introduced a modular approach for the FMC lattice with a prescribed dispersion function. Trbojevic. p.
2.32: Ma j .IV. /3p and D? are respectively the betatron amplitude and dispersion functions at the center of the focusing quadrupole for the regular FODO cell. (2. the momentum compaction factor of the accelerator can be varied. the dispersion phasespace coordinates are located on a circle as shown in Fig. 2. 2. 75The packing factor is defined as the fraction of the circumference of an accelerator that is occupied by magnets. Although not strictly necessary. We also assume reflection symmetry for all Courant-Snyder functions at symmetric points of the module. The right plot shows a similar plot in thin-lens approximation. and S's stand for dipoles. The horizontal betatron transfer matrix of the FODO cell from the marker Ma to the marker Mb is given by [see Eq. Although they look slightly different. we have chosen <&x = $ z = $ for the betatron phase advance of the FODO cell. we consider a basic module composed of two FODO cells and a dispersion matching section shown schematically in Fig. Detailed matching process is given in following subsections.c are marker locations. O's are drift spaces.297) . Then. Q's are quadrupoles. (2. Figure 2.4. reflection symmetry considerably simplifies the analysis and optical matching procedure. Figure 2.34. the dispersion function is propagated through FODO cell to obtained a dispersion vector at the marker Mj. Adjusting the initial dispersion function value Da. Since there is no dipole in this example shown in Fig. First. for simplicity. For example. If the lattice has a reflection symmetry at the marker Mc.34 shows an example of dispersion function matching for an FMC module by plotting the normalized dispersion phase-space coordinates X& vsPj. the thin-element approximation can provide essential insight in the preliminary design where dispersion matching is required. (2.34. the desired value Da of the dispersion function at the marker Ma is chosen. The procedure of betatron amplitude function matching is given as follows. ( cos$ -^sin$ /3 F sin$ cos$ DF(l-cos$)\ | f sin $ . the dispersion function is matched in the matching section.Q F B QD B -QF j Mb {Q Fl Oi QD 2 O2} M C + reflection symmetry .32. the matched dispersion phase-space coordinate is Pd = 0. A negativemomentum-compaction module requires X& < 0 in dipoles as demonstrated in the left plot of Fig.3] 0 0 I ) where. 2. and a symmetry condition /?F = 0 and D'F = 0 is assumed to simplify our transfer matrix in Eq.297).75 Because the dispersion function inside dipoles in FODO cells is mostly negative. OFF-MOMENTUM ORBIT 151 (2) a matching section that matches the optical functions.33 shows the betatron amplitude functions for a matched FMC basic module with an added dipole in dispersion matching section in order to increase the packing factor.257) and Exercise 2. the resulting momentum compaction factor can become negative. where Majb.
b = \ ^ + foDb ] = Jdj?[l .2 Dispersion matching The dispersion function at the beginning of the FODO cell is prescribed to have a negative value of Z?a with D'& = 0. TRANSVERSE MOTION Figure 2.C) cos $ + (1 . B. D'h = DF~D* Pb sin <> .£>a) cos $ . In this example. and the dispersion action is invariant in this region. Jd.C)2] . Jd. (2. The overall compaction factor can still be adjusted by a properly chosen Da.35 shows \/Jdtb/Jd.2(1 . E (2. Now we assume that there is no dipole in the matching section.297). and it increases when the initial dispersion £>a at marker Ma is chosen to be more negative.299) where £ = L>a/Dp is the ratio of the desired dispersion at marker Ma to the dispersion function of the regular FODO cell. and J^p is the dispersion action of the regular FODO cell at the focusing quadrupole location. It is preferable to have a smaller dispersion action in the matching section in order to minimize the dispersion function of the module.33: The lattice function of an FMC basic module. (2. Using the transfer matrix in Eq. we find the dispersion function at marker Mb to be Db = DF~ (DF . . Although the dipole in the matching section will contribute a positive value to the momentum compaction.F as a function of ( for various values of phase advance per cell. The ratio of the dispersion action has a minimum at ( = 1 — cos $. i. given in thin-lens approximation by v4k c o 3 E:j s i 1fi2[ sin3 f (1 + sinf) J <2'»> Figure 2. a dipole is added in the middle of dispersion matching section in order to increase the machine packing factor. As we shall see.e.c = J*.c are dispersion actions at markers Mb and Mc. the choice of D a essentially determines the dispersion excursion and the 7T value of the module.152 CHAPTER 2. Jd.b.298) where /?b is the betatron amplitude function at marker Mb with /3b = /3F.
The dispersion matching condition at marker Mc is D'c = 0. The betatron transfer matrix for the matching section is / M6_>c= y^cos^ —ri=sinV> V^ScSinV ^/¥cosip 0\ 0 .IV. and we have assumed the symmetry conditions /3b = 0 and /3'c = 0 for the Courant-Snyder parameters.A compromise choice for the phase advance of the FODO cell is between 60° and 75°. (2.298). However. we obtain 1-(1-C)cos$ v ' .300) is inversely proportional to (sin($/2)) 3 / 2 . and thus we should choose a larger phase advance for the FODO cell in order to obtain a smaller overall v/2Jd. where the normalized dispersion phase-space coordinates X& = Z)/\/S v s Pd = {otx/VPx)D + yffi^D' are shown. where horizontal steps are associated with a dipole that is divided into three segments. ij. is the betatron phase advance between Mb and Mc. OFF-MOMENTUM ORBIT 153 Figure 2.c. (2. The normalized dispersion phase-space coordinates for periodic FODO cells are marked "FODO CELLS.34: Left: an example of dispersion matching for a basic FMC module. the dispersion amplitude i/2J d F in Eq.35 that a smaller phase advance in the FODO cell would be preferred. 2. Using Eq. The dispersion functions and other Courant-Snyder parameters are then matched at the symmetry point at marker Mc with a doublet (or triplet). There is no dipole in the matching section and thus the normalized phase-space contour is a perfect circle in the matching section." The corresponding thin-lens approximation for the FMC is shown in the right plot. One might conclude from Fig. (2. This can be true if we compare the dispersion of the basic module with that of the regular FODO lattice with the same phase advance.301) V o 0 1 / where /3b and /3C are values of the betatron amplitude function at markers Mb and Mc.
17). or a low-beta insertion with doublets or triplets for the matching section. (2.35: Ratio of dispersion actions Jd. Figure 2. Figure 2. the phase advance ip and the quadrupole strengths in the dispersion matching section are constrained by the stability condition of betatron motion (see Sec. TRANSVERSE MOTION Figure 2. Furthermore. C « -0.4. The total phase advance of the whole basic module is then given by 2($ + ip). Quadrupoles QFI and QD 2 m the matching section can be adjusted to attain the required phase advance if] given by Eq. This means that the phase advance of the matching section is not a free parameter.F as a function of £ = £>a/DF for various values of phase advance $ in the FODO cell. B. II.302) and to produce low betatron amplitude functions at marker Mc.6 and Exercise 2.b/Jd. the values of the dispersion function at the midpoints of dipoles in the FODO cell are given by . This condition is independent of whether we use a FODO-type insertion.36 shows the required betatron phase advance 2ip (in unit of 2n) in the matching section as a function of phase advance $ of the FODO cell for various values of C = D^/D^.3 ~ -0. which is a function only of the desired dispersion function at marker Ma and the phase advance $ in the FODO cell.36 shows the total phase advance of the whole module as a function of the phase advance of the FODO cell for £ = —0. but is determined completely by the initial dispersion value _Da at marker Ma and the phase advance of the FODO cell. To achieve a |TT phase advance for a quarter-wave module.6.3 Evaluation of momentum compaction factor In small-angle approximation.3 to —0.4 and a phase advance per FODO cell of $ = 60° to 75° can be used.154 CHAPTER 2.
the phase advance of the FODO cell. .^(y/Sl + 8S+ + SI) + j ( v t e + 8S+ .36: Left: Phase advance in the matching section as a function of phase advance $ in the FODO cell for various values of C = D&/DF.) .303) should be modified as follows: DBl = £>. we obtain qc _2LU-2S65 2 .2S3 5 2 (5 + 25) 1 C+ ^ ^ .306) +1* [| +1(^/5!+8S + -S_)]. Note that the momentum compaction factor of the module is determined entirely by the choice of £>a. (2.L^ [ 4-2S-S 2 4-2S-H ' (2-3°5) where S = sin($/2).IV. The momentum compaction factor should be modified accordingly. Eqs.|^+] l-L6. In comparison with the momentum compaction factor of a lattice made of conventional FODO cells. [l . The momentum compaction becomes where 9 is the bending angle of each dipole and L m is the length of the half-module. Right: The total phase advance of a FMC module as a function of the phase Advance $ in the FODO cell for various values of <: = DJDF.^(y/Sl+8S+ + S_)] + < DB2 = D* [l . the momentum compaction factor depends linearly on the initial dispersion function £>a. and the ratio of the lengths of the FODO cell and the module. (2. If the horizontal phase advance <&x of the FODO cell differs from its vertical phase advance <£z. where S± = sin2 ^$x ± sin2 | $ z . OFF-MOMENTUM ORBIT 155 Figure 2. When the length of the module is constant.S.
or a triplet. D. The dispersion function inside a sector dipole is given by D(s) = p(l . In general. and rf cavities.iAn\* . Other similar FMC modules CHAPTER 2. If the achromat condition is imposed. a doublet. The dispersion matching section on the right side of the dipole can be made of a single quadrupole. a slightly smaller |£| can be used to minimize the magnitude of the dispersion function in the module.e. j3D. D F . the module is called a double-bend achromat (DBA). . FMC in double-bend (DB) lattices A double-bend module (Fig. (2./fcosf* DF-DD^ cos^ -fl-ft-sinf* ' (2-307) V o o i ) where <> is the phase advance of a FODO cell. Because the dispersion value at the defocussing quadrupole location is smaller than that at the focusing quadrupole location.308) (2.4.30) is made of two dipoles located symmetrically with respect to the center of the basic module given by Ma / Triplet \ or B {dispersion matching section} Mc + {reflection symmetry}.156 C. The betatron transfer matrix in the DOFODO cell becomes ' V^ c o s f* Ma^b= _ i s i n | $ y[hh. V Doublet/ A triplet or doublet matching section on the left side of the dipole B is the betatron amplitude matching section. D'{s) = s i n < £ . DD are the betatron 3 amplitudes and dispersion values at the focusing and defocussing quadrupoles of the FODO cell. the result will be a larger total dispersion value. TRANSVERSE MOTION The above analysis can be applied to a basic FMC module composed of two DOFO cells and a dispersion matching section (see Exercise 2. The zero dispersion region is usually used for insertion devices such as the undulator. 2. three FODO cells instead of two are placed inside a basic module. A similar analysis with a different number of FODO cells can be easily done.17).309) 76 The packing factor of a lattice is defined as the ratio of the total dipole length to the circumference.cos <£) + Do cos <f> + pD'o sin 0. and /3F.76 one may use a DOFODO in place of the FODO cell. the wiggler. i. To design a lattice with a higher packing factor.— s i n <1> + D'O c o s <j>.
In this section. Finally. a reverse-bend dipole placed at the high dispersion straight section can also be used to adjust the momentum compaction factor of a DBA lattice. the strategy of minimizing {%} inside dipoles will be discussed.cosl?) J where Lm is the length of one half of the double-bend module.312) where R is the average radius of the storage ring. The dispersion function in the rest of the module can be matched by quadrupole settings.9 Minimum {H) Modules In electron storage rings.c o s 0 ) .313) . the momentum compaction in small dipole angle approximation becomes acDBA « ^ (2. Since the DBA module has Do = 0 and D'Q = 0.2a o psin0(l . In small-angle approximation. OFF-MOMENTUM ORBIT 157 where p is the bending radius of the dipole. 8 = L/p. the momentum compaction is given by ac = -f-\e-sin9+— Lm I p sin 8 + D'0{l.11) H = n0+2(a0D0+/3oD'Q)sin(f>-2(jQD0 + aoD'0)p(l-cos(l>) +A)Sin20 + 7op2(l -coscp)2 . For a matched double-bend module.IV. and Do a n d D'o are respectively the values of the dispersion function and its derivative at s = 0. (2. IV.l . The evolution of the "H-function in a sector dipole is given by (see Exercise 2. The momentum compaction factor of a DBA lattice is independent of the betatron tune. the natural (horizontal) emittance of the beam is determined by the average of the ^-function in the dipoles (see Chap. we will consider a single dipole lattice unit where the dispersion and betatron amplitude functions can be independently controlled. <t> = s/p is the bend angle. (2. Such a lattice can provide a small-emittance negative momentum compaction lattice for synchrotron radiation sources.4. To simplify our discussion. 4). and d'o = D'o/8. the condition for negative momentum compaction is given by 6do + 3 d ' o < .311) where d0 = D0/L0. and L is the length of the dipole. Note here that the momentum compaction factor depends on the initial dispersion function at the entrance of the dipole.
158 CHAPTER 2. TRANSVERSE MOTION where "Ho = jaD^ + 2a0D0D'0 + PoD'o> a0.E -> 1. With the normalized scaling parameters do = % d'o =^ . A = -S^P°3> 77 The (2-318) average of the ti function is given by (H) = j f n(4>)ds = \ j H(4>)d4>. (2. 7o = 7o^. (H) ~ p93. and <f> = s/p is the coordinate of the bending angle inside the dipole. we obtain (H) min . the average %-function becomes (-H) = p63i[jQdl + 2aodod'o + Pod2 + (aoE-^F)do +0OE . (2. +|cj .e.B -t 1.| F ) 4 + kA-^B Note that (H) obeys a scaling law. the average ^-function is {U) = p9*^A-^B + ^c}. and Fm 2(1-cos 0) 6-8cos0 + 2cos20 = gi ' 6(fl-sinfl) C 6fl-3sin2fl p • 30^ . (2.315) where L = p9 is the length of the dipole.40sin0 + 5sin20 W = B(9) In the small-angle limit. /?0. Li JO <> Jo . A ^ 1. The average H-function in the dipole becomes77 {%) = n0 + (a0D0 + poD'0)02E(e)-l-(y0D0 + a0D'0)pe2F(9) (2-314) +^<PA(0) . i. and F ->• 1.C -t 1.^ B ( J ) + ^ C ( » ) > where 6 is the bend angle of the dipole. j 0 . Minimum (^)-function with achromat condition In the special case with the achromat condition d0 = 0 and d'o — 0. Po = ^.316) A.317) Using the condition /3o7o = (1 + do). ao=ao. Do and D'o are the Courant-Snyder parameters and dispersion functions at s = 0.
the dispersion matching quadrupole at the center is also split into two in order to leave space for a sextupole. The evolution of the betatron amplitude function in the dipole can be obtained from Eq. 2. 2. The APS lattice has 40 superperiods so that the circumference is 1104 m. XA) in Fig. Figure 2. Px is minimum inside a dipole in order to attain a minimum (H). Note that (W) is slightly smaller in a long dipole because of the 1/p2 focusing effect of the sector dipole. we show the normalized dispersion coordinates (Fd.39.28 x 10~4 in agreement with that of Eq.15B2. The momentum compaction factor is QC = 2. The tunes of this lattice are Qx = 35. In the small-angle approximation.312).38: The low emittance lattice functions for a superperiod of APS. Here. The ME lattice data are for minimum (U) without the achromat constraint. the minimum betatron amplitude and its location are given by S--^-L 4v/6Q^' s* Smin.298.37: The minimum (H) factors G = V16AC .A - - -3L g ^- Figure 2.15B2_ for the DBA (lower curve) and G = VlSAC . The factor G = \/16AC — 15B2 (see Fig.37) decreases slowly with increasing dipole bending angle 6 because of the horizontal focusing of the bending radius.IV.56). In the APS lattice. /?0 = Vl2 CJy/EG and a0 = &B/G.38 shows the betatron amplitude functions of a low-emittance DBAlattice at the advanced photon source (APS) in Argonne National Laboratory.219. Qz = 14.A Pmin.15B2 for the ME (upper curve) lattices are plotted as a function of the bending angle 0. OFF-MOMENTUM ORBIT 159 Figure 2. where G = V16AC . (2. To study the behavior of the lattice dispersion function. where the dipole has been split into 10 . (2.
TRANSVERSE MOTION slices in order to show the propagation of the normalized dispersion coordinates inside a dipole. Since the lattice is designed to minimize (H) inside dipole. the details of emittance minimization procedure will be addressed in Chapter 4.38 for the corresponding lattice functions. (2. the normalized dispersion coordinates are small (to be compared with that shown in Fig. minimization of the ^-function can be achieved through the following steps.28).a0B + ^6) . employ low emittance DBA-lattice for their storage ring. (H) can be minimized by finding the optimal dispersion functions with dd0 ' dd'o to obtain dOMa = ^F.319) (H) = ^ (fa* . APS. Note that the achromat section of the DBA lattice is located at the origin X d = P d = 0. Since we emphasize the dispersion function in this section. etc.320) where A = 4A-3E2. we obtain the minimum (H) as Using the relation $. 2. 2. III. See Fig. B.160 CHAPTER 2. {nUn=iimp93' (2'321) . The resulting (H) becomes d'0. First.£ . JSRF.39: The normalized dispersion coordinates for the low emittance APS lattice is shown in one superperiod.B = W-2EF.C = \C-\F2.7o = 1+djj. In the dispersion matching straight section.min = . ELETTRA. Many third generation high-brilliance light sources such as the ESRF. the normalized dispersion phase-space coordinates are located on a circle with the center at the origin. Sec. Figure 2.. (2. Minimum (%) without achromat constraint Without the achromat constraint.
15B2 (see Fig. i. Exercise 2. 2. Show that the solution can be expressed as fD(s)\ (Do\ (?) ""(?)• where the transfer matrix is ( cos VKs -V^sinv^s -^ sin VKs cos y/Ks ^ (1 .4 161 where G = y/l6AC . Even though the minimum (7i) is one third of that with the achromat condition. Computer codes such as MAD [19] and SYNCH [20] can be used to optimize {%). i. In actual machine design. the required minimum betatron amplitude function is less stringent.37).4.cos VKs) \ ^sin^s . The corresponding minimum betatron amplitude function at the waist location is /^ i n = L/\/60 in small-angle approximation with 9 < 1. combined-function magnets with defocussing field may be used (see Exercise 2. Let DQ and D'Q be the dispersion function and its derivative at s = 0.EXERCISE 2.4 1. s* — L/2. The dispersion function in a combined-function dipole satisfies the equation D" + KXD = 1/p . (a) Show that the solution for constant Kx = K > 0 is D = a cos VKs + 6 sin VKs + ijpK. Thus the minimum (H) without achromatic constraint is a factor of 3 smaller than that with the achromat condition.e. I o o ) .18). where we will find that (H)min is actually larger than for a separate function lattice.e. R* "min = -/?* o^rnii^A" The corresponding maximum betatron amplitude function will be reduced accordingly. The betatron amplitude function at the minimum (%) is The waist of the optimal betatron amplitude function for the minimum (7i) is located at the middle of the dipole. We have discussed the minimum (H) only in sector dipoles.
2. show that the horizontal transfer matrix is (see Exercise 2. Each FODO cell is given by [iQF B QD B ±QF].3) /I M rectangu i ar dipoie = 1 0 \0 psinfl 1 0 p(l-cosfl)\ 2 tan(6>/2) . Use the thin-lens approximation to find the phase advance per cell and the betatron and dispersion functions at the focusing quadrupole.249).162 CHAPTER 2. and B is a dipoie with bending angle 9. 2. Ja(QD) as a function of the phase advance per cell < . The bending arc of an accelerator lattice is usually composed of FODO cells. (b) Simplify your result in part (a) with $ x = $ z = $ and calculate the dispersion actions Jd(QF). TRANSVERSE MOTION (b) Show that the transfer matrix for constant Kx = K < 0 is / coshv/JK|s -J=SmhJ\K\s cosh ^/\K~\s 0 ^(_l + C osh7iK| 5 )\ Af=L/[Ffsinh v PT'S V 0 —L. (a) Using thin-lens approximation. where QF and QD are the focusing and defocussing quadrupoles with focal length f\ and —farespectively. where / is the focal length. show that the dispersion function and the betatron amplitude functions are given by R Px'F L Z 1 ^ sin($x/2)Vl-T+' ft Px'D L l^-T+ sin($x/2) V 1 + T_ ' where 5± = sin 2 -^±sin 2 -^. . T± = ±(y/Sl+&S+±sS). and $ x and $ z are the horizontal and vertical betatron phase advance per cell. and £ and 9 are the length and bending angle of the dipoie. show that the transfer matrices M for quadrupoles and dipoles become / M q u a d i o o\ o 1/ Mdipoie= /i 0 e ee/2\ 1 8 = V o . (c) Part of the SYNCH program input deck for the TEVATRON (1988) is given below. Plot & Jd(QF)/J d (QD) as a function of $.I f f1 0 .sinh v/LKTs • 1 ) (c) Show that the transfer matrix of a sector magnet is given by Eq. (2. 1 / where p and 9 are the bending radius and the bending angle. Let L be the half cell length. Vo o i / . (d) For a rectangular magnet. (e) In thin-lens (small-angle) approximation.
702 BZ = 44. Estimate the momentum beam size vs the betatron beam size in the arc. The arc section is composed of regular FODO cells with bends. kG-m. a l l quads run a t same e x c i t a t i o n a s t h e C 35 t u r n .60038 C magnet d e c l a r e : length g r a d i e n t brho dipole type B MAG BL 0. kG/m.14028 00 000 DRF 0. I AGS Lceii (m) 13.4 163 TEV RUN DOUBLER LATTICE C DOUBLER l a t t i c e u s i n g two s h e l l normal quads and s p e c i a l C l e n g t h matching quads. RHIC and SSC lattices in thin-lens approximation. (2. V 0 0 1 / .45 $ (deg) 52.67894 0 DRF 0.EXERCISE 2.6 180 97.27664 GF = 760. and Ls and L a are the length of the straight section and the arc. 3.32056 BL = 6. Show that the 3x3 transfer matrix of a repetitive cell is generally given by Eq.001 | 0.m) 30 (Ap/p o ) rms I . Show that the transfer matrix of repetitive FODO cell is / cos* M = I -7Fsin$ /3 p sin$ cos$ 2£> p sin 2 ($/2)\ 7 F D F sin$ . and the straight insertion section is composed of quadrupoles without dipoles. kG.32056 GD = -760. 21 f o o t d i p o l e s .0001 (e) A collider lattice is usually made of arcs and insertions.257).5 Energy (GeV) 25 eN(w/j.2794 DRF 2.96 90 90 90 250 1000 20000 8000 30 30 10 15 I 0.1214 QL = 1. The dispersion suppressor matches the dispersion function in the arc to a zero dispersion value in the straight section. C A l l q u a n t i t i e s i n u n i t s of m. BRHO BZ $ QF MAG QL GF BRHO QD MAG QL GD BRHO C beam l i n e declaration f o r a CELL HC BML 00 B 0 B 0 B 0 B 000 CELL B L M Q D HC QF HC (d) Use the data in the table below to estimate the dispersion function of AGS.0001 | 0. Show that the momentum compaction factor of such a lattice is given by 1 aC^a2rc(l+£a/£a)' where 27ri>arc is the total accumulated phase advance in the arcs.003 | 0.005 I RHIC I Tevatron I SSC 1 LHC 29. BRHO = 33387.
and £>F is the dispersion function at the center of the quadrupole. the dispersion suppressor is composed of two reduced bending FODO cells. and $ is t h e phase advance of the F O D O cell.1) F O D O cells are [§QF B QD B | Q F ] with dipoles. (a) Show that /-I M| =90 o = 0 V0 0 -1 0 2DF\ 0 . Adjoining the regular arc./ unit.78 Show that the conditions for zero dispersion after the dispersion suppressor are j = W^ry and ei+d2 = e' where 6 is the bending angle of each dipole in t h e regular cell.4 is thus verified. (c) To match the dispersion function from a regular FODO cell in the arc to a zero value at the straight section._ 1 = sin$ . each with 90° phase advance. 0 1/ (b) Show that two FODO cells. 4. a n d t h e bending magnets in the last F O D O cell are replaced by drift spaces. show t h a t t h e dispersion function at t h e entrance of the first F O D O cell with a dipole is 1 — cos n<& + cos $ — cos(n — 1 ) $ 2(1-cos n$) FI 1 = .164 CHAPTER 2. show that the momentum compaction factor ac of an accelerator made of N FODO cells is given by 1 ac~2irRf 78A fDx ( 2TT \ 2 1 pdS-{2Nsin%) /SFsin* ~ vV reduced bending cell can be represented by the following matrix with fi = Q\jQ: ( cos$ £ I D F ( 1 . with bending angle #2 a n d 9\ for each dipole.s i n n $ + sin(n . (d) This exercise shows the effect of dispersion mismatch. we need a dispersion suppressor. ) . 1 / /I M| = 9 0 o = 0 \0 0 0\ 1 0 . TRANSVERSE MOTION where the symmetry conditions aF = 0 and D'F = 0 are used. At $ = TT/2. T h e theorem of dispersion suppression of Section IV. which is related to the integer stopband. The resulting mismatched dispersion function can be very large at n $ w 0 (mod 2TT). match a zero dispersion region to a final dispersion of D = 2Z)F and D' = 0.cos$)\ ~7j?sin$ 0 cos$ 0 SI7F£>Fsin* 1 I . $ is the phase advance per cell.1)$ 2(1-cos n$) 7 F' where DF is the dispersion function of the regular FODO cell at the center of the focusing quadrupole and $ is the phase advance per cell. these two F O D O cells form t h e . Assuming that the accelerator lattice is made of n F O D O cells. Using thin-lens approximation. j3F and 7 F are the Courant-Snyder parameters. where (n .
D = p/(l .5) with a constant focusing index 0 < n < 1. and Xd.244) can be transformed to d2jt +Sx-v2pm where X = D/y/fi. the ao harmonic dominates. p (V3) v 2TTP J Since v = f ds/27r/3 w fl/<^>. Show that the integral representation of the dispersion function in Eq.243). Consider a weak-focusing synchrotron (Exercise 2. show that Eq. (2.cos 2irvx)Xd + sin2irvxPd}P0 where Xp = x/V3i and Pp = (axx + f5xx')ly/]5x a r e normalized betatron coordinates. show that the path-length change due to the betatron motion is79 rs+C x AL = / -ds Js p = [sin 2irvxXd .4 165 where -R is the average radius of the accelerator. 79Use the integral representation of Eq. (2. 5.n). Assuming p w constant for all magnets.n.266). (b) Using Eq. where the zeroth harmonic ao dominates. (2.2. and vx is the horizontal betatron tune. the path length depends on the betatron amplitude quadratically. ^ k=-oo show that Note here that Z)(s) can be approximated by ao^/P{s) in a regular FODO lattice. Show that the lattice and dispersion functions are j3x = p/y/1 . show that i //J1/2. (Xp) = {Pp) = 0. Using the Floquet transformation. (2.264) satisfies Eq.(1 . show that a c « l/v2.EXERCISE 2. (2. $ is the phase advance per cell.265). and the transition energy is 7 T = VI —n. . Since the time average of the betatron motion is zero.265). (2. (2. 6.cos 2nux)Pd]Xp + [(1 . Substituting the betatron coordinate into Eq. show that A:=—oo where R = C/2-rr is the mean radius. (a) Using the Fourier expansion.266). Pz = p/y/n. 7. In most accelerator design.P<j are the normalized dispersion function phase-space coordinates of Eq. and <f> = /o ds/u/3.
2ao[Mi3M2i . In general.M23M12] + po[MuM2i .M 23 M 12 ] 2 . (2.166 CHAPTER 2.^(» 0 )|).M23Mu}2 +7o[Mi3M22 . The closed orbit of a horizontal dipole field kicker at location so in a synchrotron is given by (see Sec. 11. where #o is the kick angle. is the phase of /*• 10. etc. D'2 = M21Di + M22D[. 9. The values of the dispersion function at two locations in the beam line are related by D2 = AfnDi + M12D[. The change of orbit length due to a dipole error is the product of the dipole kick angle and the dispersion function at the kicker location. Show that the evolution of the 'H-function is U = Ho + 2(a0Do + /3oD'0)[M23Mn-M13M2i} +2(7oA) + a0D'0)[MuM22 . Show that the effect of the dipole kicker on the orbit length is AC = j> —ds = D{so)6Q.M23Mii][M\3M22 . Show that the vertical dispersion function k=—oo z where The vertical dispersion function can sometimes be approximated by a simple pole Dz « vzypz(s) .\i>x(s) . and Dx is the horizontal dispersion function at the skew quadrupole location. Vertical dispersion can also be induced by vertical closed-orbit error in quadrupoles. In a straight section of an accelerator. dipole rolls. Ill) xco(s) = Vy f l )&( 8 °) e0 cos (™s . In the presence of a skew quadrupole field with ABX = (dBx/dx)x. TRANSVERSE MOTION 8. for the vertical closed orbit is 80 z» Hill's equation + Kz(s)Z=-L^Dx6. Show that H = jD2 + 2aDD' + fiD12 is invariant in the straight section. feed-down from magnetic multipoles. and ipx is the phase advance function. M13 = 0 and M23 = 0. . where k = [vz] is the integer nearest the vertical betatron tune and (. Bp ox where Bp = po/e is the momentum rigidity.M23Mi2] 80This exercise demonstrates that residual vertical dispersion can be generated by skew quadrupoles. the dispersion function transfer matrix is given by Eq.254).
i. /o (MHz) 1 1. Double-Bend Achromat: Consider an achromatic bending system with two sector magnets and a focusing quadrupole midway between two dipoles. 0 .6 PHT 1.82 m.6c] O[l2] QF[K.6 -3. calculate the horizontal and vertical dispersion function at two horizontal and two vertical BPM locations.8 6. 1 .EXERCISE 2.9 0.03268 I 1.0 I I V.4.7 -5.03168 I 1. in thin-lens approximation. (2.po.249).6 -2.Jq] O[/2] B[-p.0] O[h] QF[JC.5 -5. (2.e. and BESSY (Berlin).-er] O\h] QF[JT. The parameters for this experiment were (1) proton kinetic energy = 45 MeV. (2) circumference = 86.4 I 0. and (b) the reverse-bend DBA B[p. and 70 are Courant-Snyder parameters at the initial location.3 -10.VKlq 2 -.5 PH24 I -0.2 2.0 3. Chasman-Green lattice cell.313). TLS (Taiwan).D0 + 1<XODQDQ +POD'Q\ Mij is a matrix element of the transfer matrix.9] where the reverse bend angle 9r < S can be used to adjust the desired momentum compaction factor.lq] O[l] B[p. KLS (Korea).6 PV12 PV14 PVT PV26 0. arranged to minimize {H) in the dipole. .BPM I 1. The double-bend in the design of low emittance storage rings. Here K and lq represent the focusing strength function and the length of the quadrupole.280).9]. Using the beam position data (in mm) in the table below at three different revolution frequencies (in MHz) at the IUCF cooler ring.f q ] O[h] B[p.03268 I 1.7 1.1 0. 12.03068 H.4 167 where %Q = 70.1).9] O[l] QF[K. Table: Some Beam Positions xco or zco (mm) vs /o (MHz) of the IUCF Cooler. Other (a) the triple-bend achromat (TBA) B[p.e0] which has been used in many synchrotron radiation light sources such as the ALS (Berkeley).8 | PV24 | 13. B[p.0 1.Iq] O[/] B[p.lq] O[/] B[p.6. the matching condition reduces to Eq.4 -12.03168 MHz.3 4.3 PH14 2.0o] O[ii] Q F [tf. show that % in a sector dipole is given by Eq.2 -4. This basic achromat is also called a achromat (DBA) is commonly used where quadrupole configurations are achromat modules are .03068 Zco Zco £co_ 0. and cto.3 PH12 -0.BPM Ico -Tco 3: co PHI 0.9 | 0.0 -2.4 | 1. Show that the dispersion matching condition is given by ptan —h I = —F= cot 2 JK and that.4 -1.9 -4. (3) reference orbit frequency = 1.7 8. (b) Find ~H in a rectangular dipole (use the result of Exercise 2.1 5. (a) Using the My of Eq. and (4) transition energy 7r=4. (2.03168 I 1.
.168 CHAPTER 2.Zq] O[h] B[-p.+ li = —== .-6] O[/a] B[-p. extraction.cos 9) cos <t> Pw { -[sin0 + 2 tan 0(1 . Assuming that DQ = D'D = 0. in small angle approximation.0] is achromatic if the following condition is satisfied: I _ 2cosS + l p sin# 16.= I (1 .0] O[l] B[p. /q <C(<1+*B) 4 + 24 + £ B ' where / q is the focal length of the quadrupole and £Q is the length of the dipole. etc. internal target operation. TRANSVERSE MOTION 14. It can be used as a beam translation (chicane) unit to facilitate injection. e) 0 Lw (.0] O[h] QF[K.-6] is achromatic if the following condition is satisfied:81 0 lc cos \fKlq + -L sin VKL psin. Show the three sector dipole system B[p. is Rm = 202(<?i + \p»)( p w . 0<s<Lw —i^. f-(l-cos</>).P w . VK -^^. sure to take the edge focusing into account.(1 .cos 0)] sin <f>.Q O[lc] QF[#. It can also be used as one unit of the wiggler magnet for modifying electron beam characteristics or for producing synchrotron radiation. show that the dispersion function created by the wiggler magnet is 82 .2 cos VKlq Show that.-26) ( 3LW Pw . (a) Show that the rectangular magnet beam translation unit is achromatic to all orders. in thin-lens approximation. . A set of four rectangular dipoles with zero net bending angle B[p. 2 lcy/K sin VKlq .e] O[k] B[-p. and show that the R^e element of the transport matrix. Lw < s < 2LW 82Be 81 At the symmetry point of the antisymmetric bending section D = 0. Achromatic translating system: Show that the transport line with two sector dipoles given by B[p. 2LW 4LW s (b) A simplified compact geometry with lx = l2 = 0 (shown in the figure above) is often used as a unit of the wiggler magnet in electron storage rings.0] O[l] B[p.cos (j>) . .-0\ O[h] B[p. 15.6] has many applications. e) .
Since D' = 0 at the symmetry point. 0 < s < iw V.cos6).cos 0)] cos 0. (a) Show that the phase advance of the dispersion matching section is determined completely by the prescribed dispersion function Z>a and the phase advance of the DOFO cell. / i : focal length of QDI.) = / -s 2 /2pw. I -(2Ll .(2LW . f2'.2tan0(l . In small bending-angle approximation. (1 . An FMC basic module can also be made of two DOFO cells with a dispersion matching section (shown schematically below).betatron phase advances of one half of the dispersion matching section.cos $ + C cos $ (b) In thin-lens approximation. and 169 { Show that .1~C°*e. 2L: length of the DOFO cell. DD dispersion function at the defocussing quadrupole.sin (j>. Lm: length of half a complete module. D(s = 2Lv. L w < s < 2LW. sin 0 + (1 .'( S N = { -s/Pw.) = 2pv.s) 2 )/2p w . the wiggler is an achromat. show that the dispersion function becomes D(.focal length of QF2I i'xi'i'z'.cos 0) sin 4> -[sin0 + 2tan0(l . £. DOFO CELL Dispersion Matching Section DOFO CELL v .s)/pw.EXERCISE 2. COS0 D'(s = 2K) = 0. 0<s<L . PD> 7D = 1 //^D : values of betatron functions at the center of QD. L w < s < 2LW.C ) s i n $ tan ibx = — 1 . 6: bending angle of a half DOFO cell.s i n # .L w ) / p w . \ -(2L W . -Da: prescribed dispersion function at marker M a with C = DA/DD. Lc: half length of the matching section from Mb to Mc.4 where <j> = s/pw and (j> = (s . s = iw+ LW < s < 2LW. 0 < s < Lw QD/2 i i in I B 11—11 II QF B I hn i i Qm Q F2 T i m i I i 11 1 B In I II QF B QD/2 I In Ma Mb Mo Use the following notations <&: phase advance of the FODO cell. 17. show that -M' + £)('-&)' »2*4-7f)(i+i)' .
g.' ic ^ /1 .a a o (1 + W / i ) 2 j o~i COS^ 1px Pz. TRANSVERSE MOTION Show that the stability of the betatron motion is a necktie region bounded by four lines: Lc < » 2/2 . ^ = 7r/4. e. Lc//2 and ^ are determined. " 1 . (c) Show that the values of the betatron amplitude functions at Mc are c Px. (2.170 CHAPTER 2. The dispersion function in the combined-function dipole satisfies D" + KXD = \/p.31). . Since V"i is determined by parameters $ and £.9—.g. To simplify the design of a DBA in a synchrotron storage ring. sin2f /5 1 * \ 1 18. where Kx = 1/p2 + (l/Bp)(dBz/dx) is the effective defocussing strength function and p is the bending radius of the dipole.2(1 . e. combined-function dipole magnets have often been used. in ELETTRA in Trieste and in the UVU and X-ray rings in the National Synchrotron Light Source (NSLS) at BNL. the parameter £ c //2 is a function of Lc//i> i-eLc _ /a ~ 2 cos2 Vx > 1 + ic//i' Draw the line Lc//2 vs Lcjf\ for constant ^ x . Furthermore.$\.Lc/2f2 ' where /J^D is the horizontal betatron amplitude function at the center of QD. 2• Lc/fl) COS 2 1pz (d) Show that the dispersion action in the matching section is Jb = Jc = JD (l .' LQ_ Lc/fi 2/2-1-Le//!1 Lc Lc/fi 2/2-l+Lc//i" Plot the necktie diagram of Lc/f2 vs Lc/f\. $ .c = Pz..83 83Because the electron has a negative charge.C)2) • (e) Show that the dispersion function at the middle of the dipole is / 1 *\ L6 Show that the momentum compaction is Doe\( . This means that the dispersion matching section is a one-parameter lattice.C) cos * + (1 . Once L c //i is chosen. the gradient term in Kx has a sign opposite that in Eq.c = Pi. show that the length of the matching section is _ /8IiDsini/'iCos^3.a a o (1 .
Kl=l.685 D3 :DRIFT.e.SYMM.D5.K2=0.4018946 Q4 :QUADRUPOLE.K2=0.L=.29943942 B : RBEND.. Plot G = V16AC . J30 = /30/L. 6 .4 (a) Show that 171 <W)=p0 3 [|^)-^B( g ) + gC(g)].L=0. TITLE.Q3. Note that the dipole is declared as RBEND for a rectangular magnet. however.L=.L=.L=..L=2.D6.45.HSUP.L=. and 70 = -y0L.4132.Kl=-1.Kl=-. "NSLS X-ray RING" Ql :QUADRUPOLE.70825 D5 :DRIFT.. .L=.026848954.L=0.. i. SD :SEXTUPOLE. discuss the effect of the combined-function dipole on (H). the combined-function DBA gives rise to a larger (H).8 cosh g + 2 cosh 2g ? ' = tf ' with q = y/\Kx\L.B.225.D4. we should take into account the edge focusing of the dipole magnet. (b) Show that the minimum of (H) is <W>min = 47l5^3' where G = V16AC .7.#S/E TWISS STOP .15B2 vs the quadrupole strength and show that V16AC — 15B2 > 1. Dl :DRIFT.SUPER=8 PRINT.D5. 3(sinh2g-2g) „.L=.EXERCISE 2.3484 D4 :DRIFT.L=. Neglecting the edge focusing.25 D2 :DRIFT.33731236 Q3 :QUADRUPOLE.ANGLE=.Q1. where in principle.L=2. (c) Use thin-lens approximation to verify the strength of matching quadrupole Q4 of the NSLS lattice input data (MAD) file (shown below) for the achromat condition. a0 = a 0 .25 HSUP :LINE=(D1. where A{q) = — v — ' (9) = _. In Chapter 4 the effect of damping partition number on the natural emittance of electron beams will be discussed.50186576 Q2 :QUADRUPOLE.8.SF.L=.Q4) USE.D2)Q2. .. 30g-40sinhg + 5sinh2<j C(9) .Kl=l.39269908 SF :SEXTUPOLE.SD.D3.15B2.9 D6 :DRIFT.Kl=-1.
g. CERN 91-04. the chromatic gradient error should include the effects of dispersion functions.Equation (2. (2323) where K = B^Bp and Bx = dBz/dx.I we define chromaticity and discuss its measurement and correction. 53.£ + *<•)] * + <>«•)—IM [ AKZ = -K(s)S + O(S2) « -KXS. that the chromatic gradient error is essentially equal to the product of the momentum deviation 5 and the main focusing functions —Kx and — Kz. S. TRANSVERSE MOTION V Chromatic Aberration A particle with momentum p executes betatron oscillations around an off-momentum closed orbit xco(s)+D(s)5. fringe fields. For details see. K. where xco is the closed orbit for the on-momentum particle. p. 1965).242) is Hill's equation of the horizontal betatron motion.[ . CERN Accelerator School. AKZ= ( ± ) ' D ' + 1 * . Note. Some of these terms are includes below: AKx = \ * + K + 2° (± . Note that the higher-order gradient error depends on the betatron amplitude and dispersion functions. (2. We will neglect all chromatic effects arising from the dispersion function and fringe fields of magnets.2 we examine the nonlinear perturbation due to chromatic sex84Including the effect of off-momentum orbits. the gradient error arising from the chromatic aberration is proportional to the designed focusing functions Kx and Kz. etc. The resulting gradient errors AKX and AKZ are given by84 | A * . Similar gradient error exists in the vertical betatron motion. New York. Steffen. and thus the chromatic gradient error is a "systematic" error that can cause major perturbation in the designed betatron amplitude functions and reduce the dynamical aperture for off-momentum particles. In Sec. The dependence of the focusing strength on the momentum of a circulating particle is called "chromatic aberration. the "beta-beat" associated with the half-integer stopbands. D is the dispersion function. a lower energy particle with d < 0 has a smaller momentum rigidity and a stronger effective focusing strength.172 CHAPTER 2. A higher energy particle with 5 > 0 has a larger momentum rigidity and thus a weaker effective focusing strength.(I)' [ p2 p \p2 ) \p) 1-K+-D+ D' + ^ D ] {8 -62 + •••) 0xp J v ' -••)+•••. in particular. V. High Energy Beam Optics (Wiley.242).. This is reflected in the gradient error AKX in Eq. Guiducci." Furthermore. 1991. . V. In this section we study the effects of systematic chromatic aberration and its correction.D \ {8-82 + where K = B\ jBp is the gradient function of quadrupoles. in Sec. The effects of chromatic aberration include the chromaticity. Proc. e. +•••.K ) . etc. and S = (p-po)/Po is the fractional momentum deviation from the on-momentum po.
1 for a FODO lattice. V. Since the chromatic effect of Eq. The "specific chromaticity. A beam is composed of particles with different momenta.4. and vy = N$y/2n is the betatron tune of the machine. V. (2. The natural chromaticity of a FODO lattice is given by (see Exercise 2. the betatron tune decreases with particle momentum. in Sec. the momentum spread gives 85In mathematical language.4 we outline basic machine design strategy. <£y is the phase advance per cell. and Cy < 0.V.326) Because the focusing function is weaker for higher energy particles. (2.196). since the betatron amplitude function is always larger at a focusing location where Ky > 0.85 The magnitude of the natural chromaticity Cy.324) \^ = VJ P*AK*dS * ( 4^ / t'K-*8) 6Cy = ^ . III. and the natural chromaticity is negative.3) f-rFODO 1 N (flnax Anin \ _ tan($ y /2) where N is the number of cells.3 we study systematic half-integer stopbands and their effects on higher-order chromaticity. defined as the derivative of the betatron tunes vs fractional momentum deviation. The chromaticity arising solely from quadrupoles is called the "natural chromaticity. On the other hand.nat depends on the lattice design. / is the focal length.y = Cy/vy. we find that the gradient error can induce betatron tune shift and betatron amplitude function perturbation. and in Sec. V." Cy^tK^fpyKvds.5. the resulting betatron tune shift. the integral § /3yKyds > 0. the specific natural chromaticity can be as large as —3. . Because of the chromaticity.325) The chromaticity. for a collider lattice or a low-emittance lattice. (2. is f Avx = -?.hxAKxds 47r/ « (^ l(5xKxds) 5. The momentum spread of the beam is typically of the order of as ~ 10~5 —10~2 depending on the application and the type of accelerator. CHROMATIC ABERRATION 173 tupoles. is nearly equal to ." defined as £. is where the subscript y stands for either x or z. Cx or Cz. (2.I Chromaticity Measurement and Correction In Sec.323) gives rise to a systematic gradient error. given by Eq. U n T ' (2.
E. Bleser. we find that 1 2 $ Figure 2. the growth of transverse head-tail instabilities depends on the sign of the chromaticity (see Sec. The dashed straight line shows the theoretical expected value. R. (2.40 shows the "measured specific" chromaticities of the AGS. 276 (1987). Auerbach. as discussed in subsection C. [3]). *-$—»•& <2-328> where r\ is the phase-slip factor.8). A. AGS Tech Note No. The solid curved line is obtained by modeling the sextupole field in the dipoles. .174 CHAPTER 2. Chromaticity measurement Machine chromaticities can be derived from measurements of betatron tunes vs beam momentum. but the horizontal chromaticity becomes more negative.40: The measured chromaticities divided by the betatron tunes of the AGS vs the beam momentum. 86E. VIII and Ref.e. particle loss may imminently occur.86 Note that the vertical chromaticity becomes positive above about 22 GeV. The dashed line shows the value of — ^ t a n * of Eq. TRANSVERSE MOTION rise to tune spread in the beam. From the experimental data. E. AGS Tech Note No. and wrf is the angular frequency of the rf system (see Exercise 2. 288 (1987). Since beam momentum is related to rf frequency. where $ as 53.5.8° is the phase advance of an AGS FODO cell. If the chromaticity and the momentum spread of the beam become large enough that the betatron tunes overlap a low-order nonlinear resonance. the chromaticity can be obtained from measurements of betatron tune vs rf frequency. Them.327). Furthermore. Bleser. i. Figure 2.
called geometric aberration. (2.333) (2.330) where xp is the betatron displacement and D(s)5 is the off-momentum closed orbit.331) can produce nonlinear perturbation in betatron motion.334) J 87It is also worth pointing out that the second term of Eq. AKZ = -S(s)D(s)S (2. Since a circulating beam with such a large tune spread does not have a long storage lifetime. Note that the first term of Eq. This requires a magnet whose focusing function increases linearly with momentum in order to compensate the loss of focusing in quadrupoles. and the resulting tune spread will be Av « 0.332) depend linearly on the off-momentum deviation. Chromatic correction The natural chromaticity of a high-luminosity collider with low-/?* insertions is usually large. the chromaticity becomes Cx = ^ f 4TT P*[K*(S) .V. we obtain \ZP 2 2 (2331) [ -^f = -[S(s)D(s)5]z0 .nat ~ -250. x= ~Z)> Xp{s) -W=BpXZ> Substituting the transverse displacement of an off+ D(s)S.329). (2. into Eq. pz[Kz(s) + S{s)D{s)}ds.87 Including the contribution of sextupoles. VII. the natural chromaticity for the RHIC injection lattice is about C^nat ~ —50. (2. Since the effective quadrupole focusing functions AKX = S(s)D(s)6. (2.1 for a beam with an rms spread of 8 = ±2 x 10~4. momentum particle. the natural chromaticity for the Superconducting Super Collider (SSC) was expected to be about Cy.S(s)D(s)]ds. sextupoles can be used for chromaticity correction. where we will find that the placement of sextupoles is important in minimizing nonlinear resonance strengths. which can lead to a natural tune spread of about Av « 0. CHROMATIC ABERRATION 175 B. where S(s) = —B2/Bp is the effective sextupole strength. The magnetic flux density of a sextupole magnet is ABZ B2 2 2 ABX B2 (2329) -W=2B-P{X where B2 = d2Bz/dx2\x=z=0. Similarly.331) depends linearly on the transverse betatron displacement. to be discussed in Sec. First we examine the possibility of using sextupole magnets for chromaticity correction.5 with a beam momentum spread of 6 = ±5 x 10~3.S(s)xpzp. chromatic correction is needed to ensure good performance of a storage ring. (2. Cz = ^ l . For example.
and 9 the bending angle per half-cell. where 4f. For example. we consider a lattice of N repetitive FODO cells.3) _ 1 5F~2/^(1 sinf + Isinf)' bD~ 1 sinf 2/20(1-I sinf)' [ } where / is the focal length. where (3XDX and /3ZDX are maximum. Nonlinear modeling from chromaticity measurement The measurements of chromaticities can be used to model nonlinear sextupole fields in an accelerator. strong sextupoles are needed to correct it. C.5. 4d. For colliders or low-emittance storage rings. • In order to minimize their strength. which consist mainly of FODO cells or DBA/TBA type cells. where sextupoles are located near the focusing and defocussing quadrupoles. The second-order chromaticity and the betabeat can be simultaneously corrected by a proper chromatic stopband correction. In Sec. Let Sp = —B2{F)la[/Bp and So = —i?2(D)4d/Sp be the integrated sextupole strengths at QF and QD respectively. the chromatic sextupoles should be located near quadrupoles. TRANSVERSE MOTION This shows that sextupoles located at nonzero dispersion function locations can be used to correct chromaticity. For example. 5 2 (D) are the length and the sextupole field strength at QF and QD. • A large ratio of px/pz for the focusing sextupole and a large ratio of /3z/f3x for the defocussing sextupole are needed for optimal independent chromaticity control. located in the arcs. the simple chromatic correction scheme using two families of sextupoles may not be sufficient to correct the higher-order chromatic effects. Since the low-/?* values in these lattices give rise to a large chromaticity. Figure 2.3. The sextupole strength needed to obtain zero chromaticity is (see Exercise 2. If the intrinsic systematic half-integer stopband widths are large. V.176 CHAPTER 2. • The families of sextupoles should be arranged to minimize the the systematic half-integer stopbands and the third-order betatron resonance strengths. and B2(F). Rules for their placement are as follows. we will show that the chromatic gradient error can also create a large betatron amplitude function modulation (betabeat). Generally. $ the phase advance per cell. which in turn induces a large second-order chromaticity.41 shows an example of chromatic correction with two families of sextupoles in RHIC. chromatic sextupoles are also arranged in families. two families of sextupoles are needed to correct horizontal and vertical chromaticities. we discuss the nonlinear sextupole modeling of the . they are called chromatic sextupoles. Note that the second-order chromaticity Afx>2 ~ C^2) S2 can cause substantial tune spread in a beam with a large momentum spread.
2 x 10~4 + 5.40 represent theoretical calculations with the integrated sextupole strengths 5b = 5e -5. and proportional to B. For the long magnets (2. . 5 e and the first term in 5b may be considered as the systematic error in dipoles. and 5 e is the integrated sextupole field distributed only at the end of each dipole. which has recently attained an intensity of 6 x 1013 protons per pulse. we assume that the sextupole fields arise from systematic error at the ends of each dipole. CHROMATIC ABERRATION 177 Figure 2.7.fodo. where px « fiz. Cz. the eddy current sextupole due to the vacuum chamber wall.0 x 1 0 . A chromaticity of about —Zvx does not appear to cause difficulties in the AGS operation.6 x 10~4p .fodo. Many low energy synchrotrons do not use chromatic correction sextupoles. To model the AGS.017 (nr 2 ). = -0. which is inversely proportional to the beam momentum. The saturation term is nonlinear with respect to the momentum p. The second term in 5b is due to the eddy current on the vacuum chamber wall. AGS based on the measured chromaticities shown in Fig. the horizontally defocussing sextupoles must be located in dipoles. where 5 = 2 T/s in this experiment.3876 m) in the AGS.41: Variation of the betatron tune vs Ap/p after chromatic correction with two and four families of sextupoles in RHIC.fodo. and the iron saturation sextupole at high field. and they are momentum independent.V + 2. chromaticity correction is absolutely essential in high energy synchrotrons and storage rings. 2. However. and C^data + C2]data = CXtfodo + Cz.8 x 10"V) (m" 2 ).8 x W~2/p -(3. 5b is the integrated sextupole field in each dipole distributed in the whole dipole.40.0066 m) respectively.data > Cz.V. and the saturation sextupole field depends on a higher power of the beam momentum. the eddy current sextupole field depends inversely on the beam momentum. The systematic error is independent of the beam momentum. for the body and the ends of the short AGS bending magnets (2. 2. the integrated sextupole strength of the 5b term is assumed to be proportional to their length. The solid lines in Fig. Since CX]data < Ci. Here p is the beam momentum in unit of (GeV/c).
we will show that if chromatic sextupoles are separated by an odd multiple of 180° in the betatron phase advance. their contributions to the third-order stopband width cancel each other in the first-order perturbation theory. TRANSVERSE MOTION V.178 CHAPTER 2. (2. the nonlinear resonance strength can be minimized by properly arranged sextupole families. Such arrangements can also be used to correct the systematic half-integer stopband discussed in the next section.336) The Hamiltonian including sextupole nonlinearity is This Hamiltonian can drive third-order and higher-order nonlinear resonances at 3ux = £. V. VII. drl>1/d8 = l/fa.337).A> we obtain where AK = K\— Ko is the gradient error. In Sec.10) .. Thus. the betatron amplitude functions /?o and Pi satisfy the Floquet equation P'Q = -2a0.3 s ^ ) .To. However.f t &ff — .2 Nonlinear Effects of Chromatic Sextupoles H = | (x'j + Kxxj + 4 + Kzz2) + ^ ( ^ . the change of A across a sextupole is given by Av AA= [Jp^AKds^-^-^-.. we find that A2 + B2 = constant in regions where AK = 0. From Eq. Defining the betatron amplitude difference functions A and B as (see Exercise 2.. In thin-lens approximation.aojdi B _ Pi . ai = ffi0i-7i. Po . and six families of sextupoles can be used in a lattice with 60° phase advance per cell. vx ± 2vz = £. (2. and ipo and ipi are the unperturbed and perturbed betatron phase functions.. _ aiPo .2 a 1 . Similarly.3. four families of sextupoles can be arranged in a lattice with 90° phase advance per cell. d^0/ds = 1/A» /3'1 = .3 Chromatic Aberration and Correction The systematic chromatic gradient error can produce a large perturbation in the betatron amplitude functions for all off-momentum particles. the change of A across a quadrupole is given by J v / Po where / is the focal length of the quadrupole. a'o = KoPo . A A « . where £ is an integer.
(2. (2. Systematic chromatic half-integer stopband width We have found that the perturbation of betatron function is most sensitive to stopband integrals near p ss [2v] harmonics (see Sec. Since the phase of A or B propagates at twice the betatron phase advance (see Exercise 2. At p — 0 (Mod P).3. Here we investigate the effect of systematic chromatic stopband integrals. two identical quadrupoles (sextupoles) separated by odd multiples of 90° in betatron phase advance cancel each other. Note that Jp<y = 0.201): \jp.338) We consider a lattice made of P superperiods. The integral of Eq. The treatment is identical to the stopband integral to be discussed next. i.z = —fpzAKze-^ds. where L is the length of a superperiod with K(s + L) = K(s).y = ~{^[PyKye-^ds\ = ~{^[PyKye-^ds^ [l + e-*¥ + e"^* + e^3p* + • • •] CP(js)e-j*pE^.4). and the diffracting function CP (u) is given by Note that the diffracting structure function £P -» P as u -> integer. CHROMATIC ABERRATION 179 where ge$ = (B2As/Bp)D is the effective gradient error. Let C = PL be the circumference of the accelerator.V. two identical quadrupoles (sextupoles) separated by an integer multiple of 180° in betatron phase advance will produce additive coherent kicks. unless p = 0 (Mod P). (B2As/Bp) is the integrated sextupole strength.x = ^-hxAKxe-^ds. Similarly. each superperiod contributes additively to the chromatic stopband integral. (2. I llX J [jp. 0(s + L) = /3(s).10). A. We will show that systematic stopbands can generate a sizable second-order chromaticity.e. the global chromatic perturbation function of the lattice can be minimized. and D is the dispersion function. The effect of systematic chromatic gradient error on betatron amplitude modulation can be analyzed by using the chromatic stopband integrals of Eq.338) becomes JP. the half-integer stopband integral increases by a factor of P. . (2. By using sextupole families. III.339) where y stands for either x or z.
which is composed of JV FODO cells.341 where $ is the phase advance per cell. the AGS lattice has P = 12. the AGS lattice can be approximated by a lattice made of 60 FODO cells. In particular.Cw(f—)e~J—or-. TRANSVERSE MOTION Since the perturbation of betatron functions is most sensitive to the chromatic stopbands near p w [2ux] and [2vz]./V FODO cells is small. /3 max and /3m.88 Generally. and P = 22 for the SPRING-8 at JSRF.340). it is beneficial to design an accelerator with high super-periodicity so that the betatron tunes can be located far from the important chromatic stopbands. 18. i. and the diffracting function (N(u) is given by Eq. the chromatic stopband of the arc adds up to zero at harmonics p w 2u. Similarly. and the resulting chromatic perturbation is small. P = 40 for the APS.^ ) [l + e-** + e-^i + e-t*>i + • • •] = y ( s i n .90 • • •. the chromatic stopband integral at p s» 2v due to . The actual betatron tunes at vx/z = 8. For example.60.e. This can be achieved by choosing the betatron tunes such that [2vx] and [2vz] are not divisible by the superperiod P. If p^/2iri/ = 0 (Mod N). Trcosf \ 2 4u \. The chromatic stopband integral in thin-lens approximation is given by Jp = ~ ( £ p . or the stopband integrals of two modules cancel each other.v I 2-nv 2. B. / is the focal length of each quadrupole. which are far from the betatron tunes. 30. if N<& = integer XTT. In fact.8 are indeed far from systematic half-integer stopbands at p = 6 and 12. the diffracting function is equal to N. Thus the goal is to design an accelerator such that the chromatic stopband integral of each module is zero. 12. a basic design principle of strong-focusing synchrotrons is to avoid important systematic chromatic stopbands. a high energy accelerator or storage ring with large super-periodicity is costly. 24. 88The stopbands in a collider can also be minimized by local cancellation of various beam line modules. (2. Fortunately. the TEVATRON has a super-periodicity of P = 6.180 CHAPTER 2. etc.^ p e . since $/27r is normally about 1/4 (90° phase advance) so that p<&/2-Ki> « p/4v ss 1/2.n are values of the betatron amplitude function at the focusing and defocussing quadrupoles respectively. Chromatic stopband integrals of FODO cells Now we examine the chromatic stopband integral of the arc. etc. However. . and the betatron tune should avoid a value of 6. the stopband integrals a t p « [2i/] resulting from iV FODO cells in the arcs is small if the total phase advance of these FODO cells is iV$ = integer x 7r.+ j s i n ^ .c o s y . P = 16 for the ESRF. The important stopbands are located at p = 30. This means that each FODO cell contributes additively to the stopband integral. Some examples of high superperiod machines are P = 12 for the ALS. to be discussed in the next subsection. and the betatron tune should avoid 18.
91 A. G. J.89 cancellation of the chromatic stopband integrals between two adjacent insertions would be desirable. particularly when the beam momentum spread is large.V. 1626 (1985). . IEEE Trans. Substituting /3 = A>(1 + A/3/A0 into Eq. Thus. the chromatic stopband integrals of two adjacent insertions cancel each other. and is proportional to 5. high-/? triplets or doublets on both sides of the IP contribute additively to the systematic half-integer stopband near p « 2vx/z. .338).D. (2. Hahn. Effect of the chromatic stopbands on chromaticity The chromatic stopband integrals for large colliders. (2. private communications. Lee. the insertion may contribute a substantial amount to the chromatic stopband integral. This cancellation principle remains valid when two insertions are separated by a unit transfer matrix. The chromatic stopband integral of insertions Because of its small /?* value. Claus. Since it is difficult to design an insertion with zero chromatic half-integer stopband width. remain important even after careful manipulation of piecewise cancellation.Y. Such a procedure was extensively used in the design of the RHIC lattice90 and the SSC lattice.324). we obtain Auy = C^5 + C^52 + ---. Let 3>lns and J™s be respectively the phase advance and the chromatic stopband integral of an insertion.344) 89In fact. E. we obtain Jp = 0 if $ lns = (2n+l)ir/2. Parzen. The betatron modulation of the lattice is given by A/3_ T |JPlcos(p0 + x) 2(u-p/2) ' . 90S. CHROMATIC ABERRATION 181 where the transfer matrix of the arc becomes a unit matrix / o r a half-unit matrix -I. See also the SSC report. H. Nud.342) At the harmonic p « [2v]. called betabeat. C. NS-32. Garren. n s [l + e x p ( j ^ ) ] . such as the SSC and RHIC.91 D. Sci. The following example illustrates the effect of betatron amplitude function modulation on chromaticity. if the insertion is a quarter-wave module. (2. (2'343) where the chromatic stopband integral Jp is given by Eq. and second-order chromaticity for off-momentum particles. The total contribution of two adjacent insertions becomes Jp = j. Courant. We consider a lattice dominated by a single p harmonic half-integer chromatic stopband. They give rise to a large betatron amplitude modulation. (2.
To obtain a nonzero chromatic stopband integral.41 shows an example of the second-order chromatic tune shift with 8. Effect of sextupoles on the chromatic stopband integrals The chromatic sextupoles also contribute to the systematic chromatic stopbands. that is commonly used in FODO cells with 90° phase advance.e. Hahn. The remaining second-order tune shift C^62 can arise from the chromatic stopband integral. (1987). E. G. Conf. Here we present an example of chromatic correction for a collider lattice. the chromatic sextupole does not contribute significantly to the chromatic stopband integral if the transfer matrix of the arc is / or —/.347) where N is the number of cells.D K r W .348) Lee. the chromatic stopband integrals due to the parameters Ap and AD are given by AJp. D^ = Su — AD}.F. Sox = SQ + AD. 1328. sextupoles are organized in families. (2.W-Dp*/2".\) 92S.182 CHAPTER 2. (2. Dell.^ . and C^ and C£2) are the first. G.and second-order chromaticities c»1} = ~h / &(*» ~5^)rfSi (-4) 2 35 If the first-order chromaticity is corrected.340). then C^1^ = 0.sext = ^ C v ( | ^ . p. First we evaluate the stopband integral due to the chromatic sextupoles.339). (2. and the diffraction function £N is given by Eq.sext = £. Here the parameters 5 F and So are determined from the first-order chromaticity correction. i. H.C {Jr-) [fcSFDF + pDSDDDe-^2"} e-. However. A D will not affect the first-order chromaticity. Let S? and SD be the integrated sextupole strength at QF and QD of FODO cells in the arc. the stopband integral is zero or small if N$/n = integer. Parzen.a / ^ . We consider an example of a four-family scheme with {SFI = Sp + Ap. the parameters A F . which is proportional to the zeroth harmonic of the stopband integral. Since /3(s) and D{s) are periodic functions of s in the repetitive FODO cells. (2. We next discuss the half-integer chromatic stopband correction using sextupole families. Proc 1987 Part.92 The stopband correction that minimizes the /3-modulation also minimizes the second-order chromaticity. . The p-th harmonic stopband integral from these chromatic sextupoles is JP. Accel. SF2 = Sp — Ap. As in Eq. [/3FAFZ?P +/?DADZW*/4"] e . Figure 2.. TRANSVERSE MOTION where y stands for either x or z.Y.
. To improve the slow extraction efficiency. S F3 . particularly in the strip injection scheme. where the second-order chromaticity and the betatron amplitude modulation can be simultaneously corrected. • The betatron tunes should be chosen to avoid systematic integer and half-integer stopbands and systematic low-order nonlinear resonances. 2. • The chromatic sextupoles should be located at high dispersion function locations. the lattice design of accelerator can be summarized as follows. 5 F2 . Fig. • The betatron amplitude function and the betatron phase advance between the kicker and the septum should be optimized to minimize the kicker angle and maximize the injection or extraction efficiency. we have (w -> JV.41 shows an example of chromatic correction with four families of sextupoles in RHIC.V. Local orbit bumps can be used to alleviate the demand for a large kicker angle. The scheme works best for a nearly 90° phase advance per cell with N$ — integer x TT.349) V. the six-family sextupole scheme works for 60° phase advance FODO cells. The focusing and defocussing sextupole families should be located in regions where /3X 3> /3Z. every FODO cell contributes additively to the chromatic stopband. the stopband width should be corrected. Dm. i. and low-emittance lattice storage rings. The lattice is generally classified into three categories: low energy booster. The /3X and j3z values at the injection area.e. the j3 value at the (wire) septum location should be optimized. By adjusting Ap and AD parameters.5 D3 } has two additional parameters. and px <C Pz respectively in order to gain independent control of the chromaticities.4 Lattice Design Strategy Based on our study of linear betatron motion. CHROMATIC ABERRATION 183 At p « [2i/] and $/2TT W 1/4 (90° phase advance). the betabeat and the second-order chromaticity can be minimized. the injection line and the synchrotron optics should be properly "matched" or "mismatched" to optimize the emittance control. should be adjusted to minimize emittance blow-up due to multiple Coulomb scattering. The resulting stopband width is proportional to Ap and A D parameters. SDI. where the third-order resonance-driving term vanishes also for the four-family sextupole scheme. otherwise. where the six-family scheme {Spi. Similarly. The local vacuum pressure at the high-/? value locations should be minimized to minimize the effect of beam gas scattering. Furthermore. collider lattice. (2.
. and /3 max is the maximum betatron amplitude function at the triplet. and $ is the phase advance of the FODO cell. beam lifetime. 4. TRANSVERSE MOTION • It is advisable to avoid the transition energy for low to medium energy synchrotrons in order to minimize the beam dynamics problems during acceleration. nonlinear betatron detuning. Some of these issues will be addressed in this introductory textbook. A set of three quadrupoles ({QFI QD2 QF3} or {QDI QF2 QD3}). Exercise 2..184 CHAPTER 2. Sec. Besides these design issues. is commonly used in insertion regions to provide horizontal and vertical low-/3 squeeze. show that the betatron phase advance between the triplet and IP is 7r/2.3 (see Exercise 2. should be addressed. vacuum requirement. etc.5 1. problems regarding the dynamical aperture.3. Note that the required sextupole strength is larger at the defocussing quadrupole.2). 9 is the dipole bending angle of a half FODO cell. This criterion usually determines beam emittance and intensity. called a low-/3 triplet. (c) Show that the triplets on both sides of IP contribute additively to the stopband integral at p w 2u. (a) Show that the low-beta triplets contribute about __2As___J_ //3 max 4TT/8* ~ 2n\j /3* units of natural chromaticity. (b) If /9max > jS*. 3. where $ is the phase advance per cell and v = N$/2n is the betatron tune. Show that the strengths of two sextupole families used to correct the chromaticities of FODO cells are F 1 sin($/2) 2/ 2 0(l + isin($/2))' = D 1 sin($/2) 2/ 2 6>(l-I sin($/2))' where / is the focal length of the quadrupole in the FODO cell. where v is the betatron tune. 2. • Experience with low energy synchrotrons indicates that the Laslett space-charge tune shift should be limited to about 0. where As is the effective distance between the triplet and the interaction point (IP). rf system. /3* is the value of the betatron amplitude function at IP. III. collective beam instabilities. Show that the chromaticity of an accelerator consisting of N FODO cells in thin-lens approximation is F0D0 _ tan($/2) Onat ^ — ^ . The design of minimum emittance electron storage rings will be discussed in Chap.
( ^ 1 . (2.340). (2.0542203 ANGLE = 0.6 and C = 86.N{u) is given by Eq. and vx. where 7 T = 4.889612 Kl = 0. $ z . Show that the chromatic stopband integrals for a lattice made of JV FODO cells in thin-lens approximation are JPiX = -i.030680 Qx 3.7243 3. FNALBSTCELL : LINE = (BF S120 BF S050 BD S600 BD S050) BF : SBEND L = 2.7364 1 4. and the diffraction function C.5 185 4.0577073 ANGLE = 0.82 m. show that the chromatic stopband integral is j" . The AGS is composed of 12 superperiods with 5 nearly identical FODO cells per superperiod.889612 Kl = -0. 6.060157561 Sabc : DRIFT L = a.M E l e . Verify Eq.W * ) ^(JLje-W-i)/^ where $ x . Calculate the systematic stopband widths for harmonics 17 and 18 respectively.bc Find the systematic stopband width and discuss the choice of the betatron tunes. The Fermilab booster is a combined function synchrotron. 8.w 1 " " . (2.8 and vx = 8. The betatron tunes are vz = 8.347) and Eq. The lattice is made of 24 cells.Pz axe betatron amplitude functions.031680 I 1.EXERCISE 2. Betatron tunes vs revolution frequencies of the cooler Frequency [MHz] I 1." 5.-Jit h ?cos s + ' si ° 5} f .032680 I 1. / F and / D are focal lengths for focusing and defocussing quadrupoles.348). vz are the phase advances per cell and the betatron tunes. Assuming / F = fD with $ T = $ z = $ = 2nu/N. Use the experimental data below to calculate the chromaticity of the IUCF cooler ring.070742407 BD : SBEND L = 2.7.7080 _Qz | 4. What region of betatron tunes should be avoided to minimize the effect of systematic stopbands? 7. Px.6790 .7156 3.6913 | 4. as shown below. at 45 MeV proton kinetic energy.
(2. Here we find that the linear coupling can induce amplitude exchange between horizontal and vertical betatron motions.93 Measurement and correction of linear coupling will also be discussed. for solenoids. . (2. As = 0. 1 The Linear Coupling Hamiltonian The vector potentials for skew quadrupoles and solenoids are given by Ax = Az = 0. vertical closed-orbit error in sextupoles or horizontal closed-orbit error in skew sextupoles. 94 The skew quadrupole can also arise from "feed-down" of an off-centered vertical closed orbit in sextupoles.4. VI.350) Ax = ^Bl\(s)z. + 2gx' -(q + g')x = 0. TRANSVERSE MOTION VI Linear Coupling We have discussed uncoupled linear betatron motion.19). Substituting the components of the vector potential in Eq. IV.351) 93In Sec.—zr—) x z . The skew-quadrupole field arises from quadrupole rolls. Let zco be the closed orbit at a sextupole with sextupole strength B? = d2Bz/dx2.6. The solenoidal field exists in electron cooling storage rings. 21dz dx f° r skew quadrupoles.10). (2. and feed-downs from higher-order multipoles.g')z = 0.3): x" + Kx(s)x z" + Kz{s)z + 2gz' -(q. we show that a skew quadrupole at a high horizontal dispersion location can produce vertical dispersion.— ^-f) and B\\(s) are skew-quadrupole gradient94 and solenoid field strength. fringe field of a Lambertson septum. Linear betatron coupling is both a nuisance and a benefit in the operation of synchrotrons: the available dynamical aperture for particle motion may be reduced. but in reality betatron motions are coupled through solenoidal and skew-quadrupole fields. The effective linear coupling Hamiltonian and resonance strength will be derived based on perturbation approximation. but the vertical emittance of electron beams in storage rings can be adjusted. (2.186 CHAPTER 2. where \ir§£. which can generate vertical emittance for electron beams and result in lower luminosityforcolliders (see Exercise 2. Here we discuss the beam dynamics associated with linear betatron coupling arising from skew quadrupoles and solenoids. and in high-energy detectors at the interaction point (IP). Ag = --B\\(s)x. we obtain the linearized Hamiltonian for particle motion in accelerators as (see Exercise 2. and the Touschek lifetime limitation can be alleviated by linear coupling. x As = -(-^. The effective skew quadrupole strength becomes q = B2Zco/Bp.350) into the Hamiltonian in Eq.
351) can be derived from the following linearized Hamiltonian: -q{s)xz .356) where R is the average radius of the accelerator. and are effective solenoid and skew quadrupole strengths. (2. <j>z) are pairs of conjugate phase-space coordinates..c) + ( G l .px/p.* " . Kx and Kz are quadrupole-like focusing functions. and (Jx. $z = <i>z+ Xz (s) . (2.) 1 / 2 |[-Q + i fffe-^][cos(^ + $ 2 )+cos($ x -$ z )] (2. <j>x) and (Jz. Since V\c{s) is a periodic function of s. _ M e ^ . Xx= JO Px ^-. Xz= Jo j .g{s) \^z . it can be expanded in Fourier harmonics as VXc{9) = ^ 5 £ { ( G l .*. (2. are given by Gl'™ *** = h f V ^ ^ s ) eSbc^X'-f^-Wda. where rs f^Q ®x = <t>x+ Xx{s) ~ VXB. we obtain the coupling-potential: Vc = (^ 2 JxJ. be the conjugate phase space coordinates.^ + * .s i " .* 1 • PJ [P (2-354) Using Floquet transformation of Eq.^ J . (2.pz/p) motion of Eq. G\ ^neJX~ and G\\ie3XJr. The Fourier coefficients of the difference and sum resonances.355) +9 (jx ~ j ^ sin($x + $.+ j^j sin(*s . The betatron Let (x.> + c. (2.) J .c)} .357) . w e**-"--"^ + c.z.VI.353) Here the linear coupling potential is Vlc = -qxz .vz9. £ is an integer.) + g (J.94) for the uncoupled Hamiltonian. LINEAR COUPLING 187 where the primes are derivatives with respect to the independent variable s.
and the resonance strength is usually small. TRANSVERSE MOTION Here vx. Thus the effective Hamiltonian for betatron tunes near an isolated coupling resonance will be discussed in the following sections.358) obtained from the linear coupling potential of Eq. (2.^ + g(s)(f . P Px Pz Px Pz (2. Thus the linear-coupling resonance-strength should be minimized. The coupling resonance can cause beam size increase and decrease the beam lifetime.188 CHAPTER 2.2: Linear coupling resonances and their driving terms Resonance Driving phase AmplitudeClassification dependent factor vx + Vi=l ($ x + $ z ) ji/2ji/2 sum resonance vx . the horizontal and vertical betatron motions are coupled. • • -. vz are betatron tunes. Both skew quadrupoles and solenoids can drive the sum and difference linear coupling resonances. 16. Table 2. to minimize the effect of the systematic linear coupling resonance. each superperiod contributes additively to the linear coupling resonance strength.vz = £'.355). 95We will show. The strength of the linear coupling resonance due to random errors such as quadrupole roll and vertical closed orbit in sextupoles is smaller. %z = So ds/Pz are betatron phases. the difference between the integer part of the horizontal and vertical betatron tunes should not be 0. It occurs at all integer £. In general. If the linear coupling kernel A\CT satisfies a periodic condition similar to that in a synchrotron with P superperiods. where £ and £' are integers. Table VI. 8. Near a difference linear coupling resonance. the resonance coupling coefficient Gi iT i^ will be zero unless £ is an integer multiple of P. If £ is an integer multiple of P. in Sec. since the superperiodicity of the LEP lattice is 8. Xx = So ds/@x. and the linear coupling kernel ^4ic^(s) are A^{s) = .95 the optics of the betatron motion is normally designed to avoid sum resonances.$ z ) J]j2 J\l2 difference resonance The linear coupling potential has been decomposed into terms of the difference and sum resonances located respectively at vx — vz = £ and vx -\.v2 = £ ($ x . the coupling betatron sum-resonances are dangerous to the stable betatron motion. VII. 1 lists the corresponding driving terms. that the horizontal and vertical betatron amplitudes can grow without bound near a betatron sum resonance. This is called the systematic linear coupling resonance. For example. .Hi) + jg(s)(~ ± h .
£ and the phase factor \._i. 2 Effects of an isolated Linear Coupling Resonance Since the betatron tunes are normally near the linear coupling line. (2.353).£) ±-X (2. |(?i.£| as demonstrated in Fig. The Hamiltonian Eq.359) corresponds to two coupled linear oscillators. (2.If + |Gi. LINEAR COUPLING 189 VI.± = l£yx + vz + l)±-^\. (2.vz = I. and provide linear coupling correction. and the horizontal axis is the digital to analog conversion (DAC) unit of a COMBO power supply for a set of horizontally focusing quadrupoles. showing that the horizontal and vertical motion are coupled. in action-angle phase space coordinates.360) .a r e given by Eq.359) where the Fourier amplitude Gi. The vertical axis is the fractional part of the betatron tunes. The minimum distance between two normal modes is equal to the coupling coefficient jd?iT_i^|. Figure 2. Figure 2. (2. this section studies the effects of an isolated coupling resonance on betatron motion. the Hamiltonian Eq.VI. reaching a minimum value of tune separation.42. As the strength of a quadrupole is varied across the linear coupling resonance vx — vz + 1 « 0 (I = — 1 for the IUCF Cooler). 2.± =-(vx + vz .357).361) This means that the betatron tunes are separated by A. can be approximated by H a* vxjx + i>zJz + Gi-lttJj^Jzcos{<l>x -<pz-ie + x). u2.42 shows an example of measured betatron tunes vs quadrupole strength at the IUCF cooler ring. the betatron tunes of normal modes approach each other. This method has been commonly applied to measure the linear coupling strength. Effective Hamiltonian for a Single Linear Coupling Resonance Near an isolated coupling resonance vx . (2.42: The measured betatron normal mode tunes vs the strength of an IUCF cooler quadrupole.--i._ M | 2 . where A = yl(vx -v. A. which can be expanded in terms of two normal modes with tunes (see Exercise 2.5) V\. and the minimum separation between the normal mode tunes is |Glj_lj^|.6.
we obtain Jx + Jz = J2 = constant. where Si =vx — vz—l'vs the resonance proximity parameter. J\ = Jx. J i = Jx + Jz+ X)J\ + <t>zJ2.W J i ( J 2 . _ v > / J i ( J 2 . C. J\. (2. <t>z. H2(J2) = vzj2. (2. which corresponds to an initial horizontal betatron oscillation with Ji imax = J2. all tori can be described by a single parameter Ei that is determined from the initial condition.364 ) (2. TRANSVERSE MOTION B. 2 l cos0 x .190 CHAPTER 2. The particle trajectory that satisfies Hi = 61J2 is P2 + Q2 = 2J2. h) = ((t>x -<t>z-£0 where the new action-angle variables are </>l = <t>x . (2. ' '2y/Ji(J2-Ji) ( 2 .26 + X. Initial horizontal orbit We first consider the simple orbit with "energy" E\ = SiJ2.365) For a given J2. Since J2 is invariant.<t>z .359) into the "resonant precessing frame" by using the generating function F2{<j)x. Resonance precessing frame and Poincare surface of section We transform the Hamiltonian Eq.J i ) s i n < ^ 0! = 5 ! + G i _ 1 £ — .Ji) cos <fc. . Hamilton's equations of motion are Ji = G i . The particle motion in the resonant precessing frame is determined completely by the condition of a constant J2 and a constant Hamiltonian value Hi(Ju <j>u J2) = Ex. Hx = 6^ + G i .366) Q^ + | F 2 = 26p2.362) 02 = 4>z. J2) + H2{J2). The new Hamiltonian is H = HiiJufa.363) The horizontal and vertical betatron motions exchange their actions while the sum of actions is conserved. (2. The system is integrable with two invariants J2 and H\ = E±. (2.367) .
_i^|.367).-M /2~2Jl = 0. This means that the horizontal action can be fully converted to vertical action and vice versa. where Ji = J2. the particle inside the Courant-Snyder circle. the phase (f>i rapidly varies on the Courant-Snyder circle (see the top right plot of Fig. If there is no other noise source. If 6i = 0. The size of the ellipses depends on the initial condition. moves along the Courant-Snyder circle shown in very rapidly.-!^! 2 . 2/MJ2-. Figure 2. (2. (2. When the particle trajectory Eq.366). They are located at fa = 0 or T with T S1 ± Gx. LINEAR COUPLING where A = JS{ + IGI. (2.365).368) Figure 2.366) and (2. 191 and the normalized coordinates Q.367). (2.370) . P^-^/zhsinfa. P are Q = fehcosfa.VI. (2. As the betatron oscillation reaches follows the coupling ellipse. the coupling ellipse becomes a straight line cutting through the origin Q = 0 and P = 0.P = sfiT-i.47).369) If S\ 3> |Gi. Based on the equations of motion (2./1) (2.43: Schematic drawing of the CourantSnyder circle of Eq.365) changes rapidly on the Courant-Snyder circle. D. which is The minimum horizontal amplitude is Qmin = %fih.366) and the coupling ellipse of (2. A (2. Eq.43 shows a schematic plot of the Courant-Snyder circle and the coupling ellipse of Eqs (2. particle motion will follow the path of solid (or dashed) lines. 2.364) and (2. The phase coordinate fa of Eq. the phase fa varies Q = 0. General linear coupling solution The fixed points of the Hamiltonian are determined by the conditions j \ = 0 and fa = 0. then Qmm ~ V^h and the betatron coupling is negligible.367) with 5i = X/s/2.
where the betatron tunes are vx = 4.00628 m" 1 .364) and (2. The results are obtained from simple tracking calculations of particle motion in a synchrotron with perfect linear decoupled betatron motion everywhere except a localized skew quadrupole kick. Thus the effective resonance strength is about G\t-ifi = 0. Thus the evolution of the action coordinate at a linear coupling resonance is given by h = /7 2 -(£/A) 2 cos[A0 + <p] + J. Using Hamilton's equations [Eqs.372) J = (2Ji£ + Gl_litJ2)/2\2 with a Hamiltonian value E = 5\J\ + Gi^iti^Ji(J2 . (2. n At SFPs. The values of the betatron amplitude functions at the skew quadrupole location is fix = 10 m.44 shows 6 Poincare surfaces of section in the resonance rotating frame with a given value of J2 = Jx + Jz.368) in the resonance rotating frame obtained from numerical simulations of particle motion in a synchrotron with linear betatron motion and a localized skew quadrupole kick. Figure 2. we obtain Ji + X2Ji = A2 J.43 and 2. (2.01.44: Phase space ellipses of P vs Q given by Eq. and j3z = 10 m.366) is divided into two halves (see Figs.825 and the strength of the skew quadrupole is a\/p = 0. (2. (2. (0i = TT) -fr . the Courant-Snyder ellipse Eq. the skew quadrupole strength is a\/p = 0.192 CHAPTER 2. Figure 2. (2.00628 m"1. vz = 4. where (2. The values of betatron amplitude functions are /3X = 10 m and f}z = 10 m at the skew quadrupole location. the betatron tunes of the machine are vx = 4. Note that the structure of the phase space ellipses remains the same if J2 is varied.820.825. TRANSVERSE MOTION The stable fixed points (SFPs) of the Hamiltonian are J (2 "" °i/AA)J2' (01-0) [ \ (5 + <*i/2A) J 2 .367) rewritten as G\t-\/ \fT\ cos 4>\ — &\y/7^—~J[. With the coupling ellipse Eq. These ellipses correspond to various initial J\ and (pi values with J2 = 90TT mm-mrad.373) . 2.82 and vz = 4. the horizontal and vertical betatron motions are correlated in phase without exchange in betatron amplitudes.365)].44).Ji) cos</>i.
VI. (2.82 m. they will orbit around different fixed points at different island tunes. and the betatron tunes for this experiment were chosen to be vx = 3. we discuss an experimental study of linear coupling at the IUCF cooler ring. But." Here we find that the island tune of coupling motion around SFPs is equal to A. The cooler is a proton storage ring with electron cooling.*|2. vz = 4. The beat period were measured to be about 120 revolutions. VI. The tune of beating is equal to A = yj(vx . the bunch will resume its original shape after A"1 revolutions. The number of complete island motions in one revolution is called the "island tune. LINEAR COUPLING 193 where \E\ < XJ. The linear coupling gives rise to beating between the horizontal and vertical betatron oscillations. Note that the tune of the linear coupling motion is independent of betatron amplitude. In the following. The experiment started with a single bunch of about 5 x 108 protons with kinetic energy of 45 MeV at the Indiana University Cyclotron Facility cooler ring.96 If particles in a given bunch distribution have identical betatron tunes. However. which is independent of the betatron amplitude.45: The measured coherent betatron oscillations excited by a horizontal kicker. The SFPs of Eq. Stable islands are separated by the separatrix orbit that passes through unstablefixedpoints (UFPs).0083. and the motion will decohere after some oscillation periods. The circumference is about 86.362). . if particles have different betatron tunes. which corresponds to A ss 0.817 with ux — vz ss —1.371) correspond to the orbit with E — ±AJ = i A J ^ p . and the injected beam was electron-cooled for about 3 s before the measurement. there is no UFP for the linear coupling Hamiltonian Eq.uz)2 + |Gi. Figure 2. the horizontal and vertical betatron tunes are tuned to the linear coupling resonance line at vx — vz = t.826. and tp is an initial phase factor._i.3 Experimental Measurement of Linear Coupling To measure the effect of linear coupling. (2. linear coupling can cause bunch shape oscillations. The cycle time was 10 s. producing a full-width at half-maximum bunch length of about 9 m 96The motion about SFPs of a nonlinear Hamiltonian resembles islands in the phase space and is thus called island motion.
Figure 2.0309 MHz. The Lambertson septum magnet at the injection area also contributed a certain amount of skew quadrupole field. The linear coupling in the IUCF cooler ring arose mainly from the solenoid at the electron cooling section.45 are transformed to the Poincare map in the normalized coordinates (x. The tune separation between these two normal modes is equal to A of Eq.05 7r-mm-mrad). Note that the betatron beating between the x and z betatron motion gives rise to energy (action) exchange between the horizontal and vertical betatron oscillations. we used a kicker with rise and fall times at 100 ns and a 600 ns flat top. . the motion of the beam can be visualized as a macro-particle. The subsequent bunch transverse oscillations from a BPM are detected and recorded. TRANSVERSE MOTION Figure 2. The beat period were measured to be about 120 revolutions. The rf system used in the experiment was operating at harmonic number h = 1 with frequency 1.0083. (or 100 ns) depending on the rf voltage. For the IUCF cooler ring. which corresponds to A « 0.Vz) is out of phase with that of the horizontal map.46: The betatron oscillations of Fig.0309 MHz revolution frequency.97 The minimum tune splitting of these two normal modes is equal to the magnitude of the linear coupling constant. Figure 2. which was locally corrected. In the presence of linear coupling. the measured betatron tunes correspond to normal modes of the betatron oscillations. The vertical map (z.42 shows the normal-mode tune vs quadrupole combo. The coherent betatron oscillation of the beam was excited by a single-turn transverse dipole kicker.Vx)The amplitude modulation of betatron motion is translated into breathing motion in the Poincare map. Since the emittance of the beam in the cooler is small (0. and possibly also from quadrupole roll and vertical closed-orbit deviations in sextupoles. (2. IGi^j^.45 shows a typical example of the beating oscillations due to the linear betatron coupling following a horizontal kick.360). 97 A tune combo is a combination of power supply to a set of quadrupoles for achieving independent horizontal or vertical tune change. 2. This is sufficient for a single bunch with a bunch length less than 100 ns at 1.194 CHAPTER 2.
where the relative betatron phase advances at the locations of the horizontal and vertical BPMs were included. The solid line in the bottom right plot shows a fit by using Eq. Transforming the phase space into the resonant precessing frame. and f)x = 7.43.55 m. 2. where the particle motion follows the solid line of the Courant-Snyder invariant circle and the coupling ellipse shown in Fig.46 shows the normalized phase space x. the horizontal and vertical phase-space maps were completely smeared.z') is shown in the top right plot.364) to obtain the coupling strength Git-ite = 0. dJ\/dN (right) in [7r-mmmrad/turn].47: The top left shows the actions Jz vs Jx near a linear betatron resonance. The solid line in the bottom left plot shows a five-point running average. The Poincare surface of section in the resonant processing frame derived from (x. LINEAR COUPLING 195 Figure 2. Measurement of linear coupling phase To measure the linear coupling phase x. Vx (a similar plot can be obtained for z.0078 and the coupling phase x = 1-59 rad. II. the torus of the 2D Hamiltonian is shown in the upper right plot in Fig. (2. 2.VI. . we can transform the horizontal and the vertical Poincare maps into the resonant precessing frame discussed in Sec.45. x') and (z.J2Ji(}xsm<l>i. The orientation of the resonant line was used to determine the coupling phase x = 1-59 rad. Vz) of the data shown in Fig. P = -. Because of linear coupling.47. Figure 2. The resonance phase was fitted to obtain an upright torus. The bottom plots show the action Ji (left) in [Tr-mm-mrad] and its time derivative. where Q = yJ2J\Pxcos(j>i. 2.
2. "The difference signal or a A-signal from BPMs carries the information of betatron oscillations around the closed orbit.5 ms before the beam was coherently excited by a horizontal kicker. where the vertical betatron tune jump method is used to overcome intrinsic depolarizing resonances.48 shows the output from a spectrum analyzer using the Asignal of a horizontal beam position monitor (BPM) as the input.196 CHAPTER 2.98 The Poincare map derived from experimental data at a 2D linear coupling resonance shows invariant tori of the Hamiltonian flow (see Fig. A spectrum analyzer operating at zero span mode is a tuned receiver that measures the power of betatron motion. VI. we can obtain the magnitude and phase of the linear coupling. of skew quadrupoles. Figure 2.45 corresponds to the time interval between the dips of Fig. we can determine the magnitude and the phase of the linear betatron coupling. Such a correction method can be used for on-line diagnosis to make the choice of skew quadrupole correction families more efficient. The beat period shown in Fig. where a five-point moving average of Ji is used to obtain a better behaved time derivative of the action Ji.48. 2. 2. The data of the time derivative dJi/dN are fitted with Eq. This implies a smaller nonlinear betatron detuning for the IUCF cooler ring. The magnitude of the linear coupling obtained from the invariant tori agrees well with that obtained by the traditional method of finding the minimum separation of the betatron tunes with combos of quadrupole strengths.47 has a small curvature.4 Linear Coupling Correction with Skew Quadrupoles The linear coupling resonance is usually corrected by maximizing the beat period of the transverse betatron oscillations using a pair. The procedure for linear coupling correction is as follows 98Note that the coupling line shown in the Poincare surface section (upper left) plot of Fig.43). shown as a solid line in the lower graph of Fig.47. 2.364) to obtain Gh-he = 0. 2. 2.0006 and x — 1-59 rad. TRANSVERSE MOTION Measurement of coupling strength Git-ij The measured action J\ as a function of time and its time derivative dJi/dN = 2irJi are plotted in the bottom plots of Fig. Using these invariant tori and Hamilton's equations of motion. Knowing the dynamics of the linear coupling of a single-particle motion may also help unravel questions concerning the dynamical evolution of the bunch distribution when the betatron tunes ramp through a coupling resonance." The spectrum analyzer was tuned to a horizontal betatron sideband and was triggered 1. 2. top right plot and the solid line in Fig.47. . the increase in vertical emittance due to linear coupling may cause difficulty in later stages of polarized proton acceleration. When the betatron tunes cross each other adiabatically after the tune jump. In a single digitized measurement. Such a problem is important for polarized proton acceleration in a low to medium energy synchrotron.47.0078 ±0. or at least two families. (2.
F. Sci.. and (3) the characteristic change in features at a 17 ms interval corresponded to a strong 60 Hz ripple. which altered betatron tunes. NS20. which is evident in Fig. Nucl.6. 1990 EPAC.6.2). II can be expanded into 4x4 matrix by using transfer matrices for skew quadrupoles (Exercise 2. This reduces the coupling strength Gi.6. This procedure is however hindered by the betatron decoherence and by the 60 Hz power supply ripple. 1429 (1990).1) and solenoids (Exercise 2. the most important issue is that there is no guarantee a priori that the set of skew quadrupoles can properly correct the magnitude and phase of the linear coupling. Guignard. (2) the decay of the power spectrum corresponded to betatron decoherence.6).5 ms before a coherent horizontal kick. 100D. p. et o(. 1432 (1990). 2. The 4x4 transfer matrix in one complete revolution can be diagonalized to obtain normal-mode betatron amplitude functions. 2. Repeated iteration of the above steps can efficiently correct the linear coupling provided that the skew quadrupole families have proper phase relations. IEEE Trans. Other possible complications are closed-orbit changes due to off-center orbits in the quadrupoles and skew quadrupoles. The transfer matrix method of Sec. Figure 2. Maximize the time interval between dips (or peaks) of the spectrum by using families of skew quadrupoles. Maximize the peak to valley ratio in the spectrum by using quadrupole combos. 2. G.A.-i/. . Note that (1) the time interval between these dips corresponded to the beat period of Fig.VI.C. ibid.45. Teng. Gourber et al. Proc.48: The spectrum of the A-signal from a horizontal BPM from a spectrum analyzer tuned to a betatron sideband frequency with resolution bandwidth 30 kHz and video bandwidth 30 kHz triggered 1.48. VI. p. LINEAR COUPLING 197 1. 758 in Ref.5 Linear Coupling Using Transfer Matrix Formalism So far. Edwards and L. Willeke and G. our analysis of linear coupling has been based on single-resonance approximation in perturbation approach. [11] (1988).100 This procedure has been implemented in MAD [19] and SYNCH [20] programs (see Exercise 2. Thus measurement of the coupling phase is also important.P. J. and the coupling angle at each position in the ring. This is equivalent to setting Si = 0 for attaining 100% coupling. However. Ripken. p. 855 (1973).
where Mqua<j is the transfer matrix of a quadrupole.d »)• " < ' 2)- .—^ Bp dz (b) Show that the transfer matrix of a skew quadrupole is C+ -JqSC_ \-y/qS+ where C+= 5 + = —. where q= .6 1. z) by an angle tj> is ( i \ / a: \ / cos ( ^ 0 sin (j> 0 \ £' I p/j. z" + qx = 0. p / S+/Jq C_ C+ -JqS+ 5_/V? C+ C_ -y/qS- 5_/V9\ C_ S+/^q\ C+ I cos 6 + cosh 6 _ s i n # + sinh# = 2 ' 2 ' °-= 5 cos 6 — cosh 0 - = sin^ — sinhS 2 ' 2 ^ ' 0 = ^fqL. > > (d) In the thin-lens limit. show that the 4x4 coupling transfer matrix reduces to "-1 + T»- v . Az = 0.198 CHAPTER 2. and £ is the length of the skew quadrupole. AX = 0. i. This exercise derives the linear transfer matrix for a skew quadrupole. Bs = 0.25). z) to (x. (a) Show that the equation of motion in a skew quadrupole is x" + qz = 0. This means that a skew quadrupole is equivalent to a quadrupole rotated by 45°.~ ) 2 \ dx dz Jx=z=0 where Bo is the main dipole field strength.\ I ^ . L — 0 and qL — 1 / / . p/j-v f RW=\ 0 sin(j) C0Sl ^ 0 0 cos sin0 I • ^ 0 V 0 -sin^ 0 cos(f>J Show that the transfer matrix of a skew quadrupole is M skew quad = -R(-45°)M quad i?(45°). Bx = BoalX. where the magnetic field is with B0oi = \ (^ . where / is the focal length. (c) The coordinate rotation from {x. the skew quadrupole field satisfies Maxwell's equation dBz/dz + dBx/dx = 0. Apparently. TRANSVERSE MOTION Exercise 2.e. =W) ^ 5'/ \z'/ . The vector potential is Bl = -Boalz. As = -BQCLIXZ. and oi is the skew quadrupole coefficient in multipole expansion of Eq. (2.
y = e? y. where L is the length of the solenoid.jg'y = o. and the equation of motion becomes y" + g2y = 0. i. show that the transfer matrix for the solenoid becomes cos 2 0 isinflcosfl -sin0cos6> -isin20 \ —g sin 9 cos 9 cos2 9 gs'm92 — sin 9 cos 9 s'm9 cos 0 isin20 cos 2 0 isin^cos^ —psin2^ sin 9 cos 9 —gsin9cos9 cos2 9 ) 3. Thus both horizontal and vertical planes are focused by the solenoid. .6 199 2. the corresponding focal length is f~x = g2L = 0 2 / I . and 0 = gL is the rotating angle of the solenoid. where the solenoidal field strength is g = —2p • (a) Show that the coupled equation of motion becomes y" .353) in the presence of skew quadrupoles and solenoids are ( I01Note here that the solenoid. (b) Transforming coordinates into rotating frame with 9 = 1 gds. x" + Kx(s)x + 2gz'-(q-g')z = 0. Jo show that the system is decoupled. The focusing function is equal to g2. and j is the complex imaginary number.e.jtgy' . Linear transfer Matrix of a Solenoid: The particle equation of motion in an ideal solenoidal field is x" + 2gz' + g'z . z" . in the rotating frame. (c) Show that the transfer matrix in the rotating frame is y =ye~^"\ where ( cosS jsin0 0 0 \ -gsmO 0 0 cos0 0 0 0 cos6> -gsin9 0 | isintfl' cosfl / where 9 = gs.101 (d) Transforming the coordinate system back to the original frame. In small rotating angle approximation.g'x = 0. z" + Kz(s)z + 2gx'-(q + g')x = 0. where g = B\\(s)/2Bp and q = -{dBz/dz)/Bp = ax/p.EXERCISE 2. Show that Hamilton's equations of motion for the Hamiltonian (2. where y = x + jz.2gx' .0. acts as a quadrupole in both planes.
TRANSVERSE MOTION (a) Show that the perturbation potential due to skew quadrupoles and solenoids is V]c = -^Xz P + \P g{s)(^x-^z).Y. and Ei = <5i/i + G\t-\tiy/I\{l2 — Ii) cosrj)i is a constant of motion.359). (a) Show that the new conjugate phase-space variables are h = Jx. show that the linear coupling equation of motion for the Hamiltonian (2.<f>zJi. . Liu et al. 4.102 5.h) = (<£* -4>z-id + X)h + 4>zh.i for the ^-th harmonic is given by Eq. see J. Rev. Px =-V^sin((j)x+Xx). A = JS\ + G\ _x e. \Z = v / 2 ^ c o s ( ^ z + Xz). 102 For a general discussion on linear coupling with nonlinear detuning. Using the generating function F2(<t>x.362) in resonance rotating frame. The Hamiltonian H = uxjx + vzjz + Git-ite\/JxJzcos((t>x ~4>z + x) for a single linear coupling resonance can be transformed to the normalized phasespace coordinates by (X = %/2Jx~cos(<l>x + Xx). Pz = -^/2Tzsm{j>z + Xz). fa = <t>x ~ 4>z ~ 10 + x. 8\ = vx — vz — t is the resonance proximity parameter. (d) Discuss the solution in the resonance rotating frame. (a) Show that the Hamiltonian in the new phase-space coordinates is H = \vx(X2 + Pi) + \vz{Z2 + P2Z) + \GX^{XZ + PXPZ). 2347 (1994). h = Jx + Jz.357). Phys.200 CHAPTER 2.359) can be transformed into the Hamiltonian (2. <h = 4>z- (b) Find the invariants of the Hamiltonian (2. E 49. where the overdot corresponds to the derivative with respect to orbiting angle 6.. (c) If the accelerator lattice has P superperiods. where Xx ~ Xz = X 1S a constant linear coupling phase (Mod 2TT) that depends on the location in the ring. show that G\ _i i = 0 unless ( = 0 (Mod P). P ) (b) Expand the perturbation potential in Fourier series and show that the coupling coefficient Gi:-i. (c) Show that the equation of motion for I\ is /i + A271 = S 1 5 1 + 7 2 G ? j _ M / 2 . (2.
sin $ z I ' ]j | sin^sin^zl j ' f ~ I V where / is the focal length of the skew quadrupole. we note that the "horizontal" and "vertical" betatron oscillations carry both normalmode frequencies. 2. the particle motion in a synchrotron with linear betatron motion and a localized skew quadrupole kick. Show that the condition of linear stability for betatron motion is VM~* < M i n ( 2 /(I + " » * » ) ( ! + « » $ . i.EXERCISE 2. can you find the stability limit of a linearly coupled machine with superperiod P? . 6.6 (b) Show that the eigen-frequency of the Hamiltonian is v± = \{vx + vt)±^\ A = y/(ux . /3X and /3Z are values of betatron amplitude functions at the skew quadrupole location.) 1 I sin $3.44. ) 2 /(l-C06& a )(l-C0Bg. Based on your study of this problem. Analyze the linear stability of the simple tracking model shown in Fig. T^]f\A+ cos(l/+f + *+) + A~ ""("-V + ?-)- where ^4±.e.v2f + |G x . 201 (c) Solve X and Z in terms of the normal modes. and §x and $ 2 are betatron phase advances of the machine without the skew quadrupole. and show that { X = A+ cos(v+ip + £+) ~ J^T£AZ = COS(I/_<^ + £_).^± are obtained from the initial conditions. Particularly._ w p.
The changes of phase space . the betatron Hamiltonian [x12 + Kxx2 + z12 + Kzz2\ + V3(x. the nonlinear magnetic field can give rise to geometric aberration in the beam ellipse if a resonance condition is encountered. z. they are an integral part of accelerator lattice design.z. Including the sextupole field. Since the sextupole magnets used in accelerator are usually short. More generally. (2.202 CHAPTER 2. V shows that chromaticity correction in particle accelerators is essential for attaining particle beam stability.376) The evolution of phase space coordinates of orbiting particles can be obtained by tracking the equation of motion. s). modern high energy storage rings usually use high field (superconducting) magnets that inherently possess systematic and random multipole fields. TRANSVERSE MOTION VII Nonlinear Resonances Our discussion in Sec. For off-momentum particles. 0 (2.374) . Let S = J S(s)ds be the integrated sextupole strength. and thus the nonlinear geometric aberration due to sextupoles and higher order multipoles needs to be addressed. the sextupole strength S(s) should be replaced by B2/KI + S)Bp].375) where Vz(x.s) = ^S(s)(x3 — 3xz2) is the nonlinear perturbation potential with S(s) = —B2/BP. z" + Kz(s)z = +S{s)xz. This section provides an introduction to this important subject. Hill's equation of motion becomes x" + Kx{s)x = -^S{s)(x2 .3xz2). Careful analysis of the nonlinear beam dynamics is instrumental in determining the dynamical aperture. (2. Tracking methods In the presence of sextupole magnetic field. where B2 = d2Bz/dx2\x_z_0 is H=lAs = § ( x 3 . 1 Nonlinear Resonances Driven by Sextupoles The vector potential for a sextupole magnet is Ax = Az = 0. but is limited to first-order perturbation treatment. Since sextupole and higher order multipole magnets are needed in chromaticity correction. thin lens approximation has often be used in particle tracking.z2). where 5 — Ap/po- A. VII. Although the nonlinearity is normally of the order of 10~3 — 10~4 relative to that of the linear component.
s) becomes Vs = -^Ji / 2 J z /3y 2 ^5( S )[2cos$ I + cos($:c + 2$ 2 )+cos($ I -2$ z )] +^Jll2Pl'2S{s)[cos 3$x + 3 cos $x].49: The Poincare maps for the betatron motion perturbed by a single sextupole magnet at a tune below (left) and above (right) a third order resonance. The topology of the phase space maps rotates 120° when the tune moves across the third integer resonance. B. With Eq. z.378) 103In this mapping equation for betatron motion.49 shows the Poincare maps with one sextupole in an otherwise perfectly linear accelerator near a third order resonance. Figure 2. NONLINEAR RESONANCES coordinates at the sextupole magnet are103 Ax' = -\§(x2 .94) for coordinate transformation. (2.377) The propagation of phase space coordinates outside the sextupole magnet is given by the mapping equation (2. we disregard the effect of sextupoles on orbit length. we perform Floquet transformation to the Hamiltonian (2. The integrated sextupole strength is S = 0. (2. The region of stability decreases as the tune approaches the third order resonance. Az' = Sxz. we find AC = (xAx' + zAz').45).z2).170). Figure 2.375). .VII. The leading order resonances driven by sextupoles In order to analyze the third order resonance analytically.5 m~2. 203 (2. (2. the nonlinear perturbing potential V3(x. Using Eq.
Jl12 cos 3$x Pl/2 Jjj/z driving terms Classification sum resonance difference resonance parametric resonance parametric resonance C. .i drive vxJr2vz — £ and ux—2uz = £ resonances. <j)x) and (J 2 .^i. Table Resonance vx + 2i/z = £ ux -2vz=£ ux = £ 3vx = £ 2.£ are Fourier amplitudes. Gi.^ I1/ V. ^i.379) 1 where £ is an integer. Since V3 is a periodic function of s.i.i) + £ GwHwJl'2J. . xx= -5-.i and G\. G3. 1 lists nonlinear resonances that can be excited by sextupoles in firstorder perturbation theory._2. + J2 Gi. and • • • describes the remaining resonance driving terms at vx = integers. Third order resonance at 3^x = £ Near a third-order resonance at 2>yx = £.vxe.379) can be approximated by H w vxjx + G3fiil fj2 cos (30X -£9 + 0.-2.. the Hamiltonian (2.104 The Hamiltonian (2. (2.£ are the phase of the Fourier components.o.-2. . Jo pz Here (Jx. G\2.iJl'2 cos(30x . G\^.0. The Fourier amplitude Gz$.o. We discuss below a ID third-order resonance at 2>vx = £. Guignard. 008(0. can also be excited by strong sextupole fields through second..2$ z ) Pl/2p~z Jx/2Jz cos$ x Pl/2PZ] P'^2 Jx^2Jz.«.id + &.or higher-order perturbation expansion.0 2 ) are pairs of conjugate phase-space coordinates.£) e cos(0s + 2cj>z -M + 6.2 .-2.i drives the third order resonance at 3 ^ = £. Table VII.3: Resonances due to sextupoles and their Driving term Lattice Amplitude cos($ I + 2$ z ) Pll2Pz Jx/2Jz cos($j.£. p.20.«. etc.. [10] (1988)._w) + • • •. .375) expressed in action-angle variables becomes H = vxJx + VzJz + J2 G3.2. (1976). Other higher-order resonances. §z = <l>z + Xz{s)-Vz6. .o. Guignard. CERN 76-06. ^3.204 where CHAPTER 2. such as 4vx = £. and similarly. it can be expanded in Fourier harmonics. 822 in Ref.£6 + £3. G. G. .2. (2-380) 104 G. TRANSVERSE MOTION $x = <l>x + xx{s) . . 2vx±2vz = £. Jo p x f ds Xz= -5-. .
381) The betatron phase space of the Hamiltonian (2.M-(3*-Mds. Since the chromatic sextupoles are usually arranged according to the superperiod of the machine. (f>x are conjugate phase-space coordinates.384) 0 = 5 +\G3.eWe note that if the accelerator has a superperiod P and the sextupole field satisfies a similar periodic condition. 9 is the orbiting angle serving the time coordinate.382) the Hamiltonian Eq. Using the generating function F2 = (4>x-i-e+^)J.o.380) becomes H = 5J + G3. the resonance strength Gz$.24.VII. one should pay great attention to the systematic sextupolar nonlinear resonances.e.o. (2. (2.1). the sextupole can cause "geometric aberration" to the betatron motion. Since the Hamiltonian (2. where the new phase-space coordinates are (2.380) is distorted by the nonlinear resonance. and the Fourier amplitude G3.o. and their resonance strengths are usually weak. the systematic third-order resonance strength for the AGS will be zero except for £ = 12.i is zero unless £ is an integer multiple of P (see Exercise 2.383) Here S — vx — £/3 is the resonance proximity parameter. For example. Particle motion in the phase space follows the contour of a constant Hamiltonian.< are G3i0/ & = ^ f Pi'2 S(s) e*Px. the contribution of each superperiod is coherently additive to the resonance strength.7. Using Hamilton's equations of motion cos 30. At a systematic resonance.« and the phase f = £3io. etc. vx is the horizontal betatron tune.385) . NONLINEAR RESONANCES 205 where Jx. the "Hamiltonian" is invariant. Thus the nonlinear resonances are classified into systematic and random resonances. (2. J = 3G3. The magnitude of the geometric aberration is proportional to the resonance strength G3. the betatron tunes should avoid low-order nonlinear resonances. (2.383) is autonomous. Random sextupole fields induce nonlinear resonances at all integer £.eJ3/2 cos 30.o. Systematic nonlinear resonances are located at £ = Px integer. (2. Nevertheless. i.Q/J112 we obtain 3 + 9623 YG10/32 = 96E.£J3/2sin30.
388) These unstable fixed points can be easily verified as follows. The solid line shows aJ^/Gzja^i vs aS/G^0<l for the case a = axx > 0 and 03. When the amplitude is small. the UFP. the tune. TT. Thus a particle can stay indefinitely at a fixed point.50: The dashed line shows the third-order betatron resonance in the zero detuning limit. the Hamiltonian assumes the value *--!(«£. Figure 2. The fixed points are characterized as stable or elliptical fixed points (SFPs) and unstable or hyperbolic fixed points (UFPs). . Small-amplitude motion around a SFP is a bounded ellipse. Bifurcation of the third-order resonance occurs at a8/G\Ql = 9/16 marked by the rectangular symbol.206 CHAPTER 2.)'• K . (2. The equation of motion becomes 105Fixed points of a Hamiltonian are phase-space loci with zero velocity field.6-^—K2 = 0.383) has three unstable fixed points (UFPs)105 with de These UFPs are located at ji/2__2S_ UFP " 3G3. <2-» (2. ±2^/3. (2"386) At the fixed point. where FP stands for both stable and unstable fixed points. The SFP section and the UFP section are marked. of Stable and unstable fixed points The Hamiltonian Eq. J^l/G^i vs 6/GJM.o. if S/GW > 0. of the third-order resonance is 35 because the betatron amplitude repeats three times in every revolution.352K . U F P = 0. around an UFP it is hyperbolic. Let K = J — JUFP and E = EUFP. in the resonance rotating frame. and the overdot indicates the derivative with respect to the orbiting angle 9.0/ > 0. I 0FP = dbr/3.* with ' de~ if«5/Gw<0. TRANSVERSE MOTION where E is the Hamiltonian value.
the third-order resonance appears at all values of S.391) for the case of 5/Gz.1] [P . 2. With X and P defined as X = J-—cos(/>. the betatron tunes depend on the betatron actions. Three straight lines X = 1/2. the separatrix can be obtained from Eq.389) Separatrix The separatrix is the Hamiltonian torus that passes through the UFP. Note that the stable phase-space area is proportional to the resonance proximity parameter <52 (see Eq. the stable phase-space area in (x. The stable motion is bounded by the curve of J^P(S) shown in Fig. Without a nonlinear detuning term. and P = -^{X + 1) divide the phase space into stable region and unstable regions. beam particles can be slowly squeezed out of the stable area and extracted to achieve high duty cycle for nuclear and high energy physics experiments. V "MJFP P = -J-—sin<t>. Near a thirdorder resonance.o. i\. (i^). P)VFP is given by the intersection of these three lines: (X. Including the effect of nonlinear .390) the equation for the separatrix orbit becomes [2X .383) with the condition H = EVFP. NONLINEAR RESONANCES 207 Thus the motion near the fixed point is hyperbolic. (2. For the thirdorder resonance in the zero detuning limit. The third-order resonance can be applied to extract beam particles slowly from a synchrotron.386)). i.P)UFP = (-l. For a given aperture J. (2. nonlinear magnetic multipoles also generate nonlinear betatron detuning. Thus the separatrix is given by three intersecting straight lines.50. V "AjFP (2.~).VII. the width of the third-order betatron resonance is then Hwidth = 3G3.±(X + 1)] [p + -^(X + 1)] = 0 (2. x') is equal to %/3<J/2|G3io.e.^|. Because of the nonlinear term in Eq. the amplitude is seen to grow faster than an exponential. P = ±(X + 1). and (X.0).388).£J1/2/2(2. Beam loss may occur when particles wander beyond the separatrix.o.i > 0. If the betatron tune vx ramps slowly through a third-order resonance. The dynamical aperture is defined as the maximum phase-space area for stable betatron motion. Effect of nonlinear detuning In fact. (2.
Measurements of Poincare maps near a third-order resonance have been successful at SPEAR. (2. 7942 (1992).106 E. ±TT/3 6 < 9G^/16a 0 < 5 < 9G§p(W/16or ' (2.e < 0.= I +3/4 + (3/4) y/l . stable fixed points appear.393) The bifurcation of third-order resonance islands occurs at I6a5 < §G\ 0 1 . we can model sextupole strengths of the storage ring.51 shows a Poincare map obtained from a nonlinear beam dynamics experiment at the IUCF cooler ring.0.D. Figure 2. [ + 3 / 4 .378). Ellison et al.« J 3 / 2 cos 3<£. experimental measurements are generally difficult. D.±27r/3 A 5<0 J ^ . 572 (1996) for the vx +2vz= t resonance at the IUCF cooler ring.392) to obtain parameters G^o^ and £.(I6a5/9G2M).50 shows aJ^F2p/\G3fi/\ vs a5/G\fil for the bifurcation of third-order resonance. . 107 See 106 D.208 CHAPTER 2. Aladdin.±ir/3 cj> = TT.(3/4) y/1 . Converting into action-angle variables. and the IUCF cooler ring. Phys. Figure 2. et at. sextupoles contribute importantly to the nonlinear coupling resonances at vx ± 2vz = t with integer Lim The third-order resonance strength can generally be obtained by taking the Fourier transform of Eq.2vz = I induce betatron coupling. I i < = 0.2vz = I resonance. Budnick et al. we can fit these data by the Hamiltonian (2. Methods A368.3 / 4 + (3/4) 0 : ^ ( 1 6 ^ 7 9 0 1 ^ ) . Inst. Nucl. TRANSVERSE MOTION betatron detuning. Rev.(16aaS/9Gjji(W). It is easy to observe degradation of beam intensity and lifetime near a resonance. TEVATRON. and obtain the parameter S by measuring the betatron tune at a small betatron amplitude.392) where a = axx is the nonlinear detuning parameter. Rev. Caussyn. 4> = w. M. Using these measured nonlinear resonance parameters. Experimental measurements of nonlinear resonances are usually difficult because of short lifetime at the resonance condition. Experimental measurement of a Zvx = £ resonance Because beam particles may be unstable at a nonlinear resonance. and J. 4051 (1994) for the vx . the Hamiltonian for the third-order resonance is H = SJ+ -aj2 + G3.c f .t > 0 are aJi/2 ^-^3. Phys.0. The difference resonance at vx . A 46. The fixed points of the Hamiltonian for a > 0 and G3fi.. A similar analysis can be carried out for a < 0 or Gsfi. while the sum resonance can cause beam emittance blow-up in both horizontal and vertical planes and leads to beam loss. With nonlinear detuning. (2. Other 3rd-order resonances driven by sextupoles Besides the third-order integer resonance. E50.
Tori for particles inside the separatrix are distorted by the third order resonance. The solid lines are Hamiltonian tori of Eq. etc. Here we give an example of the fourth-order parametric resonance at 4vx = 15. Y. 33 (1992). Avz. 292. Phys. 1838 (1992). (2. et al.108 Similarly. Nonlinear beam dynamics is beyond the scope of this book. Proc. Lett. 2i/x ± 1vz. nonlinear beam dynamics studies at the IUCF cooler ring show the importance of nonlinear resonances. Rev. and dynamical aperture. Wang. 2752 (1988). 6 1 . <j>). 67. EPAC p. T. No. 170 (1992).7496. Lett. However.109 108 A. Figure 2.Y. E 4 9 . et al. 5697 (1994). The concatenation of strong sextupoles can generate high-order resonances such as \vx. 68. rotates at a rate of betatron phase advance along the ring.378) that sextupoles will not produce resonances higher than the third order ones listed in Table VII.52 shows the Poincare maps of the single sextupole model of Fig. Phys.392). T.px) of betatron motion near a third-order resonance 3wx = 11 at the IUCF cooler ring... Satogata. 5vx. Phys. The right plot shows the Poincare map in action-angle variables (J. et al. 2.1. M. Lett.. nonlinear beam dynamics experiments at Fermilab Tevatron were used to study the concept of smear. NONLINEAR RESONANCES 209 Figure 2. nonlinear detuning. Rev. N. 791 (1988). strong sextupoles are usually needed to correct chromatic aberration. Rev. Employing strong sextupoles. 3768 (1991). Lee. et al.VII. AIP Conf. 68.. decoherence. Phys. . Chen et al.49 at vx = 3. Rev. (2. Rev. VII. Proc. Phys. determined by sextupoles. The orientation of the Poincare map. Note that a single sextupole can also drive the fourth and higher order resonances.2 Higher-Order Resonances It appears from Eq. Chao. et al. Merminga... p. Lett.51: Left: The measured Poincare map of the normalized phase-space coordinates (x.. Note that particles outside the separatrix survive only about 100 turns. 109 S. Ellison et al.
The Poincare map near a fourth-order resonance Avx = 15 measured at the IUCF cooler ring is shown in Fig. Systematic experimental measurements of nonlinear resonances can be used to derive the resonance parameters. In order to overcome nonlinear resonances. In this example. Accelerator operators are keen to avoid low order strong resonances because of visibly short beam lifetime. 2.53. the fourth-order resonance islands are enclosed by stable invariant tori. (2.49 at vx = 3. 2. a few nonlinear magnets (usually up to octupole) are powered to eliminate the Fourier components of the nonlinear resonance near the machine operation condition. The solid lines shows the Hamiltonian tori of Eq. Note that when the betatron tune is exactly 15/4.53./ can be obtained from the Fourier transformation of the effective particle Hamiltonian in the synchrotron. the betatron motion will be located at fixed points of the fourth-order resonance island.394). The tune of motion around SFP of an island is called island tune.210 CHAPTER 2. Ps) phase space and the right plot shows the Poincare map in action-angle variables. Near a fourth-order ID resonance. The topic is beyond this introductory text.52: The Poincare maps for the same sextupole model shown in Fig. (2-394) where the resonance strength G^o. The chromatic sextupoles located in the ring can also be powered to eliminate the un-wanted nonlinear resonance Fourier components. Accelerator physicists are eager to apply their skill to correct or compensate the resonances for minimizing their effects on the beams. 2. and thus the Hamiltonian for particle motion in the accelerator can be modeled. .7496 show the effect of the fourth order resonance. Small deviations from the fixed points will execute motion around the stable fixed points (SFP) shown in Fig. the Hamiltonian can be approximated by H = vx3x + ±axxJ2x + GWJ2X cos{4i>x -M + X). where the left plot shows the Poincare map in the normalized (x. TRANSVERSE MOTION Figure 2.
vx) J . The solid lines are the Hamiltonian tori of Eq. SiS&j PX:j PzJ — - j.| ^ | ) ] sin -KVX J 64?r ~ .| ^ . n l ax* ~ ^\IJS^P^P*JP^J[ i V<? < B^B^R 7 R l"cos[2(7rt/z .[ ) ' '' [ sin ZTTVX cos(7ri/g .53: Left: The measured betatron Poincare map (surface of section) (x.VII. Px) of normalized phase space near a fourth-order resonance 4ux = 15 at the IUCF cooler ring. NONLINEAR RESONANCES 211 Figure 2. the sextupoles will not. concatenation of sextupole perturbation to the betatron motion can induce substantial nonlinear betatron detuning. These coefficients are „ _ 1 V o o q3/2o3/2 [cos3(7ri/J .axz. VII.ijD]! sin7r(2j/2 .<j> = ipx). Note that the phase-space ellipse is distorted into four island when the betatron tune sits exactly on resonance.394). Qz = vz + axzJx + azzjz. 3 Nonlinear Detuning from Sextupoles Because the potential resulting from sextupole fields is an odd function of the betatron coordinates.\1>x#\] S i n 7 r(2^ +^) COS[2(7TI^ . (2. Because sextupole strengths are large in high energy collider and storage rings. . in linear approximation.396) The detuning coefficients axx.2 1 .395) (2. and azz can be obtained from the phase average of the concatenated one-turn Hamiltonian in a storage ring. The dependence of the betatron tunes on the betatron amplitude can be approximated by Qx = vx + axxJx + axzjz. contribute to the betatron amplitude detuning.\ipZtjj\) .(KVX ~ IV'x. The right plot shows the Poincare map in action-angle variables (J = Jx.\$tM\) + nux . (2.
Similarly.i ~ Ad are betatron phase advances from Sj to s. where mxvx +mzuz = I with \mx\ < 1 and \mz\ < 1 and £ is an integer.212 n a" CHAPTER 2. the betatron tunes should avoid linear coupling resonances at vx ± vz = £ due to skew quadrupoles and solenoids. III. The solid lines corresponds to resonance lines associated with normal multipoles.i . 4>Z. Unfortunately.54: Left: the linear resonance lines. TRANSVERSE MOTION _ ! V ? ? /91/2/91/2/9 ft [cos[2(7r^ . The left plot of Fig.}ijjXiij\] ~ 64^ jj SiS'P" P'* P'>iP*<* [ sin7r(2i/. and the dashed lines are those associated with skew multipoles. Since the tune depends on the zeroth harmonic of a perturbed quadrupole field.. V. n are integers. higher order multipoles can drive higher order . where \mx\ + \mz\ < 4.Ipxj.e mxvx + mzvz = £.\tpXij - ip^D'i sinir(2i/z-vx) sin in>x J' where il>X. and half-integer integer betatron resonances at 2vx = m o r 2vz — n due to the linear imperfections discussed in Sec.ij = il>X. + ^ ) | COS[2(TT^ - \jiz.54 shows linear betatron resonances for the fractional parts of betatron tunes. We have shown that sextupoles and higher multipoles are important to beam stability in Sec. Figure 2. i. Right: the resonance lines up to the fourth order coupling resonances.1 ^ 1 ) + nvx . The symbol qx and qz are the fractional parts of betatron tunes vx and vz. 2. the nonlinear detuning parameter is proportional to the superperiod of the accelerator. VII.jj\) - {TTVX - \ipx. These coefficients can be evaluated from sextupole strengths distributed in one superperiod.4 Betatron Tunes and Nonlinear Resonances The betatron tunes should avoid the linear betatron resonances at vx = m or vz = n.jj\)] ^COS(TTVX . where m.ij = tl>2.
55 shows betatron resonances up to the 8th order. where P is the superperiodicity of the machine. where the solid lines correspond to resonances due to normal multipoles.110 resonance (stopband) correction becomes important for attaining beam stability.7 213 resonances discussed in this section. (2. CERN 84-15 (1985). m Figure 2. Betatron tune stability has becomes an important issue for successful operation of storage rings. where higher order betatron resonances can decrease the beam lifetime. betatron amplitude detuning. beam-beam interaction. L. in the Proceedings of the CERN Accelerator School on Antiprotons For Colliding Beam Facility. 2. Resonancefree tune space becomes very small in storage rings. 110The betatron tune spread of a beam may arise from the incoherent space charge (Laslett) tune shift. chromaticity. The solid lines corresponds to resonance lines associated with normal multipoles.o/ = 0 if (• ^ 0 (Mod P). with \mx\ + \mz\ < 8 and integer £. Koop and G. 319. .3 x 10~3 per crossing (see Eq. for systematic resonances. p. Note that the available resonance free tune space becomes small. and the dashed lines are those associated with skew multipoles. Exercise 2. When the betatron tune spread of the beam becomes large. the beam-beam interaction can cause higher order_ resonance observed very early on at the SPPS. Evans. Lifetime degradation has been observed near the 7th order resonance at the SPPS driven by beam-beam interaction with linear beam-beam tune shift parameter of ^ = 3. The lifetime of beams in many storage rings and colliders may suffer if the betatron tunes sit near a higher order betatron resonance. (4.R. while the dashed lines arise from skew multipoles. Show that the 3vx = I resonance strength is given by Eq.7 1. (a) Show that.381) in the first-order perturbation approximation. Figure 2. 4 ) .EXERCISE 2.55: Resonance lines up to the 8th order. where mxvx + mzvz = (. edited by I.g. The right plot of Fig.54 shows the betatron resonances up to the 4th order. see also the Proceedings of the third ICFA Beam Dynamics Workshop on Beam-Beam Effects in Circular Colliders.10) in Chap. Tumaikin (Novosibirsk. are shown in this figure. 1989). and many e+e~ colliders. G3. In particular. etc. 111 See e. The tune space that is free from high order resonances becomes very small.
the "geometric aberrations" of these two sextupoles cancel each other if ip2l — T and \px(Sl)f2 [S{sx)As] = [&(s 2 )] 3 / 2 [S{s2)As]. where £ is an integer. vz are the betatron tunes. g = Gi]+2. and the stable fixed points are located at for cf>i = 0 or 7 respectively. the Hamiltonian can be approximated by H = vxjx + uzJz + gJll2Jz cos{(j>x + 24>z .<h)> and show that the new Hamiltonian is H = H\ + H2. Show also that 2Jx + Jz is invariant. Near a third-order coupling difference resonance at vx — 2vz = £.J2. <j>z.ufp = "^» and <j>i = ± arccos ( * ). show that the unstable fixed points of the Hamiltonian are located at •A. AO = (s2 . where ux. Vzh. where Hi{Juh) = H2{J2. show that the geometric aberration of two chromatic sextupoles located in the arc of FODO cells separated by 180° in phase advance cancel each other.*2 ds/px is the betatron phase advance.vz are betatron tunes.-(3^-0A»i ) where ip2i = X.<t>i. and (Jxi 4>x-> Jz) 4>z) are the horizontal and vertical action-angle phase-space coordinates. the Hamiltonian can be approximated by H = vxjx + vzJz + gJll2Jz cos(0x . and 81 = vx — 2vz — £ is the resonance proximity parameter. 2. and (Jx.). (b) For a given J2.SI)/RQ. <j>x. at the 3ux = £ resonance. Discuss the difference between the sum and the difference resonances. Near a sum resonance at ux + 2vz = £. . Ju J 2 ) = ( ^ .. TRANSVERSE MOTION (b) Show that the resonance strength of the third-order resonance at 3i/x = £ due to two sextupoles at si and s2 is proportional to Wx(si)f2 [S(Si)As] + [J3x(s2)f2 [S{s2)As\ e >P*. and Ro is the average radius of the accelerator.(j>2) = <5iJi+9J 1 1 / 2 (J2-2Ji)cos«^i.2<6Z -19 + £ ) J i + <j>zJ2.f is the effective resonance strength. (c) Based on the above result. (a) Using the generating function F2{4>x.2(j>z -£9 + 0 . 4>z) are horizontal and vertical action-angle phase-space coordinates. where vx. transform the phase-space coordinates from {JxAxiJz-i^z) to (Ji. g = G\t-2.£0 + (. Show that. T 3.214 CHAPTER 2.i is the eifective resonance strength. Jz.
e. (c) Show that the action J2 is invariant. h) = W > * + n<j>z -10 + OJi + 4>zJ*. In general. (b) Show that the new Hamiltonian is invariant. Normally only low-order resonances are important. If the betatron tune of the machine is chosen such that mvx + nvz ~ I. are integers.<f>l. i. betatron motion in storage rings can encounter many nonlinear resonances.7 215 4.J2.J\. and (. n. 4>z. (a) Transform the phase-space coordinates from {Jx. This is called a sum resonance if mn > 0. Discuss the difference between the sum and the difference resonances. Neglecting the perturbing term AH that includes contributions from other resonances.EXERCISE 2.<t>x.<l>z) to (Jl. derive the invariants of the approximated Hamiltonian. njx — mJz = constant. and find the new Hamiltonian. the Hamiltonian can be approximated by H = HO(JX. . where m > 0. and a difference resonance if ran < 0. Jz) + ff4m|/24n|/2 cos(m^ + n<f>z -W + O + Atf.Jz.(fa) by using the generating function F2{<t>x.
TRANSVERSE MOTION VIII Collective Instabilities and Landau Damping So far we have discussed only single-particle motion in synchrotrons. In Sec. Since particle motion in an accelerator is classified as transverse betatron motion or longitudinal synchrotron motion.216 CHAPTER 2. which is the electromagnetic waves induced by a passing charged particle beam. vacuum ports. impart a force on the motion of each individual particle. where each particle can be described by a simple harmonic oscillator. VII. in turn. 3. Here. and narrow-band impedance due to high-Q resonance modes in rf cavities. backward. we list these impedances as follows. some properties of impedance are listed. and BPMs. septum and kicker tanks. the impedance of an accelerator is related to the voltage drop with respect to the motion of the charged particle beams. Likewise. or slow waves. the impedances are classified as longitudinal or transverse. see Ref. Sec. In Section VIII. For beams.4. VIII. The longitudinal impedance has the dimension ohm. broad-band impedance due to bellows. A self-consistent distribution function may be obtained by solving the Poisson-Vlasov equation. there are transverse and longitudinal impedances. VIII.1. where the impedance plays an important role in determining the circulating current. . Without deriving them. single-particle motion is governed by the external focusing force and the wakefield generated by the beams. The transverse impedance is related to the transverse force on betatron motion. [3]. where the waves are classified as fast. Landau damping and dispersion relation will be discussed in Sec. The induced electromagnetic field can. Thus. etc. In reality.2 we discuss transverse wave modes. and the beam distribution is determined by the motion of each particle. and has the dimension ohm/meter. space charge. In Sec. VIII. and by definition is equal to the energy loss per revolution in a unit beam current. 1 Impedance The impedance that a charged-particle beam experiences inside a vacuum chamber resembles the impedance in a transmission wire. a circulating charged particle beam resembles an electric circuit. Likewise.3. image charge on the vacuum chamber. VIII. we will show that a slow wave can become unstable in a simple impedance model. The effect of longitudinal impedances will be discussed in Chap. wakefields are classified as having transverse or longitudinal modes. The impedance is more generally defined as the Fourier transform of the wakefield. For a complete treatment of the subject. we discuss some basic aspects of transverse collective beam instabilities and Landau damping. The transverse impedance arises from accelerator components such as the resistive wall of vacuum chamber.
where c is the speed of light. the impedance per unit length becomes X'mag _ JcABz _ . Here.oIoXo/2-Kb2. . we find the induced dipole field inside the beam cross section to be Ai3Z]b = ij. the transverse impedance is related to the longitudinal impedance by 7 _ 2cZ||irw 02 LJ . Using the result of Exercise 1. fic is the permittivity in metal. (2. and 6(x) is the Dirac 5-function. (2. Resistive wall impedance AND LANDAU DAMPING 217 The resistive wall impedance is given by112 T>i7 2x.16) Jw = -(IOXQCOS<f)K/nb2)6(r . is the skin depth of the electromagnetic wave in metal. b is the vacuum chamber radius.6). The resulting beam current density is i(r. The total induced vertical dipole field due to the beam displacement becomes By definition. Let XQ be an infinitesimal displacement from the center of the cylindrical vacuum chamber. The perturbing current is a circular current sheet with cosine-theta current distribution.VIII.397) where R is the mean radius of the accelerator.9. *) = A 9 (a irar . ZQ = fioc = 377 ohm is the vacuum impedance. For resistive wall impedance. Similarly the induced image current is (see Exercise 1.399) where Q(x) = 1 if x > 0 and 0 otherwise. B. . ac is the conductivity. and the induced dipole field due to the image wall current is ABZ:VI — —/j. and the second term arises from an infinitesimal horizontal beam displacement. Space-charge impedance Let o be the radius of a uniformly distributed beam in a circular cylinder. the first term corresponds to the unperturbed beam current. (5Skjn = y 2/<7c//ca. COLLECTIVE INSTABILITIES A.a).rwH = (l+j)£^4kin. and u is the wave frequency. Here.olQXo/2Tra2.r) + I^^5(r 7ror . the resistive wall impedance consists of a resistive and an inductive component.Zo / J L _ M ~J ploxo " J 2 7 r W b V ' 112The imaginary number j = —i of engineering convention is used throughout this textbook.
bb = T T — 75—. the perturbation current of Eq. [3].399) is invalid. . when the oscillatory amplitude XQ is large. . (2. etc. The remaining space charge impedance is the image current term.. (2. The formula that takes into account the shape of the vacuum chamber can be found in Ref. Rs is the shunt resistance. R is the average radius of the accelerator. b2 UJ b2U> 1 + jQ(LJ/U)r TT» .—-. and 6 is the beam pipe radius. Note that the space charge is capacitive because the beam radius a is less than the vacuum chamber radius b.oc is the vacuum impedance. Similarly. Narrow-band impedance Narrow-band impedances are usually represented by a sum of Eq. TRANSVERSE MOTION where j3c is the velocity of the beam. b is the beam-pipe radius. • „. bellows. where the corresponding Q-factor is usually large.218 CHAPTER 2. BPMs. the space-charge impedance is important for low energy beams. and the self space-charge force term may disappear. and Zo = fj. which is inductive. can be lumped into a term called the broad-band impedance. vacuum ports. Broad-band impedance All vacuum chamber gaps and breaks. the impedance due to the electric field is 7 • Zo (l M Thus the transverse impedance due to space charge in one complete revolution is113 * ~ ~ ^ (?-?)• (2-400) where a is the beam transverse radius. septum and kicker tanks. 113Note here that the derivation of the space charge impedance assumes a uniform circular beam distribution for the direct term and a circular vacuum chamber geometry for the induced charge term.. C. D. Because of the f32j2 factor in the denominator.401) where Q « 1 is the quality factor. which is usually assumed to take the form of a RLC circuit: 2c%bb 2c Rs OAm\ ^±. However. R is the average radius of the accelerator.(*>r/U) (2. Narrow-band impedances may arise from parasitic rf cavity modes.401). and 7 is the Lorentz relativistic factor. The space-charge impedance can be considered as a broad-band impedance because it is independent of wave frequency. etc. OJT « (R/b)u>0 is the cut-off frequency of the vacuum chamber.
f°° Z^Y^du 2?r J-OO (2.56. The factor /? in the denominator is included by convention. Properties of the transverse impedance When the beam centroid is displaced from the closed orbit. The dipole current will set up a wakefield that acts on the beam. V . du' 1 W ' . For example.e.407) or ReZj_(-w) = -ReZ ± (w). and the imaginary part is an even function. COLLECTIVE INSTABILITIES AND LANDAU DAMPING 219 E. Thus. the impedance of RLC resonator circuit in Eq. (2. where P.404) (2.VIII. it may have poles in the upper half plane. Since the wake function is real. This occurs because the driving force is leading the dipole current by a phase of ir/2. means taking the principal value integral. the real and imaginary parts are related by a Hilbert transform ReZ±^) = --f 7T J P . the motion can be expressed as a dipole current.U> ^^!. Thus the real part of the transverse impedance is negative at negative frequency.405) lmZ±(uj) = -[ do/ReZl(u/). To summarize. 2.V. the impedance at a negative frequency is related to that at a positive frequency by ZL(-W) = ~Zl(u). where the real part of the impedance is an odd function of ui.403) with the causality condition W±(t) = 0 (t < 0). the impedance can not have singularities in the lower half of the complex u plane. ImZL{-uj) = +ImZ L (w). (y) is the centroid of the beam in the betatron motion. The imaginary number j included in the definition of the impedance is needed to conform a real loss for a real positive resistance. The analytic properties of impedances provide us with the Kramer-Kronig relation. The transverse impedance of the ring is defined by Z±M = Tffiv) f F^ds = ife /(^ + ^ x S)±ds = T i W±{T) e^T dT' (2'402) where I\ = Io(y) is the dipole current. and C is the circumference of the accelerator.406) (2. (2.401) has two poles located at u> =uv [±v/l-(l/2Q) 2 +j(l/2Q)] . i. (2. . various components of the transverse impedance ZJ_(CJ) are schematically shown in Fig. however. The wake function is then related to the impedance by WL{t) = -3J.
410) where Q is the betatron tune.411) There are three possible transverse wave modes: the fast wave. (2.w=(l + . A broad-band and a narrow-band impedance are represented by peaks in the real parts. ifn>0: fast wave backward wave slow wave. the betatron oscillation of the transverse motion is oo V(t. the nth mode of transverse motion is y(t. The corresponding angular phase velocity is (1 H a =)—{-. U4Another IV M y .T—r I w0.9o)= n=—oo E ynej{Q"ot-ne°\ (2. TRANSVERSE MOTION Figure 2. i. The resistive wall impedance is important in the low-frequency region. Its Fourier harmonic contains all modes with equal amplitude (see Sec. the transverse coordinate at any instant of time is given by 114 y(tO. and n is the mode number. VIII.409) Since (2. 9) = ynej[ (»+«-»'-»»]. The real paxts are shown as solid lines and the imaginary parts as dashed lines.W (2. and the slow wave.408) where 0 is the orbiting angle.56: A schematic drawing of the transverse broad-band and narrow-band impedances. yn = constant.7). At a fixed azimuth angle 9Q. (2 412) if n < 0 and In < Q : if n < 0 and In > Q : extreme case is a beam with a delta-function pulse. 1 I wo.220 CHAPTER 2.2 Transverse Wave Modes oo For a coasting (DC) beam.e.0)= n=—oo E Vne'jn9. and W is the angular revolution frequency. and the spacecharge impedance is independent of frequency. III. where particles continuously fill the accelerator. I 1 . o 9 — 80 + uot. jwo. where the angular phase velocity is 0n. the backward wave.
(2. (2. 7 is the Lorentz relativistic factor. that of a slow wave is slower than the particle velocity. and F± is the time-dependent transverse force resulting from the wakefield.nu)0)2yk. (2-414) where R is the mean radius of the accelerator.3 Effect of Wakefield on Transverse Wave Let yk be the horizontal or vertical transverse betatron displacement of thefcthparticle. The equation of motion in the presence of a wakefield can be expressed as a force oscillator equation115 Vk + (QkUofyk = ^ p (2. If beam particles encounter collective instability of mode n. We will show later that the characteristic responses of these waves to wakefields are different.VIII. they execute collective motion with a coherent frequency w: yk = Yke^t~ne\ (2. and a backward wave travels in the backward direction.413) where the overdot corresponds to the derivative with respect to time t. the betatron motion is well approximated by simple harmonic motion.417) (2. VIII. we obtain 115In a global sense. Using the relation Vk = ^+O^=j(uj-nLJo)yk. which is a sideband at rotation harmonic nu)Q. we obtain yk =-(u . COLLECTIVE INSTABILITIES AND LANDAU DAMPING 221 The phase velocity of a fast wave is higher than the particle velocity. The EM force on a charged particle [see Eq.415) where n is the mode number.413). m is the mass. (2. The signal picked up in a transverse beam position monitor (BPM) will have a single frequency located at \n + Q\uj0. and let Qkuo be the corresponding angular frequency of betatron motion. and Yk is the amplitude of collective motion for the kth particle.416) Substituting into Eq.402)] resulting from a broad-band impedance is *l(t) = -J6-§^ (V). .
sextupoles. One can also use a positive frequency approach to express the slow wave. nw (2. Now we discuss coherent frequency spread vs off-momentum variable 5. the mode number n0 of Eq. the betatron amplitude can serve as a £ parameter. (2. from Eq. .422) is a slow wave. Here. Since betatron tunes depend on betatron amplitudes due to space-charge force. then the resulting dispersion relation changes sign. and AQ = CyS. Q is betatron tune. where ui0 is the angular revolution frequency of the beam. O + [Cy ~ mj\ W0 S. we obtain the collective wave frequency as Wn.415). TRANSVERSE MOTION ^-^^iJ^M' {y) = j PiOvkdti (2'418) is the centroid of the beam. (2.422) Thus if Cy/rj < 0. It is worth noting that. r] is the phase-slip factor. Using Awo/wo = —T)6. Since Q and u>0 depend on the off-momentum parameter 6 = Ap/p.419) and we have used the relation W — TW + Qwo « Wn. |n| < Q backward wave. |n| > Q slow wave.116 Averaging over the beam distribution. the wave frequency spread vanishes at mode number n0 = (^L. (2. discussed as follows. fast wave. we obtain a dispersion relation for the collective frequency u> epiZL f PJQ dc (2420) The set of parameters f represents any variables that wn>w and the beam distribution function depend on.222 or where CHAPTER 2. u)n)W is the wave frequency given by ( n> 0 n < 0. The beam may become unstable against transverse collective instability. and Cy is chromaticity.421) where S = Ap/p0 is the fractional off-momentum coordinate. 116When the transverse coordinate of collective motion is represented by Eq.421).w — "Wo + Qwo — 2QwoJ o Note that the real part of the slow wave frequency wniW is negative. n < 0. (2.w = W . the slow wave frequency is negative. (2. 6 can also be chosen as a possible £ parameter. we have assigned the fractional off-momentum coordinate as a £ parameter. p(f) is the normalized beam distribution function with f p(£)d£ ~ 1J ? represents a set of parameters that describe the dependence of the betatron tune on its amplitude. and other higher-order magnetic multipoles.
56). the imaginary part of the impedance gives rise to a frequency shift.424) where V is related to growth rate. The remarkable thing is that there are solutions of real u> even when — U + jV is complex.o of a slow wave is negative. In general. and there is no growth of collective instability. If the distribution function . The threshold of collective instability can be obtained by finding the solution with ui = ui — j\0+\. If the imaginary part of the coherent frequency is negative.VIII. and the resistive part generates an imaginary coherent frequency ui. Since the imaginary part of the collective frequency is negative. (2. coherent oscillation is damped.U.423) Thus. (2. wniWo is positive. the dispersion relation provides a relation between U(LO) VS V(LJ). On the other hand. Defining U and V parameters as V + jU=A e%IZLn . Beam with zero frequency spread For a beam with zero frequency spread. for a given growth rate. Beam with finite frequency spread With parameters U and V. B. On the other hand. Im [wcon] = V. the collective frequency uin>v. n • (2. if the imaginary part of each eigenmodes is positive. we obtain Re [wCou] = wn. p(£) = 5(£ — £o). For fast and backward waves. i. the betatron amplitude grows exponentially with time. the growth rate can be solved from the dispersion integral with known impedance and distribution function.w0 + j 6 223 ' . thus the imaginary part of the coherent frequency is positive. If the imaginary part of the coherent frequency is negative.w0 . the dispersion relation for coherent dipole mode frequency w becomes (-U + jVV = j P{0 dg. we obtain w = wn. where the real part of the transverse impedance is negative. 2. Similarly. The real part of the impedance ZJ_(OJ) is positive (see Fig. a beam with zero frequency spread can suffer slow wave collective instability. and U is related to collective frequency shift.e. and the beam encounters collective instability. the amplitude of the coherent motion grows with time.425) The solution of the dispersion relation corresponds to a coherent eigenmode of collective motion. COLLECTIVE INSTABILITIES AND LANDAU DAMPING A. where 0 + is an infinitesimal positive number.
(2. (2. In general. TRANSVERSE MOTION is symmetric in betatron frequency. We consider the frequency spread model w». A model of collective motion We consider a macro-particle model of a beam with (Y) = YlPkYk. (2.Thus any amount of a negative real part of the impedance can produce a negative imaginary collective frequency and lead to collective instability. the collective mode disappears.w(*)n = wB|WOn + Mi{Yk .w(fc)] Yk = WYt i PiYi.427) which is identical to the solution of Eq. The requirement of a large frequency spread for Landau damping is a necessary condition but not a sufficient one. i. Eq. (2.wo is independent of k. In nuclear physics. where Afl is a constant that determines the frequency spread of the beam.425). C.224 CHAPTER 2. The corresponding eigenvector for collective mode is Yjt)COii — PkYh. the coupling between external force and beam particles is completely absorbed by the collective mode.418) becomes [w . For any beam distribution p(£) that does not have infinite tails. the collective frequency is trivially given by117 wcoii = wn. In fact. The disappearance of the collective mode due to tune spread is called Landau damping. no frequency spread. If wn>w(fc) = wn.426). This means that if the coherent frequency shift is beyond the distribution tails. if there is a frequency spread between different particles.w0 + W. (2.423). the giant dipole resonance where protons oscillate coherently against neutrons presents a similar physical picture. and there is no coherent motion.In matrix form. This is equivalent to solve the collective mode frequency from the dispersion relation of Eq. In this model. the threshold curve contains two straight vertical lines lying on the U axis. the frequency shift of a particle is proportional to the local density of the U7 The collective mode occurs frequently in almost all many-body systems.e. if the frequency spread Aw nw among beam particles is larger than the coherent frequency shift parameter W. Now. unstable against collective instability. but may not necessarily be. where Pk is the distribution function with Y*Pk = 1. All other incoherent solutions have random phase with eigenvalue wn>w0.wn. and the growth rate is proportional to the real part of the impedance. we have to diagonalize the matrix of Eq. the U vs V contour plot will have reflection symmetry with respect to the V axis.426) where W = —U + jV for a broad-band impedance.( y » . . (2. the beam can be.
the space-charge tune shift alone can not provide Landau damping against transverse collective instability. 2. CJ@ = 0. This is the essence of Landau damping. COLLECTIVE INSTABILITIES AND LANDAU DAMPING 225 beam. we see that a slow wave can suffer transverse collective instability for a beam with zero frequency spread. A. The resulting collective mode frequency is (2. Up P (2. The particle motion will be out of phase with the external force sooner or later. We are interested in the response of the particle under external force. The lower plot of Fig.VIII. Landau damping The equation of collective motion (2.76 (line) under the action of an external force w = 0. 2. The solution is v(t) — H—5 o \s\aut Up sinugt I +y0cosuist+ P — sinujgt. and ojp = Qw0 is the betatron frequency.428) Wcoiia = wn. A smaller w — wg results in a longer in-phase time.57 shows y{t) for three particles with u$ = 0.429) where w is the collective frequency. What is the effect of frequency spread on collective instabilities? The key is the Landau damping mechanism discussed below. and the energy will be transferred back to the external force.413) can be represented by a forced oscillator: y + oj20y = Fsinujt. F = 0. as shown in the lower plot of Fig. Note particularly that if up differs substantially from the driving frequency w. and the coherent motion is relative to the closed orbit of the machine. which describe unperturbed particle motion.430) LJJ-LJ2 \ J where y0.75.85 (dash-dots). This is equivalent to the argument that the space-charge tune shift can not damp the transverse collective instability.423).57. This model of tune spread resembles space-charge tune shift.8 (dashes).4 Frequency Spread and Landau Damping From Eq. Since the second and third terms. (2. y0 are initial values at t = 0. (2. . the number of particles (oscillators) that remain in phase with the external force becomes smaller and smaller. and u)g = 0. Since the space-charge tune shift is a tune shift relative to the center of a bunch. the external force can not deliver energy to the system forever.w0 + W. As time goes on. This means that the frequency spread that is proportional to the distribution function can not damp the collective motion. VIII. The examples illustrate the essential physics of Landau damping. are of little interest here. we set yo = 0 and y0 = 0.01.
85 ( d a s h dots) . Thus the average power is given by Eq. 0 • . I • • . 118In general. .431) where £ — |(w . • I • • • • I . fewer and fewer oscillators will be affected by the external force. TRANSVERSE MOTION i r l ^sin2(f)/f3 / \ _ j i .57 shows the coherent functions (sin2 () / (2.e. . the coherent frequency window decreases. and if u is in 106 rad/s.76 p i n e ) 0. = 0 5 I I I ! I I I I I ' I I / \ r\ ! ! I ^ I I ^ 10 I |_ I ! I : CJ = 0 . PT> (2.80 ( d a s h e s ) A A A : A~ ~ : OJ = 0 . 20 • I . 7 5 0^=0.432) or equivalently. i. 40 . (2. where H is the Hamiltonian of the system. 7 5 CJ ( S =0. |w-w/8|~l/T. i . This means that the external force can not coherently act on a particle if (up — u)T becomes large. t is in s. \ . (2. When the external force can not pump power into a beam with a finite frequency spread.\ / i . . 60 80 • • • : 100 t Figure 2. Here the units of w and t are related: if u is in rad/s. we are interested in the average power that the external force exerts on the particle:118 < ( ) = i f yFrin^= [ i ^ y ^ ] r + . collective instability essentially disappears.^ /-^ o8 -10 1 . then t is in /is. where V is the potential energy.0. 7 5 co0=O. i V /. .05.e.As time T increases. The lower plot shows the response of three particles vs time t to an external sinusoidal driving force F(t) = Fsinut. Here we retain only the leading term that is proportional to time T. .Note that the function becomes smaller as the £ variable increases./ . 0 ^ A -5 I I I I = . and 0. .01. the system is Landau damped.1 shown respectively as solid line. The frequency difference of these three particles is Au = 0. The upper plot of Fig.5 — CJ = 0 . 2.wp)T. the total energy the external force delivered to the system after time T is AH = JQ yF(t)dt. \ / . In fact. o F • . and dots. dashes.226 CHAPTER 2.431). when an external force F(t) is applied to a Hamiltonian system with y+dV/dy = F(t). . 1 / .57: The upper plot shows the coherent function (sin2 C)/C2. i.i . . Note that the particle with a large frequency difference will fall out of coherence with the external force.
8. 2. v= .434) (2. where the rectangular symbols in each curve represent coherent frequency shifts at Re(ui — wn)Wo) = ±CTU (inner ones) and ±2au (outer ones)./"°° —dt. 119The complex error function is w(z) = e-*2erfc(-jz) = . we observe that if the frequency of the coherent mode is within the width of the spectrum. and o& is the rms momentum width of the beam. Using Eq. . Thus the collective beam motion is damped. or phase space decoherence due to tune spread (see Exercise 2. then induction of collective beam instability requires a finite resistive impedance. Landau damping differs in essence from phase-space damping due to beam cooling. the dispersion relation of Eq. B.435) is the rms frequency spread of the beam for mode n. The curves tivsn for Gaussian distribution for Im(w/crw) = 0 and —0.425) becomes -u + jv = j[w(^^-)]~\ where <ra = y/2 \Cy . which will absorb energy from the external force and lead to collective instability.431) comes entirely from the second term in parentheses in Eq. the term y(t) that is in phase with the force is actually a reactive term.421). COLLECTIVE INSTABILITIES AND LANDAU DAMPING 227 It is worth noting that the power dissipation to the oscillator in Eq. The term that will absorb power from the external force is the second term in parentheses in Eq.5 are shown in Fig. 2.58. That is to say. (2.119 and u= .430). From Fig. which does not dissipate energy. (2.5).VIII. (2. and most of the beam particles are off resonance. (2.nrj\ ujoas (2. (2. Solutions of dispersion integral with Gaussian distribution We consider a beam with Gaussian distribution given by ^ ^ <2-433) where S = Ap/po is the fractional off-momentum coordinate. This is because the coherent mode excites only a small fraction of the particles in the beam. This corresponds to a resistive term.430). w(z) is the complex error function.58.
y = —s T sinwi u} ~ w2 \ w .Wo) = icr^ and ±2frw.426). \ smwat . TRANSVERSE MOTION Figure 2. F = 0. (2.58: The normalized u vs v for Gaussian distribution is plotted with two different growth rates with Im(u. The rectan- gular symbols represents the coherent frequency shifts at Re(w — wn. and backward waves travel faster. 120For bunched beams. Because such modes have vanishing frequency spread.8 1. . (2. then mode no with vanishing frequency spread is a fast wave.423) for the dispersion relation at zero frequency spread is identical to the collective frequency solution by matrix diagonalization of Eq. (2. p. and backward relative to particle angular velocity. slow.434). Since the real part of a fast wave is positive. Treatment of head-tail instability is beyond the scope of this book. and thus there is no collective instability.) = 0 and —0.5CTW. and 0.01 with three particles at up = 0. The solution of Eq. 2. [14]). "IS ) (a) Plot y(t) as a function of t for w = 1. 0.228 CHAPTER 2.Coii = PkYk3. (2. respectively. and show that the eigenvector of collective motion is Yjfc. We note particularly that the frequency spread can vanish for mode number n0 of Eq.422). (2.8.120 Exercise 2.433) to show that the dispersion relation becomes an algebraic equation. Using the Gaussian distribution of Eq. (2. the imaginary part of collective frequency is positive. 134 in Ref.429) with initial condition j/o = Vo = 0 is F ( . However.9.99. if chromaticity Cy is negative below transition energy or if Cy is positive above transition energy. Verify the wave angular velocity of Eq. 4. Eq. This has been commonly employed to overcome transverse collective instabilities. the head-tail instability has been observed in SPS and Fermilab Main Ring above transition energy if the chromaticity is negative (see J. and show that fast. Gareyte. Show that the solution given by Eq. (2. slower.411). collective instabilities will not be Landau damped.
A beam is usually composed of particles with different frequencies. particles are not damped to the center of phase space. otherwise.e.2 U / ( « W ) _ 2e -«/(« 2 +« 2 ) cos[t)/(u2 + v20]' ' Show that the condition that Imw = 0~ is U2 + (V + . 121This exercise illustrates the difference between Landau damping and beam decoherence (or filamentation). Note that coherent beam motion will decohere within a time range At ~ 1/(7. See also Fig. 2.58. v = V/2aw. i. 229 Show that the first term in square brackets does not absorb energy from the external force but the second term can. where a is the rms frequency spread. where |e| is a small frequency deviation.8 (b) Let u)p = iij + e. Show that F ri-cos(et) . 5. and compare your result with that shown in Fig. otherwise. i. The first term corresponds to a reactive coupling and the second term is related to a resistive coupling. and 0. As the coherent motion is damped. . Consider a beam with uniform momentum distribution ( J ) fl/(2A) \0 if|5|<A. (b) Show that the imaginary part of the coherent frequency is T ..121 Let p(up) be the frequency distribution of the beam with J p(u)p)dwfj = 1. find the centroid of beam motion as a function of time with the following frequency distribution functions. and begin coherent betatron motion. we choose £ = e and p(e) = 1/2A. and aw = \Cy — nr)\uit)A. (y(t)} = fy(t)p(£)d£. 2. where a is the rms frequency spread of the beam. 1 ) 2 2TT = JL. (a) Show that the dispersion relation Eq. For example. .e.15 for coherent betatron oscillation induced by an rf dipole kicker. (c) If a beam has a distribution function given by p(£) with J p(£)d£ = 1.. . if |e| < A.EXERCISE 2. 4TT2 Plot u vs v. y{t) = jlcosu^t. If initially all particles are located at y = y = 0. { n w e-"sinfr/(«2+^)1 1 + e . where u = U/2aw. discuss the centroid of beam motion.425) becomes (—u+jv) = In • . show that (y) = AeT" * I2 cos uiot. (a) If the frequency distribution of the beam is p(W/J) = -J=^ c -<"*-o) i /*' 2 . 6. and at time t = 0 all particles are kicked to an amplitude A. . (2. y(t) as — ^-^-sinurt sin(ei) 1 —-cosurf .
ifcjo-r<w0<uJo otherwise. TRANSVERSE (b) If the frequency distribution is a one-sided exponential P(UR) p/ = MOTION / (l/ff)c-to-"»>/*. otherwise. PK 0> •.230 CHAPTER 2. show that . where F = \f%o and cr is the rms width. where Jo a n d Ji are Bessel functions. + T. otherwise.r < w / 8 < w o + r. (0.sinFt (e) If the frequency distribution is parabolic with piijff) = {( 3 /( 4r )) I1 ~«w/>-^)/r)2]. . ifw O -r<w / . where F = 2<r and a is the rms width.w o )a + r2]' where F is the width.wo)/rf. otherwise.<« o + r.((up . show that (y) = 2 2 [cos wi . = / i / ( 2 r ) . . show that {y) = A [Jo (Ft) + J2(Fi)] coswoi. where <x is the rms frequency spread. show that . (d) If the frequency distribution is uniform with n(u. i f w O . where F = y/ia and cr is the rms width. and (cup) = u)g + a. . /sinrt cosFA <2/> = M W .T r 7 F J C O S W 0 < (f) If the frequency distribution is quadratic with p(Up) = I (2/TTF) y/l . . \ 0. \0 if W / J >wo.o-tsinwt]. [ 0. (c) If the frequency distribution is a Lorentzian p(w'} = »[( W / J . show that (y) = Ae~vt cos wi.
/V I!\\ / : \\ 1 1 .—•—i—i—i—•—i—i—•—•—'—. . and A = 1.—i—.EXERCISE 2. 1 1 1 r- A o n e —side e x p o n e n t i a l : Gaussian : d o t s uniform: d a s h e s line -i.—. 1 . .1.—i—. Compare your result with the Gaussian.—iO 1O 2O 3O 4O cot . a = 0. 1. and uniform distribution results shown in the plot below.—.o L—J—. 1 . . 1 1 .—. .0 r i — j -i -| . one-sided exponential. 1 .8 231 (g) Plot (y(t)) vs t for the parabolic and Lorentzian distribution functions with wo = 1.
Thus the rf accelerating field can be represented by ^.px.sk) sin(wrf* + fok).437) where 6p(s — s^) = Yin <HS ~ s * ~ 2nnR) is a periodic delta function with period 2nR.pz. The terminology of synchrotron motion is derived from the synchronization of particle motion with rf electric field. —H) or (t.px) phase-space coordinates. The name "synchrotron" has been broadly used for all circular accelerators that employ rf electric field for beam acceleration. This unified description has the advantage of treating synchrotron motion and betatron motion on an equal footing. (2. Lett. we disregard vertical betatron coordinates (z. Vk is the rf voltage. 5133 (1998) for the effects of space charge dominated beams. —E). (2. As is the longitudinal vector potential.rf = — E W w rf k s " sk) cos(wrft + <j>Qk). z. Suzuki. The static transverse magnetic field is 1 dAs x 1 8AS 1 + (x/p) dx' 1 + [x/p) Dz ' and the longitudinal varying electric field can be obtained from r)A Es = —gf = Yl W 5 . Rev. $ is the sealer potential.122 To simplify algebra. Here we will study the "synchrotron" equation of motion for phase-space coordinates (t.Y.438) 122 See T. wrf is the angular frequency of the rf field. we have discussed particle motion only in (x.232 CHAPTER 2. (2. p = ^J(E/c)2 — (me)2 is the momentum of a particle. and <j>ok is the initial phase of the /cth cavity.px. TRANSVERSE MOTION IX Synchro-Betatron Hamiltonian So far.) U „ x. Part. 115 (1985). the Hamiltonian is [see Eq. —E) have not been mentioned.—E) are canonical phase-space coordinates.pz) and consider only a planar synchrotron. \/E-e$\2 . 18. p is the bending radius of the Frenet-Serret coordinate system. see also S. Lee and H. Okamoto. . Neglecting vertical betatron phase-space coordinates.-<4H) +(1+ .z. Accel. and (x. The remaining phase-space coordinates (t. It is particularly useful in the study of synchrobetatron coupling resonances.18)] Ho = .>(^H-' <» "> o __ z J -m2c2-p2-p2 22 2 211/2 -eAs where the orbital length s is used as an independent variable. Phys. 80.t.( 1 + .
we obtain „ AE 1 . P (2. (2.* . and the dispersion function D satisfies (^. 2 AE x (2 4 4 1 ) Ho = .ft_|i)> D" + KXD = -. . Using the generating function F2{x. = Box + | V + iBi(x 2 . — + „ . Expanding the dipole field 5 Z in power series with Bz = Bo + B\X + • • •.438) stands for the vector potential of rf cavities. _ ( _ ) -Po^+PI+PI + P±{KXX2 + Kzz2) _ eAsT{ zp0 z up to second order in phase-space coordinates. (2. Substituting the sealer and vector potentials into the Hamiltonian.440) where AStIi given by Eq.* . SYNCHRO-BETATRON HAMILTONIAN 233 The Hamiltonian is an implicit function of energy E. The next step is to transform the coordinate system onto the closed orbit for a particle with off-energy AE. where Si = dBz/dx. This procedure cancels the cross-term proportional to (AE/P2E0) • x in the Hamiltonian.D*Hr f=t+ AE = E-E0.442) .z2) + • • • + As. Let E = EQ + AE and p = p0 4.Ap. which signifies the expansion of x around the closed orbit at the reference energy.£>— r )p s . AE . that can be represented by a sealer and vector potentials $ = Vsc and ASiSC with J4SJSC = P2Vsc/pc. The space charge force of the beam particles gives rise to a mean field. where Eg and po a r e the energy and momentum of the reference particle.ri + AIJK. and we used the identity condition BQ = —Po/ep. We obtain then PO ~ ^ 2 s 0 27* > V • P2E{i ~ po + 2 7 2i Po) • ^-«».-AE) = (x .(E + AE)t P -bo where the new phase-space coordinates are PX=PX-D'^ -x = x. we obtain ^ . where Kx = 1/p2 — B\/Bp is the focusing function for the horizontal plane.IX.pX:t.
px.448) Making a scale change to canonical phase-space coordinates with {x. Using the corresponding function hs F2 = xfx + (wrff .s*) cos(wrf£ + (j>ok) = -^—j^ cos(wrft .W) -+ (x. (2. i. W = Ulrf (2.447) . <Sp(s-s*)cos(wrf«-|-0o*) = 7 .—)W. (2. 4>.—).{ i .hs <5P(s .444) Now we expand the standing wave of the rf field into a traveling wave.234 we obtain a new Hamiltonian.$. we obtain the Hamiltonian x = x. H (2.p E [e J ( n e + W r f ' + ^-^) + e j ( n e . TRANSVERSE MOTION 1\(AE\2 px2 p- 2 . Wf V r L V P &o Pc I J (2-443) Note that x is the betatron phase-space coordinate around the off-momentum closed orbit. AE 1 (D CHAPTER 2.445) ^ti n-_oo Keeping only terms that synchronize the beam arrival time with n = ±h.x' = Po —. where Px=Px.^ + Tc")-eA^ eASid = — ]T eVk Sp(s .sk) cos L r f ( i .-^zrPx + -^-x) + ^ok \.W r f '-^.446) 1 °° - where <j>ok + h9k should be an integer multiple of 2?r.^ .d S . </> = w r f i .n f l t )]. and the rf vector potential is (2..— + 4>0k + h0k). Po . we obtain 1 .e.
it is called synchro-betatron resonance (SBR). Jta where E and B are electromagnetic fields. can generally be expressed as a function of 6D phase-space coordinates. sj are the entrance and exit azimuthal coordinates of the kicker. Proc. then the energy gain depends on the transverse coordinates. The synchrotron phase-space coordinates are chosen naturally to be (R(f>/h. 249.A. Lambertson. x') are coupled through dispersion function (D. If a resonance condition is encountered. No. Consider a particle of charge e and velocity v = ds/dt experiencing a kick from a component in an accelerator. 537 (1992). . and Vj_ is the transverse gradient. D') in rf cavities. E-ds. AIP Con}. SYNCHRO-BETATRON we obtain the Hamiltonian HAMILTONIAN 235 -Ssk? rti ~(*-5 I * + jH Since < and (a. R(j>/h is the longitudinal phase-space coordinate of the particle. which relates the transverse kicks to the longitudinal energy gain.L. the SBC potential must satisfy the Panofsky-Wenzel theorem. The total transverse momentum change is rh Ap± = e / (E + v x B)i_dt.IX. syn> / chrotron and betatron motions are coupled. p. (2. The total energy change will be AE = e rsb -. which satisfies the Panofsky-Wenzel theorem Eq. This synchro-betatron coupling potential. This is called synchro-betatron coupling (SBC). Thus if the transverse kick depends on the longitudinal coordinates. and 4 — ta is the transit time of the kicker component.449). where sa. Then the Panofsky-Wenzel theorem yields a relation between the transverse kick and the energy gain123 AAfA^vJ-^) Rd4>[po ) H W ' (2 449) ( ' where Apx/p 0 is the transverse kick. Goldberg and G. In general. and the Hamiltonian in 6D phase-space coordinates becomes 123D. — Ap/po)..
4 lists some properties of electron storage rings. Neglecting synchro-betatron coupling.452) = ^ i .W-9k).^ ) s i n ^ ] ' lh\r)\eV Vs"i2*P2EQ (2-454) where rj is the phase slip factor. TRANSVERSE MOTION - | ^ E ^ - ^ + ^ ) . the phase factor -Dx' + D'x will be the same for all cavities (see Exercise 2. (2. Thus it is beneficial to put rf cavities in dispersion free regions. the Hamiltonian for canonical phase-space variables (x.T ) ( — ) 2 " £ lS^[ C 0 S ^+(^. (2. z'.453) where rf cavities are assumed to be at dispersion free locations.450) It is worth noting that if RF cavities are located in a straight section. and is the synchrotron tune of the stationary bucket with <f>s = 0. The action of the synchrotron oscillations and the linearized betatron oscillations can be defined on an equal footing as 7» = A / — W' Ix = ^-fx'dx.1). .*') sin 4>. R<j>/h.) = . The driving terms for the synchro-betatron coupling in Eq. -Ap/p0) is H = H±(x.z') with HL H° + Hs(^^) h p0 (2.^I { — ? ~ 0 l o 2 EJQ I c o s ^ ~ cos<f>*+ {<!>.9.455) Table 2. Z7T/1 J Po Z7T 7 Iz = ^-fz'dz. where the transverse emittances and the longitudinal phase-space area are determined by the equilibrium between the quantum fluctuation of photon emission and the phase-space damping due to beam acceleration to compensate energy loss in the longitudinal direction.236 CHAPTER 2. x'. the Hamiltonian for synchrotron motion becomes (H. Z7T J (2. Averaging over one revolution around the ring.451) = ^{x12 + Kxx2 + z12 + Kzz2) + • • • (2.450) coherently add up in all cavities arranged in one straight section.<t>s) sin<f>s\ p po Zixhp 1 / \ 27 j/2 = -2 7 7 (~) 2 -/^l[ C 0 S < ? ! > ~ C 0 S ^ + W . (2.x\z. z.
51 0. Show that rf cavities located in a straight section contribute coherently to SBC if the dispersion function is not zero.6 -11.7 ^ 34 48 ass [m-1] -3.0 eV-s.28 76.8 98 127 267 699 -6.96 0.3 -25. and the primes are derivative with respect to the longitudinal coordinate s. x is the horizontal betatron function.0 6.35 60 a[xHT4] 400 152 C [m] 240.064 (AE/Eo) [xlO" 4 ] 4.6 3. Table 2.5 6.18 24.2 -0. the corresponding longitudinal action is 100-1000 times as large as the transverse action. —Ap/po). it satisfies the Panofsky-Wenzel theorem.28 25.8 9.1 9 55 7 32.9 237 The synchrotron action Is (Tr-mm-mrad) is related to the commonly used phase area A (eV-s) of the phase-space coordinates {<j>.65 0.4 574 -0.18 70.2x10^ M - (2-456) Since the typical longitudinal phase-space area is about 0.8 0.3 A [xlO" 4 eV-s] 3.7 3.61 Exercise 2.0522 0.5 7.36 ex [nm] 450 240 ez [nm] 35 8 p [m] 10.4 9.5 -10.7 78.6 165.3 196.8 328 499. .2 14.z'.0082 7.38 vz 6.4: Parameters BEPC CESR E [GeV] 2.86 1.1.48 4. AE/huo) by is = ^ A = 3. LER HER LEP APS 3.3 96 48 51 8 3.9 1.18 4. Show that if the SBC potential is an analytic function of 6D phase space coordinates (x.5 of some electron storage rings.8 2.016 0.3 26658.085 0.x'.08 30.z.22 35.4 h 160 1281 / rf [MHz] 199.27 ALS 1.96 14.5 14. This result has important implications for the synchro-betatron coupling resonances.7 5.0498 0.5 499. where D is the dispersion function.4 3.1 . 4.EXERCISE 2.2 38.0061 9.43 1.9 1104 3492 3492 31320 1296 476 476 352.1 5. 2.2 6 ux 5.3 2199.866 2.8 9.9 24.1 4.8 vs 0.2 35.4 768.1 8. Show that the function -Dx' + D'x in the Hamiltonian H4 is invariant in a straight section.0 3096.2 352.R<j>/h.01 14.374 2199.93 0.1 0.0 6.28 8.2 Is [103 nm] 7.
.
particles gain energy from the electric field in the longitudinal direction. 2. Alternatively. what happens to a particle with a slightly different momentum when the synchronous particle is accelerated? 1This statement also applies to beam acceleration in the betatron and the induction linac. Particles with different betatron amplitudes execute betatron motion around this ideal closed orbit. for simplicity.1 Since the electric field strength of an electrostatic accelerator is limited by field breakdown and by the length of the acceleration column. Since the energy gain depends sensitively on the synchronization of rf field and particle arrival time. D5. A synchronous particle will gain or lose energy. In this chapter we study particle dynamics in the presence of rf accelerating voltage waves. A particle with momentum p has its own off-momentum closed orbit. A particle synchronized with rf phase <f> = <j>a at revolution period To and momentum po is called a synchronous particle. 239 . where D is the dispersion function and 5 = (p — po)/po is the fractional momentum deviation. we will derive the synchrotron Hamiltonian based only on the revolution frequency and energy gain relations. electrostatic accelerators have been used mainly for low energy acceleration. <ps is a phase factor. A beam bunch consists of particles with slightly different momenta. eVsin(f>s. Sec. This formalism lacks the essential connection between synchrotron and betatron motions. here. a radio-frequency (rf) cavity operating in a resonance condition can be used to provide accelerating voltage with V sm(4>s + uijft).Chapter 3 Synchrotron Motion In general. IX). in which the induced electromotive force is given by the time derivative of the magnetic flux. per passage through the rf cavity. and wrf is the angular frequency synchronized with the arrival time of beam particles. but it simplifies the choice of synchrotron phase-space coordinates. Normally the magnetic field is ideally arranged in such a way that the synchronous particle moves on a closed orbit that passes through the center of all magnets. Although we can derive a 6D Hamiltonian for both synchrotron and betatron oscillations (see Chap. where V is the amplitude of the rf voltage.
Particle acceleration without phase stability is limited to low energy accelerators. I. In Sec. bunched beams can be shortened. In Sec. etc. bunch rotation. we study beam injection. In the case of df/d5 < 0. Section II deals with adiabatic synchrotron motion. cf>d = hcjot. etc. etc. VIII. h is an integer called the harmonic number.240 CHAPTER 3. we derive the synchrotron equation of motion in various phase-space coordinates.e. df/dS > 0. Section VI treats fundamental aspects of rf cavity design. called "synchrotrons. higher energy particles will receive less energy gain from the rf gap. Cockcroft-Walton. If the revolution frequency / is higher for a higher momentum particle. it remains the cornerstone of modern accelerators. i." and after half a century of research and development. we introduce collective longitudinal instabilities. where the Hamiltonian is not invariant. extraction.g/2l3c) (n = integer). Therefore if the rf wave synchronous phase is chosen such that 0 < <> < TT/2. (3. Ill. or stacked to achieve many advanced applications by using rf manipulation schemes. beam manipulations with double rf systems and barrier rf systems. (j> < <j>s. ground vibration. In Sec. In Sec. where g is the rf cavity gap width. We assume that the reference particle passes through the cavity gap in time t € nT0 + {-g/2pc. /s Similarly. combined.e. and 4>s is the phase angle for a synchronous particle with respect to the rf wave. elongated. the higher energy particle will arrive at the rf gap earlier. lower energy particles will arrive at the same rf gap later and gain more energy than the synchronous particle. The discovery of phase stability paved the way for all modern high energy accelerators. stacking.g.1) We assume that the longitudinal electric field at an rf gap is where wo = PQC/RO is the angular revolution frequency of the reference (synchronous) particle. we study the perturbation of synchrotron motion resulting from rf phase and amplitude modulation. Furthermore. . e. In Sec. I Longitudinal Equation of Motion £ = £0 sin(<?!>rf (t) + & ) . phase stability requires TT/2 < cj>s < TT. Van de Graaff. In this chapter we study the dynamics of synchrotron motion. we provide an introduction to the linac. phase displacement acceleration. V. VII. Phase-space gymnastics have become essential tools in the operation of high energy storage rings. £0 is the amplitude of the electric field. we treat non-adiabatic synchrotron motion near transition energy. synchro-betatron coupling through dipole field error. /30c and Ro are respectively the speed and the average radius of the reference orbiting particle. In Sec. i. This process provides the phase stability of synchrotron motion. betatron. SYNCHROTRON MOTION The key answer is the discovery of the phase stability of synchrotron motion by McMillan and Veksler [17]. where an invariant torus corresponds to a constant Hamiltonian value. IV.
(j> = <f>s + A<f>.ujo. ( w = wo + Aw. E = Eo + AE.cj. However. and T is the transit time factor: _ sin(/ig/2fio) . and the energy of the synchronous particle. where <f> is the rf phase angle. i. Here (j)s. it encounters the rf voltage at the same phase angle 4>s every revolution. If the gap length is small. the azimuthal orbital angle. the momentum. Thus the acceleration rate of a non-synchronous particle is E=~eVsin(t>. The acceleration rate for this synchronous particle is £0 = ^ s i n i .p.1).6s.E are the corresponding parameters for an off-momentum particle.(hg/2RQ) • (3-3) 1 The effective voltage seen by the orbiting particle is V — SogT.e. a high electric field associated with a small gap may cause sparking and electric field breakdown.6) .9. The phase coordinate is related to the orbital angle by A / = <f> — <f>s = —hA9. the angular revolution frequency. (3.I LONGITUDINAL EQUATION OF MOTION The energy gain for the reference particle per passage is AE = eSofic / /•9/2/Soc 241 sm(huot + <j>s)dt = e£ogT sin 0S. (3. or <> Au = ±A9 = -l±A^=J-dA.po. (3.2) J-g/2poc where e is the charge of the circulating particles. Since a synchronous particle synchronizes with the rf wave with a frequency of Uri = hu>o. I p = po + Ap. where u>0 = pQc/R0 is the revolution frequency and h is an integer. Now we consider a non-synchronous particle with small deviations of rf parameters from the synchronous particle. the transit time factor is approximately equal to 1.Eo are respectively the synchronous phase angle.5) dt hdt v hdt y ' The energy gain per revolution for this non-synchronous particle is eV sin <j>. 2TT (3.4) where the dot indicates the derivative with respect to time t. e = 9s + A0. The transit time factor arises from the fact that a particle passes through the rf gap within a finite time interval so that the energy gain is the time average of the electric field in the gap during the transit time (see also Exercise 3. and <f>.
where the momentum compaction factor is3 a c = . Sec. is called the transition energy. up to first order. 2. (3. Most accelerator lattices have Qo > 0 a n d the closed-orbit length for a higher energy particle is longer than the reference orbit. the mean radius of a circular accelerator is R = RO{1 + a o 6 + a x 5 2 + a 2 6 3 + •••). is independent of particle momentum.8). SYNCHROTRON MOTION and the equation of motion for the energy difference is2 d {AE\ 1 Using the fractional off-momentum variable d-^~ Ap _ u0 AE PE~W' (3-8) we obtain S=^^eV(smct>-smcl>s). 2.12) and 7Tmc2.w0) = -hAoj. (3.£ = a o + 2 a x 5 + 3a262 + ••• = —. where. 2 We use the relation u UJQ Uo UQ WO L A£> J at \ OJO / 3Typically. .5).J T . Sec. The orbit length in a negative compaction lattice is shorter for a higher energy particle. using Eq.242 CHAPTER 3. IV. we have 4> = -h(u . (3. medium energy proton synchrotrons have been designed to have an imaginary j r or a negative momentum compaction (see Chap.9) The next task is to find the time evolution of the phase angle variable 0. The ai term depends on the sextupole field in the accelerator. we obtain — " ^ . (3. Some specially designed synchrotrons can achieve the condition a0 = 0. Recently.10) Using the relation LUR/LJQRQ = /3/Po. we have Qi72 ss -^-^f =s 1 for accelerators without chromatic corrections. or simply 7 T .1 (3 11) Following the result in Chap. where the circumference.13) (3. IV.
I. (3. IV. i.21) form the "synchrotron equation of motion. 7o /7o y9o(5/8n . a higher energy particle with 5 > 0 has a higher revolution frequency. LONGITUDINAL EQUATION OF MOTION 243 Let p — mcP^f = po + Ap be the momentum of a non-synchronous particle.18) (3-19) (3. (3. with 7 > 7 T .1 . we obtain 2. the particle appears to have a "negative mass.7) and (3.11) and (3. The AVF cyclotron operates in this isochronous condition.17) j)." . % = ~ + ai . Above the transition energy. The nonlinear term in Eq. S) are pairs of conjugate phase-space coordinates.17). Equations (3.(5)5 = . (3. (3. the revolution frequency is independent of particle momentum.17) becomes important near transition energy.15) A> yji + 2ffi6 + /?0252 To 27o2 27o2 Combining Eqs. AE/u>0) or equivalently (4>.a0r/o.10) and (3.^ r i + Q 2 ~ 2 a ° a i + " I + "o7?" ~ -TT•^7o 7o ^7o In the linear approximation.20) Below the transition energy. The speed of the higher energy particle compensates more than the difference in path length. and (3. r)0 = (a0 1 3/32 (3. we have Aw = -T)OUJO6 = ( — .— )wod. At the transition energy.e.( % + ViS + V262 + • • -)6. a higher energy particle with 6 > 0 has a smaller revolution frequency.12). the phase equation becomes where ((/>." With Eqs.= Jl+ 20$6 +ffiP. we obtain — where = -r. (3. with 7 < 7 T .1 ) Qi 2 3^o Q 0 %= . The fractional off-momentum coordinate 5 is 8=^ =^ . to be addressed in Sec.14) Expressing /? and 7 in terms of the off-momentum coordinate 5.
although legitimate. AE/LOO) or H = -hujor]o52 + 2 "°f 2 _g[cos (j> .c o s & + (<£-&) sin &] (3. Phys. SYNCHROTRON MOTION I.I The Synchrotron Hamiltonian The synchrotron equations of motion (3. the synchronous phase angle should be 0 < fa < TT/2. 5 This Lee. E49.7) and (3. Below the transition energy. . This Hamiltonian. The synchrotron tune. To simplify our discussion. we now discuss the stability condition for small amplitude oscillations. 4 S. can be accomplished by a phase shift of TT — 2<j>s to the rf wave.5 The angular synchrotron frequency is [heV\rfycosfa\ Ws c IheV\r]cosfa\ =i?V = WoV WE 2*E ' (3-26) where c is the speed of light and R is the average radius of the synchrotron. where the linearized equation of motion is The stability condition for synchrotron oscillation is r]0cosfa<0 (3. where s is the independent coordinate. with 7 < 7 T or 70 < 0. A fully consistent treatment is needed in the study of synchro-betatron coupling resonances. is inconsistent with the Hamiltonian for transverse betatron oscillations.25) discovered by McMillan and Veksler [17]. Rev. Similarly the 7 synchronous phase angle should be shifted to n — fa above the transition energy. we will disregard the inconsistency and study only the synchrotron motion.cos fa+ (</>-fa)sin fa] (3.22) for phase-space coordinates (</>.244 CHAPTER 3.4 With this simplified Hamiltonian. defined as the number of synchrotron oscillations per revolution. 6) with time t as an independent variable. is UJS Qs = ^0=i lheV\r]0cosfa] 2K(PE • (3 ' 27) Typically the synchrotron tune is of the order of < 10~3 for proton synchrotrons and 10"1 for electron storage rings.21) can be derived from a "Hamiltonian" H=lj^E{w) + ^ : [ c o s ^ . 5706 (1994).23) for phase-space coordinates (<fr.Y.
2 The Synchrotron Mapping Equation In Hamiltonian formalism. It is no surprise that Eq. where the rf potentials for (f>s = 0 and cj>s = TT/6 are shown. then the rf phase 0 n + 1 depends on the new off-momentum coordinate <5n+1. (3. The potential well near the synchronous phase angle provides restoring force for quasiharmonic oscillations. Figure 3. the rf electric field is considered to be uniformly distributed in an accelerator.1: The left plot shows schematically the rf potential for (f>s = 0 and vr/6. 3. and therefore synchrotron motion is more realistically described by the symplectic mapping equation ( I Sn+l =5n+ eV o^g (sin (j>n .Esx vs cf>. The right plot shows the corresponding separatrix orbits in {ixh\r)\leV02E)1l2di. 1.28) . The dashed line shows the maximum "energy" for stable synchrotron motion. 2nhr}(6n+i)6n+i.I LONGITUDINAL EQUATION OF MOTION 245 The stability of particle motion in the rf force potential can be understood from the left plot of Fig.g 2g.1. The corresponding stable phase-space (bucket) area is shown in the right plots. . First. The horizontal dashed line shows the maximum Hamiltonian value for a stable synchrotron orbit. The phase (f>u is the turning point of the separatrix orbit. In reality. I 4>n+i = K + The physics of the mapping equation can be visualized as follows. rf cavities are localized in a short section of a synchrotron.sin <j)s). the particle gains or loses energy at its nth passage through the rf cavity.
During beam acceleration. Figure 3. AE/u>0) is invariant.30) (3.28) is independent of energy. AE//32E) with parameters V = 100 kV.<5n+i) preserves phase-space area. 3.1 is a closed curve. When the acceleration rate is high. they are usually used in particle tracking calculations. AE/LUQ) should be used. The phase-space tori change from a fish-like to a golf-club-like shape.6 Note that the separatrix is not a closed curve when the acceleration rate is high. tori of the synchrotron mapping equations are not closed curves. In a rapid cycling synchrotron or electron linac where the acceleration gradient is high. (3.n + eV sin fa. and the separatrix orbit shown in Fig.5). Eq. Hamiltonian formalism and mapping equations are equivalent.246 satisfies the symplectic condition: CHAPTER 3.n+i = £o. Since the acceleration rate for proton (ion) beams is normally low. (3. the separatrix torus shown 6The actual rf voltage V is about 200 V in a low energy proton synchrotron. h = 1. It is worth pointing out that Eq. If the acceleration rate is low. AE) are AEn+l = AEn + eV(sin <f>n . 1.04340. 6) obtained from Eq. Because synchrotron motion is usually slow. Therefore. j = E/mc2.I/7 2 . In reality.2 shows two tori in phase-space coordinates (cj). (3.30) depends on the acceleration rate according to E = £0. the rf cavities may be distributed non-uniformly. 3.1 can be considered as a closed curve. a c = 0.3 Evolution of Synchrotron Phase-Space Ellipse The phase-space area enclosed by a trajectory (</>. The rf phase change between different cavities may not be uniform. (3. /3 = y/l . AE/LJ0) to {(j). The adiabatic damping of phase-space area can be obtained by transforming phase-space coordinates (<j>.I/7 2 . the phase-space area in (4>. SYNCHROTRON MOTION Jacobian= 9(w^ = 1 (329) Thus the mapping from {cj>n.28) can not be used in tracking simulations of beam acceleration. the separatrix is not a closed curve.sin fa). The mapping equations for synchrotron phase-space coordinates (<j).31) 0n+l = <t>n + 2^AEn+u where the quantity rj//32E in Eq. the factor hr]/{32E is nearly constant. This is equivalent to the adiabatic damping of phase-space area discussed earlier.28) treats the rf cavity as a single lumped element in an accelerator. . 5n) to (<^n+i. and r) = ac . (3. fa = 30° at 45 MeV proton kinetic energy. The separatrix for the rf bucket shown in Fig. Because of the simplicity of the mapping equations. The phase-space mapping equation for phase-space coordinates {<f>.
3.I LONGITUDINAL EQUATION OF MOTION 247 in Fig. The voltage requirement becomes Fsin0 9 = 1.30) and (3. Similarly.04340. where p « 2. ac = 0. 1.31) with parameters V = 100 kV.1 is a good approximation. the tori near the separatrix may resemble those in Fig. = 30° at 45 MeV proton kinetic energy. . we obtain V sin <j>s ss 240 Volts.4 m.6 Tesla/s.2MV. which is independent of the harmonic number used. 3. The circumference is 3319. (3.2: Two tori in phasespace coordinates ((f>. e.4 m with p = 235 m.g. and &. proton acceleration in the IUCF cooler ring from 45 MeV to 500 MeV in one second requires B = M ^ l „ l. in many electron accelerators. When the acceleration rate is high. h = 1. Figure 3.2. AE//32E) obtained from mapping equations (3. acceleration of protons from 9 GeV to 120 GeV in 1 s at the Fermilab Main Injector would require B « 1.4 Some Practical Examples Using the basic relations e we find a basic rf cavity requirement: Vsm(j)s = 27rRpB.l Tesla/sec. Using R ss 14 m.32) pc 2nR For example.
7 . where 7 > 0. . .34) • Using (<f>. —Ap/p0) as phase-space coordinates. Using t as independent variable Using time t as an independent variable. 6) as phase-space coordinates: d(j> Tt=huoV6. (3.5 CHAPTER 3./iwor) . V = — {h\r)\/vs)5) as the normalized phase-space coordinates: — = uavj>.40) The corresponding normalized phase space is (T.38) ^ I7?! • Using ( r = (0 — (j>s)/hwo. AE/UIQ) as phase-space coordinates: d<p hLoZn fAE\ H=l^(—) d(AE/u0) 1 .-W)--«./IW0T sin <£s]. the negative sign in the first term corresponds to negative mass above the transition energy.sin^ s ). — = —7W0^s(sini/> . .<£s) sin <j>s]. In particular. (3. • Using {4>.cos 4>s + {4>.41) where ua = Jh\r]\eV/2irf32Eo is the synchrotron tune at <j>s = 0. This synchrotron Hamiltonian is on an equal footing with the transverse betatron motion. d5 ojoeV .T>\—) ^ \Po / -T571[cos^-cos& + (^-&)sin&]) M \V\ (3.35) (3. 2 J^ 6 (3-39) [cos(0s . B. (3. .248 1. f/w s ). the Hamiltonian is H = -T. Using longitudinal distance s as independent variable • Using (R<f>/h. + ^[cos0-cos0s + (0-0s)sin0s]. • Using (<f>. (3.t) as phase-space coordinates: I"* H =-i2+ 5-^"-<*.37) 1 Tj H = -ZOJQI/SV2 + ~UJ0^S[COS (/> . .cos 0S . .„ ocx _ = 5-^(sin0-sm0s).36) H = -huj0r]62 + ^ [cos <(> . (3. the equations of motion and the Hamiltonian are listed as follows.<t>s) sin 0 S ]. SYNCHROTRON Summary of Synchrotron Equations of Motion MOTION A.cos 4>s + {<(> .
e. Calculate synchrotron tunes for the proton synchrotrons listed in the following table with <f>s = 0.5 24. (3.E.1 249 1. 2.12 3833. e and m are the charge and mass of the particle.6 86. Thus the transit time factor T is the same for all particles.8 5.446 C [m] 807. An rf cavity consists of an insulating gap g across which the rf voltage is applied. and x = 0/2TT. (Mod 1). The total energy gain of a particle passing through the gap is the time average of the rf voltage during the transit time. 4. Redefine y = h\r)\8.95 h 12 342 588 84 7r 8.2 K.8 ^s 1 1 1 1 I I ~ 3. RF parameters of some synchrotrons P-synchrotron I AGS I RHIC I FNAL-MI 1 FNAL-BST 28 8 0. Show that the effective voltage is V _ V T sin(hg/2R) where R is the mean radius of the accelerator. xn+i = xn + yn+\ where ws is the synchrotron tune.28) of the symplectic mapping equation for a stationary bucket synchrotron motion can be transformed into the standard map: 2M+1 = Vn .0001 1 4.2 Vrf [MV] 0.3 2 0.4 474.5 21. The gap length is finite and the rf field changes with time during transit time At. .84 3319. and c is the speed of light.045 0.3 0. [GeV/u] 0.EXERCISE 3. i. and <j> the rf phase of the particle. _ hi \ g2w 1/2 (a) Show that Eq.1 Exercise 3. p is the bending radius of the dipoles. RQ is the mean radius of the accelerator. At J-At/2 e fA*/2 V(t) = Vg sm(<j> + huot) where Vs is the peak gap voltage. Show that the relation between the rf frequency of an accelerator and the magnetic flux density B(t) during particle acceleration at a constant radius is given by " r f ~ Ro [B2(t) + (mc2/ecp)2 where h is the harmonic number. AE= — V(t)dt.2 I SSC 2000 10 17424 140 87120 I Cooler 0.IKVI sin27rzn.
62 4>s [deg] I BEPC 240.12 I TRISTAN 3018 30.2. The energy loss per turn is given by UQ = Cj^Et/p.35 0.81 ux j^ 70.8 160 5.8 38.4 2.39324366 [2ws2c = 0.2 7.y) such that x e [—1.5 10 1. (c) Explore the phase-space evolution at us > i/SjC.91 46. explore the dependence of the separatrix on the acceleration rate.1].98 p [m] 3096.28 35.18 7. /3c is the beam velocity. Uo = eVTfSm<j>s.3 8. 2/£[-1. and C 7 = 8.8 1060 223 Energy [GeV] 50 1.85 x 10" 5 m/(GeV) 3 . where the critical synchrotron tune is i/SiC = 0.5 400 5120 25.44 64.1 76. I LEP I ALS I APS 1 NLC P R ? H 26658.2 4.01 38. The energy loss due to synchrotron radiation is compensated by the rf accelerating field.86 26.35 Vrf [MV] 400 1.2 8. SYNCHROTRON MOTION (b) Write a program to track the phase-space points (x.18 14. where Eo is the beam energy.0 1.0 6.1]. i.9 196.2 14.7 _& 1 1 1 1 1 I . 6.22 23.96 4.971635406]. Write a computer program to track synchrotron motion near the separatrix. Calculate synchrotron tunes for the electron storage rings listed in the following table. p is the bending radius of dipoles. and verify the golf-club-like tori in Fig.2 10. 246.e. Electrons in storage rings emit synchrotron radiation. 5. 3.0 h 31320 328 1248 531 7r 50.5 36.250 CHAPTER 3.
The typical synchrotron tune in electron storage rings is of the order of 10"1.§ « 1. when a^ < 0." For simplicity in notation. i. Typically. (3.„ (3 ' 27) . ADIABATIC SYNCHROTRON MOTION 251 II Adiabatic Synchrotron Motion With time t as an independent variable. 6= " (sin <j> .05. the subscript of the energy Eo of the beam has been neglected.</>s)sinc/>s]. =V'fi™**1 ~ .coscj>s + {(j>. This corresponds to adiabatic synchrotron motion. if the acceleration rate is low. During beam acceleration. The synchrotron period is TS = TO/QS! where To is the revolution period. 6) is H = ^hu0r]52 + ^^[cos<f> . where parameters in the synchrotron Hamiltonian change slowly so that the particle orbit is a torus of constant Hamiltonian value. the small amplitude synchrotron tune is ^ Qs = lheV\ricos(/>s\ r. However. The typical synchrotron tune in proton synchrotrons is of the order of 10~3. the Hamiltonian (3.II.s u n f e ) . the Hamiltonian is time independent or nearly time independent. If }rj\ ^ 0. If the rf parameters V and 4>s vary only slowly with time so that the gain in beam energy in each revolution 7 is small. Figure 3. the synchrotron Hamiltonian for phase-space coordinates (<j>. (3. aa = \*£ = i.23) generally depends on time. The condition for adiabatic synchrotron motion is wj dt 2?r dt where a/s is the angular synchrotron frequency and a ^ is called the adiabaticity coefficient. V 2*fPE where "s-y2nP*E \h\q\eV is the synchrotron tune at j cos0 s | = 1. a (3. the Hamiltonian can be considered as quasistatic. and 7 differs substantially from 0.42) where the over-dots indicate derivatives with respect to time t. it takes about 1000 revolutions to complete one synchrotron oscillation. Hamilton's equations of motion are 4> = hr)bjQ5.e.23) where the first term can be considered as "kinetic energy" and the second as "potential energy.43) .1 illustrates schematically the potential energy as a function of cj> for <f>s = 0 and </>a = TT/6. hereafter. the time variation of synchrotron period is small and the trajectories of particle motion can be approximately described by tori of constant Hamiltonian values.
Small amplitude phase-space trajectories around the stable fixed point are ellipses.7r/6. The phasespace area enclosed by a Hamiltonian torus is A = J8{<t>)dct>. 5TT/6.252 CHAPTER 3. Particles outside the rf bucket drift along the longitudinal direction. and only particles in the stable region can be accelerated to high energy. the phase-space trajectories near the unstable fixed point (UFP) (ir . . Thus the UFP is also called a hyperbolic fixed point. and particles inside the rf bucket execute quasi-harmonic motion within the bucket. The synchrotron phase space is divided into stable and unstable regions.7r/3 and for r] > 0 with (f>s = 2?r/3.4>s. Figure 3. A beam in which particles are grouped together forming bunches is called a bunched beam.4>s. 1 Fixed Points The Hamiltonian for adiabatic synchrotron motion has two fixed points (<^s. TT.0) and (TT . The stationary buckets that have largest phase space areas correspond to cj>s = 0 and n respectively. which is also a contour (a torus) of constant Hamiltonian value. where 0 = 0 and 5 = 0. The phase space area enclosed by the separatrix is called the bucket area. it separates phase space into regions of bound and unbound oscillations.7T. and for r\ < 0 with <j>3 = 0.7r/6. The torus that passes through the UFP is called the separatrix. SYNCHROTRON MOTION II. Therefore the SFP is also called an elliptical fixed point. Thus particles in synchrotrons are naturally bunched.57r/6. Figure 3. 0) is the stable fixed point (SFP). particle motion adiabatically follows a phase-space ellipse. The maximum momentum deviation of the separatrix orbit is called the bucket height.3: The separatrix orbits for r) > 0 with <j)s = 2?r/3. The phase-space point (cf>s. For a slowly time-varying Hamiltonian.7r/3. 0).3 shows the separatrix orbit for r) < 0 with <f>s = 0. 0) are hyperbola. On the other hand. The phase-space area enclosed by the separatrix orbit is called the bucket area.
e.1 6 v S £ « ' * » = ^ = 3 »<«• (3-47) where the factor ab(</!|s) is the ratio of the bucket area between a running bucket (4>s ^ 0) and a stationary bucket (</>s = 0).1 lists Qb((/>s) as a function of the synchronous phase angle cj>s. (3. 1 /•*-«.49) The corresponding invariant bucket area in (</>.(?r . Table 3.^ .46) A . AE/UJ0) phase-space variables. The phase-space area enclosed by the separatrix is called the bucket area. i.2 lists some relevant formulas for rf bucket properties. where <fiu is c o s (j>u + <j>n s i n 4>s = — c o s <^>s + (n — (j>s) s i n 4>s. [cos0 + cos <f>s . AE is the bucket energy height (eV). and Qb(7r/2) = 0.<j>a) sin0 S ] = 0.45) The separatrix has two turning points. (3. Therefore the Hamiltonian value of the separatrix is Hsx = ^—^ [-2 cos <j>s + (TT . i. (3.44) The phase-space trajectory of the separatrix is H = Hsx.e. ADIABATIC SYNCHROTRON MOTION 253 II.50) W o is the phase-space area of h buckets in the entire accelerator. where At is the bucket width in time (s). or *« + rKpitihr\ eV « n . the bucket area vanishes at 90° synchronous phase angle./ « • ) * .II. 1 + sin <> /s (3. B2E AB = -—. Note that Qb(0s) can be approximated by a simple function aM « £?£. Naturally ab(0) = 1. . and the resulting bucket phase-space area is in eV-s.e.2&) sin 0J. 0). For </)s = 0. i.4B « hAtAE. (3. the turning points are — ir and ir. 2 Bucket Area The separatrix passes through the unstable fixed point (TT — </>s.48) [ *7 J Table 3. f In I 11/2 ah{<ps) = -r7= \—^ [cos0 + cos(/> s -(7r-0-(k)sin0 s ] df 4 v ^ '^u (3. <fiu and TT — <fis.
4286 0.45 0.1429 0.75 0.2903 0.00 -136.37 168. 1 Bucket Area (Affi 1 (<P.00 I 0. and the bucket height or the maximum momentum width is s-'{^mfY^'mtiU (351) . 1 0.8888 0.00 [ 90.6729 0.4832 0.2919 0. SYNCHROTRON MOTION Table 3.7992 0.0408 0.8182 0.4815 0.7577 0.2: Formula for bucket area in conjugate phase space variables.4459 0.3806 0.46 165.80 0.6611 0.48 4.6667 0.19 1.51 156.1765 0.9048 0.0526 0.4305 0.3793 0. bucket height and bucket area factors.79 115.54 159.2058 0.42 180.2885 0.85 0.1731 0.00 0.10 0.0000 0.3967 0.S) a 1 (4.84 108.1: Bucket length.5852 0.1111 0.26 171.25 0.2349 0.95 4>u 1 7T .88 -13.90 0.0000 0.3333 0.35 0.8402 0.38 37.9208 0. Table 3.0170 1.0256 1.26 150.0811 0.15 0.7391 0.& I y ( & ) I qb(<fe) I jgjffi -180.5385 0.1323 0.11 -38.5399 0.05 0.00 | 90.47 -118. | 0.59 25.2500 0.254 CHAPTER 3.1679 0.5388 0.9606 0.59 -64.90 -105.8041 0.^6) ^ 16 Q b ^> 16 (£§f^) V * Qb(&) 16 ( a f e f f ) 1 " Bucket Height 1 2 ( g g j ^ y f o ) [ ^^J)1''^^) | 2 YQfeT The bucket length is \(n — <j>s) — <f>u\.7294 0.70 0.55 0.13 174.50 0.45 -55.32 -93.6000 0.69 -30.66 -47.87 121.7156 0.3333 0.77 53.75 14.5980 0.0000 0.8807 0.1028 1.63 143.00 177.3455 0.46 135.31 -4.20 0. sin& I 0.2460 0.41 126.31 -21.71 -83.26 -73.30 0.0685 0.52 162.40 0.65 0.42 153.2121 0.13 139.60 0.6295 0.00 146.57 131.4936 0.0991 0.
e. The corresponding rms phase-space area is . Gaussian beam distribution The equilibrium beam distribution is a function of the invariant ellipse of Eq. In many beam applications.53) where ip = 6 — 6S. The phase-space area that contains 95% of the particles in a Gaussian beam distribution is 6-4rms. (3.52) Table 3. The phase-space ellipse of a particle /^V+M 2 -! * __(eV\cos6s\\1/2_ Qs [dj + U J ~ ' 4>-WpEh\v\) -h\n\' (3'55) where 5 and 6 are maximum amplitudes of the phase-space ellipse. Y{6S) = 1 n -26 ^ 1/2 tan 6S .Arms = naga^.26 1/2 ^ sin 6S . Y{6S) = cos 6S . (3. II. and the bucket height factor.55). .27).4 7 r / g 2 g V?2. (3.1 lists the turning points.II. where UJS = becomes QSLJ0 5= -~-J>sH^ h\r]\ + X). (3-54) is the angular synchrotron tune. we use the normalized Gaussian distribution given by ***>-5^-H[$ + £]}-495% = (-6 35 ) where as and o"^ are rms momentum spread and bunch length respectively. Y(6S). ADIABATIC SYNCHROTRON MOTION 255 Here the bucket height factor Y((j)s) and Y(6S) are the ratios of the maximum momentum height to the height of the stationary bucket. A. where the factor 6 depends on the distribution function.3 Small-Amplitude Oscillations and Bunch Area 2 w0eVcos6s 2 1 H = -h^rjS2 . The phase-space area of the ellipse is ir5(f>. of an rf bucket. and (p = (f>cos(ujst + x). i. The linearized synchrotron Hamiltonian around the SFP is (3. The synchrotron frequency is given by Eq.
i.58) become S~ ^ 1 /2yl/4 fe l/4| r? |-l/4 7 -3/4 ! fl~>tVV-l/4/l-1/4|r?|l/47-l/4) (3 6 Q ) where the adiabatic damping is also shown.J-fr where 5 and 9 are maximum amplitudes of the phase-space ellipse.57) S = AW (j±. Thus the phase-space area is commonly defined as phase-space area per amu expressed as [eV-s/u] for heavy ion beams. It will be discussed in Sec.5) space becomes 7Since the energy of a heavy ion beam is usually expressed as [MeV/u] or [GeV/u]. e -> Ze.. the E in the denominator of Eq.\V2 \*PEJ (heV\coS(f>s\\1/4 { 2TT/32E\V\ ) ' h \np2EJ \heV\ cos <f>s\J Qs 5_(heV\coscl>s\y/2 Note that here the phase-space area A is the invariant phase-space area for one bunch in eV-s. Similarly. where Zis the charge number of the ion beam. and 8 —> 0. and u = 0. (E/A). As the energy approaches the transition energy with rj — 0. SYNCHROTRON MOTION Since the synchrotron phase-space area A.7 it is related to A by A = *W = hA(j^y Using Eq. The scaling properties of the bunch length and bunch height of Eq. the invariant phase-space ellipse in (9. (3. we expect that 5 —> oo.55). where A is the atomic mass number or the number of nucleons in a nucleus. The factor eV/E in this chapter should be modified by a charge-to-mass ratio of Z/A for heavy ion beams. (3.57) can be expressed as A x (E/A). IV. is defined as the area in the phase space (c/>/h. and E -^ Ay. E/A = juc2. (3. S) phase space is W .e. The normalized Gaussian distribution in (9. usually measured in eV-s. This is not true because the > synchrotron motion around the transition energy is non-adiabatic.931494 GeV/c2 is the atomic mass unit. . AE/u0) for one bunch. the maximum momentum width and the bunch length are (3.256 CHAPTER 3.
y/2-Kag (3.e -*/*} V27T(T^ o r m = -T=-e-"»i. where R is the average radius of the accelerator. or at = crg/uio in s. = fsini/>. See Eq. the synchrotron Hamiltonian becomes (see Exercise 3. i.II. and the solution can be expressed as r = fcos?/). Now we consider NB particles distributed in a bunch.68) where f and ip are respectively the synchrotron amplitude and phase.333) for its application.| tan <^s ^ . (3. Synchrotron motion in reference time coordinates In the discussion of collective beam instabilities.67 ) The equation of motion is f 4.63) The peak current (in Amperes) of the bunch is ~ = NBe = NBeuj0 = f 2ir \ NBe \p2mat y/^ag \\f2KOg) To B. ADIABATIC SYNCHROTRON MOTION 257 Here og and as are respectively the rms bunch angular width and the rms fractional momentum spread.e.65) The linearized synchrotron Hamiltonian becomes where uis is the angular synchrotron frequency shown in Eq. The phase-space ellipse that corresponds to a constant Hamiltonian is T' + ~2 = f^ ( 3 .69) . (3. The line distribution is p(fl =-*l. ip = ip0 + ujst (3.2. it is sometimes useful to use the particle arrival time r and its time derivative f for the synchrotron phase-space coordinates.52 + ~^Ql [v>2 .-^ p* + • • •] . r = -lzlli and r = — =+tf. The bunch length is as = Rag in meters. where JVB may vary from 108 to 1013 particles. Approximate action-angle variables Expanding the phase coordinate around the SFP with <j> = <f>s + <p.26). (3. C.w|r = 0. (3.11) H = -hu)0r.
w e o b t a i n • = u)0eVcos<i>s 2TT/32E J = w2<5. With normalized coordinates . 5 8In reality. V Vs V nv (3.258 CHAPTER 3. V 0 = -2^O¥>2tanV>. If we use the approximate action-angle variables. the beam does not move in the phase space. tp) are approximate action-angle variables of the Hamiltonian in (<f>. the Hamiltonian for synchrotron motion becomes H = UJOQSJ + CJ0^2yB 12 tan4>s J3'2 [cos3</> + 3cosV>] . the averaged synchrotron Hamiltonian becomes H = uj0QsJ . <P' . (3. Using the generating function for linearized synchrotron motion ^ . we assume r\ > 0 in this section.11).73) (3. 6) phase space. the phase of the rf wave is being shifted so that the UFP is located at the center of the bunch. ^ = VU)° ^ ^ or ^5 = waV(3. we study the evolution of an elliptical torus of Eq.74) QS(J) * Qs [l . 6 = -J^-^i>.^ II. Now.55) at the UFP.uj—^cos^</>• 6 (3.71) where the conjugate variables (J. (j> x o = S-.4 Small-Amplitude Synchrotron Motion at the UFP Small amplitude synchrotron motion around the unstable fixed point (UFP) is also of interest in accelerator physics.2. We would like to find the evolution of bunch shape when the center of the beam bunch is instantaneously kicked8 onto the UFP at time t = 0.72) If we apply the canonical perturbation method (see Exercise 3. SYNCHROTRON MOTION where Qs = ij heV\r] cos <j>s\/2nft2E = vs^j\ cos</is| is the small amplitude synchrotron tune. For simplicity.. Expanding the Hamiltonian around the UFP. ip = (j> — (TT — <f>s). (l + | tan2 0S) j ] . .V ip = —. (3. instead. i.^ Thus the synchrotron tune becomes ( l + jj tan 2 ^ J 2 + • • •.76) Thus the particle motion is hyperbolic around the UFP. the synchrotron phase-space coordinates are transformed according to (fi= (3.e.70) /2§ZcosV.
tanh 2uist ) <p6 + 52 = (cosh 2ujst)-1.II. we find the synchrotron oscillation period as T = j \2tiLJoV \Ho .&) sin 0 j j j #.be'"3''. . The width and height of the phase-space ellipse increase or decrease at a rate e±u>st.79) where Ho is the Hamiltonian value of a torus.6) is the synchrotron Hamiltonian in Eq. 6 = ae"'1 .78) \\v\ ) Thus the upright phase-space ellipse will become a tilted phase-space ellipse encompassing the same phase-space area.S) = Ho. (3. where H(4>.23) and the constant Hamiltonian value Ho is Ho = -hu}Or/S2 = 2n32E^C°S ^ ~ C ° S ^+ ^ ~ ^ Sm^' Here 0 and <5 are respectively the maximum phase coordinate and fractional momentum deviation of synchrotron motion.76) are <p = ae1"*' + be-Ust. This scheme of bunch deformation can be used for bunch rotation or bunch compression.77) where a and b are determined from the initial condition. where t is the length of time the bunch stays at the UFP (see Exercise 3. The action of the torus is J=hi[h^[Ho~S^E{COS*~cos0s+{<t>~0s)sin^s)]) #. With the constants a and 6 eliminated. II. At ust 3> 1.cos & + {</.(3-80) The synchrotron period of Eq.| ^ [ « > s <f> .2.5 Synchrotron Motion for Large-Amplitude Particles The phase space trajectory of adiabatic synchrotron motion follows a Hamiltonian torus H(cj). Using Hamilton's equation 0 = huiorj8. the ellipse becomes a line (p ± 5 — 0. 259 (3.. ADIABATIC SYNCHROTRON MOTION the solutions of Eq.80) with respect to J. (3. However.5).79) can also be derived by differentiating Eq. (3. The angular synchrotron frequency is 2TT/T. (3. the evolution of the bunch shape ellipse is (p1 -2\rL. (3. (3. and using dH0/dJ = UJ(J) to find the synchrotron frequency. the nonlinear part of the synchrotron Hamiltonian will distort the ellipse.
.74) at <f>s . When the value of the Hamiltonian Ho approaches that of the separatrix Hsx of Eq. .85) Figure 3. where 5{<f>) = ^ . (3.81) (3-82) (3. Stationary synchrotron motion CHAPTER 3.85) with a measured synchrotron tune at the IUCF Cooler. we find Qs(4>) « (1 .2. In the small angle approximation. Synchrotron tune I-K/2 //.. where the modulus of these integrals k = sin(</>/2) and .4 compares the theoretical curve of Eq. ±6((j>)). Since the synchrotron tune is nonlinear.P s i n io •'0 are the complete elliptic integrals of the first and second kind. fit/2 1 (3.k2) K{k)] . _ .[-B(fc) .83) The Hamiltonian torus is the phase space trajectory given by ((/>. If the bunch area is . or the action J enclosed by the Hamiltonian torus is A = 2TT J = 2 / 0 5{<t>)d4> = 1 6 ^ .44). the synchrotron tune becomes zero and the synchrotron period becomes infinite. Note that the maximum off-momentum coordinate 5 is related to the maximum phase coordinate (j> a Hamiltonian torus by *-*4 (. which is identical to Eq. we consider the stationary synchrotron motion above the transition energy with 7 > 0. (3. SYNCHROTRON MOTION For simplicity.(1 . •/o VI .cos0). particles having different synchrotron amplitudes in a beam bunch can have different synchrotron tunes.8) Q8(0) = 7r*V2#(sin(^/2)) (3. or <f>s = n.260 A. The Hamiltonian value for a torus with a maximum ? phase coordinate 4> (or maximum off-momentum coordinate 5) is Ho = ihuovS2 = % s 2 ( l . E{k)= Jl-k2sin2wdw..V ' 2 ( C 0 S ( ? i .84) The synchrotron tune of the Hamiltonian torus with maximum phase amplitude 4> becomes (see Exercise 3.C 0 S ^ ) The phase space area A. K{k)= B.YQ4>2)US.0. (3.
synchrotron tune spread is useful in providing Landau damping for collective beam instabilities. For a mismatched beam bunch. The final bunch area is determined by the initial beam distribution and parameters of the rf system (see Fig. The zero amplitude synchrotron tune was ua = 5. or rf voltage and phase modulations. Since synchrotron oscillation is rela- . Beam filamentation causes a mismatched beam bunch to evolve into spirals bounded by a Hamiltonian torus. In this section we discuss the methods of measuring the off-momentum and rf phase coordinates of a beam. The fractional off-momentum coordinate of a beam can be derived by measuring the closed orbit of transverse displacement Axco at a high dispersion function location. V). the synchrotron tune spread may be large. 3.II. The solid line shows the theoretical prediction of Eq. Filamentation can dilute the phase-space density of the beam. a phase detector is needed in implementing a phase feedback loop to damp dipole or higher-order synchrotron modes.4: The measured synchrotron tune obtained by taking the FFT of the synchrotron phase coordinate is plotted as a function of the maximum phase amplitude of the synchrotron oscillations. When the beam encounters longitudinal collective beam instability. and the corresponding FFT spectrum. The off-momentum coordinate is Ap Axco T = "#' (3-86) where D is the horizontal dispersion function. The inset shows an example of the synchrotron phase-space map measured at the IUCF Cooler.6 Experimental Tracking of Synchrotron Motion Experimental measurements of synchrotron phase-space coordinates are important in improving the performance of synchrotrons.2 x 10-4. or mis-injection in the rf bucket. a filamentation process. where beam particles spread out in the synchrotron phase space. synchrotron tune spread can cause beam decoherence. ADIABATIC SYNCHROTRON MOTION 261 Figure 3. etc. This process is important to rf capture in low energy synchrotrons during injection. a substantial fraction of the bucket area. For example.. (3.85). II. On the other hand.19 in Sec. the mismatched phase-space distribution will decohere and result in beam dilution.
and the horizontal dispersion function D is about 4. Equation (3. the signal from a beam position monitor (BPM) or a wall gap monitor (WGM)9 is OO 00 I(t.4 shows a synchrotron phase space ellipse measured at the IUCF Cooler Ring.262 CHAPTER 3. The BPM system had an rms position resolution of about 0.87) shows that the periodic delta-function pulse. The wall current that flows through a resistor. To measure the phase coordinate or equivalently the relative arrival time r. is equivalent to sinusoidal waves at all integer harmonics of the revolution frequency. the type III utilizes the edge triggered JK-master-slaveflip-flopcircuit. The 10 See Roland E. The position signals from the BPM were passed through a 3 kHz low-pass filter before digitization to remove effects due to coherent betatron oscillations and high frequency noise. we examine the characteristics of beam current signal from a beam position monitor. We assume that the bunch length is much shorter than the circumference of an accelerator. To is the revolution period. or we select the fundamental harmonic with a low-pass filter including only the fundamental harmonic. can then be measured. typically about 50 Ohms with a stray capacitance of about 30 pF. and the fractional off-momentum coordinate is obtained from the displacement of the beam centroid measured with a beam position monitor (BPM). The synchrotron phase coordinate can be measured by comparing the bunch arrival time with the rf cavity wave. the signal-to-noise ratio can be enhanced by using a low-pass filter at a frequency slightly higher than the synchrotron frequency.0 m at a high-dispersion location. The bandwidth is about 100 MHz. The type II phase detector utilizes XOR logic. and the phase between the beam and the reference rf wave can be obtained by using phase detectors. With the beam bunch approximated by an ideal 5-function pulse. 7-9 (McGraw- . By averaging the position measurements the stability of the horizontal closed orbit was measured to be within 0. Hill. Phase Locked Loops. pp. Design. by Aa. The momentum deviation is related to the off-momentum closed orbit. r) = NBe £ fc-oo 5{t + r . and Applications. Axco. The phase coordinate is obtained by a phase detector. we first select a sinusoidal wave by using the band-pass filter. a 3 kHz low-pass filter could be used to average out betatron oscillations of a few hundred kHz. The sinusoidal signal is compared with the rf wave. Best. the 9A wall gap monitor consists of a break in the vacuum chamber. Since the synchrotron frequency at the IUCF Cooler in this experiment was less than 1 kHz for an rf system with h = 1. SYNCHROTRON MOTION tively slow in proton synchrotrons. u>0 = 2TT/T0 is the angular revolution frequency. 3. Theory.87) where NB is the number of particles in a bunch. and has a range of ±180°.1 mm.co = D5. The inset in Fig. 1984). in time domain. where 5 = Ap/po is the fractional momentum deviation.02 mm. and r = (8 — OS)/UIQ is the arrival time relative to the synchronous particle. and has a range of ±90°.10 Normally. First.ITO) = NBe £ n=-oo ein^t+T\ (3. New York.
(b) The bunch area is determined by several factors. Phys. 4051 (1994).5 GeV? (d) How does the rf bucket area change during the acceleration process? 3. the acceleration time is 160 ms. Rotate the phase-space ellipse of Eq. where t is the time the bunch stays at the UFP. the transition energy is ~fT = 4. what is the minimum rf voltage needed? (c) What is the rf frequency swing needed to accelerate protons from 200 MeV to 1. Rev.5 GeV kinetic energy.2 263 BPM sum signal or the WGM signal can be used to measure the relative phase of the beam.The top inset in Fig. To extend the range of beam phase detection. etc. show that the width and height of the bunch change by a factor e±Ust. show that the rf bucket area in the (AE/UJQ.2 1. and estimate the time needed to double the bunch height. Verify Eq. the synchrotron motion can be tracked at N revolution intervals. such as line charge density. the extraction energy is 1. If we need a bunch area of about 1 eV-s per bunch in (AE/uiQ. 4>/h) phase space has a minimum at 7 = \ / 3 7 r . Exercise 3. the type II can be used.5.<f>/h) phase space. 4.EXERCISE 3. the bottom inset shows the FFT of the phase data. (3. This exercise concerns the acceleration of protons in the AGS booster. 3. and the harmonic number is h = 3. transition crossing in the AGS. (3. The resulting synchrotron tune as a function of peak phase amplitude is compared with the theoretical prediction in Fig. but can adequately measure the synchrotron tune. Ellison et al.78 m.78) into the upright position. n M . where N -C l/vs. Particle acceleration at a constant bucket is a possible "rf program" in synchrotrons.4 MHz low-pass filter to eliminate high harmonics noise before it was compared with an rf signal in a phase detector. Find the relation between rf voltage and beam energy. For more accurate measurement of phase amplitude response.03 MHz for the 45 MeV protons in this measurement at the IUCF Cooler. a type IV phase detector with a range of ±360° can be used. The phase-space map of synchrotron oscillations can be obtained by plotting Ap/po vs <j> in each revolution.76). Write a simple program to calculate a^((j>o)2. the BPM signal was passed through a 1. . For a constant rf voltage and synchronous phase angle. microwave instabilities. 3. and the bucket area is about 1.4.11 Since the rf frequency was 1. E 50. (a) Find the rf voltage needed for acceleration of a proton bunch in the booster. Since the synchrotron tune of a proton synchrotron is small.4 shows the Poincare map of the longitudinal phase space at 10 turn intervals. The circumference of the booster ring is 201. type III has a phase error of about ±10° near 0°.2 times as large as the bunch area. 5. The injection kinetic energy is 200 MeV from the linac.
7.V. (b) At 1/4 of the synchrotron period after antiproton injection.4 474.5 0. and R = 83 m. Dugan.045 Vrf [MV] 0. [13] (1983). Show that the final rf voltage V2 is related to the initial voltage V\ by Find the final matched rf voltage for the Debuncher.0001 h 12 342 588 84 1 7T 8. (3. SYNCHROTRON MOTION 6.84 3319.95 0. The anti-protons produced from the Main Injector (Main Ring) pulses have the following characteristics: p 0 = 8. 9.84).5 0. Assuming stationary bucket.3 0. Tollestrup and G. aE = 180 MeV.2 86.15 ns. The antiprotons are captured in the Debuncher into the 53. or Ap/p 0 = ±2%.6 C [m] 807. 954 in Ref.E. Define p^ = hur]5. . P0 = o w 2 ( s i n ^ . and show that the synchrotron equations of motion become <I> = P4>. and synchrotron period of the Debuncher ring for the rf system.s i n ^ s ) .446 4. 12 (a) Find the bucket height.5 21.8 A [eV-s] 1. 7 T = 7.0001 aE [MeV] <Jt [ns] I I I I I 8. 12See A. P-synchrotron I AGS I RHIC I FNAL-MI I FNAL-BST I Cooler K. fill out the beam properties of the proton synchrotrons in the table below. Note that 5 MV is the maximum voltage that the Debuncher rf system can deliver. synchrotron tune. at = 0.264 CHAPTER 3.1 MHz (h = 90) rf bucket with V = 5 MV.5 24.12 3833.8 5. (c) Show that the final energy spread in this debunching process is Find the final energy spread of the antiproton beams. the rf voltage is lowered suddenly to match the bunch shape. p. [GeV] 25 250 120 8 0.3 2 0.15 0JJ5 0.9 GeV/c. Show that the synchrotron tune of a particle with phase amplitude <j> in a stationary bucket is where K(x) is the complete elliptical integral of the first kind given in Eq.7.
w o Q 2 p . r —- . Let (j> be the maximum synchrotron phase amplitude.8. Show that the Hamiltonian below the transition energy becomes H = usJ + w2 1 cos 2 tb .EXERCISE 3.. where //tey|T?cos^sJ For simplicity. + ^s2(cos^-l). and the overdot indicates the derivative with respect to time t. Expanding the phase coordinate around the SFP with (j) = <j>s + ip. the synchrotron Hamiltonian becomes H = -hw0r. show that the synchrotron tune is approximately given by [2 where J\ (w) is the Bessel function. we assume r\ > 0 in this exercise.82 + ^ . V<S> = —\/2Jussm4>. w J V ws (c) Compare the accuracy of the above approximated synchrotron tune to that of the exact formula given by Exercise 3.tan fa <pz . Show that the maximum offmomentum deviation is 11.2 265 where us = w$y/h\q\eV/2-Kp1E. where J and ip are action-angle coordinates.—<pA + • • -j . L ws Vv w s j \ (b) Using the phase averaging method. 10.2. The Hamiltonian for a stationary rf system with fa becomes ff=^ (a) Using the generating function i^-y^tan^ show that phase-space coordinates are <j> = \J2Jjus cos-0.cos I A — cos ip I .
the rms fractional momentum spread of the electron beam is ag = 0. Compare your result with that of Eq. show that terms proportional to J 3 / 2 in the Hamiltonian can be canceled if Gz and G\ are chosen to be G3 = ^ t a n ^ / 3 / 2 i G l = ^ t a n ^ s 7 3/2. p is the bending radius. (> where <p is the maximum synchrotron amplitude in the quasi-harmonic approximation. (c) Show that the new Hamiltonian is H = u0QsI uiohrj —I* cos4 ip 6 tan 2 4 .266 (a) Using the generating function CHAPTER 3. where Cq = 3. . 12 Un ^ j3/2 [CQS ^ + 3 CQS ^ _ "skn ji 6 C0S4 ^ (b) Using the generating function F2(ip.."frv^Qa 8 0 s /i/2[ c o s 3 ^ + 3cos ^][3Gr3 c o s 3^.3. 8— — y2QsJ/hijsinip.. + G l CDS ^]_ Now the perturbation in the new action variable is proportional to 7 2 .. 12. (d) Show that the average Hamiltonian is <tf>="oQs/-^(l + ^tan^s)/2 + .85) for <ps = 0. I)=ipl + G3(I) sin 3^ + Gi(J) sin V. SYNCHROTRON MOTION show that the phase-space coordinates are related to action angle (J. Show that the synchrotron tune for a particle with a synchrotron amplitude (p is Q¥ ) = Q s [ l .83 x 10~13 m. The natural rms fractional momentum spread of electron beams in a storage ring is OE/E = JCq~f2/JEP. and the Hamiltonian in action-angle coordinates is H = u^j _" o v y .1. Find the bunch length and rms phase-space area in eV-s. and JE ~ 2 is the damping partition.000813. In the NLC damping ring (DR) parameter list shown in Exercise 3. Finding new canonical variables to cancel low-order perturbation terms is called the canonical perturbation technique.^ ( l + 5tan 2 <^ 2 ]. ip) by ip = J2hr] J/Qs cos ip. (3.
EXERCISE 3. the resulting coherent beam motion is called the quadrupole synchrotron mode. Make coordinate transformation into the synchrotron rotating frame. .61). AV is the mismatched voltage. show that a2e « al{\ .—^ H 5. Because the bunch tumbles at twice the synchrotron frequency. When a mismatched Gaussian beam p(5. Show that the peak current for the weakly mismatched beam is Discuss your result. where 6 = \r)\6/uB. The nonlinear synchrotron tune will cause the mismatched injection to filament and the resulting phase-space area will be larger. (b) For a weakly mismatched beam. whereCTO= Jag + {r]as/i/s)2 is the matched rms beam width. (3. The equilibrium distribution in linearized synchrotron phase space is a function of the invariant ellipse given by Eq.p)dp. t) = J^Le-8*l2*2 V27TCT ~a2 = CT| CQS2 ^ + fl^/^2 gin 2 ^ Show that the peak current is I(t) = N^euol{\p2/na•. The beam profile in the x plane is equal to p{x) — J p(x. and the mismatch condition for the linearized synchrotron motion is given by ag / \r)\as/vs(a) Show that the projection of the beam distribution function onto the 0 axis is 13 p(0..> ' 27r<T5CT9 F \ 2 [o-92 as2\ J is injected into the synchrotron at time t = 0.2 267 13. (va5/us)2 « «rg(l + AV/2V). and V is the voltage for the matched beam profile. ^Transform the (9.AV/2V).8) coordinate system into the normalized coordinate system (x = 9 and P — M^/"«)> where the matched beam profile is a circle.0) = exp < . what is the time evolution of the beam? Here ag and ag are respectively the initial rms bunch angular width and fractional momentum spread.
In the normalized phase-space coordinates.l Normalized Phase-Space Coordinates Using normalized momentum deviation coordinate V = —(/i|7?|/^s)(Ap/p). The synchrotron Hamiltonian is autonomous (time independent).2).90) where the complete elliptical function integrals are [25] E{k) = [n/2 Jl-k25m2w Jo dw. (3. (3. we study the effects of a single frequency sinusoidal rf phase and voltage modulation on particle motion and beam distribution. The understanding of the beam response to a single frequency modulation can be applied to the analysis of more complicated situations. V) are normalized conjugate phase-space coordinates.88) where i/s = *Jh\r)\eV/2n/32E is the synchrotron tune at |cos^>s| = 1.COSUJ. and the maximum bucket area is A — 27rJmax = 16 (see Table 3.. and medium frequency from mechanical vibration etc. .(1 . Expressing the synchrotron coordinates in parameters k and w as d> sin— = ksinw. the orbital angle 9 is the independent variable. The action is J^~ivd4>=[E(k) . power supply ripple. III. (3. where k = 0 corresponds to the SFP and k = 1 corresponds to the separatrix orbit that passes through the UFP. s i n 2 ^ . the Hamiltonian for a stationary synchrotron motion is #o = ^ P 2 + 2 M . In this section. SYNCHROTRON MOTION III RF Phase and Voltage Modulations Particle motion in accelerators experiences perturbations from rf phase and amplitude noise.89) we obtain Ho = 2i/sk2.k2)K(k)}. wakefields.0) and (<fr. = dw.O). These perturbation sources cause rf phase or voltage modulations. etc. low frequency from power supply ripple and ground motion. the frequency spectrum of rf noise may contain high frequency arising from random thermal (white) noise. and (<fi. V — = A. In general.V)UFP = (TT. and thus the Hamiltonian value is a constant of motion.k2 sin w . The Hamiltonian has fixed points at ( ^ ) S F P = (0. the maximum action (k = 1) is J max = 8/TT. K(k) = r / 2 Jo l VI .268 CHAPTER 3.
can be obtained simply by integrating Eq. in Fourier harmonics of the conjugate angle parameter ip. (3. ^W ~ 2[1"(2j T"(2-4j 3 (2-4. which is conjugate to the action J.6} 5" J" Thus the action is related to the parameter k by J = 2k2(l + l-k2 + ^ + • • •). RF PHASE AND VOLTAGE MODULATIONS 269 For synchrotron motion with a small action. (3. Jo we obtain the angle coordinate ^ v = 4G7W*Jt. using Hamilton's equation 4> = vaV.94) Using the generating function Fa{<l>. (3. (3. we can relate the orbital angle 0 to the w parameter of Eq. .93). the power series expansions of elliptical integrals are K(k) = \ 1 + (\?e + ( I ^ | ) V + ( ^ | ) 2 * 6 + • • • ] . The synchrotron tune becomes 9 J ~ 2K(k) ~Ml ys(<7) where we have used the identities 2k^ = E(k) -K(k).e.91) 2fc2 = J^-TeJ-iej2 -•)• 8 256 }' ^ (3-93) In terms of the action. ^dJ§t = ^m-K(k). vs k V((/>) dJ The angle variable ip. and the synchrotron phase coordinate <j>.III.89) as vs{0 . First. i.e0) = f* § = u .96) The next task is to express the normalized off-momentum coordinate V.u0. V> = ^ 0 + ift>(3.J)= [*V($) dj>. the Hamiltonian is Ho{J).
dn2(w|A. u0 = I . v = 2toH. n=—oo (3.100) Gn(J) = ^. en and sn.98) Thus the expansion of V and sin(0/2) in Fourier harmonics of ijj is equivalent to the expansion of cn(u\k) and sn(u\k) in V = vu/2K.97) and the synchrotron phase-space coordinates are related to the Jacobian elliptical function by V = 2k cn(u\k). Sum rule theorem The solutions of many dynamical systems can be obtained by expanding the perturbation potential in action angle variables.cos^e-^di/j. sin | = k sn(u\k). (3. [25]. SYNCHROTRON . are then defined as sinw = sn(u|fc).cos 5V + ---.k2).e.102) . Jo VI — k2sin w Jo v l — k2sin w The Jacobian elliptical functions.c o s 2 V . The expansion of normalized coordinates in action-angle variables is useful for evaluating the effect of perturbation on synchrotron motion. aw. aw. i. For the case of rf phase modulation.— c o s # + • • •.99) where ip is the synchrotron phase with the q parameter given by with K' = K(\/l . This can be achieved by using Eq. (3. (3.2) in Ref. (3.u>o 1 MOTION .) = _v_^_^__ c o s ( 2 n + 1)^ (oj)3/2 cos3V' « (2J) 1 / 2 cosV + ^ — + (2J)5/2 i ^ . .23. (16. we <h 2sin2r= °° 1 J2 J2 Gn{J)e>n* « .[2n{l . (3.101) 2?r JO where G_n = G*. Gn = 0 for odd n.270 where u= rw I CHAPTER 3..). using the identity k2sn2(u\k) obtain = 1 . discussed below. . cosw = cn(u\k). Similarly. the expansion of the normalized off-momentum coordinate is oo ?= E W)eini>. Because 1 — cos<f> is an even function.
719 (1993). from Eq. Rev. RF PHASE AND VOLTAGE MODULATIONS where /_„ = f* and. Phys. Furthermore. the sum of all strength functions is (see Exercise 3. The resulting rf phase difference in every revolution is Aip = 2TTi/macos{vm9 + xo)For simplicity. In this section. we consider the case of a stationary bucket with 4>s = 0 for r\ < 0. Rev. the strength functions /„ are hm+1~ 271 2nVkqm+^ K(k)(i + q*"+iy _ hm~ Because V is an odd function. and xo is an arbitrary phase factor. / > where vm is the modulation tune. where 9 = u>ot is the orbiting angle serving as time coordinate. M. H. (3. only odd harmonics exist.107) 14 M. (3. 4678 (1993). 70. E 4 8 . 71.103) We observe that the strength functions are zero at the center of the rf bucket where J — 0 and at the separatrix where Qs(JSx) = 0.3. Using the normalized off-momentum coordinate V = — {h\rj\/vs)8. E 4 9 . Phys.2) E \fn\2 = ^~J^s n=-oo (3. Rev. 591 (1993). 1610 (1994). Ellison et al. eV 5n+1 = 5n + —(sm4>n+1-sixi(t>s). the synchrotron mapping equation is <l>n+l = <Pn + 27ThVSn + Aip(e). we consider only a sinusoidal rf phase modulation with14 < = <zsin(j/m0 + Xo). (3.2 RF Phase Modulation and Parametric Resonances If the phase of the rf wave changes by an amount <p(6). . Wang et al. Lett.106) where the perturbation potential of rf phase modulation is Hi = vmaV cos(i/m0 + xo).99).105) where A(p(9) = ip(9n + 2TT) — ip(9n) is the difference in rf phase error between successive turns in the accelerator.104) (3. Huang et al. a is the modulation amplitude. Rev. Phys.III. Y. Phys. III. we obtain the perturbed Hamiltonian H^H0 + Hl = -vj>2 + 2iss sin2 ^ + vmaV cos(t<m0 + Xo). Syphers et al. (3.. Lett.
$•$ — a/32\/2.n.I) = (Tl>-vm0-xo-n)IThe new phase-space coordinates become X = 1> . In the following. SYNCHROTRON MOTION Expressing the phase-space coordinate V in action-angle coordinates with Eq.vme . we consider only the dominant dipole mode below. B. In this section. Dipole mode If the phase modulation amplitude is small.1H) . particularly the 1:1 dipole mode. (3. A. vm = (2m + l)fs. Near the first-order synchrotron sideband with vm sa vs.xo)] (2J)3/2 +v™a IOQ [C0S(3^ + vm0 + Xo) + cos(3V> .110) The Hamiltonian can be transformed into the resonance rotating frame by using the generating function F2{il>.99). etc. However. VII. (3.xo . For example.108) where J and ip are conjugate action-angle variables. The rf phase error generates only odd order parametric resonances because V is an odd function.vm6 . i.109) where / i = a/y/2. Chap. we can expand the perturbation in action-angle variables Hi = vmcn/j/2 [cos(V> + vm6 + Xo) + cos(^ . This primary parametric resonance is called (2m +1):1 resonance. stationary phase condition exists for a parametric resonance term.e. two nearby strong parametric resonances can drive secondary and tertiary resonances. The effect of rf phase modulation on phase-space distortion can be solved by using the effective parametric resonance Hamiltonian. (3. the 1:1 and 3:1 parametric resonances driving by a strong phase modulation can produce a secondary 4:2 resonance at vm ~ 2vs. We neglect all non-resonance terms in Hi to obtain an approximate synchrotron Hamiltonian HKVSJ^ysJ2 + vmf2m+xJm+1'2 cos ((2m + l)tf . (3.272 CHAPTER 3.xo)] + • • •.vmd Xo). I = J. we discuss only the primary parametric resonances. 2. that resembles the Hamiltonian for 1-D betatron resonances discussed in Sec. (3.112) (3. the Hamiltonian for the dipole mode is HKVJ±-VSJ2 ID + ^J1'2 V2 cos(V .vm6 - Xo). Effective Hamiltonian near a parametric resonance When the modulation tune is near an odd multiple of synchrotron sideband.vm6 . the dominant contribution arises from the m = 0 sideband.
115).1 .III.115) \ vs J When the modulation tune is below the bifurcation tune i/\.1/2. with x = 0 or 7T. 273 (3.. which characterize the structure of resonant islands. where x = l-vm/vs. Eq. xbif = l-uhi{/vs.^ I 1 / 2 cosx. solid lines) and UFP (\gc\.114) The fixed points of the Hamiltonian.^( 4 «) 2 / 3 ] . 16 3 V VZbif/ Here ga and gt. The reason that ga and gt. 15Find the root of the discriminant of the cubic equation (3. RF PHASE AND VOLTAGE MODULATIONS and the new Hamiltonian is H=(vsvm)I . x bif = — (4o) 2 / 3 . where Hamilton's equations of motion are X = vs-vm--vsl-vs—j==cosx. and gb -)• 0.\f given by15 vw = *. we have f -> 7r/2. [l . are respectively the outer and the inner stable fixed points (SFPs) and gc is the unstable fixed point (UFP). In the limit vm <C vbif. (V = 0) [7~~x~\^ .±vj2 .113) Since the new Hamiltonian is time independent in the resonance rotating frame. Using g = %/2Jcosx. x = 0./ = x1'2 cos . dashed line) shown in the left plot of Fig. • gb(x) = ^=x"2 sin(^ . are given by the solution of 7 = 0.5 vs the modulation frequency is a characteristic property of the dipole mode excitation with nonlinear detuning. . 3. we obtain the equation for g as g3-ie(l-—)g + 8a = 0. Particle motion in the phase space can be described by tori of constant Hamiltonian around SFPs. to represent the phase coordinate of a fixed point.. are SFPs and gc is the UFP will be discussed shortly. thus ga -> -4x 1 / 2 . The lambda-shaped phase amplitudes of the SFPs (\ga\ and \gb\. (3. gc ->• 4a. i=-vs-V2Isinx(3.115) has three solutions: 8 £ (ip = ir) (V = 0) (3-116) 9a(x) = . ^ = arctanW (3.117) ^ ( x H ^ a ^ s m ^ + i). (3.| ) . a torus of particle motion will follow a constant Hamiltonian contour.
and they disappear together.-(jL)+ij -^-(JL). and ga = — 2(4a) x / 3 at x = lyfThe characteristics of bifurcation appear in all orders of resonances with nonlinear detuning. i. the UFP and the outer SFP move in and the inner SFP moves out. the phase space ellipses return to this structure in l/ivm revolutions. SYNCHROTRON MOTION The Hamiltonian tori in phase space coordinates V — — %/2/sin x vs X = %/2/cos x are shown in the right plot of Fig.r^/. and the unstable fixed point is gc. As the modulation tune approaches the bifurcation tune. (3.02. vm > z/bjf (x < Xbif). The intercept of the the separatrix with the phase axis is denoted by g\ and <2 ?When the modulation frequency approaches the bifurcation frequency from below. and right plot: Poincare surfaces of section for fm = 245 Hz and / s = 262 Hz at a = 0. resonance islands can be created or annihilated.i8) In particular.5: Left plot: fixed point amplitudes |ga|.115): / / 3 \ 1 / 3 / / 3 \ 1 / 3 fcW-(<. Figure 3..py be the local coordinates about a fixed point of the Hamiltonian.e. (3. y = V2IcosX-g. The stable fixed points are ga and <?(. The actual Hamiltonian tori rotate about the center of the phase space at the modulation tune vm. the UFP collides with the inner SFP with 9b — 9c = (4a) 1 / 3 . \gi. The torus passing through the UFP is the separatrix. C. Island tune Let y. which separates the phase space into two stable islands. ga = —(8a)1/3 at x — 0 (vm = vs). py = -V2IsmX. 3.119) . At the bifurcation frequency. i.) j . there is only one real solution to Eq.274 CHAPTER 3. (3.\.5.e. where x = rcbif and f = 0.. Beyond the bifurcation frequency. and \gc\ (in unit of (4a)1/3).
are useful in determining the maximum 16 M. These intercepts. (3. The equilibrium beam distribution (see Appendix A.122) where x — 1 — vmjv&.4). When the modulation frequency becomes larger than the bifurcation frequency so that [1 — (p3/4o)]1/2 -¥ 1.5 and 3. we obtain again ^isiand — K ( l .) = us \±xgl .113). Because gl/Aa < 0 and 0 < gl/Aa < 1. With a local coordinate expansion.(1 — j^g2) — vm\.jj-g2) . the Hamiltonian (3.120) can also provide information on the local distortion of the bunch profile. the linear superposition principle fails. 3. RF PHASE AND VOLTAGE MODULATIONS 275 where g is a fixed point of the Hamiltonian. In this region of the modulation frequency. When the modulation tune vm approaches fbif.a)xl2 —¥ 0. Sec.\ag^ . (3. gc is the UFP. at vm -C Vbu is approximately given by island ~ |fs(l —-j^ff2) —"m|.This means that the solution of the equations of motion can be approximated by a linear combination of the solution of the homogeneous equation with tune f s (l — j^g2) and the particular solution with tune ^m. With the UFP gc substitutes into the Hamiltonian (3.±g*c . the separatrix torus is H{J. (3.16 Thus the island tune is the beat frequency between these two solutions. Ellison et al. and the linear superposition principle is again applicable. which satisfies the Fokker-Planck-Vlasov equation. D. II. . Rev. Eq. shown in Figs. is generally a function of the local Hamiltonian.III. Since g\jAa > 1. ga and g^ are SFPs. 591 (1993). Lett. The separatrix orbit intersects the phase axis at g\ and g2. The island tune for large-amplitude motion about a SFP can be obtained by integrating the equation of motion along the corresponding torus of the Hamiltonian in Eq. 70.£)V2 + 7 ^ + • • • • Ag 4a Ag y (3-120) Therefore the fixed point g is a stable fixed point if (1 — g3/4a) > 0. Phys.113) becomes island = ^ ( 1 . Separatrix of resonant islands The Hamiltonian torus that passes through the UFP is the separatrix.6. The island tune for the small-amplitude oscillations is ( 2\ / 3 \ ^/^ '-fej-M'-y • (3-m) The island tune around the inner SFP given by gi. with (1 — gl/4. i.113). the island tune for small-amplitude oscillation about the inner SFP approaches 0 and the small-amplitude island tune for the outer SFP at vm = i^jf is i^and = 3|K.vm\.
hi and hi. = g. satisfies the equation H(J. h2 = -he + —==. At x = x0. and hc are shown in 3. (3.6: Thefixedpoints in units of (4a)1/3 are plotted as a function of the modulation frequency in x/xbn. This means that there are two non-intersecting tori with the same zero Hamiltonian value. hi. E.276 CHAPTER 3.123) besides the solution <j>0 = 0. and other tori orbit about the outer SFP.6.-hc 2 2 •==.123) is Uxo) = -2 5 / 3 (4a) 1 / 3 . The torus passing through the origin For a beam with small bunch area. which is the torus-O. With the notation h. (3. if)) = 0. there are three solutions to Eq.21/3:Cbif). Figure 3. two solutions of Eq. called the torus-O.123) where x = 1 . This means that the torus-O is also the separatrix of islands. The SFPs are K = ffa/(4a)1/3 and hb = ff6/(4a)1/3 and the UFP is hc = pe/^a) 1 / 3 . When the separatrix passes through the origin. or vm < v0 = K. SYNCHROTRON MOTION phase amplitude of synchrotron motion with external phase modulation. where x = 1 — vmlvs and Zbif = ^(4a) 2 / 3 with a as the amplitude of the phase modulation. (3.124) .{vm/vs).(1 . ha. The Hamiltonian torus passing through the origin.When x > x0 = 21/3Zbif. (3. the intercepts of the separatrix are hi . the phase axis intercept of Eq. The intercepts of the separatrix with the phase axis are shown as hi = ffl/(4a)1/3and/l2=fl2/(4a)1/3. all particles can be approximately described as having initial phase-space coordinates at the origin. The intercepts (j>0 of the torus-0 with the phase axis are then 4>0{<j>l .32x(j>0 + 3 2 a ) = 0. (3. One of the tori orbits about the inner SFP. and the fixed points. The intercepts of the separatrix with the phase axis. /(4a) 1 / 3 .123) become degenerate.
5. A separate function generator produces two modulating voltages. A. The intercept is then ^'--("•qK'-SE) III. The corresponding revolution period was 969 ns with an rf frequency of 1. initially at 0. or about 1910 revolutions (turns) in the accelerator.m0.4 m (or 60 ns). II. resulting in an rf phase shift ymod in the rf wave.123) besides <j>0 = 0. experiences the rf phase sinusoidal modulation with ipmod = asinz. 3.6. The phase lock feedback loop was switched off in our experiment. RF PHASE AND VOLTAGE MODULATIONS 277 At a higher modulation frequency with a. At 1 MHz. The low-frequency rf system of the IUCF Cooler at h = 1 was used in this experiment. Measurements of subsequent beam-centroid displacements have been discussed in Chap. The torus-0 orbits around the outer SFP. = 0.III. The experimental procedure started with a single bunch of about 3 x 108 protons with kinetic energy 45 MeV. The principle of the phase shifter used is as follows.= 0. Sinusoidal rf phase modulation When the bunch. resulting in a half-power bandwidth of about 25 kHz.3 U/ x* \"' +1J l"s -[('-££) "'] )• <3'125> \i x> v" 1"*] Measurements of Synchrotron Phase Modulation Here we discuss an example of experimental measurements of rf phase modulation at the IUCF Cooler. the synchrotron oscillation frequency was chosen to be about 540 Hz. the beam was kicked longitudinally by a phase shifter and the data acquisition system was started 2000 turns before the phase kick. The corresponding response time for a step rf phase shift was about 40~50 revolutions. These two modulated signals were added. each proportional to the sine and cosine of the intended phase shift <pmod. Both the phase error due to control nonlinearity and the parasitic amplitude modulation of the IUCF Cooler rf systems were controlled to less than 10%.. The control voltage versus actual phase shift linearity was experimentally calibrated. The cycle time was 10 s. the two rf channels are multiplied by sin (pmod and cos tpmOd respectively. using an rf power combiner.03148 MHz. The injected beam was electron-cooled for about 3 s. Sec.As a result of the amplitude modulation. the quality factor Q of the rf cavity was about 40. The full width at half maximum bunch length was about 5. < x0. In this experiment. where um is the modulation tune and a the modulation . there is only one real root to Eq. For the longitudinal rf phase shift. The response time of the step phase shift was limited primarily by the inertia of the resonant cavity. The rf signal from an rf source is split into a 90° phase shifted channel and a non-phase shifted channel. (3.
The modulation amplitude was a = 1. Thus the synchrotron equation of motion becomes (j> H LOQ (j> + v l s i n 4> = . and the initial phase kick amplitude was 45°. and A is the damping decrement due to electron cooling.7 show examples of measured <j) and V = ^-^ vs turn number at 10-turn intervals for an rf phase modulation amplitude of 1. (3.126) where <f> is the particle phase angle relative to the modulated rf phase. Since the measurement time was typically within 150 ms after the phase kick or the start of rf phase modulation. Figure 3. 3. the effect of electron cooling was not important in these measurements.278 CHAPTER 3. The synchrotron motion. The corresponding Poincare surfaces of section are shown in the right plots. 5 = sin</>. The upper and lower plots correspond to fm = 490 and 520 Hz respectively.127) The measured damping coefficient a at the IUCF Cooler was a = uj0X/4n « 3 ± 1 s"1.a v 2 m s i n v m 6 H UJQ vma cos vm6. is eV <f> = hr]d + uma cos vm6. (3.45° after an initial phase kick of 42° at modulation frequencies of 490 Hz (upper) and 520 . The subsequent beam centroid phase-space coordinates are tracked at 10 revolution intervals. in terms of a differential equation. The left plots of Fig. SYNCHROTRON MOTION amplitude with a -C 1.110).7: The left plots show the normalized off-momentum coordinate V and the phase 0 as functions of revolutions at 10-turn intervals. The solid line shows the Hamiltonian torus of Eq.45°. the overdot indicates the derivative with respect to the variable 9.X5. (3.
This procedure can improve the accuracy of data analysis. V) space by a constant factor h\q\l{ys^J\ cos0s|).129) is used to calculate k.^ 0 (3. . 1. (3.e.130). (3.E(k) and q functions in order to obtain the action J and ip. the action J is obtained from Eq. 2 tan^ = . Poincare surface of section The Poincare map in the resonance frame is then formed by phase-space points in (^/2Jcos(•0 . -V2Jsm{tjj .vm6)). B. and finally.128) in the {<j>. S) phase space is related to the action in the (0. can be obtained from the expansion i£i = £ tan( f " *>" 1 1 T T ^ s i n ^ (3-130) For synchrotron motion with relatively large k. Eq.90) or Eq.V). i.34) and (17. Eq. 2.36) of Ref. C.vm9).III.129) The action can be obtained from Eq. (17. [25] to evaluate K(k).91).3. V) phase space. (3. (3. (3.71) can be used to deduce the action and angle variables. The resulting response can be characterized by the beating amplitude and period. (3. The k value at the phase-space coordinates {<t>. and the beating amplitude is equal to the maximum intercept of Poincare surface of section with the phase axis.V) is *2 = ^ + sin2|. (3.3. The beating period is equal to To/VjSiand> where To is the revolution period and z/jSiand is the island tune. RF PHASE AND VOLTAGE MODULATIONS 279 Hz (lower). a better approximation for data analysis can be obtained through the polynomial approximation of Eqs. 17Note that the action in the (<j>.90).17 J = I ( 0 2 + p 2 ). Action angle derived from measurements For small-amplitude synchrotron motion. For large-amplitude synchrotron motion. we need to use the following procedure to deduce the action-angle variables from the measured synchrotron phase-space coordinates. For each data point {(j>. tjj. The synchrotron phase. The corresponding angle variable ip is obtained from Eq.
SYNCHROTRON MOTION The resulting invariant tori are shown in the right plots in Fig.113).858 to avoid nonlinear betatron resonances. Equation (2. The trajectory of a beam bunch in the presence of external rf phase modulation traces out a torus determined by the initial phase-space coordinates of the bunch.03168 MHz at 45 MeV proton kinetic energy. Sec.173) in Chap. (3. Sec. the phase slip factor was r) « —0.7 shows invariant tori deduced from experimental data. where the synchrotron frequency was fitted to be about 535±3 Hz. The change in the circumference is AC = D6(t) = D§ sin(wmi + xo). if the dispersion function at the modulating dipole location is not zero. (3. III. depends on the rf phase modulation frequency. 3. and the revolution frequency was /o = 1. the stable phase angle was 4>0 = 0.7.<f>x(s0)\). the horizontal closed-orbit deviation xco(t) becomes (see Chap. This effect is equivalent to rf phase modulation. 0 = Bm£/Bp. The solid lines are invariant tori of the Hamiltonian in Eq. and the response amplitude is the intercept of the invariant torus with the phase axis. the path length and thus the arrival time at rf cavities of the reference (synchronous) particle are also modulated. shows that the path length of a reference orbit is changed by dipole field errors at nonzero dispersion function locations. With horizontal dipole (vertical field) modulation at location s0. which passes through fixed initial phase-space coordinates. The synchrotron tune was vs = u>s/u)0 = 2.131) . 2. If the dipole field is modulated.280 CHAPTER 3. The rf voltage was chosen to be 41 V to obtain a synchrotron frequency of / s = WS/2TT = 262 Hz in order to avoid harmonics of the 60 Hz ripple. Here we discuss experimental measurements of dipole field modulation at the IUCF Cooler.4 Effects of Dipole Field Modulation Ground motion of quadrupoles and power supply ripple in dipoles can cause dipole field modulation.86.| ^ ( s ) . 2. the path length is also modulated. which gives rise to parametric resonances in synchrotron motion. and Bm is the peak modulation dipole field. Figure 3. The effect is a special type of "synchro-betatron coupling" that may limit the performance of high energy colliders. where 9{t) = 0sin(wm£ + xo). For this experiment. Since the torus. The corresponding smallest horizontal and vertical betatron sideband frequencies were 177 and 146 kHz respectively. It becomes clear that the measured response period corresponds to the period of island motion around a SFP. vz = 4. We chose vx = 3. Ill) ico(<) = VP*{s)Px{So)9(t) 2 sin •KVX c o s ( ^ . Furthermore.54 x 10~4. the harmonic number was h = 1. the measured tori depend on the driven frequency. Ill.828.
Expanding the term sin</> in Eq. the damping time for the 45 MeV protons was measured to be about 0. which was indeed small compared with us = 1646 s .134) where the damping coefficient is a = \u)a/kn. (3.132) where the fractional momentum deviation of particles (Ap/po) is the conjugate coordinate to synchrotron phase angle <j>.33 ± 0. where we measured the transient solutions. Eq.1 .82 m is the circumference of the IUCF Cooler. Because the synchrotron frequency is much smaller than the revolution frequency in proton storage rings. i. (3. where C = 86.134). We therefore measured the steady state solution. we obtain the equation for the modulation amplitude g as [-u2mg + 2ujJ1(g)]2 + [2aumgf = [wmw8a]2 + [2awsa]2 (3. Let the steady state solution of the nonlinear parametric dissipative resonant system.75 A. (3. the effective phase modulation amplitude parameter a is « = ^ = ^ A 0 . (3. where the magnetic field Bm is in Gauss.135) Although the cooling was weak. the transient solution of Eq. Thus the synchrotron equation of motion. Ap/po) at the nth and the (n + l)th revolutions are transformed according to the mapping equations <£ n+1 = 4>n + 2-Khr] ( ^ J +Acf>.136) where we used the approximation of a single harmonic.134) was damped out by the time of measurement. The longitudinal phase-space coordinates (0. in the presence of transverse dipole field modulation.1 s or a = 3 ± 1 s""1. — g (XL (XL (3.e. (3.134) up to the first harmonic. The equivalent phase modulation amplitude is enhanced by a factor wo/27rwm. With an electron current of 0.78 x 10~5i?m radians. (3. the maximum rf phase shift per turn A~4> was 0. In our experiment. be </>xgsin(Ljmt-x). becomes — + 2a— + W sin</> = w^a coscomt + 2aojsasinujmt. The corresponding rf phase difference becomes A<j> = 2TTII(AC/C).III. the phase errors of each turn accumulate. RF PHASE AND VOLTAGE MODULATIONS 281 where D is the dispersion function at the modulation dipole location. and A is the phase-space damping parameter related to electron cooling. in contrast to the experiment discussed in the previous section.137) .
When the modulation frequency is larger than the bifurcation frequency. with Xb w —TT/2 is the inner attractor. Xc) approach each other. only the outer attractor solution exists. ( ( W ) 2 + (2aa. they collide and disappear. i.s)2 \1/2 A. while the attractors rotate about the center of the bucket at the modulation frequency.137). Particles in the phase space are damped incoherently toward these attractors. At a large damping parameter. T h e stable solution at a smaller phase amplitude <?(. (3. T h e existence of a unique phase factor x f ° r solutions of the dissipative pararrietric resonant equation implies that the attractor is a single phase-space point rotating at modulation frequency w m . these fixed points of the Hamiltonian become attractors.137) are called attractors for t h e dissipative system. Eq. Xa) and the unstable solution (gc. (3. When the modulation frequency is far from the bifurcation frequency. 3. A stable solution with a large phase amplitude ga and phase factor Xa ~ T / 2 is the outer attractor. A weak damping force does not destroy the resonance island created by external rf phase modulation.e. or for the outer attractor at uim S> wt>if. as shown in Fig. the response amplitude for the inner attractor at ujm -C oJuf. can be approximated by solving the linearized equation (3. (3. Chaotic nature of parametric resonances In the presence of a weak damping force. the outer SFP and the UFP may collide and disappear. T h e third solution gc with Xc ~ —TT/2 corresponds to the unstable (hyperbolic) solution. When the damping parameter a is small. fixed points of the time-averaged Hamiltonian become attractors. Wbif. given by the condition OUm =0. Because of phase-space damping.e. i. the outer attractor solution disappears. They rotate about t h e origin at the modulation frequency [see Eq. these two stable solutions are nearly equal to the SFPs of the effective Hamiltonian.137) has three solutions. which is associated with the U F P of the effective non-dissipative Hamiltonian.5. When the modulation frequency is below t h e bifurcation frequency. and are almost opposite to each other in the synchrotron phase space. When the damping parameter a is increased. Steady state solutions of Eq. . the stable solution (flo. SYNCHROTRON MOTION l 9 ^i + ^)-2^mJl{g)l where J\ is the Bessel function [25] of order 1. As the damping force becomes larger.282 with the phase x given by = arctan CHAPTER 3.136)].
(3. Numerical simulations indicate that all particles located initially inside the rf bucket will converge either to the inner or to the outer attractor. . especially for particles outside the bucket. 250. Observation of attractors Since the injected beam from the IUCF K200 AVF cyclotron is uniformly distributed in the synchrotron phase space within a momentum spread of about (Ap/p) w ±3 x 10~~4. 3. where each black dot corresponds to initial phase-space coordinates that converge toward the outer attractor. (3. Figure 3. it also shows the rf waveform for reference.8. However.133). 240. 220. which converge to the outer attractor are shown for Bm = 4 Gauss and / m = 230 Hz. The synchrotron frequency is 262 Hz. Complementary phase-space coordinates converge mostly to the inner attractor except for a small patch of phase-space coordinates located on the boundary of the separatrix. The number of phase-space points that converge to the inner or the outer attractors can be used to determine the beamlet intensity.131). RF PHASE AND VOLTAGE MODULATIONS 283 Numerical simulations based on Eq. obtained from a numerical simulation of Eq. B. To which attractor a particle will converge depends sensitively on the initial phase-space coordinates.133) were done to demonstrate the coherent and incoherent nature of the single particle dynamics of the parametric resonance system. The orientation of initial phase-space coordinates converging toward the inner or the outer attractor depends on the initial driving phase xo of the dipole field in Eq.9 shows the longitudinal beam profile accumulated through many synchrotron periods with modulation field Bm — 4 G for modulation frequencies of 210. Figure 3. initial phase-space coordinates in a small patch located at the separatrix of the rf bucket converge toward two attractors moving along the separatrix. One of the results is shown in Fig. and 260 Hz.III. which will converge toward two attractors located near the separatrix. all attractors can be populated. (3. 230. The phase coordinates of these attractors could be measured by observing the longitudinal beam profile from BPM sum signals on an oscilloscope.8: Initial normalized phase-space coordinates. The basin of attraction for the inner and the outer attractors forms non-intersecting intervolving spiral rings.
as if there were no synchrotron motion for the beam bunch located at a relatively large phase amplitude. we found that the beam profile was not made of particles distributed in a ring of large synchrotron amplitude. The sine waves are the rf waveform. The modulation amplitude was Bm = 4 G. using a fast sampling digital oscilloscope (HP54510A) for a single trace. The relative populations of the inner and outer attractors can be understood qualitatively from numerical simulations of the attractor basin. which measured the peak current of the beam. 14 o) V27TCT2 where pi and p% represent the populations of the two beamlets with px + p 2 = 1. Both beamlets rotated in the synchrotron phase space at the modulating frequency. then the sum signal. the beam profile became flat with a smaller peak current. would show a large signal at both extremes of its phase coordinate. the current density distribution function becomes p^t\ = fj V27T<Ti e-[4>-Mt)?l2°l + P^e-[*-Mt)?lz°^ ( 3 . It was puzzling at first why the longitudinal profile exhibited gaps in time domain. If the equilibrium distribution of the beamlet was elongated. Therefore the profile observed with the oscilloscope offered an opportunity to study the equilibrium distribution of charges in these attractors.9: Oscilloscope traces of accumulated BPM sum signals showing the splitting of a beam bunch into beamlets below the bifurcation frequency. When the beamlet rotated to the central position in the phase coordinate.Since each particle in the two beamlets rotates in the phase space at modulating frequency .284 CHAPTER 3. If we assume an equilibrium elliptical beamlet profile with Gaussian distribution. However. as measured from the fast Fourier transform (FFT) of the phase signal. SYNCHROTRON MOTION Figure 3. where the peak current was large. but was composed of two beamlets.
the hysteresis depended also on the dissipative force. the parameters </>i]2 and <Tii2 are <j>\(t)=gasin(umt . For example. these profiles were not sensitive to the relative positions of the two beamlets. obtained by solving Eqs. Since the profile observed on the oscilloscope was obtained by accumulation through many synchrotron periods. .x 6 ).III. the center peak disappeared (see 260 Hz data of Fig. if the modulation frequency. where the amplitudes of the coherent 7r-mode oscillations showed hysteretic phenomena. On the other hand. 3. given by 1 : 1 + r\ of the outer beamlet at modulation frequency 220 Hz was found to be about 1:3 from the profile in Fig. At a modulation frequency near the bifurcation frequency. RF PHASE AND VOLTAGE MODULATIONS u)m. obtained by fitting the data. (3. cr2. the phase amplitude jumped from the outer attractor to the inner attractor solution. The hysteretic phenomena of attractors The phase amplitudes of attractors shown in Fig.137). Hirata.9. The relative populations of the two beamlets was about 75% for the inner and 25% for the outer.Xa).137) and (3.138). i-e. 3. was ramped downward. Accel. Similar hysteretic phenomena have been observed in electron-positron colliders. the phase amplitude of the synchrotron oscillations increased along the outer attractor solution. This means that the peak current for the outer beamlet was reduced by a factor of 3 when this beamlet rotated to the center of the phase coordinate.i>. p.9). and <7io and <72Q represent the average rms bunch length. Since a large damping parameter could destroy the outer attractor.18 At a large beam18See T.10 also exhibited hysteresis phenomena. the amplitude of the synchrotron oscillations jumped from the inner to the outer attractor solution. When the modulating frequency was higher than the bifurcation frequency w^f. 4>2(*) = 56sin(wmt . 3. As the modulating frequency increased toward the synchrotron frequency. related to beam-beam interactions.b are the amplitudes and phases of the two beamlets. 285 Here <7aii) and Xa. (3. the phase amplitude of the outer beamlet became smaller and its population increased. The eccentricity parameters r\ and r2 signify the aspect ratio of the two beamlets. which was initially above the bifurcation frequency. the amplitude of the phase oscillations followed the inner attractor solution. When the modulation frequency. Ieiri and K. Conf. When it reached a frequency far below the bifurcation frequency. 1989). 926 (IEEE. = cr20 (1 + r2 sin2 wm£). Proc. The observed phase amplitudes were found to agree well with the solutions of Eq. it did not depend on the parameters Xo. the aspect ratio. New York. was ramped up toward the bifurcation frequency. 1989 Part. The hysteresis depended on beam current and modulation amplitude a. originally far below the bifurcation frequency. and CT? = <7io (1 + r i sin2 wmi). C.
The solid lines show the synchrotron tune and its third harmonic. On the basis of the KAM theorem. Systematic property of parametric resonances The formalism discussed so far seems complicated by the transformation of phasespace coordinates into action-angle variables. In this section. The circles in Fig. the essential physics is rather simple. 1987 Part. 3580 (1979).R. Thus the external perturbation excites only particles locally in the phase space where the amplitude dependent synchrotron tune falls exactly at the modulation tune. are compared with the theoretical synchrotron tune. 3. the perturbed Hamiltonian contains a perturbing term similar to that in Eq.M.P. . Proc. and the measured third order 3:1 resonance islands fall on the curve of the third harmonic of the synchrotron tune. However. measured from the oscilloscope trace.19 D. IEEE Trans.108). Accel. New York. Sci.H. most of the Hamiltonian tori are not perturbed except those encountering a resonance condition. When an external time dependence perturbation is applied to a Hamiltonian system. The third order resonance island falls also on the third harmonic of the synchrotron tune. Jackson and R.286 CHAPTER 3. Because the rf phase modulation does not excite 2:1 resonance. 1987). Paterson. NS-26. Con}. The sideband around the first order synchrotron tune corresponds to the 60 Hz power supply ripple. Donald and J. The bifurcation of the resonance islands follows the unperturbed tune of the synchrotron Hamiltonian. 3. (3. G. Siemann. SYNCHROTRON MOTION beam tune shift. shown as the lower solid line (see also Fig. the vertical beam size exhibited a flip-flop effect with respect to the relative horizontal displacement of two colliding beams.10: The phase amplitudes of beamlets excited by rf phase modulation. we will show that the global property of parametric resonances can be understood simply from Hamiltonian dynamics. 1011 (IEEE. where the particle 19See M.10 show a compilation of beamlet phase amplitude vs modulation frequency for four different experimental phase modulation amplitudes. We note that the bifurcation of the 1:1 resonance islands follows the tune of the unperturbed Hamiltonian system.H. p. Figure 3. we did not find parametric resonances at the second synchrotron harmonic.5 on the bifurcation of 1:1 parametric resonance). Nucl.
An important implication of the above parametric excitation theorem is that chaos at the separatrix orbit is induced by overlapping parametric resonances. can be excited by time dependent perturbation.109). ground vibration. and the local potential well becomes the basin for stable particle motion. However. 3. rf phase error. we do not observe a 2:1 attractor in Fig. Now. where higher order nonlinear resonances serve as the source of time dependent modulation. In reality. however. Thus a beam inside an rf bucket can split into beamlets. a perturbation with low frequency modulation can produce many overlapping parametric resonances near the separatrix and lead to local chaos. In fact. . 3.III. a stronger phase modulation is applied to the dynamical system. forming islands within the bucket. If the beam size is relatively small. as clearly seen in Fig. i. the perturbation arising from wakefields. This result can be applied to synchrotron motion as well as to betatron motion. Let Q(J) be the tune of a dynamical system. If. particle orbits near the center of the bucket will be strongly perturbed. The 20The remaining terms play the role of time dependent perturbations to the effective Hamiltonian of Eq.. a 2:l-like (4:2) parametric resonance can be formed by 1:1 and 3:1 resonances through second order perturbation. Now a time dependent perturbation can induce a series of parametric resonances in the perturbed Hamiltonian. when the modulation frequency approaches the tune of particles at the center of the bucket. These parametric resonances. dipole field error. This can be understood as follows. When the modulation frequency is varied. SFPs (attractors) are formed along the tune of the unperturbed Hamiltonian. Q(JSx) = 0. etc. we apply this result to evaluate the effect of low frequency modulations on particle motion. where the tune is zero at the separatrix. the SFP becomes an attractor. When a weak damping force is applied to the dynamical system. Since nQ(Jsx) ~ 0 for all n near the separatrix.10. Since the rf phase modulation does not excite even synchrotron harmonics. vm = n&(J S F P ). located at nQ(J) with integer n.109). it will induce overlapping parametric resonances only near the separatrix. If the amplitude of low frequency modulation is not large. i.10. a n d the amplitude of perturbation.141) The measurement of attractor amplitude vs modulation tune is equivalent to the measurement of synchrotron tune vs synchrotron amplitude. the external perturbation creates a local minimum in the potential energy at the SFP locations. consists of a spectrum of frequency distributions. the strength function gn(J).e. RF PHASE AND VOLTAGE MODULATIONS 287 motion can be described by the effective parametric resonance Hamiltonian (3. the stochasticity at the separatrix will do little harm to the beam motion.20 The size of the resonance island depends on the slope of tune vs amplitude. (3. (3.e. The KAM theorem produces a hierarchy of higher order resonance islands within parametric resonance islands.
N. et al. For details see M. Shih and A. Byrd. Part. Instru. 11th Int. S. there has been some interest in employing rf voltage modulation to induce super slow extraction through a bent crystal for very high energy beams. power supply ripple. (1997). SYNCHROTRON MOTION mean field of the perturbation gives rise to the effect called potential well distortion. 92. The complicated collective instability phenomenon is in fact closely related to nonlinear beam dynamics. Accel 42. Caussyn et al. Li et al. (3. 107 (1982). 1980).J. 3484 (1979). Li et al. Conf. the synchrotron equations of motion are (/>n+1 = </>n-2vvaj^Pn. M. 620 (Birkhauser.21 III.. The remaining time dependent perturbation can generate further bunch deformation. Proc. 25 M. Lee. bunch splitting.. W.22 Recently. (1997). 235 (1993). Sweeping the rf phase modulation frequency and measuring the response by measuring either the centroid of the beam.g. Boussard. Rosenzweig. Since the rf voltage modulation may be used for enhancing a desired beam quality. Dome.G. 1993). 555 (2002). modifies the unperturbed tune of the system. Accel. NS-26. depending on its frequency spectrum. Phys.143) a series of beam transfer function measurements were made at electron storage rings.M. SSCL-389 (1991). Rice. Nucl. Kick. solved self-consistently in the Vlasov equation. etc. p. Basel. Conf. (1997). 2 4 D. et al. J. Minty et al. CERN 87-03. and S.23 rf voltage modulation to stabilize collective beam instabilities. R. Wang. Ace. . Peggs. Part. ibid. 5 RF Voltage Modulation The beam lifetime limitation due to rf noise has been observed in many synchrotrons. E 48. Piscataway. etc. the super proton synchrotron (SPS) in CER.M.24 The rf voltage modulation has been implemented to stabilize coupled bunch instabilities induced by parasitic rf cavity modes with high brightness beams at the Taiwan Light Source. Krinsky and J. 205 (1995). on High Energy Accelerators. rf voltage modulation for extracting beam with a short bunch length. (3. Accel 12. Methods A 364. Part. 370 (1987). D.142) Vn+i 21Recently. ibid. p.288 CHAPTER 3.Y. Boussard. D. wakefields. IEEE Trans. et al. D. 29 (IEEE. Gabella. 2 2 D. Con}. Wang. etc. 2 3 H. the response of the beam to external rf phase modulation can be obtained. J. G. Nucl. Proc.H. Sci. and S. The equation of motion with rf voltage modulation In the presence of rf voltage modulation. NJ. D.25 A. p. R1638 (1993). Beam response to externally applied rf voltage modulation has been measured at the IUCF Cooler. 1997 Part. hysteresis. that may arise from rf noise.D. we will study the physics of synchrotron motion with rf voltage modulation. Rev. e. = Pn-2irvs[l + bsm{vm9n+1+X)}sm<l>n+1-—Vn. private communications. or the beam profile from a synchrotron light monitor using a streak camera.H. Wang. Journal of Applied Physics. Proc. Taratin. which.
. W = 2TT/O is the angular revolution frequency. equivalently. 2i/«(l . we need to consider only the lowest order Mathieu instability.III. i. r\ < 0. (3.146) Similarly. In other words. 7 is the phase slip factor. b = AV/V is the fractional rf voltage modulation strength (b > 0).D. Thus stable solutions of Mathieu's equation are obtained with the condition that the parameter p is bounded by the characteristic roots aT(q) and bT+i(q). the phase-space damping rate was measured to be about a « 3. vs = Jh\r]\eV/2Tr/32Eo is the synchrotron tune at zero 7 amplitude. 9 is the orbital angle used as time variable. Eq. Neglecting the damping term.1. and q = 2fo/ s2 /^. Without loss of generality.M.0 ± 1.147) where the second order Mathieu resonance can be obtained from second order perturbation theory.26 The width of the instability decreases rapidly with increasing order for small b. the second order unstable region is ". Eo is the beam energy. o and a is the phase-space damping factor resulting from phase-space cooling. (3. In our application.144) where the overdot indicates the time derivative with respect to 9. which is much smaller than wo^si typically about 1500 s" 1 for the h = 1 harmonic system.(1 . p and q are real with 5 < 1. Landau and E. Oxford.2. • • •. where r = 1. (3.144) reduces to Mathieu's equation. 8 = Ap/po is the fractional momentum deviation from the synchronous particle. we can linearize Eq. 1976). • • •. ed. x is a phase factor. a = 0.145) In accelerator physics applications. the equation of motion for phase variable (f> is ij> + v2s{\ + bsm{vm0 + x)] sin<£ = 0. 26L. 3rd. (3.144) into Mathieu's equation [25] ^ 4 + (p . we discuss the case for a particle energy below the transition energy. i.2q cos 2z)(j> = 0. Lifschitz.e.2. By choosing X = -TT/2 and z = \vm9. vm is the rf voltage modulation tune.. In linear approximation with sin<jf> « <j>. (Pergamon Press.(l + ^& 2 ). p = 4^/j/^. At the IUCF Cooler. RF PHASE AND VOLTAGE MODULATIONS 289 where V = -h\ri\5/vs is the normalized off-momentum coordinate conjugate to <> /. where r = 0. Since synchrotron motion is nonlinear.\b) <vm< 2PS{1 + hb).e.0 s"1. the linear Mathieu instability analysis should be extended to nonlinear synchrotron motion as follows. (3. (3. Z Thus the first unstable region is obtained from 61 (q) < p < < i (q) or. unstable solutions are in the region br(q) < p < ar(q). Mechanics.^& 2 ) < ^m < «/.
cos <t>]. (3.n]. induced by the external harmonic modulation of the rf voltage.150) where we choose x — 0 for simplicity. and |G n (J)| is the Fourier amplitude of the factor (1 — cos</>) with 7 n its phase. (3.101). particle motion is coherently perturbed by the rf voltage modulation resulting from a resonance driving term (stationary phase condition). defined in Eq. vm fa nQs (n = even integers). (3. SYNCHROTRON MOTION The synchrotron equation of motion with rf voltage modulation can be derived from the Hamiltonian H = Ho + Hi with Ho = ^usV2 + vs{l-cos<f>).n^ 7n).149) where Ho is the unperturbed Hamiltonian and Hi the perturbation.101) is zero except for n even with G_n = G*n. The resonances.148) (3. Since (1 — cos(j>) is an even function of T/J in [—ir. are called parametric resonances. The perturbed Hamiltonian CHAPTER 3. i.151) . Using the generating function we obtain the Hamiltonian in the resonance rotating frame as H = E{J) . the Fourier integral for Gn from Eq. we expand Hi in action-angle coordinates of the unperturbed Hamiltonian Hi = vjb £ \Gn{J)\ sm(i/m0 . we obtain U0 ~ 2 J + 2048J + ' "" ' U 2 ~ ~ 4J + 128J + ' U i 64J + 2048 J ° 6 4096J +"""- The GQ(J) term in the perturbation can contribute to synchrotron tune modulation with a modulation depth AQS « -J/s6sinfm^.290 B. Expanding Gn(J) in power series. 6). Hi = psb sin(z/m0 + x) [1 . For a weakly perturbed Hamiltonian system. $. C. (3. Thus rf voltage modulation generates only even-order synchrotron harmonics in H\.e. n=—oo (3.—J + vsb\Gn{J)\cosn4> + AH(J. Parametric resonances When the modulation frequency is near an even harmonic of the synchrotron frequency.
158) vm = 2i/. Note here that \Gn+2/Gn\ ~ J for n > 0. •••.11. RF PHASE AND VOLTAGE MODULATIONS 291 where the remaining small time dependent perturbation term AH oscillates at frequencies um. 0. if i/m < 21/. .153) is autonomous. (3.155) o 4 j> = us-^f-^J+^bcos2ip.^ L ) .154) (3. 2vm.s + I^ (3-15?) with ip = 7r/2 and 3TT/2.III. particle motion is governed by the n = 2 parametric resonance Hamiltonian (H) = (!/„ -V-f)J . according to (3. Thus the time averaged Hamiltonian (H) for the nth order parametric resonance becomes (H) = E{J) . Quadrupole mode When the rf voltage modulation frequency is near the second harmonic of the synchrotron frequency.146).s-I^<. For simplicity. We note that the second harmonic rf voltage modulation can induce an instability at J UFP = 0 in the frequency domain 2us . Nonlinear synchrotron motion extends the instability to lower modulation frequency at larger synchrotron amplitude.^ J 2 + £&Jcos2tf Z ID 4 (3.\vtb. Tori of the Hamiltonian flow around SFPs are shown in Fig. This is the first order Mathieu resonance of Eq. In the time average.| 0. (3. The system is most sensitive to the rf voltage modulation at the second synchrotron harmonic. Since the Hamiltonian (3. we drop the tilde notations.— J + vsb\Gn{J)\cosni>. . The unstable fixed points (UFPs) are located at j J U F P /8(l-£)-2&. we have (AH) « 0. the Hamiltonian is a constant of motion.i&i/. The resonance strength is greatest at the lowest harmonic for particles with small phase amplitude.(l . D.152) n The phase-space contour may be strongly perturbed by the parametric resonance. Li (3.153) in the resonance rotating frame. if2. o z .\bvs < vm < 2vs + \bvs. 3. tp = 0. Hamilton's equations are j = ^-bJsm2if).m<2. if vm > 2vs + ~bvs with tp = 0 and TT. Zi The fixed points that determine the locations of the islands and separatrix of the Hamiltonian are obtained from J = 0. The stable fixed points (SFPs) are (.
To obtain the island tune. rotating about the longitudinal phase space at half the modulation frequency. Proc. Since electrons are damped incoherently into the SFP by the synchrotron radiation damping. European Part. Island tune and equilibrium beamlet profile The island tune i/isiand. p. The synchrotron frequency is / s = 263 Hz. where /iSiand is the frequency with which a particle rotates around a SFP in the resonant precessing frame. the Mathieu resonance gives rise to an UFP at the origin of the phase space and the SFP is displaced to JSFP = 26. Accel. Con}.D. 1992). The damping mechanism may be understood as follows. Py = — v2JsinV> + y2Josim/>oj 27J. Fox and P. Modulation of the rf voltage at the second harmonic of the synchrotron tune has been found useful in damping the multi-bunch instabilities for the damping ring of the Stanford linear collider (SLC). defined by /isiand//o. the beam distribution becomes dumbbell-shaped in phase space.153) in the resonance rotating frame. the collective instability of high brightness electron beams in the SLC damping ring can be controlled. SYNCHROTRON MOTION Figure 3. 1079 (Springer-Verlag. and the modulation frequencies are / m = 526 Hz (left plot) and fm = 490 Hz (right plot).292 CHAPTER 3. Corredoura. with y = \/2J cosij) — y2J 0 cosi/'o. Heidelberg. which is a nonlinear extension of Mathieu instability. the voltage modulation amplitude is b = 0. we expand the phase-space coordinates around a fixed point of the Hamiltonian. The size and orientation of the dumbbell can be controlled by parameter b and phase xE. . or equivalently the synchrotron frequency.05. When the voltage modulation at vm = 2v& is applied.27 By adjusting the amplitude and phase of the rf voltage modulation.11: The separatrix and tori of the Hamiltonian (3. is an important property of a resonant system.
which satisfies the Vlasov equation. as shown in Fig. (3. it is the line density of the bunch or the projection of the density distribution function onto the phase coordinate P(y) = I p(y.161) are ay = <TQ/yJ\B\.163) (3. The small amplitude island tune is inland = \/AB. Eq. the beam profile will retain its shape except for exchange of its local coordinates. the . Since the rms widths of the distribution function (3. In terms of the local coordinates py and y. aPy = ao/y/\A\. For b > 0.-vaJo sin2 Vo .y .153).159) T ^ O COS2 Vo + -Ms ~ -l/aJ04 4 o It is clear that the fixed point is stable if the parameters A and B satisfy AB > 0. and Jo = JSFP — 8(1 — vm/2vs) + 2b are stable because A = -\vsb and B — —\VSJSFP SO that AB > 0.^ K . RF PHASE AND VOLTAGE MODULATIONS 293 where (v^^JoCosip0.162) the aspect ratio of the phase-space distribution <JPy/oy is ^J\B/A\ evaluated at the SFP.III. The longitudinal profile monitor measures the image current on a wall gap monitor. TT/2.. Since the resonance rotating frame rotates in the phase space at half the modulation frequency. n. the fixed points associated with i/i = 0.g^s Jo. Assuming a Gaussian distribution. the Hamiltonian. becomes H =\A with A = vs .J Z ^ J U F P and B = \bvs. where nal/VAB is the rms phase-space area of the beamlet. we obtain *y^»1BM-^-^M (-6) 3 11 in the resonance precessing frame. When the beamlet rotates to the phase coordinate. (3. The small amplitude island tune at the SFP becomes v^BA = y/AB = va^^-. and so AB < 0. (3.11.159).Py)dpy. is a function of the Hamiltonian (3.3IT/2.160) JUFP and Jo = are unstable since The equilibrium beam profile. The fixed points associated with tp = A = . -^/2Josin^o) are the phase-space coordinates of a fixed point of the Hamiltonian in the resonance precessing frame. B = l/s--^Z V \ + l-By2 + ••• (3. 3.
/max = (1 + \fl-x)JS¥V.VFP). which passes through the UFPs. G. F. SYNCHROTRON MOTION aspect ratio of the beamlet becomes yjJSFP/26.\bvs if 2vs . and the peak current will be large. is given by H(J.11 shows also the intercepts of separatrix with phase axis. The intercepts can be used to determine the maximum synchrotron phase oscillation due to rf voltage modulation.n and J max given by i n = (1 . Figure 3. On the other hand.E). The separatrix intersects the phase axis at the actions Ji and J 2 given by j and J.164) = l JSFP + \/JIFP ~ JUFP \ 2JSFP ^ ^m < 2vs .\bvs <vm< ^ 2vs + \bvs ^ ^m < 2i-s . The island size A</>jSiand is \f2J[ — y/2J2.^UFP if (3. the Hamiltonian is a constant of motion.\bvs . = ] ^ S F P ~ V^SFP . the action J is limited by Jm. with x € [^UFP/^SFP > !]• Note that JSFP = |(^min + Jmrn). Using Hamilton's equations of motion. where x = E/Es. the aspect ratio becomes y26/JsFP and the line density becomes small. when the beamlet rotates to a position 90° from that of Fig. The Hamiltonian value is Es = ^ ^ S J | F P at SFP.153). and Eu = J^S-'UFP a^ UFP.294 CHAPTER 3. \ 0 if 2vs .The island tune becomes .3 16g. . which is normally much larger than 1.iP) = H(JVFP.i.11. The amplitude dependent island tune of 2:1 parametric resonance For an autonomous dynamical system governed by the Hamiltonian (3. 3.\bvs <vm< 2vs + \bvs. where For a given Hamiltonian value E.X) JSFP.Vl . The separatrix The separatrix torus. we obtain J = f(J.
with a non-vanishing perturbation strength function Gn(J). The . where J is the invariant action of the particle motion. producing a full width at half maximum bunch length of about 9 m (or 100 ns) depending on rf voltage. the perturbation creates n resonance islands in the longitudinal phase space around J = J r . Near the resonance island.160) at x = 1. the island tune of the separatrix orbit is zero. the island tune is zero at the separatrix with x = JUFP/^SFPH.03168 MHz. (3. Voltage modulation control loop The voltage control feedback of the IUCF Cooler rf system works as follows. The cavity rf voltage is picked up and rectified into DC via synchronous detection. RF PHASE AND VOLTAGE MODULATIONS where K(k) is the complete elliptical integral of the first kind [25] with . where J r is obtained from the resonance condition: vm — nQs(JT). Physical interpretation In the longitudinal phase space. (3. This oscillation tune is equal to the island tune. These islands precess in the phase space at 1/n of the modulation tune.168) reduces to Eq. the action J is no longer invariant and the synchrotron tune is likewise perturbed. The experiment started with a single bunch of about 5 x 108 protons with kinetic energy 45 MeV. Particles located far from resonance islands experience little effect on their synchrotron motion if the voltage modulation amplitude is small. In other words. For particles located at the "separatrix" of the parametric resonance (not the separatrix of the rf bucket) the period of this amplitude oscillation becomes infinite. The synchrotron tune of a particle on the separatrix is exactly vm/n. The low frequency rf system used in the experiment was operating at harmonic number h = 1 with frequency 1.6 Measurement of RF Voltage Modulation We describe here an rf voltage modulation measurement at the IUCF Cooler. The cycle time was 10 s. When the rf system is perturbed by a harmonic voltage modulation at vm. Similarly. _ 1 / % JSFP — JUFP 295 It is easy to verify that Eq. or equivalently the synchrotron tune at the resonance action J r . The synchrotron tune for particles at or near SFP is also vm/n with an amplitude modulation whose tune is equal to the island tune.III. A. particles execute synchrotron motion with an amplitude dependent tune QS(J). III. with the injected beam electron-cooled for about 3 s. A particle executing synchrotron motion within the nth order resonance island will have a characteristic tune f m /n with a regular amplitude oscillation due to the island motion.
Then the steady state bunch distribution was measured. The error found goes through a nearly ideal integrator that has very high DC gain.296 CHAPTER 3. A fast 1 x 109 sample per second oscilloscope was used to measure the profile of the beam in a single pass.2. i. the effect of its inertia can be ignored if the loop gain is rolled off to unity well before /o/2Q. The amplitude modulation is summed with the reference and compared to the cavity sample signal. The maximum modulation rate is limited by the loop response time of about 10 kHz. The overall loop response exhibits the exponential behavior prescribed by a first order differential equation. the rf voltage was modulated. The beam was injected.05 at modulation frequency fm = 480 Hz with synchrotron tune / s = 263 Hz.75 A. and the beam was cooled with electron current 0. The modulation amplitude was measured and calibrated. s. The modulation rates in our experiments are well within this limit. Thus the phase .200 fj.12 shows that the sum signals from a beam position monitor (BPM) on a fast oscilloscope triggered at the rf frequency exhibited two peaks around a central peak. Note that the outer two beamlets rotated around the center beamlet at a frequency equal to half the modulation frequency. SYNCHROTRON MOTION rectified DC signal is compared to a preset voltage. Thus. The voltage modulation amplitude is 6 = 0. Because of the relatively low Q of the cavity at the IUCF Cooler. The beam particles were damped to attractors of the dissipative parametric resonant system. we first measured the phase oscillation amplitude of the steady state solution by using the oscilloscope. dV/dt = —V/T. Figure 3. The modulation causes a change in the error voltage sensed by the control loop and results in modulation of the attenuator around a preset cavity voltage. Figure 3.12 indicated that there were three beamlets in the h = 1 rf bucket. where V is the rf voltage and the characteristic relaxation time r is about 10 .12: The beam bunch was observed to split into three beamlets in a single rf bucket measured from a fast sampling scope. 3. Observations of the island structure Knowing that the beam bunch will be split into beamlets.e. B. The profile shown in Fig. no proportional error feedback is needed to stabilize the loop. as shown in Sec. where /o is the resonant frequency of the rf cavity and Q ~ 50 is the cavity Q value. III. The integrated signal is then used to control an attenuator regulating the level of rf signal being fed to rf amplifiers.
A possible explanation is that the actual beam size was larger than the separation of islands. (3. the observed beam profile in an oscilloscope is a time average of the BPM sum signal. the resulting beam profile will exhibit two peaks at the maximum phase amplitude. Experimentally. Once fm reached 2/ a . where we did not observe beam splitting. On the other hand.EXERCISE 3.13. Figure 3.3 1. .90). where JUFP = 0. The actions of UFP J UFP and intercepts J\ and J2 of the separatrix with the phase axis are also shown. Since the attractors (or islands) rotate around the origin of the rf bucket with half the modulation frequency. the synchrotron frequency was determined more accurately to be about 263±1 Hz for this run. JSFP is also a linear function of modulation frequency. 3. It was also clearly observed that all parametric resonance islands ceased to exist at / m = 2/ s + | 6 / s « 532 Hz. Exercise 3. when a beamlet rotates to the flat position.153) is also shown for comparison.156) fits data with /„ = 263 Hz. 3. Using this sensitivity. resembling that in Fig. The measured action J of the outer beamlets as a function of modulation frequency is shown in Fig. Here J ~ j ^ 2 with 0 as the peak phase amplitude of attractors. where J SFP of the Hamiltonian (3.3 297 amplitude of the outer peaks measured from the oscilloscope can be identified as the phase amplitude of the SFP. a larger peak current can be observed. The solid line for JSFP obtained from Eq.\bfs « 520 Hz. the aspect ratio becomes small and the line density is also small. where the slope depends sensitively on synchrotron frequency.520] Hz. This implies that when a beamlet rotates to the upright position in the phase coordinate. where the SFPs are located on the V axis. Different symbols correspond to measurements at different times for an almost identical rf voltage. Because the equilibrium beamlet distribution in a resonance island has a large aspect ratio in the local phase-space coordinates. In this case. Prove the identity of the action integral in Eq. (3. Similarly.12. we found that the action of the outer attractor varied linearly with modulation frequency. the beam was observed to split into only two beamlets. the SFPs were about 100 ns from the center of the bucket. Our experimental results agreed well with the theoretical prediction except in the region / m € [510.13: The measured action J of outer beamlets as a function of modulation frequency.
Jo show that the coordinate transformation between phase variable ip and coordinate (/> where ip is the conjugate phase variable to the action J. n=—oo 00 prove the sum rule theorem n=-oo "» 3. where D(SQ) is the dispersion function at the dipole location. Prom Exercise 2. m t + xo). Expanding V in action-angle variables with P = £ fnein*.4. The modulating tune is um = cjm/u)Q. Give a physical argument that the amplitude of the equivalent rf wave phase error a is amplified as t h e modulation t u n e vm becomes smaller. we find that the change of orbit length due to a modulating dipole kicker is given by AC = D{s0) 6(t) = D{sQ) 6 sm{umt + X o). However. (b) Show that the amplitude of the equivalent rf wave phase error is o = A(/)/2in/m. where wo is the angular revolution frequency. and h is the harmonic number. fs is the small amplitude synchrotron tune. . (a) Show that the modulating dipole field produces an equivalent rf phase error A0 = cS° sin(cjmt + xo) = A>sin(o. 28 In linear approximation. small amplitude behavior of the potential is not a necessary condition for the sum rule theorem stated in this exercise. Using the generating function F2 = / Vd<t>. and V(<j>) is the potential. 9 is the maximum dipole kick angle. where C is the circumference of the synchrotron.298 2. uim is the modulating angular frequency. V) are conjugate phase-space variables with orbiting angle 8 as time variable. where ((j>. and xo is an arbitrary initial phase. the potential can be expressed as V(cj>) = \vs<j>2 + •••. We consider a general Hamiltonian CHAPTER 3. 28 The action is J = (l/2ir) §Vd4>. SYNCHROTRON MOTION H = \vsV2 + V(<j>).8.
and octupoles are used to provide nonlinear detuning otxx. B% is the octupole strength.{vm/2) and. Show that the separatrix for vm 5: 2vs — Vsb/2 is given by two circles (Q_Q c )2 with Qc = \/46. The resulting effective Hamiltonian is 1 £ -Heff = vxjx + -axxJ% + bJx cos(tl>x .4 1 1 1 1 262 60 60 4 4 1 1 1 1. PUFP = 0 (2J/S .8 3319.4 3319. where axx = (1/WnBp) § 0^. (3. 2vs + usb/2) PUFP = ^/l6(l .5 1 342 588 1 4. + vab/2) QUFP = 0. C (m) ABl (Gm) / m o d (Hz) D (m) 7 h.3 299 (c) Evaluate the effective rf modulation amplitude a for the accelerators listed in the table below. Quadrupoles are used to provide resonance driving term b. P= -V2Jsm(ip--vm9).EXERCISE 3. The separatrix in the betatron phase space for slow beam extraction that employs a half integer stopband is identical to that given in this exercise.8 3833. + p2=r2. D is the dispersion function at the dipole.157). Using the conjugate phase space coordinates Q = V2Jcos(i>--vme).vj2vs) .46 Compare this result with Eqs. AB£ is the integrated dipole field error. without loss of generality. show that the Hamiltonian (3.04796 24 21. Show that the fixed points of the Hamiltonian are located at -PSFP = 0.-0). / mo d is the modulation frequency. 7 is the Lorentz relativistic factor. r = yfl66/vs.um/2us)+4b (i/m > 2M. and h is the harmonic number. QUFP = 0. a 1 IUCF Cooler I RHIC [ MI I Recycler 86.153) for the quadrupole mode is H=\{5+ f)Q> + I(« . we assume b > 0.vsb/2 <um< (um < 2vs -wsb). QSFP = 0 (vm> 2us + uab/2) PSFP = 0.156) and (3. QSFP = ^16{1 .B^ds is the detuning parameter. where C is the circumference. (Q + Qc) 2+p2=r2 .8 9. and 6 is the half integer stopband width.f)P> " g(O2 + P2)\ where 8 = vs.
182) near a betatron sideband can be casted into an effective Hamiltonian HeS = uJ+ -aJ2 + gjll2 cos(V> . and a. <f>). and i/m is the rf dipole modulation tune. (2. Show that the equation of motion for rf dipole on betatron motion in Eq. the action-angle coordinates. Find the fixed points of the Hamiltonian and discuss the dependence of the fixed point on parameters vm — v.300 CHAPTER 3. SYNCHROTRON MOTION 5. where v.vm0 + x). . (J. g is proportional to the rf dipole field strength. a are the tune. and the detuning parameter of the betatron motion.
g. and in Sec. 29Since the bunch width becomes very short and the momentum spread becomes large at transition energy. IV. IV. . when the phase slip factor % of Eq. The integral of the linearized Hamiltonian is also an ellipse. and the circulating beams can suffer microwave instabilities and other collective instabilities for lack of Landau damping.4 we study the effects of nonlinear phase slip factor and examine the properties of the so-called a-bucket. Oxygen and sulfur ions have been filtered at transition energy in the CERN PS.e. Although the action of a Hamiltonian flow is invariant. Sci. to be discussed in Sec. NS-28. In Sec. If the phase slip factor is independent of the off-momentum variable. 2389 (1981). The linearized rf potential is a good approximation. We will discuss the scaling properties of the beam at the transition energy crossing. J. VII.H.P. we will obtain analytic solutions for the linearized synchrotron motion near transition energy in Sec. i. This results in non-adiabatic synchrotron motion. the adiabaticity condition (3. NONADIABATIC AND NONLINEAR SYNCHROTRON MOTION 301 IV Nonadiabatic and Nonlinear Synchrotron Motion Transition energy has been both a nuisance in machine operation and a possible blessing for attaining beam bunches with some desired properties. and K. parts of a beam bunch can encounter a defocussing force during transition energy crossing. IV. See e. Near the transition energy region. 1. (3. which may provide beam bunches with ultra-short bunch length. the synchrotron frequency spread vanishes at transition energy. where the bucket area increases dramatically. the Hamiltonian is time dependent and is not a constant of motion. Delahye.18) becomes small. and the action is a constant of motion.29 However. such as enhanced beam separation for filtering ion beams having nearly equal charge to mass ratios. Using the sensitivity of the closed orbit to beam momentum at transition energy. IV. IEEE Trans.IV.3 we examine beam manipulation techniques for particle acceleration through transition energy. the torus is highly distorted and particles in a beam may be driven out of the rf bucket after crossing the transition energy. i. Nucl. This again raises another nonlinear prob7 lem in synchrotron motion.5 we study problems associated with quasi-isochronous (QI) storage rings. In Sec. the nonlinear phase slip factor term 7 1 can be important. However. one can filter beam momentum from nearly identical Z/A (charge to mass ratio) ion beams.2 we study nonlinear synchrotron motion due to nonlinearity in phase slip factor. In Sec. transition energy may be used to generate short bunches. IV. R.e.. and the phase-space area occupied by the beam bunch is a small fraction of the bucket area. Reich. Cappi.43) is not satisfied. and beam bunches with ultra-small beam width.
(3.7 4.5 Tad (ms) 02 JTO I AGS I RHIC I KEKPS I CPS 339. 1 Linear Synchrotron Motion Near Transition Energy Since the energy gain per revolution in rf cavities is small.4 20.5 30When \t\ > 4raa the adiabatic condition is approximately fulfilled because aad = IdC^f 1 )/^! = |(rad/l*l) 3/2 « 0.3: The adiabatic and nonlinear times of some proton synchrotrons.5 2. 31 Note that the beam parameters for RHIC correspond to those of a typical gold beam injected from the AGS with charge number Z = 79.30 Table 3. and assume that all particles in a bunch pass through transition energy at the same time. The injection energy for proton beams in RHIC is above transition energy.3 lists the adiabatic time for some proton synchrotrons.12 3833. .302 CHAPTER 3.19 | 0.27).06.2 3319.169 ) Here we have neglected the dependence of the phase slip factor on the off-momentum coordinate 5. 0.4 T8 628. r ad (3-170) where r ^ is the adiabatic time given by T a d = Uo 2 ^|cos0 s |J • (3-171) At |i| S> Tacj.32 200 6-20 6.5 70 1.6 L5 r nl (ms) I 0.29 90 9 6.4 7 (s" 1 ) 200 190 A (eVs/u) 0.4 V (kV) 950 4000 h 84 588 7T 5. SYNCHROTRON MOTION IV.2 «^r7T ( 3 .5 22.61 807.4 2.43) is satisfied. Substituting Eq.5 60 0. I FNAL I FNAL Booster MI C (m) 474.8 300 300 12 360 8. Table 3.3 5. and atomic mass number A = 197.5 6. where j = d^/dt is the acceleration rate.3 6.13 | 0.5 36 [ 63 | 0.76 40 0. we obtain Ws2 = 4 .6 1. The phase slip factor becomes % = a o . the adiabaticity condition (3. 31 Typically r ad is about 1-10 ms.04 0. we assume 7 = 7T + jt. (3.169) into Eq.04 5 (xlO~ 3 ) 6.7 | 0.7 . and t is the time coordinate.
177) = g^2 [(|^5/3 . • 2hoJojt A<f> bx2?3 ( [2J 2 / 3 1 [2iV2/3 }\ Combining this with Eq. ^ = £ {^r) M~962 K.176).175) can be written readily as32 A<f> = bx [cosx J2/3(y) + sinxN 2/3 (yj\ ./.175) where <p = y~2'3A<f>. (3.2^/3) . (3.^^-)<p y y = 0.^/a (2j2/3 .6A(/)5 + assS2 = 1. (3. The offmomentum coordinate 5 can be obtained from Eq. The notation used for Nu{z) is Yu(z) in Ref.172). the synchrotron equations of motion near the transition energy region become where the overdot indicates the derivative with respect to time t.IV. i./-^(zJJ/sinTri/. It is also called the Bessel function of the second kind. NONADIABATIC AND NONLINEAR SYNCHROTRON MOTION 303 In linear approximation. (3. . and the primes indicate derivatives with respect to time variable y.\y^)].173) can be transformed into Bessel's equation of order 2/3.e. Taking into account the synchronous phase change from <j>a to IT — <j>s across transition energy. (3. /3 2/sJ" V 7^ J 32 The Neumann function is defined as JV^(z) = [J r (z)cosTTI/. where au (3. and 6 = Ap/po and A</> = <f> — 4>s are the fractional off-momentum and phase coordinates of a particle.174) Eq. . </ + V + (1 . we obtain the constant of motion a 0 0 (A0) 2 + 2a<j. (3. we obtain Defining a new time variable y as Jo £(i|| A ') + A ' = 03 (3173) r^ y= fx x ' 2 dx=\x % l 2 with x=$-. [25].2^2/3) + (2J2/3 .\yJ^) ] .(|^.176) where \ a n d b are constants to be determined from the initial condition. The solution of Eq. (3.
183) atjuj.177) is a constant of motion given by Ja^ass . SYNCHROTRON MOTION There is no surprise that the constant of motion for a time dependent linear Hamiltonian is an ellipse. (3.1/a 7T~ 1/3 - (3-186) .7-*.184). /?27Tmc2 where A is the phase-space area of the bunch in eV-s.ofa 2/1704. the maximum momentum width of the beam will be smaller. The asymptotic properties of the phase space ellipse The phase-space ellipse is tilted in the transition energy region. we obtain 7T2 4 a ^ = 9^3V3[r(§)]2> (3'180) ^ = -gj2 {—^—) V (3 ' 181) aw = 9^l^T"J ^ ^ ~ ' . (3 8) The tilt angle. In (0. Using a Taylor series expansion around y = 0. Thus the parameter b is (2AhMirlV12 b ~ \ 3 m c ^ ) • (3-179) A. we obtain the following scaling property: 5 7=7T ~ /i 1 ' 3 V 1 / ! U 1/2 7. At a higher acceleration rate. — ass i^^f(f)(3^J .T -v / 2A V2 v ( A \1/2 - / a« _32/3r(|)/2^^7^d\1/2 ~ y <*„<*„-<% ~ ~r~ \ zmtp-t) • (3-185j Note that 6 is finite at 7 = 7T for a nonzero acceleration rate.a502 ^(^j ' (3^ X 0 7=.171) into Eq. the maximum momentum spread. The phase-space area enclosed in the ellipse of Eq. (3.304 CHAPTER 3. the shape of the ellipse changes with time. S) phase-space coordinates. Substituting the adiabatic time r ^ of Eq. (3. and the maximum bunch width of the ellipse are ^=^tan-1 2 2a** (3.
and.17) becomes quite important.JA4>)2+2a^6(A4. we can evaluate the evolution of the peak <> current at the transition energy crossing. 1. the synchrotron equations of motion become A> = h. This is not true. all particles were assumed to cross transition energy at the same time.3 ( a » Q W . NONADIABATIC AND NONLINEAR SYNCHROTRON MOTION 305 The scaling property is important in the choice of operational conditions. and the normalized distribution functions Gi(A(f>) and C?2(<5) are Gl(A fl = i 3(°»"« ~ als) exp{. the nonlinear phase slip factor of Eq. 2 Nonlinear Synchrotron Motion at 7 fa 7T In Sec.IV. (3. (3. (3. At time . (3. the phase slip factor has been truncated to second order in 6. Near the transition energy region. where NB is the number of particles in the bunch.o ( ^ + V1S) 5. we can use asymptotic expansion of Bessel functions to obtain "•2 1/9 „ ft2 (2/*7^o'T?d\ 1/2 The phase-space ellipse is restored to the upright position. B.O ftW} \| nagg ass V 7T a Ss G2{6) = J**H exp{-3ass(5 + ^A<^) 2 }. IV.178) corresponds to 95% of the beam particles. we obtain #0(A<M) = = 3NB(Oi4"i'aS6 ~ als)1/2 c-3\ad. 6 = ^ g g ^ (A*). the factor 3 is chosen to ensure that the phase-space area A of Eq. Expanding the phase slip factor up to first order in 5. to a good approximation.187) NBG1{Ac/>)G2(6). and the peak current is still located at A / = 0. IV. because the phase slip factor depends on the off-momentum coordinate 5.188) where the synchronous particle crosses transition energy at time t = 0.177). Note here that Gi(A(j>) is the line charge density.)+ali/i62] (3. Using the ellipse of Eq. The equilibrium Gaussian distribution function at transition energy The distribution function that satisfies the Vlasov equation is a function of the invariant ellipse (3. Using the Gaussian distribution function model.177). In the adiabatic region where x 3> 1.
Note that the nonlinear time for RHIC is particularly long because superconducting magnets can tolerate only a slow acceleration rate.17). 7 1 is obtained from 7 Eq. we hope that the unstable motion does not give rise to too much bunch distortion before particles are recaptured into a stable bucket. (3. and the a\ term can be adjusted by sextupoles.14. A beam bunch is represented by a line of r](5) vs S.306 CHAPTER 3. which is below the transition energy of 5. some portions of the beam could experience unstable synchrotron motion. particles at 5 > 0.e. 7?c*i = 0. At the beam synchronous energy of E = 5. a portion of the beam particles can cross transition energy and this leads to unstable synchrotron motion. SYNCHROTRON MOTION Figure 3. particles are projected onto the off-momentum axis represented by this line. The synchrotron motion corresponds to particle motion along this line.446. rjo + Vi$ = — (2771. i. Table 3. or where 8 is the maximum fractional momentum spread of the beam.14: Schematic plot of n vs 5 near the transition energy region for the Fermilab Booster. A beam bunch with momentum width ±6 is represented by a short tilted line. 3. Note that the nonlinear time depends on the off-momentum width of the beam. At a given time (or beam energy). where 7T = 5.05 eV-s are used to calculate TJ(S) for the beam. To characterize nonlinear synchrotron motion.11 GeV. the synchronous phase is also shifted from <f>s to w — (j>s in order to achieve stable synchrotron motion. For a lattice without sextupole correction.14 shows the phase slip factor 7 vs the fractional off-momentum coor7 dinate 5 near transition energy for a beam in the Fermilab Booster. as shown in the example at 5. Figure 3. When the beam is accelerated (or decelerated) toward transition energy. Since the phase slip factor is nonlinear. the line is tilted. we typically have 7^«i ~ 1 (see footnote 3). where ct\ = 0 is assumed. Since the synchrotron motion is slow.1/7') + Vi$ = 0.3 lists the nonlinear time of some accelerators.1 GeV. Within the nonlinear time ±r n i.0018 will experience unstable synchrotron motion due to the nonlinear phase slip factor. .1 GeV beam energy in Fig.5. we define the nonlinear time rn\ as the time when the phase slip factor changes sign for the particle at the maximum momentum width 5 of the beam. t = 0. and a phasespace area 01 0.
IV. 1. Since the solution of the nonlinear equation is not available. When the bunch is 33 The microwave instability will be discussed in Sec.188). The relative importance of non-adiabatic and nonlinear synchrotron motions depends on the adiabatic time of Eq. we estimate the growth of momentum width by integrating the unstable exponent. Note that when the nonlinear time rni vanishes. However. Depending on the adiabatic and nonlinear times. Using Eq. The problem is most severe for accelerators with a slow acceleration rate.171) that governs the adiabaticity of the synchrotron motion. Tad 5 (3. the solution of Eq. the effects of nonlinear synchrotron motion and of microwave instability can be analyzed. Therefore these particles experience defocussing synchrotron motion. (3-192) where r)0 = 2-yt/j^.33 We have seen that the momentum width will increase due to the nonlinear phase slip factor. Therefore the area of the phase-space ellipse of each particle is conserved. The growth factor is for a particle with 5 = 5.190) is an Airy function. NONADIABATIC AND NONLINEAR SYNCHROTRON MOTION When the beam is accelerated toward transition energy to within the range 7 T . The action integral is a distorted curve in phase-space coordinates. discussed in Sec. lower momentum portions of the bunch will experience defocussing synchrotron motion. we can easily prove that the Jacobian is 1. within which some portion of the beam particles experiences unstable synchrotron motion. (3. After the synchronous energy of the bunch reaches transition energy and the synchronous phase has also been shifted from </>„ to IT — 4>s. we obtain S" = -X5 + ^C. we should bear in mind that the synchrotron motion can be derived from a Hamiltonian H = iha. The maximum momentum height is increased by the growth factor G. IV. (3. Expressing Hamilton's equation as a difference mapping equation. which depends exponentially on T^/T^. VII. 307 the phase equation begins to change sign for particles at higher momenta while the phase angle (j>s has not yet been shifted. (3. and the nonlinear time rnl. and the ID dynamical system is integrable."Km < 7 < 7 T + 7Tm.£ J | ^ cos U^)2.0 [% + | m< j] <52 . .190) where the primes indicate derivatives with respect to x = |t|/r a( j.
if j r is changed by one unit in 1 ms. For a modern high intensity hadron facility. The 5% beam loss at transition energy found for proton synchrotrons built in the 60's and 70's may arise mainly from this nonlinear effect.34 the effective transition energy crossing rate is 1000 s~1. the KEK PS. Bunched beam manipulation are usually needed to minimize beam loss and uncontrollable emittance growth.308 CHAPTER 3. Minimizing both Tad and TD\ provides cleaner beam acceleration through the transition energy. Transition energy jump By applying a set of quadrupoles. . The effective 7T crossing rate is jes = 7 — j r . The scheme has also been studied in the Fermilab Booster and Main Injector. which is much larger than the beam acceleration rates listed in Table 3. Use of momentum aperture for attaining faster beam acceleration The synchronization of dipole field with synchronous energy is usually accomplished by a "radial loop. The tolerance of microwave instability near transition energy will be discussed in Sec. IV. Since there is no frequency spread for Landau damping. The growth of the bunch area is approximately G2 = exp{|(r n i/r ad ) 3 '' 2 } shown in Eq. Transition jT jump has been employed routinely in the CERN PS. (3. However. the revolution frequencies of all particles are nearly identical.3 Beam Manipulation Near Transition Energy Near the transition energy.193) B. therefore efforts to eliminate transition energy loss are important. i. the beam is isochronous or quasi-isochronous.e. They may be captured by other empty buckets of the rf system.3. The nonlinear phase slip factor can cause defocussing synchrotron motion for a portion of the bunch. the beam can suffer microwave instability. SYNCHROTRON MOTION accelerated through transition energy.r n . VII. and the AGS. In most accelerators. The minimum 7T jump width is A 7 T = 2 7 x Max(r ad . some portions of the phase-space torus may lie outside the stable ellipse of the synchrotron Hamiltonian. the maximum B is usually limited. Sec. IV.191). the time scale can be considered as adiabatic in betatron motion so that particles adiabatically follow the new betatron orbit. but the rf voltage and synchronous phase angle can be adjusted to move the beam across the 34The 7T jump time scale is non-adiabatic with respect to synchrotron motion. (3. or may be lost because of the aperture limitation. A.)." which provides a feedback loop for rf voltage and synchronous phase angle.8). the loss would cause radiation problems. For example. 2. transition energy can be changed suddenly in order to attain fast transition energy crossing (see Chap.
M. Phys.t\) + Vlhco052 (t . u0 = 4. E54. C.<t>\. (3. all particles gain an equal amount of energy each turn. Rev. {/S. CM. Inst. Bhat et al. have many applications such as time resolved experiments with synchrotron light sources. Flatten the rf wave near transition energy Near transition energy. The radial loop can be programmed to keep the beam closed orbit inside the nominal closed orbit below transition energy. Phys. Nucl. 36 A. and thus 5 of each particle is approximately constant in a small energy range. This concept was patented by G. very short proton bunches are needed for attaining small emittance. Note that the ellipse evolves into a boomerang shaped distribution function with an equal phase-space area.194) 7T where t\ is the rf flattening period. partial loss of focusing force in synchrotron motion can be alleviated by flattening the rf wave.5. Figure 3. U. 35see e.4 Synchrotron Motion with Nonlinear Phase Slip Factor In the production of secondary beams. and 7^771 « 2. or to reduce the momentum compaction factor for electron storage rings.15 shows the evolution of the phase-space torus when the rf wave is flattened across the transition energy region.36 employed this method for beam acceleration. j . Very short electron bunches. The rf flattening scheme is commonly employed in isochronous cyclotrons. the parameters used in this calculations are 7T = 22. D.35 In the flattened rf wave. Because of its potential benefit of the low r\ condition.IV. (3.g. For an experienced machine operator to minimize the beam loss with a radial loop. E55. we carefully study the physics of the QI dynamical system. and damping rings for the next linear colliders. Jeon et al. Bai et al. A 301. sub-millimeter in bunch length. h = 360. 3493 (1997).188) with 5 = 0 is A<t> = A0! + ^ ^ ( i 2 . and 8 — Si. E54..917 x 105 rad/s. This can be done by choosing <f>s = n/2 or employing a second or third harmonic cavity.h). IV. Phys. 815 (1996). E55. the essential trick is to attain a faster transition energy crossing rate. ti = —63 ms. Pellegrini and D. coherent synchrotron radiation. e.1. Riabko et al. Methods.B. Patent 2778937 (1954). Rev. Rossi. Rev. Since the ratio of bunch length to bunch height is proportional to J\ri\. Rev. NONADIABATIC AND NONLINEAR SYNCHROTRON MOTION 309 momentum aperture. The solution of Eq. and to attain faster acceleration across transition energy so that the beam closed orbit is outside the nominal closed orbit above transition energy. 4192 (1996). C.6 s"1. a possible method of producing short bunches is to operate the accelerator in an isochronous condition for proton synchrotrons. The AVF cyclotron has routinely . Robin.S. 1028 (1997).g. Phys.5\) are the initial phase-space coordinates of the particle.
we obtain the Hamiltonian for synchrotron motion as H=\h (r]0 + ^ms"j S2 + ^ S ^ [ c o s 0 . We define vs = Jh\rio\eV/2-KJ32E for small amplitude synchrotron tune.2 and Eq. Proc. LBL-37758 (1995). The fixed points with 5pp = 0 are the nominal fixed points. 9 (1993). Robin.0S. D.V ? i ) . 0). <5)UFP = (JT . Expanding the phase slip factor as r\ = r)0 + rjiS H and using the orbiting angle 6 as the independent variable. <5)SFP = (0s. 822 (1973). Brack et al. (3. H. Takano and B. (3-196) (0. Note that the off-momentum coordinates of each particle are unchanged. Takano. . Appl.0s)sin0s]. 0). and use the normalized phase space coordinates 0 and V = (hrjo/i/s)6. A. . 29 (1993)..60) show that the synchrotron bucket height and momentum spread become very large when |7y| is small. Hama. (3. Robin. while the bunch length elongates along the tj> axis. Isoyama. A329. 32. NS20. Phys. The synchrotron Hamiltonian needs to take into account the effects of nonlinear phase slip factor. et. Nucl. 128 (1994). 1285 (1993).310 CHAPTER 3. Nadji.197) Note that the nonlinear phase slip factor introduces another set of fixed points in the phase space. Methods. Japan J. H. The fixed points of the nonlinear synchrotron Hamiltonian are (0. EPAC94 p. Table 3. (0S. E48. L. and A. Hama and G. Sci. Nadji et al. Hama. Liu et al. H. SYNCHROTRON MOTION Figure 3. (TT . H. The fixed points with (5pp = — ^o/^i arising from the nonlinear-phase-slip factor are called nonlinear-phase-slip-factor (NPSF) fixed points. Methods.15: The evolution of a phase-space ellipse in the flattened rf wave near the transition energy region. -Tfy/rn). These fixed points play important role in determining the dynamics of synchrotron motion. The Hamiltonian of 27 (1991). S.195) where we have truncated the phase slip factor to the second order in 5. al.0s. Phys. IEEE Trans. D. S.cos0s + (0 . (3. A329. its dependence on the fractional momentum deviation 5 becomes important. Rev. Nucl. Isoyama. Instru. 2149 (1993). This requires careful examination because when the phase slip factor T) is small. Nucl. Instru.
The separatrix that passes through the nominal fixed points are nominal separatrix. Particle motion can be well described by neglecting the V3 term in the Hamiltonian. Figure 3.200) For y . In this example. (3. the phase space tori will be deformed. Right: Separatrix with parameters <f>s = 180° and y = 8 (top).1962 (middle). Note the dependence of the Hamiltonian tori on the parameter y.cos &].198) 2/ = 3 H 2 / 2 % ^ (3. the stable buckets of the upper and lower branches are separated by a distance of AV = 2y/3.4. we have assumed r/o > 0 and T?I > 0.4>s) sin fa . the nonlinear phase slip factor is not important.16 shows the separatrix of the nonlinear Hamiltonian in normalized phase space coordinates for <> = 150° and 180° respectively. and if \y\ is small.IV. given by 2/cr = v/27[(?r/2 .0406 (middle) and 1 (bottom). where. P = V) for the synchrotron Hamiltonian with parameters tps = 150° and y = 5 (top). and 3 (bottom).16 and Exercise 3.199) signifies the relative importance of the linear and nonlinear parts of the phase slip factor. For y < yCI. 311 (3. They are called "a-bucket.</>s) sin <f>s}. y = yCT = 3. 3. the separatrix of two branches will become one (see the middle plots of Fig. If \y\ » 1." Since the a-bucket is limited in a small region . we assume T)Q > 0 and ?7i > 0. the separatrix ("fish") is deformed into up-down shape (see lower plots).cos 4>s+ ((/>. This condition occurs at y = yCI.16: Left: Separatrix in the normalized phase space ((/).6). Figure 3. When the nominal separatrix crosses the unstable NPSF fixed point. without loss of gener/s ality.§> ycr. y = ycr = 5. NONADIABATIC AND NONLINEAR SYNCHROTRON MOTION synchrotron motion becomes H = \vsT2 z The parameter + ^-vsV3 Zy + ua[coa cf> .
202) where % and r]i are the first order and second order phase slip factors. the equation of motion for the fractional off-momentum deviation is i -5^^'* + «. (3.204) the synchrotron Hamiltonian for particle motion in QI storage rings becomes H0 = \p2 + \x2-l-x\ (3. truncation of the phase slip factor at the ??i term is a good approximation. (^r) =4(^-ei)(p-e2)(P-e3). IV.312 CHAPTER 3. With t = vs6 as the time variable. and r\ is the phase slip factor given by V = Vo + ViS + • • •. The particle motion inside such a quasi-isochronous (QI) dynamical system can be analytically solved as follows. (3. \] for particles inside the bucket. In many storage rings.206) .5 The QI Dynamical Systems 4> = hrjd.205) This universal Hamiltonian is autonomous and the Hamiltonian value E is a constant of motion with E € [0.16). the overdot indicates the derivative with respect to the orbiting angle 0 = s/Ro. and Eo is the energy of the beam. Similarly. where * =^ . /3c is the speed. 3.201) The synchrotron equation of motion for the rf phase coordinate 0 of a particle is where h is the harmonic number. (3. where vs = JheV0\r]0 cos <j>s\/2-nP2Ea is the small amplitude synchrotron tune.203) is a good approximation because the (up-down) synchrotron bucket is limited in a small range of the phase coordinate (see Fig. and with (x. SYNCHROTRON MOTION of the phase coordinate <j>. Here.p) as conjugate phase-space coordinates. (3. small angle expansion is valid. The equation of motion for the QI Hamiltonian with Hg = E is the standard Weierstrass equation.t a *"-W* (3-203) where Vo and <j>B are the rf voltage and synchronous phase angle. 5 = Ap/p0 is the fractional momentum deviation from a synchronous particle. (3. the linearized phase coordinate in Eq. P = ^ .
is ^ ( t ) = 1 -^h7TT' p^ = ( i h W The tune of the QI Hamiltonian is < 3 . e2 = ^ + cos(£-120°). 3. The separatrix of the QI bucket is one of the separatrix. and the Weierstrass p function can be expressed in terms of the Jacobian elliptic function [25] x(t) = e3 + (e2 .16. the discriminant A = 648^(1 — 6E) is positive. The Weierstrass elliptic p-function is a single valued doubly periodic function of a single complex variable. Figure 3. Figure 3.^ * M m = .17: Schematic plots of the QI bucket (left) and the QI potential (right). The separatrix orbit. The turning points e\. NONADIABATIC AND NONLINEAR SYNCHROTRON MOTION where u = t/y/6.IV. where e2 and e3 are turning points for stable particle motion. p = x.e3 sin(f + 60°) v . The £ parameter for particles inside the bucket varies from 0 to TT/3.17 shows the separatrix of the QI bucket QI potential. and the turning points. and the turning points are ei = ^ + cos(0. shown in Fig. (3-207) e2_-es= sine ei .209 ) qw-"*•%* r r ) r V6K(m) (3^o) .e3) sn2 ( \ p . plotted sideway. e3 . For particle motion inside the separatrix. and e3 are also shown.\ + cos(£ + 120°) 313 with £ = |arccos(l — 12E). which corresponds to m = 1. e2.
Wang. A364. M. E49. et al. 3. Here.18: The synchrotron tune of the QI dynamical system (upper curve) is compared with that of a single rf potential (lower curve). Methods.18. 591 (1993). et al. we note that the synchrotron tune decreases to zero very sharply near the separatrix.314 CHAPTER 3. M. Phys. 70. Phys. D. The resulting Hamiltonian is H0(J)*J-^J2 + ---. the equation of motion for QI electron storage rings is x" + Ax' + x . 4678 (1993). Phys. (3. Rev.e3)^F ( j . Y. Note that the sharp drop of the QI synchrotron tune at the separatrix can cause chaotic motion for particles with large synchrotron amplitudes under the influence of the lowfrequency time-dependent perturbation. 7 1 . (3. E48. The action of the separatrix orbit is Jsx = 3/5?r. Huang. Ellison. Lett. E48.m) . J)= f p dx. R1638 (1993).x2 = 0. et al. Syphers. SYNCHROTRON MOTION Figure 3. Li.211) where F is the hypergeometric function [25]. et al. Rev. Using the generating function F2(x. D. et al.212) the angle variable is ip = dFz/dJ = Qt.213) 3 7 H. 205 (1995). Because of the synchrotron radiation damping. Nud. Rev. Lett. 719 (1993). et al. Inst.37 The action of a torus is J = ^fpdx = | y | ( e 2 . Rev. Rev. (3. Phys. . -^3. Li. or equivalently the bucket area id 6/5. The tune of the QI Hamiltonian is compared with that of the normal synchrotron Hamiltonian in Fig. time dependent perturbation will cause overlapping parametric resonances and chaos near the separatrix. Phys. Because of the sharp decrease in synchrotron tune. 1610 (1994).e3)2(ei .
wm = vm/i>B is the normalized modulation tune.4 1. the Hamiltonian in normalized phase-space coordinates is H = j + -x2 . Rev.5.4 where the effective damping coefficient is 315 A = A = 3lE_. 38A. the nonlinear time. where B =— (3. and vm is the modulation tune of the original accelerator coordinate system. Jeon et al. E55. Show that Eq. (a) Express the solution in terms of Airy functions and find the equation for the invariant torus. 815 (1996). (3. Rev.216) is the effective modulation amplitude. Exercises 3.215) The stochasticity of such a dynamical system has been extensively studied. Detailed discussions of this topic is beyond this introductory textbook. 2. Verify the adiabatic time. 4192 (1996).EXERCISE 3. where T ^ is the adiabatic time of Eq.. i.e.e. Rev. . and the momentum spread of the beam 8 at 7 = 7 r for the accelerators listed in Table 3. B ~ |T?I|/|T?O|3''2Including the damping force.171). M. E54. (3.3.j. (3. Phys.38 Experimental verification of the QI dynamical system has not been fully explored. Note that the effective modulation amplitude B is greatly enhanced for QI storage rings by the smallness of ?j0. the equation of motion becomes x" + Ax'+ x-x2 = -um B cos umt. Bai et al. Phys. Riabko et al. (3.217) (3. 3493 (1997). Uo is the energy loss per revolution.-x3 + ujmBx cos ojmt. E54. Including the rf phase noise. i. D. the effective damping coefficient is enhanced by a corresponding decrease in synchrotron tune. A ~ |J7O|~1/'2J where the value of A can vary from 0 to 0. a is the rf phase modulation amplitude. Phys.214) Here A is the damping decrement. In QI storage rings. and JE is the damping partition number.172) can be reduced to 6" + x5 = 0 where the primes indicate derivatives with respect to the variable x = |t|/Ta.
Show that when y = ycr of Eq. Show that the QI Hamiltonian can be reduced to Eq. calculate the characteristic time and the maximum momentum spread for a phase space area of 0.5) of Eq.177).<fe) sin 4>s] . (3. (3.04 eV-s. SYNCHROTRON MOTION 3. (3.200) the separatrix of the upper branch passes through the UFP of the lower branch. Si) of the initial ellipse. Assuming 7 T = 20.4. (3. 7. Show that T^/T^ OC 7~ 5 / 6 7~ 2 / 3 . .n) sin 4>s) = 0. show that the Hamiltonian (3. Using the normalized phase space coordinates <f> and V. Show that the phase space area enclosed by (A<f>. Discuss the effects of high vs low yT lattices on the dynamics of synchrotron motion near the transition energy.—y3 = 0. vsV* + yV2 + 1y[cos <t> .cos <f>s+ {(}>.316 (b) Verify Eq. 6.205) and that the solution is given by the Weierstrass elliptical function. Show that the separatrices of the Hamiltonian are vsV% + yV2 + 2j/[cos 4> + cos </>s + ((f> + <f>s .195) with nonlinear phase slip factor depends only on a single parameter y = 3hrjQ/2usr)i. 4. The Fermilab Main Injector accelerates protons from 8.9 GeV to 120 GeV in 1 s. 5.194) is equal to the phase space area enclosed by (A^i. CHAPTER 3.
the fractional change of beam velocity in low energy boosters can be large. The beam is accumulated. electrons are almost relativistic at energies above 10 MeV. etc. and the energy compensation of the rf field is along the longitudinal direction. The synchrotron radiation emitted by a relativistic electron is essentially concentrated in a cone with an angular divergence of I/7 along its path. The effects of synchrotron radiation must be taken into account in beam manipulation. where the space-charge tune shift. The mean-field Coulomb force can also have a large effect on the stability of low energy beams in boosters. phase-space painted. is limited to about 0. called booster synchrotrons or boosters. During beam acceleration. For acceleration of ion beams. The resulting momentum spread is independent of the rf voltage. However. phase-space stacking. the range of beam energy for a synchrotron is limited. innovative bunched . electrons emit synchrotron radiation. pre-accelerated by an electrostatic Cockcroft-Walton or an RFQ. and the mean energy loss of a beam is compensated by the rf field. and the required range of rf frequency change is small. Careful consideration is thus needed to optimize the operation and construction costs of accelerators. Equilibrium is reached when the quantum fluctuation due to the emission of photons and the synchrotron radiation damping are balanced. particle motion in synchrotron phase-space is damped. including phase displacement acceleration. that have often been applied in the space charge dominated beams and high brilliance electron storage rings for providing a larger tune spread for Landau damping. The reasons for this complicated scheme are economics and beam dynamics issues. Since dipole and quadrupole magnets have low and high field operational limits. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 317 V Beam Manipulation in Synchrotron Phase Space A charged particle beam is usually produced by an intense ion source.. adiabatic capture. Since synchrotron radiation power depends on particle energy. and debunching. which must be compensated by the longitudinal rf electric field in a storage ring. and accelerated toward higher energies by a chain of synchrotrons of various sizes. and the transverse emittance depends essentially on the lattice arrangement. and the beam dynamics issues of minimum momentum aperture and phase-space area. accordingly. We also study the barrier rf systems that have been proposed for low energy proton synchrotrons. rf frequency for a low energy booster has to be tuned in a wide range. In this section we examine applications of the rf systems in the bunched beam manipulations.3-0. prebunched and injected into a linac to reach an injection energy for low energy synchrotrons.4. stacked in a low energy booster.V. the betatron motion is also damped. bunch rotation. proportional to circumference of the synchrotron. bucket to bucket transfer. rf power source. The rf voltage requirement is determined by technical issues such as rf cavity design. In general. The beam distribution function can thus be manipulated to attain desirable properties for experiments. phase-space area is normally conserved. We carefully study the double rf systems. On the other hand.
electrons are nearly relativistic at all energies. K. the rf system is usually limited by the range of required frequency swing. [GeV] / r f [MHz] Av.30 7x84 Kf [kV] |_90 I 300 1 300 j_950 | 4000 In some applications. and the angular revolution frequency UJQ is a function of the magnetic field B and the average radius of the synchrotron RQ. On the other hand. Thus the rf frequency is an integer multiple of the revolution frequency ui^ = hLjo(B. High frequency pill-box-like cavities are usually used. and e is the particle's charge. with ramping frequency w/2?r varying from 1 Hz to 50 Hz.Rags 6x60 I FNALBST 0.E. Normally the .4-4.218).001703/p[m] Tesla for electrons. Rate [GeV/s] Max.2(0.E. SYNCHROTRON MOTION beam manipulation schemes can enhance beam quality for experiments.1273/p[m] Tesla for protons. or resonantly as B = (B/2)(l — cosu)t) = Bsin2(cut/2). The momentum po of a particle is related to the magnetic field by po = epB. Radius [m] h I AGS BST 1 AGS 0.1 2. [GeV/u] Ace. K. (3. The rf frequency is a function of the dipole magnetic field. In low to medium energy synchrotrons.47 84 1 FNALMI 8. the magnetic field can be ramped linearly as B = a + bt.68-26. V. for which cavities with ferrite tuners are usually used.4) 200 8 30. Table 3. where h is the harmonic number.5 30 0. the rf frequency should follow the magnetic field ramp according to Eq.6 128. \A-iVi) rf frequency ramping is particularly important for low energy proton or ion accelerators.8-53.74 (19/4).4: RF parameters of some proton synchrotrons Inj.7 250 26.I lists parameters of some proton synchrotrons. where p is the bending radius. and the rf frequency swing is small.8 75.2 0.0-52. Since rru?_ _ f 3.Ro).001/0.I RF Frequency Requirements Particle acceleration in synchrotrons requires synchronism between rf frequency and particle revolution frequency.2(1. Thus the rf frequency is Pc _ hepB _ he [ B*{t) |1/2 where m is the particle's mass.318 CHAPTER 3. Table V.18-4.5) 100 60 1.457 (l/4)-Rags 1-3 (2) 12 (8) I RHIC 12 3. ~^p~~\ 0.1 528.0 100 500 52.
The choice of harmonic number for high energy electron storage rings is determined mainly by the availability of the rf power source. and the ratio of harmonic numbers is 4. 350. Since the damping time of electron beams in electron storage rings (see Table 1. acceleration rate is less important in storage rings used for internal target experiments. a large angle Coulomb scattering process converting the horizontal momenta of two electrons into longitudinal momenta. Similar reasoning applies to the chain of accelerators. 1. 350.4) is short. the injection scheme of damping accumulation at full energy is usually employed in high performance electron storage rings. and 700 MHz regions. most of the rf cavities of electron storage rings are operating at these frequencies. Requirements of rf systems depend on their applications. efficient high quality cavity design. In recent years. B. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 319 frequency range can be in the 200. 4. the choice of rf voltage is important in determining the beam lifetime because of quantum fluctuation and Touschek scattering. the average radius of the AGS Booster is 1/4 that of the AGS. New methods of beam manipulation can be employed. 40A box-car injection scheme is equivalent to bucket to bucket transfer from one accelerator to another accelerator. the rf voltage is limited by the rf power source and the Kilpatrick limit of sparking at the rf gap. Chap.4. 500. The total rf voltage of synchrotrons and storage rings is 39The ISIS at the Rutherford Appleton Laboratory has a 50 Hz ramp rate. Since the rf bucket area and height are proportional to \A^f. where rf power sources are readily available. 500. In electron storage rings. To achieve high beam power in meson factories and proton drivers for spallation neutron sources. . 700 MHz regions. The choice of rf voltage High intensity beams usually require a larger bunch area to control beam instabilities. which can be important for colliding beam facilities. whereas the rf systems in the Spallation Neutron Source (SNS) provide only beam capture.V. In general. A. and the size of the machine.39 On the other hand.40 For example. a minimum voltage is needed to capture and accelerate charged particles efficiently. The choice of harmonic number The harmonic number determines the bunch spacing and the maximum number of particles per bunch obtainable from a given source. a fast acceleration rate is important. Sec. The harmonic number is then determined by the rf frequency and circumference of the storage ring. The harmonic numbers are related by the mean radii of the chain of accelerators needed to reach an efficient box-car injection scheme. wideband solid state rf power sources and narrowband klystron power sources have been steadily improved. Since rf power sources are available at 200.
5% instead of 0. .2 Capture and Acceleration of Proton and Ion Beams At low energy. intrabeam scattering. The right plots. (e) to (h). A possible solution is to install a debuncher in the injection transfer line for lowering the momentum spread of the injected beam. the intensity and brightness of an injected beam are usually limited by space-charge forces. very little beam loss in the synchrotron phase-space can theoretically be achieved by adiabatically ramping the rf voltage with <j>a = 0. the initial rf voltage should have a small finite initial voltage Vo. phase-space painting for beam distribution manipulation can be used to alleviate some of these problems (see Chap.6%.320 CHAPTER 3.1). After beam capture.. SYNCHROTRON MOTION usually limited by the available space for the installation of rf cavities. for transverse phase-space painting). A. The maximum voltage is only barely able to hold the momentum spread of the injected beam from linac. The actual capture efficiency is much lower. microwave instability. 41 The peak voltage is usually limited by the power supply and electric field breakdown at the rf cavity gap. The proton beam was accelerated from 7 to 200 MeV at 1 Hz repetition rate. the synchronous phase was ramped adiabatically to attain a desired acceleration rate. III.41 The resulting captured beam brightness depends on the rf voltage manipulation. we find that the capture efficiency is about 99. Sec. In order to satisfy the adiabatic condition. The following example illustrates the difference between adiabatic capture and non-adiabatic capture processes. and the rf voltage is ramped to a final voltage smoothly (see also Exercise 3. Good acceleration efficiency requires adiabatic ramping of V and 4>s while providing enough bucket area during beam acceleration.1% shown in this example. V. the momentum spread of the injected beam is about 0. while the synchronous phase was kept at zero. of Fig.5.43) becomes Ts dVd_ Ts dAB (3 ' 220) a a d -«^T-2^^T' where Ts is the synchrotron period and AB is the bucket area. (3.8.19 show an example of adiabatic capture in the IUCF cooler injector synchrotron (CIS). Since the injected beam from a linac normally has a large energy spread. the rf voltage requirement in booster synchrotrons needs enough bucket height for beam injection. A debuncher or a bunch rotator in the transfer line can be used to lower the momentum width of injected beams. Adiabatic capture During multi-turn injection (transverse or longitudinal phase-space painting or charge exchange strip injection). The adiabaticity coefficient of Eq. 3. In reality. 2. and the rf voltage VTi(t) was increased from a small value to 240 V adiabatically. etc. In this numerical example. plots (e) and (f).
1%. Indiana University. and thus the actual adiabatic capture efficiency is substantially lower. The actual momentum spread of the injected beam is about ±0. (e) to (h). Spacecharge force and microwave instability are not included in the calculation. . 1998). The rf synchronous phase is then ramped adiabatically to achieve the required acceleration rate.19: The left plots. show adiabatic capture of the injected beam: the rf voltage is ramped from 0 to 240 V adiabatically to capture the injected beam with a momentum spread of 0.5%. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 321 Figure 3. Kang (Ph. The right plots. (a) to (d). This calculation was done by X. Thesis. show non-adiabatic beam capture during injection and acceleration.D.V.
As seen in plots (b) and (f). The decoherence results in emittance growth. When the beam is transferred from one accelerator to another. the bunch length of a beam may need to be shortened in many applications. the peak voltage of an rf system is limited by the breakdown of electric field at the acceleration gap. However. 3. of Fig. Similarly. the rf frequency / [MHz] is related to the peak electric field gradient EK [MV/m] by f = 1.223) l*v¥LrL*v¥L- _ [ i [W] (-2) 322 . In this case.l L-R^Jacc. Non-adiabatic capture CHAPTER 3. According to the empirical Kilpatrick criterion. located at the source. Beam loss occurs during acceleration. a beam chopper consisting of mechanical or electromagnetic deflecting devices. the injected beam particles decohere and fill up the entire bucket area because of synchrotron tune spread.64 El e-8-5/EK. and the capture efficiency is low. (3. (3-221) fi [W] This matching condition may be higher than the limit of a low frequency rf system. Chopped beam at the source Many fast cycling synchrotrons require nonzero rf voltage and nonzero rf synchronous phase (j>s > 0 to achieve the desired acceleration rate.3 Bunch Compression and Rotation When a bunch is accelerated to its final energy.2 or equivalently. the beam profile matching condition is L-RflJacc. V. C. A simple approach is to raise the voltage of the accelerator rf system. capture efficiency is reduced by the nonadiabatic capture process. To circumvent low efficiency. (a) to (d). the capture efficiency may be even lower. SYNCHROTRON MOTION The left plots. it may be transferred to another accelerator or used for research. When the if voltage is set to 240 V to capture the injected beam. can be used to paint the phase-space of the injected beam and eliminate beam loss at high energy. the beam fills up the entire phase-space. as shown in plot (b). the final phase-space area is larger. the rf parameter matching condition is \l] =\l] .19 show an example of non-adiabatic capture with nonzero initial rf voltage.322 B. With microwave instability and space-charge effects included.
etc. defined as the ratio of the bunch lengths at (Vo = V2) -> Vi ->• V2.3. the longitudinal emittance of the secondary antiproton beam becomes smaller.3). the bunch height will become bunch width according to Eq. Generally.g. The antiproton beam can be further debunched through phase-space rotation in a debuncher by converting momentum spread to phase spread. the emittance of secondary beams is equal to the product of the momentum aperture of the secondary-beam capture channel and the bunch length of the primary beam. the rms bunch width and height are obtained from Eq. becomes " where we have used the properties that the bunch area supposedly fills up the bucket area at Vxi = V\. e. and the final antiproton beam is transported to an accumulator for cooling accumulation (see Exercise 3. VIII). A few techniques of bunch compression are described below. For a given bunch area.5. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 323 Because of this limitation. the maximum attainable rf voltage is limited. Vo -* V1. . we have to use different beam manipulation techniques such as bunch compression by rf gymnastics. or a kicker is fired to extract beams out of the synchrotron.A Bunch compression by rf voltage manipulation The first step it to lower the rf voltage adiabatically.58) during the adiabatic rf voltage compression from Vo to Vx. Figure 3. and the fact that the bucket area is 16 (see Table 3.20 shows schematic phase-space ellipses during the bunch compression process. Beam bunch compression is also important in shortening the electron bunch in order to minimize the beam breakup head-tail instabilities in a linac (see Sec. and the bunch area containing 95% of the beam is 6TT(T^ in the normalized synchrotron phase space coordinates. The lower-left plot shows the final phase-space ellipse in an idealized linear synchrotron motion. At 1/4 or 3/4 of the synchrotron period. Bunched beam gymnastics are particularly important for shortening the proton bunch before the protons hit their target in antiproton or secondary beam production.The unmatched beam bunch rotates in synchrotron phase-space. so that the bucket area is about the same as the bunch area. The maximum bunch compression ratio.2). and the final phase-space ellipse is distorted by the nonlinear synchrotron motion that causes emittance dilution. (3.59). (3. After the rf voltage is jumped to V"2. When the bunch length of a primary proton beam is shortened. In reality. Then the rf voltage is increased non-adiabatically from VI to Vi. V. a second rf system at a higher harmonic number is excited to capture the bunch.V.
As the voltage is nonadiabatically raised to four times the original rf voltage.88) can be approximated by two straight lines crossing at 45° angles with the horizontal axis cj>.3. The rate of growth is equal to exp(wsiufP)The maximum rf phase coordinate 0 max that a bunch width can increase and still stay . II.324 CHAPTER 3. The bunch area is initially assumed to be about 1/5 of the bucket area (top-left). (3.5). When the SFP of the rf potential is shifted back to the center of the bunch. V. In the normalized phase-space coordinates. the bunch can be captured by a matched high frequency rf system or kicked out of the accelerator by fast extraction.B Bunch compression using unstable fixed point If the rf phase is shifted so that the unstable fixed point (UFP) is located at the center of the bunch. where ws is the small amplitude synchrotron angular frequency. a kicker can be fired to extract the beam. the mismatched bunch begins to rotate. Ts is the synchrotron period.20: Schematic drawings (clockwise) of bunch compression scheme using rf voltage manipulation. When the bunch length is shortened (lower-left) at 1/4 of the synchrotron period. When the rf phase is shifted so that the beam sits on the UFP. the bunch width and height will stretch and compress along the separatrix. the bunch length and bunch height change according to exp{±wsiUfp} = exp{±27rtufp/Ts}. At 3/8 of the synchrotron period. Near the UFP. SYNCHROTRON MOTION Figure 3. The length of stay at the UFP can be adjusted to attain a required aspect ratio of the beam ellipse. the Hamiltonian for stationary synchrotron motion is given by Eq.4 and Exercise 3. the separatrix of the Hamiltonian in Eq. (3. In linear approximation.2. the bunch will begin compressing in one direction and stretching in the other direction along the separatrix orbit (see Sec.88). <j> and V = — (h\ri\/vs)(Ap/p). We now derive the ultimate bunch compression ratio for the rf phase shift method as follows. The voltage is adiabatically reduced by 16 times so that the bunch is almost fill the bucket area (topright). and iufp is the time-duration that the bunch stays at the UFP. The mis-matched bunch profile will begin to execute synchrotron motion.
the beam is kicked out of the synchrotron and the R^ transport matrix element will compress bunch. By employing a cavity to accelerate and decelerate parts of the beam bunch. The difficulty of nonlinear synchrotron motion in the final stage of bunch rotation can be solved by using the buncher in the transport line.3.ZZI) The time needed to reach this maximum compression ratio is ujsiufp = In — . lower energy particles travel shorter path. conservation of phase-space area.228) A difficulty associated with bunch compression using rf phase-shift is that the rf voltage may remain at a relatively low value during the bunch rotation stage.max — —„ ~ /» > (6. However. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 325 within the bucket after the rf phase is shifted back to SFP is given approximately by I ^ + 2sin2(^)«2. (3.\ = 7rc7-p.0. First. V. we find / mx ^a\. This method is commonly used in the beam transfer line. Thus we obtain <> a ~ V%. let the beam drift a distance L so that higher energy particles are ahead of lower .' (3.203. i.C Bunch rotation using buncher/debuncher cavity The principle of bunch rotation by using a buncher/debuncher cavity is based on the correlation of the time and off-momentum coordinates (the transport element R$&).Since there is no constraint that the final bunch size should fit into the bucket. and the higher energy particle travel a longer path. a simple debuncher used to decrease the energy spread of a non-relativistic beam out of a linac can function as follows. After proper bunch compression.e.Using Liouville's theorem. one can regain the factor of y/2 in staying longer at the UFP.226) Assuming that 95% of the beam particles reach 0 max = y/2 so that a-p.225) where we assume linear approximation for particle motion near SFP. The effect of non-linear synchrotron motion will be more important because the ratio of bucket-area to the bunch-area is small. = a4>.i — r^max — y/2/3.fcr0. the bunch length and the momentum spread can be adjusted.V.f(3. the resulting compression ratio is reduced by a factor of l/\/2. For example. we find the compression ratio as r c.
Note that the momentum spread of the entire beam remains the same in this non-adiabatic debunching process. It requires bending magnets for generating local dispersion functions so that the path length is correlated with the off-momentum coordinate. 58 (CERN. 1956. In a successful example of beam stacking in the ISR pp collider. 635 (1993). V.g. which increases the density and the collision rate. . therefore particles outside the bucket can not be captured during acceleration. the momentum spread will not change. of Int. For relativistic particles. L. Neglecting synchrotron radiation loss. p. 44.42 A buncher/debuncher cavity can then be used to shorten or lengthen the bunch. A cavity that decelerates leading particles and accelerates trailing particles can effectively decrease the energy spread of the beam. and subsequent groups are accelerated and deposited adjacent to each other. a single beam current of 57.5 Beam Stacking and Phase Displacement Acceleration The concept of beam stacking is that groups of particles are accelerated to a desired energy and left to circulate in a fixed magnetic field. p. The accumulated beam will overlap in physical space at special locations. (3. C. the resulting debunched beam has a smaller momentum spread. In this case. Methods of radio-frequency acceleration in fixed field accelerators with applications to high current and intersecting beam accelerators. Since the magnetic field 42 See e. a drift space can not provide the correlation for the transport element R^ because all particles travel at almost the same speed. we can adiabatically lower the rf voltage. Proc. 1993 Part. 43 K. T. Symon and A. Terwilliger. and S. SYNCHROTRON MOTION energy particles. Raubenheimer. Jones. The phase-space area remains the same if we can avoid collective beam instability. Emma. particles drift and fill up the entire ring because the rotation frequency depends on the off-momentum variable.229) where 5 is the maximum momentum spread of a beam. P. The debunching rate is <j> = huior]S. Con]. Accel.R. in Proc. V. e. p. Conf.g. Kheifets. Sessler. Pruett. small ft and zero D(s) locations.M.M.R. particles can not cross the separatrix.H.W.326 CHAPTER 3.43 In a Hamiltonian system.. To reduce the momentum spread in the debunching process. phase displacement acceleration is usually employed. 1959).5 A was attained. K. on High-Energy Accelerators and Instrumentation. CERN Symp. Symon and K. To accomplish phase-space stacking.. The bunch shape will be distorted because particles of higher and lower momenta drift in different directions. The debunching time can be expressed as T d b = 2Tr/hLj0r]5.4 Debunching When rf systems are non-adiabatically turned off.
giving its way to a fully operational SPPS. 8th Int. which may lower the thresholds for transverse and longitudinal collective instabilities. The machine stopped operation in December 1983.4 GeV without loss of luminosity. Space charge induces potential well distortion and generates coherent and incoherent betatron tune shifts. Proc. V. Averill et al.D. This method has been successfully used to accumulate polarized protons at low energy cooling storage rings. The installation of low-/? superconducting quadrupoles in 1981 brought a record luminosity of 1. a double rf system has often been used to increase the synchrotron frequency spread. As early as 1971. the cooling stacking method can enhance polarized proton intensity by a factor of 1000 in the IUCF Cooler. p.4 x 1032 cm 2 s -1 .6 Double rf Systems Space charge has been an important limitation to beam intensity in many low energy proton synchrotrons. a newly injected beam accelerated by phase displacement can be moved toward the cooling stack to achieve a high cooling rate. 301 (CERN 1971). only particles inside the stable rf bucket are accelerated toward high energy. which enhances Landau damping in collective beam instabilities. Particlesflowalong lines of constant action. and their energies are lowered. that observed its first pp collision at the center of mass energy of 540 GeV on July 10. the circulating beams in ISR were accelerated from 26 GeV to 31.45 Similarly. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 327 depends on rf frequency. the beam energy will be displaced upward in phase-space. In a storage rings with electron cooling or stochastic cooling. Conf. Thesis. antiprotons can be moved to the cooling stack for cooling accumulation. The change in energy is AE = LJoA/2n. To increase the threshold beam intensity. when a bucket is decelerated toward lower energy. Fast beam loss may occur during accumulation and storage when the peak beam current exceeds a threshold value. Particles outside the rf bucket are lost in the vacuum chamber because of the finite magnet aperture. Pei. For example. that can be handled by a low power rf system in the ISR. it may not be captured into the bucket if the rf bucket acceleration is adiabatic. Ph.46 This technique was also successfully applied to cure coupled bunch mode instabilities at ISR.V. 45A. with phase displacement acceleration. 1981. accelerated. . Indiana University (1993). By employing the phase displacement acceleration. and to accumulate antiprotons at antiproton accumulators. where 44A high current stack at the ISR has a momentum spread of about 3%. Phase displacement acceleration has been used to accelerate coasting beams in the Intersecting Storage Ring (ISR) at CERN44 and to compensate synchrotron radiation loss in electron storage rings. on High Energy Accelerators. an attempt was made to increase Landau damping by installing a cavity operating at the third harmonic of the accelerating frequency in the Cambridge Electron Accelerator (CEA) . Similarly.e. i. 46 R. What happens to the unbunched coasting beam outside the separatrix when an empty moving bucket is accelerated through the beam? Since the beam is outside the separatrix.
defined as the fraction of the circumference occupied by a beam or the ratio of average current to peak current.sin^ ls + ^ fsin L s + ^ ( 0 . 49 See S. on High Energy Accel. Baillod et al. p. E50. (3. IEEE Trans. and hi is the harmonic number for the primary rf system.230) where < is the phase coordinate relative to the primary rf cavity. Bramham et al. a double rf system with harmonics 5 and 10 was successfully used in the Proton Synchrotron Booster (PSB) at CERN to increase the beam intensity by 25 — 30% when the coherent longitudinal sextupole and decapole mode instabilities were suppressed by beam feedback systems. 221-251 (1995).M. Nucl. Rev. Lee et al.232) are Hamilton's equations of motion for a double rf system. and V\ and V2 are the voltages of the rf cavities. 5717 (1994). Conf. 9th Int. For example. Part. NS-30. 1974). Liu et al. PAC. 3499 (1983). Accel. Synchrotron equation of motion in a double rf system For a given particle at angular position 9 relative to the synchronous angle 9S. the peak current and consequently the incoherent space-charge tune shift are reduced.328 CHAPTER 3. .232) where the overdot is the derivative with respect to orbiting angle 9. Bramham et al.Y.231) where h2 is the harmonic number for the second rf system and <j>2S is the corresponding synchronous phase angle. 1490 (1977). R3349 (1994).sin02sj j . The equation of motion becomes 5 = ^ | jsin<£ . (CERN. the rf phase angle for the second rf system is 02 = fas -h2(69S) = <t>2s + ^ .47 Adding a higher harmonic rf voltage to the main rf voltage can flatten the potential well. Phys. J.0ls).48 At the Indiana University Cyclotron Facility (IUCF). P. Galato et al. the phase angle of the primary rf system can be expressed as 4> = <t>is-h1(6-6s). Phys. Therefore.49 A. Proc. Nucl. a recent beam dynamics experiment showed that with optimized electron cooling the beam intensity in the cooler ring was quadrupled when two rf cavities were used. than that of a single rf system. Liu et al. for a given DC beam current in a synchrotron. Since the equilibrium beam profile follows the shape of the potential well. (3. Equations (3. NS-24.Y..0ls)l . (3. (pis is the phase > / angle for the synchronous particle. SYNCHROTRON MOTION an additional cavity was operated at the sixth harmonic of the primary rf frequency. Sci. Similarly. a double rf system can provide a larger bunching factor. 47 P. IEEE Trans. Sci. E49. Rev. 1298 (1987). Proc.21) and (3.Y. J. G. 48 see J. 49.
cos(02s + h{(f> .V. the method presented in this section can be extended to more general cases with 0i s ^ 0 and 02S ^ 0. However.0). there are two inner buckets on the <j> axis. we study the double rf system with ft = 2.O) \<r | (±arccos(j).236) .^Vd<t>. Table 3.T.0 ls )) . r = -V2/Vu and <j)ls and (j>2s are the corresponding rf phase angles of a synchronous particle.cos 4>) + (<f>is .0) sin 0 l s -T-r [cos02s . the net acceleration is zero and the Hamiltonian becomes H = ^V2 + vs [(1 .233) + V(cf>)> where the potential V(<f>) is V{4>) = ^{(cos (j>is . The fixed points (0FP.235) are listed in Table V. The effective acceleration rate for the beam is AE = eVi(sincj)is — r sin02a) per revolution.cos 20)1 .h((j> .0) | (0. ft = ft2/fti.0) I UFP (TT.(1 . ISFP 0<r < i (0. Here.5: SFP and UFP of a double rf system. The action is J{E) = . 7T J-cj> (3. we study a stationary bucket with 0is = 02s = 0°. (3. To simplify our discussion. tonian is H=1-psV2 329 the Hamil(3.234) ft Here i/s = ^hleVi\T]\/2K/32Eo is the synchrotron tune at zero amplitude for the primary rf system.0) Since the Hamiltonian is autonomous. For r > I/ft. Action and synchrotron tune When the synchrotron is operating at 0 l s = 0 2s = 0.6.2].4>ls)sin02s]}. the Hamiltonian value E is a constant of motion with E/vs G [0. the conditions r = I/ft and ft sin 02s = sin</>is are needed to obtain a flattened potential well. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE Using the normalized momentum coordinate V = -(hi\r)\/vs)(Ap/p0). B. (±7r. Because the rf bucket is largest at the lowest harmonic ratio.^FP) (3.cos 0) .
the phase-space area is 2irJ. this formula is also valid for r > 0. C. ( l-2r + tl YT1* Thus the synchrotron tune becomes [26] Qs__^(l-2r) + 2tl + (l + 2r)tt vs 2(1+tl)K(k1) ' where K(k\) is the complete elliptic integral of the first kind with modulus [6-2M) 1 /(l-2r) + 2(g + (l+2r)«J In fact. The value E is related to 4> by E = 2vs(l . we obtain 9J 2(1+^) ri[ 2 d<j) = —-dt. The inner separatrix.2r cos2(</>/2)) sin 2 (0/2). which passes through the origin. r = —.. and the synchrotron tune is Qs = (dJ/dE)"1. intersects the phase axis at ±(j)h with cos(0b/2) = l/-v/2r. A given torus inside the inner bucket corresponds to a Hamiltonian flow of constant Hamiltonian value.5 case Changing the variables with t = tan .2). t0 = tan ..5 case For r > 0. L V2r J (3. The r < 0. The r > 0. where 0b is the intercept of the inner separatrix with the phase axis. the origin of phase-space V = <j> = 0 becomes a UFP of the unperturbed Hamiltonian. Let <j>\ and 0U be the lower and upper intercepts of a torus with the .330 CHAPTER 3.237) which is a monotonic increasing function of the ratio r. Two SFPs are located at V — 0 and 0 = ±<fo.5. SYNCHROTRON MOTION where 0 is the maximum phase angle for a given Hamiltonian torus. where cos(0f/2) = l/2r.5 and 4> > 0b. The bucket area «4b is A = 27rJ = 8k/rT2r-|--i=ln(\/rT2f+\/2r)l . The corresponding bucket area for the single rf system is Ab(r -» 0) = 16 (see Table 3. D.
and # = 2dt/(l +1 2 ).5.057^). BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 331 phase axis. Figure 3. and two small potential wells are formed inside the inner separatrix.5. dE nvsy/2f h dt ^(*5-t2)(t2-t?)' where tu = tan(0 u /2). the synchrotron tune becomes Qs = nt0 ^2{1+t20)K{k) = 7r(g/2i/ 5 ) 1 / 4 V2K{k) V* with modulus where to = tan(</>/2). For small amplitude synchrotron motion.21 shows the synchrotron tune as a function of the amplitude of synchrotron oscillation for various voltage ratios. Thus the synchrotron tune is Qs "= V277Ttu 1 ^/(l + ig) (1 + if) ^( fc 2> ' where modulus hi = \Jt\ — t2/tn.V. are valid only for the case with r = 0. In this case. Jtp (3. The derivative of the action with respect to the energy for the torus becomes 8J_ = V ( l + <g)(l + <?) rt.244) . where the synchrotron tune spread of the beam is maximized for a given bunch area. located at 0 = 117° (or E = 1.7786i/s. Action-angle coordinates Although analytic solutions for action-angle variables. the maximum synchrotron tune is Qs = 0.sin2(</>u/2). i = tan(<£/2). As r increases. = tan(^/2). the derivative of synchrotron tune vs action becomes large near the origin. a dip in QS(J) appears at the inner separatrix of inner buckets. where </>u = 0 and sin(<fo/2) = ^/sin2((/>b/2) . t. t0 = 0 and k0 = l/%/2. presented in this section. Since large tune spread of the beam is essential for Landau damping of collective beam instabilities. At r = 0.5. the system reduces to a single primary rf cavity. E. dQs/d<j) is very small or zero. an optimal rf voltage ratio is r = 0.5. the method can be extended to obtain similar solutions for other voltage ratios. When the voltage ratio is r > 0. At r = 0. where the synchrotron tune is Qs/vs = 1 at zero amplitude. Using the generating function F2(<M)= fvWW. Near this region.
dj-^JU r*ffP Q.. we obtain the angle coordinate as . and two SPPs are produced.21: The normalized synchrotron tune as a function of the peak phase <j> = 0 for various values of voltage ratio r. V ( W > the angle coordinate becomes (3-246) * ~ dJ ~~dJk BET ~ vJi'V _ dF2 _dE r*dV. Using the generating function ^(<M) = / .5._dF±_dE v.. SYNCHROTRON MOTION Figure 3. the center of the bucket becomes an UPP. (3. Note that when r > 0.236)._Qs [*d<t> where and the Jacobian elliptical function cnw is .332 CHAPTER 3.r+W dEd<p ~ us h V { ' where the action variable is given in Eq.
The formulas for small amplitude approximation are summarized as follows: elli cnu V . F. (3.8541. K' = A ^ / T ^ F ) . 2 50) \2J \2J 1 + [tan (0/2) cn«]2 V .250) remain valid provided that the modulus is replaced by kx of Eq. Small amplitude approximation A tightly bunched beam occupies a small phase-space area. we obtain </> = 2 arctan I tan . (3251) "s-^-fy. V) to the action-angle variables (J. (3. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE with q = e-«K'lK.249) V = . and a-p and a^ the rms conjugate phase-space coordinates.249) and (3..240) or k2 of Eq. We then obtain . or tan — = tan -cnu. (3 .^- T-^'-d-^^y J~^rte! -^r~ sin 2' + (3-252) ^^EoTT?^cos(2n l^ ^ ..250) or equivalently <j> where i = E/2vs. \ 2 ) II and from Hamilton's equation of motion. and q « e~*. I cn% . and 333 ^wir^wy' From Eq. Let A be the rms phase-space area of the bunch. Eqs. %))) can be accomplished by using Eqs.^ S ^ ^ " 1 ' 2 " ^ (-5) 32 3 where k ss l/%/2. Thus the transformation of the phase-space coordinates (<f>.cnu .248) to (3. (3.5. When the voltage ratio is not 0. K = K{k) « 1.242).2 v ^ s i n (+) tan (+) ^ ^ . we get (3-248) (3.247).V. (3. (3.
Harding. The beam may be susceptible to collective instabilities. E55. V.7 The Barrier RF Bucket Bunch beam gymnastics have been important in antiproton production.A. The demand for higher beam brightness in storage rings and higher luminosity in high energy colliders requires intricate beam manipulations at various stages of beam acceleration.Y. Nucl Sci.J. A most critical situation arises when the synchrotron amplitude of the beam reaches the region where Qs is maximum or near the rf bucket boundary. NS-34. The sum rule can be used to identify the region of phase-space that is sensitive to rf phase modulation (see Exercise 3. and A. particle motion near the center of the bucket may become chaotic because of overlapping resonances.1). IEEE Trans. the chaotic region is bounded by invariant tori.103). the Recycler would also accumulate newly produced. 1025 (1987). V. NS30. Ng. Sci. Phys. The extreme of the flattened rf wave form is the barrier bucket. accumulation. and J.E. a Recycler has been built.A. it may be of concern that large amplitude particles can become unstable against collective instabilities. Griffin. a flattened rf wave form can be employed to shape the bunch distribution in order to alleviate space-charge problems in low energy proton synchrotrons and to increase the tune spread in electron storage rings. Lee and K. and the effect on beam dilution may not be important. 5992 (1997). S. However. J. To maintain the antiproton bunch structure. Expanding V in action-angle coordinates as V = En fn{J)ejnt.Y. Ankenbrandt. Rev.334 CHAPTER 3. When an rf phase or voltage noise is applied to beams in a double rf system. and feedback systems may be needed for a high intensity beam that occupies a sizable phase-space area. phasespace painting. MacLachlan. multi-turn injection. SYNCHROTRON MOTION The rms tune spread of the beam is then AQ'7^{^T»G. IEEE Trans.3. where the tune spread is small. which would recycle unused antiprotons from the Tevatron.50 For achieving high luminosity in the Fermilab TeV collider Tevatron. Nucl. a barrier rf wave form can be used to confine the 50See J. we find that the strength functions fn(J) satisfy the sum rule shown in Eq. beam coalescence for attaining high bunch intensity. Bharadwaj. Griffin. (3. J.250). D. . In particular. At the same time. etc. cooled antiprotons from the antiproton Accumulator. Since dQ/dJ = 0 occurs inside the bucket. MacLachlan. The recycled antiprotons can be cooled by stochastic cooling or electron cooling to attain high phase-space density. Sum rule theorem and collective instabilities (3255) The perturbing potential due to rf phase modulation is linearly proportional to V of Eq. (3. C.K. 3502 (1983). Moretti.
For example. and a /"\ The barrier rf wave is normally generated by a solid state power amplifier. The rf wave form is applied to a wideband cavity with frequency hf0. we . which has intrinsic wide bandwidth characteristics. /3c and Eo are the speed and the energy of a synchronous particle. Acceleration or deceleration of the beam can be achieved by employing a biased voltage wave in addition to the bunchconfining positive and negative voltage pulses. a pulse width To. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 335 beam bunch and shape the bunch distribution waiting for the next collider refill. its energy can increase or decrease depending on the sign of the voltage it encounters. and / 0 is the revolution frequency of synchronous particles. and To is its revolution period. Equation of motion in a barrier bucket For a particle with energy deviation AE.22 shows some possible barrier rf waves with half sine. Thus the wide bandwidth rf wave can create a barrier bucket to confine orbiting particles. A. The required bunch length and the momentum spread of the beam can be adjusted more easily by gymnastics with barrier rf waves than with the usual rf cavities. and square function forms. the fractional change of the orbiting time AT/To is — To ~ V i V -n— f3 256) ( 56) where rj is the phase slip factor.T. orbiting particles see no cavity field in passing through the cavity gap. When a particle travels in the time range where the rf voltage is not zero. Most of the time.. Without loss of generality. the accelerator is divided into stable and unstable regions. A . and integrated pulse strength / V(r)d. rf wave is characterized by a voltage height Vo. These wave forms are characterized by voltage amplitude V(r). triangular.22: Possible wave forms for the barrier bucket. An arbitrary voltage wave form can be generated across a wideband cavity gap. pulse gap T2 between positive and negative voltage pulses. The barrier pulse gap Ti.V. the integrated pulse strength for a square wave form is VoTi. Figure 3. pulse duration 7\. Figure 3. where h is an integer. The effect on the beam depends mainly on the integrated voltage of the rf pulse.. In this way.
„ 1 fT-2/2+W ^=r{AE)2 = -f eV(r)dr. Since the barrier rf Hamiltonian is time independent.257) and (3. the wave form of the barrier bucket is reversed. an invariant torus has a constant Hamiltonian value. The equation of motion for the phase-space coordinate r is Passing through a barrier wave. For T] > 0. . all physical quantities depend essentially on the integral / V(r)dT. dJ^r B. The time coordinate for an off-momentum particle —r is given by the difference between the arrival time of this particle and that of a synchronous particle at the center of the bucket. the particle gains energy at a rate of .258). SYNCHROTRON MOTION consider here synchrotron motion with r\ < 0. We define the W parameter for a torus from the equation below: hi . Thus.336 CHAPTER 3.2— (^A where Tc is + 4T (3 262) Clearly. the essential physics is independent of the exact shape of the barrier rf wave.257) and (3. Synchrotron Hamiltonian for general rf wave form From the equations of motion (3. (3.258). we obtain the general synchrotron Hamiltonian for an arbitrary barrier rf wave form: H=-2h^2-T-SeV{T)dTThus the maximum off-energy bucket height can be easily derived: (2B2En rT2/2+Ti \1/2 (3-259) where 7\ is the width of the barrier rf wave form.^ • <3-258> at l0 The equations of particle motion in a barrier rf wave are governed by Eqs. (3.261) The synchrotron period of a Hamiltonian torus becomes T .
^ where u>0 = 2TT/0 is the angular revolution frequency of the beam.T2/2<lr\<{T. The mathematical minimum synchrotron period of Eq. we consider only the square wave forms with voltage heights ±V0 and pulse width Ti in time. and To is the revolution period of the beam. it gains (loses) an equal amount of energy eVQ. separated by a gap of T2.V.264) Thus the phase-space trajectory for a particle with maximum off-energy AE is f (AE)2 if \T\ < T 2 /2 ^ 2 = {{AEf-(\r\-T{)^^ . d(AE)/dt = ±eV0/TQ every turn. The phase-space area of the invariant phase-space ellipse is ^ = 2 T 2 A ^+3^^(A^3- (3-266) The maximum energy deviation or the barrier height that a barrier rf wave can provide is where 7\ is the pulse width of the rf voltage wave. When the particle passes through the cavity gap at voltage ±V0. The bucket height depends on VQTJ. which is the integrated rf voltage strength / V(T)CLT./2)+Tl. (3. The number of cavity passages before the particle loses all its off-energy value AE is JV=^L (3. The synchrotron period is A£b=hrTj (eV0T12pE0V/2 ' (3-267) for particles inside the bucket. The phase-space ellipse is composed of a straight line in the rf gap region and a parabola in the square rf wave region.e. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE C. Square wave barrier bucket 337 Since the effect of the barrier rf wave on particle motion depends essentially on the integrated rf voltage wave.268) is S'min=l T =2 w l ^ J + 4 ^ T o s \v\eV0 ) ' T2(p*E0\ \AE\ (-6) 328 (3'269) and the corresponding maximum synchrotron tune is _(T1WV^V" \l2 6lfi'-tiQ ) . i.
TuT2). Hamiltonian formalism The Hamiltonian for the phase-space coordinates (r. Note that if T2 > 4Ti.3 Hz.23 shows va vs AE with Fermilab Recycler parameters Eo = 8. This feature is similar to that of a double rf system. SYNCHROTRON MOTION Note here that TTTO/(16T2) plays the role of harmonic number h of a regular rf system.^)0(r .. Lee et al. the synchrotron tune is peaked at an off-energy AE smaller than the bucket height AE\. D. The synchrotron tune is a function of the off-energy parameter AE given by *-**-i/i||(1+« [Ill's)"'- w Note that when the rf pulse gap width decreases to T2/T1 < 4. -yT = 20. AE) is Ho = ^ r ( A £ ) where MT.338 CHAPTER 3.2.| ) + ( r Tl + '^) (3.5 us.. . Parameters used are EQ = 8. (3.7 x 10"5 for T2 = Tu i. the synchrotron frequency is 3.51 Figure 3. and Ti = 0. For example. J.9 GeV.8 kHz. the synchrotron tune is a monotonic function of AE. Liu et al. 7\ = 0.e. Rev.23: Synchrotron tune vs off-energy parameter AE.TUT2) 2 + ^^fO(T.Y.4.Y. Phys.| ) ] . the synchrotron tune becomes peaked at an amplitude within the bucket height. /o = 89.8 kHz. 5717 (1994). On the other hand. Figure 3. and 8. and T2/Ti = 1. E50. if T2 < 4Ti. /„ = 89.5 /JS.272) = -l + Yi[(r + Ti + ~)e(r + T1 + ^)-(r+^)9(r _ ( r _ Tl)0{T .9 GeV. E49. -yT = 20. i/s>niax = 3. Phys. Vo = 2 kV.7. Vo = 2 kV.Tx . 51 S. R3349 (1994).273) .7. Rev.
B = 2nJ = (2T2 + ^Ti) AEh. AE) to the actionangle variable can be achieved by using the generating function F2(J. J—t (3. and W = 7\ is associated with particles on the bucket boundary. For a constant Ti.e. W has the physical meaning that it is equal to the maximum phase excursion \T\ in the rf wave region.TuT2)dr.T)= fT AEdr.274) The parameter PV with a dimension of time is related to the Hamiltonian value by *-f»^K' <3-275> For a given Hamiltonian torus.277) V(T)CIT Again. E./ AEdr = i R p 27T J 2ir \ 7r|r?| / /W + Mr. Action-angle coordinates = Canonical transformation from the phase-space coordinates (r. i. The action of a Hamiltonian torus is J = i. \ o / (3. Therefore W = 0 corresponds to an on-momentum particle. T2 and Vo. J " (3. the bucket area depends only on the integrated rf voltage strength / VQTX.V. BEAM MANIPULATION IN SYNCHROTRON PHASE SPACE 339 Here 9(x) is the standard step function with 9(x) = 1 for x > 0 and 9(x) = 0 for x <0. the Hamiltonian Ho is a constant of motion. The action for a particle torus inside the bucket is The bucket area is related to the maximum action with W = 7\. The angle variable ip is .278) where f = W + (T 2 /2).
340 CHAPTER 3. AE < 0 2ir\/W / 1 4Vc + 2V. The momentum compaction factor is ac = 0. (a) Assuming that the rf voltage is ramped according to Vrf (i) . SYNCHROTRON MOTION The integral can be evaluated easily to obtain ^mJW A + + \T* + T \1-W-\T2<T<-\T2. and that the motion of a stable particle orbit in the barrier bucket with 7 < 0 is clockwise. We choose the convention of ip > 0 corresponding to ? a clockwise motion in synchrotron phase-space. The resonance strength function decreases slowly with mode number. Active compensation may be used to compensate the effect of rf phase modulation. .W . . stable bucket area may be reduced. The resonance strength functions and their associated sum rules can be derived analytically.V0 + {V.V0) U^ 52 See . Lee and K. T!]. The rf system operates at h = 1 with a maximum voltage 240 V.5 1. Note that 2ipc + ips = n for one half of the synchrotron orbit.52 Exercise 3.6191. 5992 (1997). if 2 c + A .^ 7 w iW V I 2^ A£>0 1-K\fW 3A + A + I 1 + ^ + T2 + 4W\]W+2T2 2T2 < r <W+±T2: AE < 0 nTtW {lT2 ~ T) -5 T 2 < ^ < \T2. it is important to avoid a large reduction of stable phase-space area. The rf phase and voltage modulation can severely dilute bunch area if the modulation frequency is near the top of the synchrotron tune and its harmonics.Y.T2 + 4W\]W + -T2 + r if . such as rf noise. Phys. Rev. S.AE>0 T^hw(T+lT2) + if-|T 2 <r<lT 2 . where ^ = l^w> ^ = T7^W <3-280) are respectively the synchrotron phase advances for a half orbit in the rf wave region and in the region between two rf pulses. AE < 0. The rf phase modulation due to orbit length modulation resulting from ground vibration can be important. . The circumference is 17. When a perturbation. t e [0. The Cooler Injector Synchrotron (CIS) accelerates protons from 7 MeV to 200 MeV in 1. The momentum spread of the injection linac is about ±5.AS>0 lT2~T ~T if if l T 2<r<W+iT 2 . Because the solid state amplifier is a low power device.\T2 < T < -\T2.Y.0 x 10~3.0 Hz. is applied to the barrier rf system.2^) .364 m. E55. Ng.
show that the bunch length in the final step is where initial is the initial bunch length in orbital angle variable.2. it is given by (E] -C<JsP> where Js is the damping partition number with Js w 2 for separate function machines.4 m. the rf gymnastics for bunch rotation is performed by adiabatically lowering the voltage from Vi to V2 and suddenly raising the voltage from V2 to V\ (see also Exercise 3.EXERCISE 3.5 341 where VQ and V\ = 240 V are the initial and the maximum final rf voltages. and find the maximum B. find the frequency ramping relation of the rf cavity. 2. the harmonic number is h = 588.15 ns with an initial voltage V\ = 4 MV. Using Eq. and the phase-space area is A = 0.4. (3.1. and vs\ and uS2 are the synchrotron tune at voltages V\ and V2.05 eV-s for 6 x 1010 protons. Neglecting wakefield and other diffusion mechanisms.43). (b) If the magnetic field of a proton synchrotron is ramped according to B(t) = B0+ (£ . -Bo).58) and conservation of phase-space area. (3. Calculate the adiabaticity coefficient of Eq.8. In proton accelerators. Ti is the voltage ramp time. For a flattened potential well in the double rf system with (6ls = <j>2S = 0.83xl0-i3m . where the circumference is 3319. For an isomagnetic ring. and c 4mc 32V3 me Using the electron storage ring parameters listed in Exercise 3. and the rf bucket height during the rf voltage ramping as functions of time t with VQ = 10 V and T\ = 10 ms. calculate the phasespace area in eV-s. show that the Hamiltonian for small amplitude synchrotron motion is 3Cy» = J5 A = 3. Apply the bunch rotation scheme to proton beams at E = 120 GeV in the Fermilab Main Injector.5). 4. The energy of the secondary antiprotons is 8.2~\ (B. the transition energy is yT = 21.9 GeV. Find the voltage V2 such that the final bunch length is 0. h] where Bo and B\ are magnetic field at the injection and at the flat top. Change these parameters to see the variation of the adiabaticity coefficient. If the acceptance of the antiproton beam is ±3%. what is the phase-space density of the antiproton beams? 3. Verify Eq. and t = 0 and t = ti are the time at the beginning of ramp and at the flat top. te [0.232) and derive the Hamiltonian for the double rf system. what is the phase-space area of the antiproton beams? If the antiproton production efficiency is 10~5. the momentum spread of an electron beam in a storage ring is determined mainly by the equilibrium between the quantum fluctuation of photon emission and the radiation damping. (3.
We solve the synchrotron motion for the quartic potential below.-V»-(^)d.j .85407468 is the complete elliptical integral with modulus k = l/v/2. h is the ratio of the harmonic numbers.342 CHAPTER 3. Compare your results with that of Eq. the Hamiltonian value E is a constant of motion. 41 is the amplitude of the phase oscillation.253) for the h = 2 case.J)= ( Td<t>. Show that the action variable is related to the Hamiltonian value by where K = K{J\) = 1. sn. . (b) Show that the synchrotron tune is (c) Define the generating function F2{<I>. and the independent "time" variable is the orbital angle 6. (3.(^Hi). SYNCHROTRON MOTION where b = (h2 — l)/24. (a) Since the Hamiltonian is time independent. where en. and dn are elliptical functions with modulus k = l/\/2. Jo and show that the solution of the synchrotron motion is given by (IK 1\ <t> = 4> en ( — ip\. .
defined as the ratio of the square of the rf voltage seen by the beam to the dissipated power. Some fundamental parameters of cavities are transit time factor. V-E = 0. but a smaller gap can cause electric field breakdown due to the Kilpatrick limit (see Sec. Beam loading and Robinson dipole-mode instability will be addressed. The transit time factor reduces the effective voltage seen by passing particles.VI FUNDAMENTALS OF RF SYSTEMS 343 VI Fundamentals of RF Systems The basic function of rf cavities is to provide a source of electric field for beam acceleration.3) reflects the finite passage time for a particle to traverse the rf cavity. Maxwell's equations (see Appendix B Sec. Properties of pillbox and coaxial-geometry cavities will be discussed. We may reduce the accelerating voltage gap to increase the transit time factor.281) where e and \i are dielectric permittivity and permeability of the medium. The longitudinal electric field must be synchronized with the particle arrival time. Further properties of high frequency cavities used in linacs will be discussed in Sec. VI. Thus a cavity with a higher Q-factor has a higher shunt impedance. V. coaxial geometry is commonly employed. Generally. VIII. 1 Pillbox Cavity We first consider a cylindrically symmetric pillbox cavity [18] of radius b and length I (left plot of Fig. of a resonance cavity will be defined and discussed. In this section we examine some basic principles in cavity design. and the filling time. VxB =^ . V) for electromagnetic fields inside the cavity are V . The quality factor (Q-factor) depends on the resistance of the cavity wall and the characteristic impedance of the rf cavity structure. The EM waves in the cavity can conveniently be classified into transverse magnetic (TM) . shunt impedance. 3. the shunt impedance. At lower frequencies. is an important figure of merit in cavity design. pillbox cavities with nose-cone or disk loaded geometry can be used. (3. Vxfi = ~ (3. and quality factor. For cavities operating at a few hundred MHz or higher. Some fundamental characteristic parameters. where only electromagnetic fields at resonance frequencies can propagate. the Q-factor. while the accelerating field varies with time. At a given resonance frequency.B = 0. the ratio of shunt impedance to Q-factor depends only on the geometry of the cavity and the characteristic impedance. It is defined as the ratio of the rf power stored in the cavity to the power dissipated on the cavity wall.24). Cavities are classified according to their operational frequencies. Resonance cavities.3). The shunt impedance. are a natural choice in rf cavity design. we will show that a resonance cavity can be well approximated by an equivalent RLC circuit. The transit time factor of Eq.
ks.344 CHAPTER 3. the TM standing wave modes in cylindrical coordinates (r.283) Similarly the radial wave number is determined by the boundary condition with Es = 0 and E. for which the longitudinal electric field is zero.f. the electromagnetic fields satisfy the boundary condition: n x E = 0. Assuming a time dependence factor e?"* for electric and magnetic fields. h • H = 0. The longitudinal wave number k is determined by the boundary condition that Er = 0 and £ 0 = 0 at s = 0 and t. V) 53 Es = A k2 Jm{krr) cos rruj)cos ks Er = — AkkT J'm{kTr) cosm(t>sinks E.284) standing wave can be decomposed into traveling waves in the +s and -s directions. There is no tangential component of electric field.l.mn = 3jy. (3. Left: pill-box cavity with disk load. are zeros of Bessel functions Jm(jmn) = 0.P = Y> P = 0. . i. The TM modes are of interest for beam acceleration in the rf cavity. <j>. Figure 3. SYNCHROTRON MOTION mode. ' Bs = 0 (3'282) BT = -jA(mui/c2r) Jm{krr) sin mcj) cos ks .p — A (mk/r)Jm{kTr) sinm(j>sinks . and UJ/C = Jk2 + k2.24: Schematic drawings of high frequency cavities.2. m is the azimuthal mode number. where j m n . kr are wave numbers in the longitudinal and radial modes. i.---. k. listed in Table V. and no normal component of magnetic field. kr.2. = 0 at r = b.e. . these high frequency cavities have similar basic features. 53A (3. right: nosecone cavity. for which the longitudinal magnetic field is zero. and transverse electric (TE) mode.e. In an ideal acceleration cavity. B$ = -jA (bjkr/c2) J'm{krr) cos m<f> cos ks where A is a constant. Although their names and shapes axe different. s = 0 and I correspond to the beginning and end of the pillbox cavity. s) are (see the Appendix B Sec. where h is the vector normal to the conducting surface.
To slow down the phase velocity. fcoio = ^ . p) is 345 kmnp = ^mn + k?. (3.25 shows an example of a coaxial cavity. VIII. it requires a very small amount of ferrite for tuning. the cavity is loaded with one beam hole with an array of cavity geometries and shapes. The EM field of the lowest mode TMoio (kSiP = 0) is Es = EoJo(kr). All cavities convert TEM wave energy into TM mode to attain a longitudinal electric field. VI. FUNDAMENTALS OF RF SYSTEMS The resonance wave number k for mode number (m.285) The lowest frequency mode is usually called the fundamental mode.25: Schematic drawing of a low frequency coaxial cavity. The TEM wave in the coaxial wave guide section is converted to the TM mode at the cavity gap through the capacitive load. B* = j ^ J i ( f c r ) . The material is made of double oxide spinel Fe2O3MO. = J%" + P~ = ^ = ^L.24 shows high frequency cavities with disk and nose-cone loaded geometries.8 cm. The phase velocity. where M can . Other resonance frequencies are called high order modes (HOM). Note that the TEM wave is matched to a TM wave at the capacitive loaded gap for the acceleration electric field. n. where the length is much larger than the width. We will return to this subject in Sec. (3. Thus beam particles traveling at speed v < c do not synchronize with the electromagnetic wave. Many different geometric shapes are used in the design of high frequency cavities. Figure 3. Figure 3.2 Low Frequency Coaxial Cavities Lower frequency rf systems usually resemble coaxial wave guides. for the traveling wave component of the TMOio mode with kSiP = 0 is infinite. a 3 GHz structure corresponds to A = 10 cm and b — 3. A= ^ . Such a structure is usually used for high frequency cavities.286) For example. w//cSlP. When the cavity is operating in 50 to 200 MHz range. The art (science) of cavity design is to damp HOMs without affecting the fundamental mode. Figure 3. but their function and analysis are quite similar.54 When the cavity 54Ferrite is magnetic ceramic material that combines the property of high magnetic permeability and high electric resistivity.VI.
Touch-Tone telephone.3) I(s. At lower rf frequency. and £ is the length of the structure. Neglecting the flux penetration in the conductor.346 CHAPTER 3. 2173 (1983). Assuming a time dependent factor e?ui. Ferrites are commonly used in frequency synthesis devices. the rf cavities must be ferrite loaded in order to fit into the available free space in an accelerator. V{s. kickers. + 2?r ^ 1 + i TX 4?r V I r2 C=^-y ln^/n) (3. ferrite rings in the cavity are needed to slow down EM waves. Let ri and r-i be the inner and outer radii of a wave guide. the TEM wave guide is usually ferrite loaded with magnetic dipole or quadrupole fields for bias frequency tuning. Cr.291) be Mn.6. The characteristic impedance of a wave guide is Zc = Rc = JL/C = LjL»-l-<[^\n-. To understand the capacitive loading that converts the TEM wave into the TM wave at the cavity gap. we consider an ideal lossless transmission line. we study the rf electromagnetic wave in the wave guide. Application in accelerator can be found in induction linac. Smythe. t) = Io cos ks + j(V0/Rc) sin ks. frequency tuning for rf cavities. the resonating frequency is w= Vrc = ijw * Tfifo' 47"7" • /[MHz]0i//* o (3'288) where e = e0 is assumed for the dielectric permittivity.Vo cos ks + jI0Rc sin ks.290) Now.ca is the skin depth of flux penetration.289) For a cavity operating beyond 20 MHz. Thus the required cavity length for the fundamental mode is t = . t) . ferrite can be used only for tuning purposes. etc. IEEE Trans. NS-30. Ni..When a biased field is applied to the ferrite core. Sci. the magnetic permeability can be tuned to match the change of the particle revolution frequency. The inductance and the capacitance of the concentric coaxial wave guides are L = ^ l n r. (3.55 Using the wave guide transmission line theory. low loss microwave devices. where the electromagnetic field has no longitudinal component. 55 W. 5skin = ^2/ujfj.^ = = ^V^/Mo (3.287) where /ic is the permeability of the conductor. etc. the current and voltage across the rf structure are (see Exercise 3. characteristic properties of rf systems can be analyzed. (3. At frequencies below tens of MHz. etc. . Zn. Typically the magnetic permeability of ferrite is about 1500/io. Nucl.R. SYNCHROTRON MOTION is operating at a few MHz range.
£T = A/4: the length of the coaxial cavity is equal to 1/4 of the wavelength of the TEM wave in the coaxial wave guide.e.293) V(s.294). For example. and I is the length. (3. and C gap .294) becomes k£T = TT/2.e. Shunt impedance and Q-factor (3. for a given £T:RC. Thus the quality factor becomes Q = £ = !£„ * ™ _ £ h I*. a total capacitance of 10 pF implies that C gap = 20 pF.t) = I0{t)cosks. I{s. FUNDAMENTALS OF RF SYSTEMS 347 where s is the distance from one end of the transmission line. and the wave number of the line is 2TT _ w For a standing wave. ri and r-i are the inner and outer radii of the transmission line. (3. i. the boundary condition at the shorted side is V = 0 at s = 0. where the end of the transmission line is shorted.VI.t) = +jI0(t)Rcsinks. VQ and IQ are the voltage and current at the end of the line where s = 0. The gap voltage of the coaxial cavity is Zin + Zgap = 0. In principle. The lowest frequency is called the fundamental TEM mode. tan kl (3. the resonance condition of Eq. and Cgap is the capacitance of a half gap. The length of the line is chosen to match the gap capacitance at a required resonance frequency: = -. Z gap = — j/(wC gap ) is the gap impedance. The line input impedance becomes Zin = j ^ . The length £T of one-half cavity.297) . there are many resonance frequencies that satisfy Eq. Thus such a structure is also called a quarter-wave cavity. (3.= +JR. or tan Uv = Vrt = +jI(0)Rc sin k£r = + j 4 = f t vl + g A. (3. u> is the rf frequency. (3. i. the gap capacitance.294) w/i c c gap g where g is the geometry factor of the cavity.295) The surface resistivity Rs of the conductor and the resistance R of a transmission line are ^ = V!7' R = ~2^in+72)> (3 ' 296) where a is the conductivity of the material.292) The line impedance is inductive if kl < ir/2. the biased current. and the external loading capacitance can be designed to attain a resonance condition for a given frequency range. If the loading capacitance is small.
The total power of dissipation P& is ^ = IH^R JK cog2 x dx = J ! i ^ _ [ ( 1 + f) ^ .e. 0.05 11000 An important quantity in the design and operation of rf cavities is the shunt impedance. The input impedance of the wave guide is represented by an equivalent inductance.Bottom: Plot of the impedance of Eq. The wave guide is loaded with capacitive cavity-gap and real shunt impedance. 298) Table 3. As the gap capacitance increases.i g + g]_ (3 . This is the resistance presented by the structure to the beam current at the resonance condition. i. If Cgap = 0.300) •K L(l + ff2)cot 1g + g\ where the expression in brackets is a shunt impedance reduction factor due to the gap capacitance loading. Figure 3. i. the shunt impedance becomes (3. and <rcu ~ 5.e. .348 CHAPTER 3. 5 = 0.2 lists typical Q-factors for a copper cavity as a function of cavity frequency. where we have used In fo/ri) ~ 1 a n d r\ « 0.05 m.301). The resonance frequency and the Q-factor of the equivalent RLC-circuit are UJT = 1/VLeqCeq and Q = iish^Ceq/ieq.8 x 107 [flm]"1 at room temperature.05 3500 I 100 6.05 _Q 1100 I 10 21. The solid lines are the real and the imaginary impedances for Q=l. n 0.6: Some characteristic properties of copper RF cavities / [MHz] 11 <5skin H 66.26: Top: Schematic drawing of an equivalent circuit of a cavity. the shunt impedance decreases.6 0.299) #sh = . tf* = ~fFor a transmission line cavity. (3. (3.M • 2 ^ !-i _L ^ Q . the capacitance loading factor is 1. and the dashed lines are the corresponding impedances for Q=30. SYNCHROTRON MOTION Table VI.
becomes Z=(~+JuCeq + -±-) = .VI.' Ws = WsOe-"^. we obtain ^JT = .304) The time for the electric field or voltage to decay to 1/e of its original value is equal to the unloaded filling time 2Q Tm = — .294) implies that the reactance of the cavity is zero on resonance. B. represented by a parallel RLC circuit. where Ws is the stored energy. ^ --^hCosVe"^.305) . Q = Rsh^/Ceq/Leq.301) for Q = l and Q=30. Accelerator cavities usually contain also many parasitic HOMs.297) is equal to the ratio of the stored power P s t to the dissipated power P^. i. (3.301) h where Leq and Ceq are the equivalent inductance and capacitance.p o = . The right plot of Fig. the cavity gap presents a capacitance and resistive load shown in Fig. uiT = (L^C^)" 1 / 2 .. At the resonance frequency wr particles see a pure resistive load with an effective resistance Rgt. and ^ = tan _ 1 2QK-a^) (3302) Here tp is the cavity detuning angle. Each HOM has its shunt impedance and Q-value. 3.The rf system becomes capacitive at w > w n and inductive at w < wr. where Z-lTi = jwL eq .e. and the effective impedance is R^.^ W . (3. (3. and lowering the Q-factor of these sidebands are very important in rf cavity design and operation. (3. The impedance of the rf system. detuning.. and Ceq = Cgap.26. Using energy conservation. Correction. If the frequency of one of the HOMs is equal to that of a synchrotron or betatron sideband.26 shows the real and imaginary parts of the impedance of Eq. Filling time The quality factor defined in Eq.The matching condition of Eq. the beam can be strongly affected by the parasitic rf driven resonance. (3. 3. (3. FUNDAMENTALS OF RF SYSTEMS 349 From the transmission-line point of view.
.300. the rf voltage is the time derivative of the total magnetic flux linking orbit (Faraday's law of induction). The dissipation power q. By adjusting the bias current and the bias field direction.306) where A = lr\ In(r2/ri) « £(r2 — ri) is the effective area of the ferrite core and Bi is the peak magnetic flux at r = r x . At an rf frequency above tens of MHz. The peak magnetic flux in Eq. The shunt impedance of an rf structure is the resistance presented to the beam current at the resonance condition.e. where A is the effective area of the ferrite core. Frequency tuning can be achieved by inducing a DC magnetic field in the ferrite core. Assuming that the magnetic flux density varies as \/r in a coaxial structure and assuming a sinusoidal time dependent magnetic flux density. ^b ~ T P ~ ^^d p \Vrt\ = ~ 5 ~ ~ ~ KcQ' ^d RcPst r> n (3. Since the Q-value of ferrite is relatively low. SYNCHROTRON MOTION C. (3.308) which alone is not adequate for the required frequency tuning range. Qferrite « 50 . (3. power dissipation in ferrite is an important consideration. i. Note that increasing the outer radius of the ferrite core is an inefficient way of increasing the rf voltage.350 CHAPTER 3.rfBM „ Vr] in a cavity should be efficiently removed by cooling methods. Qualitative feature of rf cavities Qualitatively.307) /q on^l where Pst is the power stored in the cavity and Pj is the dissipated power. The quality factor Q of the ferrite loaded accelerating cavity is dominated by the Q value of the ferrite material itself. i. the effective permeability for rf field can be changed. using an external magnet or bias current to encircle the ferrite without contributing a net rf flux. we need a large volume of ferrite to decrease the flux density in order to minimize energy loss. the cavity size (normally 1/2 or 1/4 wavelength) becomes small enough that a resonant structure containing little or no ferrite may be built with significantly lower power loss at Q « 104 with a narrower bandwidth. we obtain Kf = wrf^ / 2 Jri B{r)dr = uJlf£B1rl In — « UJV(BIA. 7"i (3.e.306) depends on the ferrite material. To obtain high rf voltage at low frequencies.
It accelerates protons (or light ions) from 7 MeV to 225 MeV. . where / is the surface current and R of Eq. The cavity is a quarter-wave coaxial cavity with heavy capacitance loading. The fi of the ferrite material is changed by a superimposed DC magnetic field provided by an external quadrupole magnet.290) is about 60 fi. The external quadrupole magnet provides biased field in ferrite rings to change the effective permeability. ten Phillips 4C12 type ferrite rings are used. Example: The rf cavity of the IUCF cooler injector synchrotron The IUCF cooler injector synchrotron (CIS) is a low energy booster for the IUCF cooler ring.296) is the surface resistance of the structure. resonance frequency can be tuned only by a slotted tuner or by physically changing the size of the cavity. 3.27). Analysis of such a field shows that the field direction is mostly parallel to the rf field. The cavity can still be considered as a coaxial wave guide. i. the main portion of the rf cavity can be made of copper or aluminum with a small amount of ferrite used for tuning.56 To make the cavity length reasonably short and to achieve rapid tuning. The stored power is I2RC and the power dissipation is I2R. Figure 3. where the biased fields in the ferrite rings are perpendicular to the rf field.287) to (3. D. At frequencies above a few hundred MHz.e. (3. The ferrite rings return the magnet flux between the two adjacent quadrupole tips (Fig. required for synchrotron acceleration. FUNDAMENTALS OF RF SYSTEMS 351 At frequencies of a few hundred MHz. private communications. Pei. except near the tips of the quadrupole.306) remain valid. (3. where adequate and efficient rf power sources are commercially available. The characteristic impedance Rc of Eq. In the working 56A. and Eqs. (3.27: The cross section (left) and the longitudinal view of the CIS rf cavity. along the azimuthal direction.VI.
the number of windings is usually limited to no more than a few turns because of possible resonance and arcing. The CIS cavity is thus able to operate with a 10:1 frequency ratio with high efficiency. rather than B/H. The phenomenon of gyromagnetic resonance associated with perpendicular biasing. It usually takes 1000 A or more to bias such a cavity. the energy loss due to the passage of an rf gap. needs to be considered and avoided in the design of the cavity. is At/ = 11 J(w) \2Z(oj)dtJ = 2krq2. E. is determined by dB/dH. it has been difficult to feed the rf generator power to the cavity efficiently because of the high voltage standing wave ratio (VSWR) caused by impedance mismatch (see Appendix B. The advantages of using an external biasing magnet include making it possible to separate the rf field from the biasing elements.5 MHz or 1 . the power loss in ferrite material varies (usually increasing as frequency increases). as in perpendicular analysis.10 MHz by varying its loading capacitor. (3. and the rf field in the cavity will not be affected by the biasing structure. As a result. to first order in wave propagation. For example. the strength of the biasing magnet in each section can be adjusted by the coupling loop.352 CHAPTER 3. and the input impedance can be maintained constant to match the transmission line impedance of the rf amplifier. The loading capacitor reduces the length requirement of coaxial cavities and can also be used conveniently to switch frequency bands. . SYNCHROTRON MOTION region of the ferrite biasing strength. As the frequency changes. represented by an RLC resonator model. The coupling coefficient can be used to compensate the change in the gap impedance.310) where is the loss factor of the impedance at frequency wr. Wake-function and impedance of an RLC resonator model If we represent a charged particle of charge q by I(t) = qS(t) = (1/2TT) / qe?utdu). This means that the passing particle loses energy and induces a wakefield in the cavity.V. In the CIS cavity this problem was solved by dividing the ferrite rings into two sections. the effect of the perpendicular component on ferrite rf-fi is small.3). however. the bias supplies for these external quadrupole biasing magnet type cavities are rated at only 20 A. If the biasing field is to be produced only by a bias winding threaded through the rf cavity.5 . In CIS and the IUCF cooler ring. the CIS cavity can be operated at 0. due to the higher impedance of a resonant structure and optimized amplifier coupling. as in parallel biasing analysis. As many windings of the bias coils as practical can be used — resulting in a smaller amperage requirement for the bias supply. The effective rf-/i.
oscillating at frequency uvf. If the filling time is long. The effective voltage at the rf gap is a superposition of voltage due to generator current and induced voltage due to induced rf current. the resulting rf voltage acting on the passing beam may cause beam deceleration in an uncontrollable manner. Thus beam loading needs to be considered in the operation of rf cavities. i. and 0 if t < 0. where </)s is the synchronous phase angle. (3. The representation of sinusoidal waves in a rotating frame is called a phasor diagram and is particularly useful in analyzing beam loading compensation problems. Z{u) = / fOO W{t)e-jutdt = / roo W(t)e~jutdt.3 Beam Loading A passing beam charge can induce wakefields in an rf cavity.e. By definition. (3. The magnitude of the vector is equal to the amplitude of the rf voltage. When beams pass repetitively through the cavity. 2ir J L wrJfo J (3.305). given by the inverse Fourier transform of the impedance. can be considered as a vector rotating in the complex plane at an angular frequency u>rf. The rf voltage.311) JO J-oo the wake function.312) where 0(i) = 1 if t > 0. we have V0cos6 = Vosin^s. and thus the rf voltage is stationary in this rotating coordinate system. for the RLC resonator model becomes W(t) = •£. W(t) « 2kTe-t/T{0 coswrt. the effective voltage is the sum of the voltage supplied by the generator current and the wakefields of all beams. Now. then the wake potential is a sinusoidal function with angular frequency u>T.^ r sinwril e-t/Tf° Q(t). . and the rf voltage seen by the beam is the projection of the rotating vector on the real axis.(1/4Q 2 )) 1/2 . Beam loading is important in the design and operation of rf cavities. Let Vo and 6 be respectively the amplitude and the angle with respect to the real axis of the rf voltage vector. Thus the filling time corresponds also to the wakefield decay time. Tf0 = 2Q/wr is unloaded filling time denned in Eq. Without proper compensation.VI FUNDAMENTALS OF RF SYSTEMS 353 Since the longitudinal impedance is defined as the Fourier transform of the wake function. VI.[ Z{u)e>utdu) = 2kT fcoswTt . we choose a coordinate system that rotates with the rf frequency. and wr = wr (1 .
passing through the cavity. . 87. the voltage is expressed as a phasor V = Veje.354 A. we obtain X = 0. (3.. V* = \V*' f = \- ( 3 .313) Phasors are manipulated by using usual rules of complex vector algebra. SYNCHROTRON MOTION The electromagnetic fields and voltages in a standing wave rf structure are normally expressed as complex quantities. The total energy deposited in the cavity is Wc = a|H(l) + Kb(2)|2 = af2V b cos-J The energy loss by these two particles is AU = [qVe] + [qVe + qVh cos(x + 9)]. (3. 5 7 P. Proc. we assume that the stored energy in a cavity in any given mode is W = aV2. Phasor CHAPTER 3. From the conservation of energy. Now. We assume further that the induced voltage lies at phase angle \ with respect to the inducing current or charge. V = Vej(-ut+e\ where u is the frequency and 9 is a phase angle.317 ) The result can be summarized as follows: B. In the rotating coordinate system. Vh = ±.316) = 2aVb2(l + cosfl). and the effective voltage is Ve = /VJ. 452 (1981). The properties of rf fields can be studied by using graphic reconstruction in phasor diagrams. when a charged particle passes through the cavity. separated by phase angle 9.314) We assume that a fraction / of the induced voltage is seen by the inducing particle. (3. the image current on the cavity wall creates an electric field that opposes the particle motion. Fundamental theorem of beam loading The cavity provides a longitudinal electric field for particle acceleration. To prove this fundamental theorem. B. Wilson. (3. AU = Wc. is the induced voltage in each passage. where V}. AIP Con}. The question arises: what fraction of the electric field or voltage created by the beam affects the beam motion? The question can be addressed by the fundamental theorem of beam loading due to Wilson. However.315) where the first and second brackets are the energy losses due to the first and second particles respectively. we consider two identical charged particles of charge q.57 Theorem: A charged particle sees exactly | of its own induced voltage.
the rf phase shift is > <j>-{u. the phase angle \ = 02. and the term . The induced voltage of a beam must have a phase maximally opposite the motion of the charge. Steady state solution of multiple bunch passage Consider an infinite train of bunches. and T[ = 2Qi. (3./k>r is the cavity time constant or the cavity filling time. is the rf image current.^ (A->0). passing through an rf cavity gap.1 / 2 is neglected. separated by time Tb. When the cavity is on resonance.0)) « J i ^ ^ c o s ^ .e.321) For rf cavities used in accelerators.l pQL(^-^r)l = t a n _i [(w _ Wf)Tf] ^ (3 320) where to is the cavity operation frequency. When the cavity is detuned by a detuning angle rp. we have A = Tb/T{ = wrTb/2QL <C 1. 3. Ve = 14/2. k = Vb2/(4W/C). and the induced voltage seen by the beam is Vb = IAhX(-l + 1 _ g_1(A+. <P = t a n . FUNDAMENTALS OF RF SYSTEMS 355 1. where k is the loss factor. Vb = 2kq or Ve = kq. The particle sees exactly 1/2 of its own induced voltage. Vbo = IiR^hTb/T{. (3. i. 4.e. QL-(RA + Rs)Rc-lTd' d " V (3 . Wc = aV£ = q2/4a = kq2. and A = T\.cor)Tb = +(T b /T f ) tan ip = +A tan V ./Tf is the decay factor of the induced voltage between successive bunch passages. wr is the resonance frequency of the rf cavity. . taking into account the generator resistance Rs in parallel with the RLC circuit of the cavity. C. Here QL is the loaded cavity quality factor.319) The filling time of the loaded cavity is reduced by a factor 1/(1 + d). the induced voltage seen by the particle is Vb = \vb0 + Vb0(e-^ +e" 2 ^ + •••) = VM(~ + 1_e_(A+J>)).322) where I. The beam induced voltage across the rf gap at the steady state is exactly the rf image current times the impedance of the rf system.VI. i. (3-318) where 0 = — (u — wr)Tb [Mod 2?r] is the relative bunch arrival phase with respect to the cavity phase at the rf gap.
Right: When the cavity is detuned to a detuning angle ip. The combination of generator voltage Vg and induced voltage V\ gives rise to a decelerating field Vo. The voltage seen by the beam is the sum of the voltage produced by the generator current and the beam induced current. AIP Conf. p. It appears that the rf system would be optimally tuned if it were tuned to on-resonance so that it had a resistive load with Vg = IoRsh. generated by the beam is twice the DC current. The beam will induce I\Rsh across the voltage gap (see dashed line in Fig. When a short beam bunch passes through the rf system. 3. we will find shortly that the effect of beam loading would render such a scheme unusable.58 The detuning angle and the generator current are adjusted so that the resultant voltage has a correct 58J. Such a large rf generator current at a phase angle other than that of the rf acceleration voltage is costly and unnecessary. One way to compensate the image current is to superimpose. F. IEEE Tran.E. and results in deceleration of the beam. (2. we need a generator rf current IQ = Ioe>e with phase angle 9 so that the voltage acting on the beam is Vacc = Vg cos 9 = Vg sin <j>s.28).211).28 (left). Sci.However. Left: The beam loading voltage for a cavity tuned on resonance. This is shown schematically in Fig. 2138 (1985). where the required gap voltage IoRsh and the synchronous phase angle 4>s are altered by the voltage induced by the image current. Boussard. the image rf current /.4 Beam Loading Compensation and Robinson Instability To provide particle acceleration in a cavity. An alternative solution is to detune the accelerating structure. 3. VI. . current directly opposite to the image current. Nucl. CERN 91-04. the superposition of the generator voltage Vs and the beam loading voltage V\ gives a proper cavity voltage Vo for beam acceleration. where (j>s is the synchronous phase angle. Proc. Griffin. 294 (1991). The projection of the resultant vector Vo on the real axis is negative.28: Phasor diagrams for beam loading compensation.356 CHAPTER 3. D. SYNCHROTRON MOTION Figure 3. as shown in Eq. 87. 564 (1981). on the generator current. Pedersen. NS-32. Thus the stable phase angle 4>s of the synchrotron motion will be changed by the induced voltage.
mis-injection. mismatched beam profile. we obtain tanfl g = s tan ip .]iU costfe " * . cos(9g . we discuss only the dipole mode stability condition related to beam loading. FUNDAMENTALS OF RF SYSTEMS 357 magnitude and phase for beam acceleration. = — 7b Ve = Vge?e rf beam image current.£) = VQ cos 9 + £VQ sin 9 = Vo sin <ps + £Vo cos <f>s.324) where 9% is the phase angle of the generator current relative to the ideal To. We define the following phasor currents and voltages for the analysis of this problem. and Eq. 1 + Fcos# 7g = 70 s T 1 + Y cos 6 .e. This scheme will minimize the generator current. i. /„ = 7oe3' Ig = Ige^B+9^ generator current necessary for accelerating voltage in the absence of beam required generator current with beam < /. The optimal operating condition normally corresponds to 9S = 0. Vacc = Vo cos(0 . The accelerating rf voltage will be perturbed by the same phase factor.325) Figure 3. A. (3. rf noise. By equating the real and imaginary parts. (3.W.323) Here the induced voltage is derived from the steady state beam loading. Here.Y sin 0 y —. studied by Robinson in 1964.7. Robinson dipole mode instability In accelerators. CEA report CEAL-1010 (1964). (3. ' (3. voltage error. the generator current is optimally chosen to be parallel to 70. . which minimizes 7g.326) Robinson. The resultant vector of the generator voltage and the image current voltage is the effective accelerating voltage for the beam.324) reduces to 7g = 7O(1 + Fsin0 s ). (3.59 We consider a small perturbation by shifting the arrival time of all bunches by a phase factor f. 7. Beam stability may sometimes need sophisticated active feedback systems. = —7. The topic of control and feedback is beyond the scope of this textbook. beams experience many sources of perturbation such as power supply ripple. 59 K.28 (right) shows the beam loading phasor diagram with a detuned cavity angle ifi. tan-0 = Ycosfc.VI. is a positive quantity required rf accelerating voltage I/J = tan"12Q(^~"r) Y = 7j//o detuning angle ratio of image current to unloaded generator current The equation for a proper accelerating voltage is % = 7 0 7 W = [IgeW+*> . etc.
60 60See D.Y—^-r1. instability is a self-adjusting process. Since the stability condition is a function of bunch intensity. Boussard. the cavity frequency is detuned with u < ur.^ ° or * TT ^ °3-332) cos <ps cos 2 <pB This means that Robinson stability requires ip < 9 = \^TT-(/>S\. 626 (1987). the bunches in the accelerator will execute synchrotron motion.# / j .332) is applicable to all higher order modes. with cos</>s < 0. p. Robinson stability can be described as follows. sin ib cos ib sin2 ib . (3. . Active feedback systems are used to enhance the stability of bunched beam acceleration. (3. Thus the equation of motion for the phasor error £ is (see Exercise 3. In general.328) (3. Eq. Kc (3.e.325). 294 (1991).330) A small perturbation in arrival time causes a perturbation in acceleration voltage proportional to the phase shift. For those modes. and its induced rf voltage is AVg = -j£IAh cos </>e-**.f rv 0 cos V sin ip. p. CERN 91-04.329) The net change in accelerating voltage seen by the bunch becomes A c = ^ocos0s[l-F^^]. Above transition energy.327) The induced accelerating voltage is equal to the projection of the phasor voltage onto the real axis: AV0 = . The wrong arrival time shifts the image beam current by a phase angle f. . CERN 87-03. SYNCHROTRON MOTION where the first term is the intended accelerating voltage and the second term is the effect of phase perturbation due to an error in arrival time. Robinson stability can be attained by choosing s i n ^ < 0. = j£ii = . i. with cos 0 S > 0. If the voltage induced by the image charge is not significant.6. (3. the cavity should be detuned so that sin ip > 0 or UJ > Ljr in order to gain Robinson stability. Below transition energy. (3. we find that the Robinson stability condition becomes 1 . Beam loss will appear until the Robinson stability condition can be achieved.358 CHAPTER 3.7) Using Eq. The perturbation to the image rf current is A7.
Robinson stability will be attained below transition energy. Exercise 3. (a) Show that the surface resistivity defined as Rs = l/o"5skin [in Ohm] is given by Rs = \l2cT- . if the cavity is detuned such that hoJo < wr. \ y<yT : III CO L_l_l : CO Figure 3. cor hcoo hw 0 coT y>yT / : i \J I i . where the impedance of the cavity is assumed to be real. If the cavity is detuned so that HLOQ > wr. Qualitative feature of Robinson instability Robinson instability can be qualitatively understood as follows.29: A schematic drawing of the real part of impedance arising from a wakefield induced by the circulating beam. Thus the centroid of the beam bunch will damp in the presence of beam loading. and the beam bunch at lower beam energy sees a lower shunt impedance and loses less energy. The wakefield produced in a cavity by a circulating bunch is represented qualitatively in Fig. higher energy particles have a smaller revolution frequency and thus lose more energy if the cavity detuning is huio > uiT. A similar argument applies to rf cavities operating below transition energy. 3. 3. left). where wr is the resonance frequency of the cavity (Fig.29. the beam bunch at higher energy sees a higher shunt impedance and loses more energy.29.EXERCISE 3. Since the revolution frequency is related to the fractional momentum spread by Aw _ AE a higher beam energy has a smaller revolution frequency above the transition energy.6 359 B. Similarly. Above transition energy. where \i is the permeability. and the dipole mode of beam motion is Robinson damped. To avoid Robinson instability.6 1. > ur above transition energy and huo < wr below transition energy. the cavity should be detuned to h<jj(. The skin depth <5skin of an AC current with angular frequency w traveling on a conductor of bulk conductivity a is <5skjn = I/2/^CTW.
(3.e. Use the following steps to derive Eq. (3. t) = [Vo cosks + satisfies the TEM wave equation. (3.282). SYNCHROTRON MOTION Note that the surface resistivity does not depend on the geometry of the conductor.t) = [Io COS ks + j{V0/Rc) sinks]ejut I V(s. (a) Show that the general solution of the right/left traveling TEM wave is given by V = f(t^sVW).291) (I{s. Q = 104. 2. Show that the solution of Maxwell's equation in the cylindrical coordinate is given by Eq.322) 7. and wT is the resonance frequency.310) and show that the loss factor of a parallel RLC resonator is given by61 _ t^r-Rsh 47rQ where R^ is the shunt resistance. (3. •n-c where Rc = \JLjC is the characteristic impedance of the line. 6. Verify Eq. Verify the Fourier integral of Eq. the characteristic impedance is the impedance seen by traveling waves with V = IRc(b) Show that the current and voltage of Eq.331). (3. (b) Show that the resistance of a coaxial structure is given by Eq.360 CHAPTER 3. 3. Verify Eqs. 5. In a lossless transverse electromagnetic (TEM) wave transmission line.1/x)2]"1 yi-(l/4Q2)±j(l/2Q). Q is the Q-factor. I=±^-f(tTsVLC). (3.312).296) with where i is the length of the structure and ri. 61Use the identity [l + Q2(x .301) and plot Z vs UJ for wr = 200 MHz. i. (3. jIoRcsinks]^1 4. (3. ri are the inner and outer radii of the coaxial wave guide. = x2 [Q2{x2-r21+)(x2-rj_)]~\ where r 1 ± = . and i U = 25 Mfi. the equation for the current and voltage is dV_ !te~ 5/ 3? dl__ BV ~d~s~~ ~dt' where L and C are respectively the inductance and capacitance per unit length. Evaluate the integral of Eq.
EXERCISE 3. in {Y. where the overdot indicates the derivative with respect to orbiting angle 0. and 5/. 1 > (sin 2^/2 cos (j>s)Y.6O°.331). (c) Draw the Robinson stability region. Show that £ = hrt&b. (3.3O°. is the momentum error of the beam centroid. 180°.e.ip) for 0S = O°. 120°. . i.6 361 (a) Let £ be the rf phase associated with the error in beam arrival time. 150°. (b) Show that " i _ eV0 cos <fe I _ b~ 2-K^E [ sin 2^ 1 2cos4>s\^ Thus you have arrived at Eq.
4. we discuss only single bunch effects without mode coupling.63 Longitudinal collective instabilities have many modes. coherent and incoherent tune shift. 1st EPAC. 273 (Edition Frontiere. luminosity degradation. Ill. VII. 1988). Mode coupling and coupled bunch collective instabilities and other advanced topics can be found in a specialized advanced textbooks [3]. The results of collective instabilities are bunch lengthening. the physics of collective instabilities becomes more important. and T. In Sec. since the growth rate of the microwave instability is very large. Zotter.2. and discuss the Keil-Schnell criterion and the turbulent bunch lengthening. and problems in machine operation. Ref. p. This causes a beam bunch to form microbunches. This section provides an introduction to the collective instability in synchrotron motion induced by the wakefield. similar to the transverse collective dipole mode instability discussed in Chap. which is the Fourier transform of the wakefield. the collective motion is governed by the impedance. The impedance responsible for collective instabilities can be experimentally derived from beam transfer function measurements. and equilibrium momentum spread and emittance.362 CHAPTER 3.62 or from passive measurements of beam loss. Chin and K. In this introduction to the collective beam instability. Part. Suzuki. we list possible sources of longitudinal impedances. we study the linearized Vlasov equation with a coasting beam.3. In Sec. 179 (1983). Sec. An experimental measurement of coherent synchrotron mode will be discussed. as discussed in Sec. VIII. VII. where the phase space are split into resonance islands. beam brightness dilution. beam loss. Gif-sur-Yvette Cedex. p. 3rd EPAC. Knowledge of coherent synchrotron modes provides useful information about possible sources. Detecting the onset of instabilities and measuring coherent synchrotron modes can help us understand the mechanism of collective instabilities. The collective synchrotron motion can be classified according to synchrotron modes. 2. and about the signature at onset. Proc. Hofmann. 181 (World Scientific. VII. Proc. 1. B. SYNCHROTRON MOTION VII Longitudinal Collective Instabilities As the demand for beam brightness increases. In Sec. we examine the microwave instability for a beam with zero momentum spread and for a beam with Gaussian momentum spread. Y. [3]. VII. we discuss the coherent frequency spectra of beams in a synchrotron. On the other hand. it can be classified according to the longitudinal mode with density fluctuation. Indeed. The wakefield generated by the beam bunch can further induce collective motion of beam particles. Singapore. In the frequency domain. 13. 1992). Accel. of collective instabilities. almost all accelerators and storage rings have suffered some type of collective instability that limits beam intensity. and derive a dispersion relation for the collective frequency in single mode approximation. and decoherence due to nonlinear synchrotron motion generates emittance dilution. . 63 See 62 A. In Sec. Satoh.
Coherent synchrotron modes The synchrotron motion of beam particles introduces a modulation in the periodic arrival time. For B equally spaced short bunches.211). The amplitude of the mth synchrotron sideband is proportional to the Bessel function Jm. the coherent rf signal is invisible. Such a beam is called a coasting or DC beam because only the DC signal is visible. To is the revolution period. (3. for the effect of a finite bunch length.e. there are synchrotron sidebands around each orbital harmonic n. This analysis is applicable to a single short bunch or equally spaced short bunches. where TVB is the number of particles in a bunch. and the corresponding frequency spectra occur at all harmonics of the revolution frequency fa. (2.Similarly. The resulting spectra of the particle motion are classified into synchrotron modes. For a •^-independent particle distribution function p(f. Ie(t) = e Y.VII. We expand the current Eq. the beam current becomes Ia{t)=jle{t)Pa{T)TdfdiP = Im f ) V ™ * ' . Nf0 is well above the bandwidths of BPMs and detection instruments. i. LONGITUDINAL COLLECTIVE INSTABILITIES 363 VII. S(t-fcos(u}st + rP)-£T0) fc-oo oo oo oo = Y £ £ j-mJm(™of)e>Kn*>+m*»+m*\ (3. A bunch is made of particles with different synchrotron amplitudes and phases.333) where e is the charge. The frequency spectra of a single short bunch occur at all harmonics of the revolution frequency /o. f and tp are the synchrotron amplitude and phase of the particle. u>s — u>0^heV\r]cos(t>s\/2n/32E is the synchrotron angular frequency with 4>s as the rf phase of the synchronous particle. where the Fourier spectra are separated by Nf0. Since N ~ 108 -10 1 3 .tp) = PO(T). 1 Longitudinal Spectra The current observed at a wall gap monitor or a BPM from a circulating charged particle is represented by a periodic 5-function in Eq. n=—oo 64 The power of a coherent signal is proportional to . (2. the coherent synchrotron modes of the bunch can be obtained by averaging the synchrotron mode over the bunch distribution.215). (2. the current of N equally spaced circulating particles is described by Eq./Vg. (3-334) .221). and Jm is the Bessel function of order m. the coherent frequency spectra are located at harmonics of BfoM A. see Eq.87) of an orbiting particle with a linear synchrotron motion in Fourier series.
and Anfi is the Hankel transformation of Po.336) (3. we can deduce the beam distribution function from the amplitudes of coherent modes An.e. (3.m. Nucl. IEEE Trans. we measure the power of a synchrotron mode of a longitudinally kicked beam. 6 5 F. Note also that po can be obtained from the inverse Hankel transformation. As an illustrative example. p(f. all synchrotron sidebands of individual particles are averaged to zero. where fic is the coherent frequency. p. ibid NS-24.e. An.65 The coherent synchrotron sidebands can be measured by taking the fast Fourier transform (FFT) spectrum of the longitudinal beam profile digitized at fixed times during the onset of coherent mode instability. e. the coherent current signal becomes / Ie{t)Ap{f. However. Sci. (3.m = 2TT J™ Jm{nuj0T)pm(f)fdT. i>) with Ap(f.L. see also J. whose power is about 1/iV of the coherent ones. These coherent mode integrals form the kernel for the Sacherer integral equation in determining the longitudinal collective instability. although one can detect incoherent synchrotron sidebands from the Schottky signal. (1973). i. p. NS-20.^)=pm(f)e^t-m*). the inverse Hankel transformation can be used to reconstruct the coherent longitudinal distribution p m . Jackson. a spectrum analyzer (SA). IEEE PAC. (1977).66 This requires a digital oscilloscope with a sizable memory. can also measure beam power arising from the coherent synchrotron mode excitation. Using the inverse Hankel transformation. Piscataway. i. which is important in identifying the source of coherent excitations.meXn"°+r™°+n^. Sacherer. if there is a coherent synchrotron mode in the bunch distribution. Proc. It is a coherent synchrotron mode. NJ. SYNCHROTRON MOTION where / av = iVBe/0 is the average current. 1993). |>ln)m|2.337) where j4n>m is the mth order Hankel transformation. n=-oo oo (3. 264 (1987). 4>) = po(f) + Ap(f.335) Note that Eq. Anfi = 2nJ Jo{nuof)po{f)fdf. Now. Laclare.334) contains only orbital harmonics nu>o. CERN 87-03.338) The mth order synchrotron sideband appears around all coherent revolution harmonics. With the power spectrum of the the mth synchrotron mode known. 66 X. Lu and G. . tuned to a synchrotron sideband.ij)fdfdij = I0{t) + Iav J2 An.364 CHAPTER 3.g. 3366 (IEEE. (3.
which could be adjusted by varying the rf voltage. [26] . Measurements of coherent synchrotron modes The experiment started with a single bunched beam of about 5 x 108 protons at a kinetic energy of 45 MeV and harmonic number h = l. was about 4. The resulting phase oscillations of the bunch relative to the rf wave form were measured by a phase detector.VII. III).2 ^ . which normally locks the rf cavity to the beam.S f ) 7 " ^ ' An. The typical bunch length. The cycle time was 5 s.338) becomes67 (3. i.633. which was used to 67See e. The rf phase lock feedback loop. while the injected beam was electron-cooled for about 3 s.^ r ^ Sj-irimQym*' where /_ m = Im. Coherent synchrotron modes of a kicked beam 365 For simplicity.342) The power of the mth sideband of a kicked beam is proportional to the square of the mth order Bessel function. 6. the initial beam distribution becomes f2 TTk = ^ ( .03168 MHz and the phase slip factor was rj = -0. We describe below an experiment measuring the coherent mode power at the IUCF Cooler. The revolution frequency was /o = 1. (3.86.e. Thus the coherent distribution is ( — A\m (-4) 330 (3"341) T2 f2 TTi ^ =1 ^ . (3. A function generator was used to generate a 0 to 10 V square wave to control the phase kick.^ . For non-Gaussian beams. (3.4 of Ref. T 1 / Tk < 3 ^ When the beam is phase kicked by T^. LONGITUDINAL COLLECTIVE INSTABILITIES B.40) for the phase space coordinates): »te)-5=?->P^). the power spectrum is a weighted average of Bessel functions in Eq. C.g. was switched off. we assume an initial Gaussian equilibrium beam distribution (see Eq.5 m (50 ns) FWHM. The bunched beam was kicked longitudinally by phase-shifting the rf cavity wave form (see Sec.338). and the coherent mode integral of Eq. the rms bunch length oy was about 20 ns.m = e'^OTk)HV2Jm(nuj0Tk).
On the other hand. video bandwidth 100 Hz.^). 3.03168 MHz.58 and uT = 20 ns for the case shown in Fig. The magnitude of the phase shift was varied by the size of the applied step voltage.30: The synchrotron sideband power of a kicked beam observed from a spectrum analyzer tuned to the first revolution sideband (upper trace) and the 6th revolution sideband (lower trace) as a function of time. SYNCHROTRON MOTION calibrate the control voltage for the phase shifters versus the actual phase shift. the spectrum analyzer (SA) was triggered about 5 ms before the phase shift.30. 3. Ji(6woTa) increases. The setting of the SA was resolution bandwidth 100 Hz. The vertical axis is coherent synchrotron power in dB. where ra is the synchrotron amplitude.58 rad. The kicked amplitude was 90 ns. A6tl ~ e-0-299. The sideband power shown in Fig. or equivalently w0Tk = 0. Since UJQT^ « 0. The response time of the step phase shifts was limited primarily by the inertia of the rf cavities. . thus the measurement of the sideband power was taken at 10 ms after the phase kick. 3. and frequency span 0 Hz. Figure 3.366 CHAPTER 3. probably because of electron cooling in the IUCF Cooler. and the horizontal axis is time at 10 ms per division. The power observed at a synchrotron sideband from the SA is shown in Fig.30 was proportional to |^4i.30 increases with time. Therefore the power spectrum shown in the lower plot of Fig. Note that the sideband power decreased with time for the first harmonic and increased for the 6th harmonic. which had a quality factor Q of about 40. The revolution frequency was 1.30. where the top and bottom traces show the SA responses at the sidebands of the first harmonic f0 — fs and the sixth harmonic 6/o — fs vs time. Both the phase error due to control nonlinearity and the parasitic amplitude modulation of the IUCF Cooler rf systems were kept to less than 10%.i|2. Thus the initial power at the fundamental harmonic sideband. as 6cooTa decreases./!^. The resolution bandwidth of SA was 100 Hz. To measure the FFT spectrum of the coherent signal.i| 2 for the lower trace. we obtain Altl ~ e-00083^^). after the phase kick will be a factor of 6 larger than that of the 6th orbital harmonic. the power Aiti decreases because Ji(woTa) decreases with decreasing ujQTa.i|2 for the upper trace and |^6. which is proportional to |^4i. 3. As the synchrotron phase amplitude decreases because of electron cooling.
(3. 100 and 150 ns. For a kicked Gaussian beam.31 shows the power of the m = 1 sideband.343) normalized to the peak. Similarly. the power P n l is proportional to |^4n. Figure 3. These data were normalized to the peak. LONGITUDINAL COLLECTIVE INSTABILITIES 367 Figure 3.VII.343) Because the actual power depends on the beam intensity.i ~ \AnA\2 = e . The curves were theoretical predictions with no adjustable parameter except the normalization constant. which in turn perturb particle motion. 3. Measurement of AUtl for all orbital harmonics can be used to obtain the coherent mode distribution function.< " ^ ) 2 ^ ^ ) 2 | J i ( « ^ o r k ) | 2 . Because of the impedance of the ring. Plots from top to bottom correspond to a kicking amplitude (time) of 53. Let ^o(5) be the normalized distribution function with f^od5= 1. A self-consistent distribution function obeys the Vlasov equation d* d$ -a* -a* . setting up the central frequency at the second synchrotron harmonic.i|2: Pn.31. (3. we can measure the m = 2 synchrotron modes for the kicked beam. there is no rf cavity and the unperturbed distribution function is a function only of the off-momentum coordinate 5 = Ap/p0. where n is the revolution harmonic. as a function of UIT ~ WJJQTY.2 Collective Microwave Instability in Coasting Beams For coasting beams. When a bunched beam encounters collective instability. for various kicking times. Solid curves are obtained from Eq. 90. VII.31: Measured m = 1 synchrotron sideband power vs frequency for different phase kicked amplitudes is compared with theory based on a Gaussian beam distribution.8. (3. The effect of finite bunch length is visible in Fig. all data are normalized at the first peak around nw0Ty « 1. the beam generates wakefields.342). the observed sideband power \AnX\2 is proportional to the weighted average of the coherent mode density p(r) shown in Eq.
Im fl < 0. If the imaginary part of the coherent mode frequency is negative.e.e. the energy gain/loss per revolution due to the wakefield is equal to the current times the longitudinal impedance: AE per turn = Z« (eIQ I AVnd5) \ J ) e^nt-nB\ where the impedance is evaluated at the collective frequency Q. . i. (3. The perturbation causes density fluctuation along the machine. ~J 1*PE J Q^^dd ~3 2^E J ({l-ruo)*d6 ' (3'3 8) where partial integration has been carried out in the second equality. i.e/OTi(Jo(Z||/n) r dVp/dd JX _ . the perturbing distribution function should be written as a linear superposition of all possible modes.345) where < > is the unperturbed distribution. First we examine possible sources of longitudinal impedance. and A\Pn(<5) is the perturbation amplitude for the longitudinal mode n.eI0n2ujQ{Z^n) f % du JS. the time derivative of the 5 coordinate in a coasting beam becomes 6=^ (e/o^i / A*nd5) e* 0 *-*'. SYNCHROTRON MOTION where the overdot is the derivative with respect to time t. the collective instability of mode number n can cause a coasting beam to split into n microbunches. With the reo lation (j = W — u)oT]S. The terminology is derived from the fact that the coherent frequency observed is in the microwave frequency range. Q is the coherent frequency. In general. By definition. The eigenfrequency fl of the collective motion is given by the solution of the dispersion relation. 68 Here we assume a single longitudinal mode. we obtain the dispersion relation 1 _ . the perturbation amplitude grows exponentially.346) Since |A$ n | -C \l/o at the onset threshold of collective instability. In the presence of a wakefield. and the beam encounters the collective microwave instability. (3. we assume that the distribution function is approximated by68 tf = tf 0 (g) + A<bnej{Qt-n^. we linearize the Vlasov equation to obtain Using 0 = u) and integrating Eq. The frequencies of the collective motion are eigenfrequencies of the coupled system.368 CHAPTER 3. Thus. the dispersion integral can be analytically obtained for some distribution functions of the beam.347). 9 is the or30 biting angle. (3.
(3. we list below some sources of commonly used impedance models.|( ) e U =. the impedance has the symmetry property Zn{-u) = Z.i / do/^M.354) ! 2-Ktr l. LONGITUDINAL COLLECTIVE INSTABILITIES 369 VII.32). the wake function is related to the impedance by Wn(t) = — 1 r00 Z7T J— oo Z^e^du. Now we consider a small fluctuation in the line density and current with A = Ao + Axe*"*-"*).353) Thus the real part of the longitudinal impedance is positive and is a symmetric function of the frequency. e is the charge.Y. p.3 Longitudinal Impedance Z. 2TIT where A is the particle's line density.69 and the real and imaginary parts of the impedance are related by the Hilbert transform R Z. Space-charge impedance Let a be the radius of a uniformly distributed coasting beam. [12].352) where P. In fact. (3. and e0 and Ho are the permittivity and permeability of the vacuum.V. 3.fieis the speed. The electromagnetic fields of the coasting beam are eXr 2-Kta2 \ R B<t>-\ ( fj. (3.350) Since the wakefield obeys the causality principle. (3. . 472 in Ref. stands for the principal value integral.351) I Z( ) = +-/" m |W d^Z^'\ (3. 89See / = 70 + Iie^nt-n6\ K. and let b be the radius of a beam pipe (Fig.349) and similarly. see also Appendix 2.|H = r J-oo The impedance of an accelerator is related to the wake function by Wn(t)e-jutdLj. the property of Z\\(u))/u is similar to that of Z±(u). Ng. A. the impedance must not have singularities in the lower complex plane. (3. Without making the effort to derive them.VII.(u>). Because the wake function is real.QeXl3cr J 2na2 xa ~ r >a . .
dX/dt = —j3c(d\/ds). SYNCHROTRON MOTION Figure 3. 3. we obtain (E. if the perturbation is on the surface of the beam. For most accelerators. where Es and Ev are the electric fields at the center of the beam pipe and at the vacuum chamber wall. the geometric factor becomes go = I + 2 In (b/a). The perturbation generates an electric field on the beam.AW] = - A s ^ § . On the other hand. the vacuum chamber wall is inductive at low and medium frequency range. and the geometry factor g0 = 1 + 2 ln(6/a) is obtained from the integral along the radial paths from the beam center to the vacuum chamber wall. The induced electric fields that arise from impedance are shown schematically.370 CHAPTER 3.e. where Io = e/3cA0 and 7i = e/3cAi.F eg° dX (1W\ t. — r—. 70If [47r£o72 . (6. .s — hy. Let L/2-KR be the inductance per unit length.Ew)As + f*L[\{8 + As) . then the induced wall electric field is L dIw_ep<?Ld\ Thus we have K-2^Rlf- 47reO7 as arises from cancellation between the electric and magnetic ^rTTd? 27ri? J 9 s ' (3'356) _ [ g _ 0W\ d\ o 6 the impedance is averaged over the beam cross section.32: The geometry of a uniformly distributed beam with radius a in a beam pipe of radius b.600) where the factor I/7 2 fields. where da is the surface integral.70 Assuming that the disturbance is propagating at the same speed as the orbiting beam particles. i.32. Using Faraday's law £ Ed£=J — fB-da at J along the loop shown in Fig. The rectangular loop is used for the path integral of Faraday's law. the geometry factor becomes go = 21n(fc/a). the electric field acting on the circulating beam becomes W .
Typical values of the space-charge impedance at transition energy are listed in Table VII. (3. microwave frequency.7 22. (3. the vacuum chamber wall is not perfectly conducting. the displacement current contribution to Maxwell's equation is small. . . B.e / 3 c / ? S[l7. 7T I AGS I RHIC I Fermilab BST [ Fermilab MI I KEKPS" 8. Penetration of electromagnetic wave into the vacuum chamber can be described by Maxwell's equations O 77 VxE = -(jt-—. part of the wakefield can penetrate the vacuum chamber and cause energy loss to the beam. and Ew has a resistive contribution that depends on the conductivity.4 20. as Z.5 5.358) is the space-charge impedance and the second term is the inductance of the vacuum chamber wall.3.VII. and Zo = l/eoc = 377 ohms is the vacuum impedance. Here we have used Ohm's law. The first term in Eq. Table 3. and neglected the contribution of the displacement current for electromagnetic wave with not so high frequencies. we find V2E = fiac^.7: Typical space-charge impedance at 7 = 7 T . we obtain the impedance.71 From Eq. LONGITUDINAL COLLECTIVE INSTABILITIES The total voltage drop in one revolution on the beam is 371 A[/ = . (3.5 I 30 [23 | 20 In fact.359) whereCTCis the conductivity and /J.8 |Z||.4 6. Using R(d\/ds) = (dX/d6).| At/ \g0Z0 1 . Resistive wall impedance Because the resistivity of the vacuum chamber wall is finite. and skin depth. defined as the voltage drop per unit current.scl/rc [ft] 1 13 1 1.W o L ]' ('5) 33 7 where /3c = us0R is the speed of the orbiting particles.360) 71 For frequencies OJ <C <rc/e fa <rcZoc « 10 19 Hz.359). (3. V x H = J = acE. is the permeability. where e is the permittivity. and ej3c\\ = I\.
Since the magnetic energy is equal to the electric energy. (3. SYNCHROTRON MOTION Substituting the ansatz of the electric field E = s Eo exp{j(ujt — kx)} into Eq.360). i^h is the shunt impedance. and the resonance frequency to be the cut-off frequency w r . The longitudinal narrowband and broadband impedances can conveniently be represented by an equivalent RLC circuit Z(w) = 7" ^. ZO \w0/ (3. The imaginary part of the wave number is the inverse of the penetration depth. we obtain the wave number k = (1 . the magnitude of the reactance is equal to the resistance.362) where the sign function.372 CHAPTER 3. <5Skin. is added so that the impedance satisfies the symmetry property. /3 is Lorentz's relativistic velocity factor. The high order mode (HOM) of rf cavities is a major source of narrowband impedances. sgn(w) = +1 if UJ > 0 and — 1 if u < 0. and Q is the quality factor.j)y/\u\acn/2. the Q-factor is usually taken to be 1. bb = LOoR/b = Pc/b. Narrowband and broadband impedances Narrowband impedances arise from parasitic modes in rf cavities and cavity-like structures in accelerators. and other discontinuities in accelerator components. The electromagnetic fields penetrate a skin depth inside the vacuum chamber wall.361) where Zo is the vacuum impedance. The resistance due to the electric field becomes ZH X2^^- = ^{-0) *"»•»' (3 . For a broadband impedance.363) where cuT is the resonance frequency. The resistive wall impedance becomes Z||(W) = ( l + j s g n H ) ^ (M^ 5skin0 . Parameters for narrowband impedances depend on the geometry and material of cavity-like structures.364) . C. where x is the distance in the vacuum chamber wall. (3. the skin depth is Ss^n = ^J2/iJ. or equivalently. (3. Broadband impedances arise from vacuum chamber breaks.acLO. and ac is the conductivity of the vacuum chamber. bellows. b is the vacuum chamber radius.o = ^2/nacu>o is the skin depth at frequency uio. H is the permeability.
The magnitude of the broadband shunt impedance can range from 50 ohms for machines constructed in the 60's and 70's to less than 1 ohm for recently constructed machines. Figure 3. In the absence of momentum spread with \&o(<5) = 5A{5).VII. and space-charge impedances. LONGITUDINAL COLLECTIVE INSTABILITIES 373 where LJO is the revolution frequency.33: Schematic of a longitudinal impedance that includes broadband. VII. Experimental observations were obtained in the intersecting storage rings (ISR). where 5 = Ap/po and Sd(x) is the Dirac 5-function. where the solid and dashed lines correspond to the real and imaginary parts respectively. To summarize. The symmetry of the impedance as a function of w is also shown.4 Microwave Single Bunch Instability The negative mass instability was predicted in 1960's. we consider negative mass instability. and b is the vacuum chamber size.348) is / fi \ 2 eI0Zn/n U) =-'iyK (-6) 335 The condition for having a real fi is -j(Z^/n)rj > 0.33. or an indue- . R is the average radius of the accelerator. 3. This condition is only satisfied for a space-charge (capacitive) impedance below the transition energy. In this section. A. the solution of Eq. it was observed in almost all existing high intensity accelerators. we discuss the single bunch microwave instability. we find that |Re(Z||/o. where microwave signal was detected in the beam debunching process. where the vacuum chamber is carefully smoothed. Subsequently. the longitudinal impedances Z\\(UJ)/U> or Z\\/n are schematically shown in Fig. Including the resistive wall impedance in the longitudinal impedances. narrowband. (3.)| becomes large at u « 0. Negative mass instability without momentum spread First.
the coherent mode frequency 7 can be obtained by solving the dispersion relation. A higher energy particle takes 7 longer to complete one revolution. Vaccaro. Sec.72 72A. the threshold impedance for microwave instability is reflectively symmetric with respect to the real part of the impedance. On the other hand. the collective frequency becomes a complex number below the transition energy.&) S i n *J> ( 3 . it is also called negative mass instability. . in the longitudinal Hamiltonian (see Chap. a beam with a small 7 frequency spread can also encounter microwave instability at 7 < j r if the impedance is inductive.374 CHAPTER 3. CERN ISR TH/68-33 (1968). or resistive. see also Ref. and collective motion is Landau damped. e. Ruggiero and V. and the solution with a negative imaginary part gives rise to collective instability. Since the "microwave instability" resulting from the space-charge impedance occurs when 7 > 0. the threshold of collective instability can be estimated from the dispersion relation. The mass. SYNCHROTRON MOTION tive impedance above the transition energy. 2. If the distribution function is a symmetric function of momentum deviation 5. For resistive impedance. space-charge impedance. In this case. This results in a collective frequency shift without producing collective insta7 bilities. [3].g. Table VII.366 ) H C0S is negative above the transition energy with 7 > 0. there is a finite region of impedance value where the growth rate of collective instability is zero. Depending on the actual distribution function. Landau damping with finite frequency spread For a beam with a finite momentum spread with 7 ^ 0. Table 3. B. the beam with a zero momentum spread is unstable.G. —77. However. the collective frequency is a real number below the transition energy with 7 < 0. or it appears to have a negative mass.G.o7? " O UP2J? tC0S ^ * « + ( * .8: Characteristic behavior of collective instability without Landau damping. if the impedance is inductive. If Z\\/n is capacitive.4 shows the characteristic behavior of microwave collective instability. Below transition Above transition Z\\/n capacitive r\ < 0 stable 7 > 0 unstable 7 inductive unstable stable resistive unstable unstable The terminology of negative mass instability is derived from a pure space charge effect. IX) 1 /Az>\ 2 eV = .
except the Gaussian distribution.VII.4. C. or U' and V parameters as For the Gaussian beam. a simplified estimation of the stability condition is to draw a circle around the origin in the impedance plane Z. In the limit of small frequency spread. Note that a distribution function with a softer tail. The rms frequency spread of the beam becomes au = tJor/as. Eq. (3.367) reduces to Eq. All distribution functions. for the normalized distribution functions ^o(x) = 3(1 — £ 2 )/4./ V 7r J-oo x + il/(nujo'r]as) —rKTf \dx = 2[1 + J^*vwW> (3-368) fi = CI — nu>o. Eq. the distribution becomes the Dirac ^-function.z 2 ) 2 /16.0.34 shows the threshold V vs U' parameters. i.x2f'2/2. and the dashed lines outside the threshold curve are unstable with growth rates — (lmQ. we find <5FWHM = \/8 In 2 ag. Thus in the limit of zero detuning (or > zero frequency spread). 8(1 .5 respectively. 15(1 . f ? \ I 2cr <S J where 5 = Ap/po and as is the rms momentum spread.367) becomes -j(U + jV)JG/2 = 1.2.0. The dispersion relation can be integrated to obtain where JG [o r°° re~x2l2 =V.z 2 ) 4 /32. Keil-Schnell criterion Figure 3.3. Based on experimental observations and numerical calculations of the dispersion relation.365). gives a larger stability region in the parametric space. a less sudden cutoff. 3. we have JQ — y~2 as y —> oo. (3.0.34 shows the threshold V vs U' parameters of collective microwave instability with Im(fi) = 0. n < 2^Ea2MF elo v . Dashed lines inside the threshold curve correspond to stable motion.)/y/2 In2wor?aj = 0. In terms of U and V parameters.e. The right plot of Fig. (3. from inside outward. The solid line in the left plot of Fig. 3. we consider a Gaussian beam model of a coasting beam given by 1 V27TCT.34 show that the stability region depends on beam distribution.1. We usually define the effective U and V parameters.-K. LONGITUDINAL COLLECTIVE INSTABILITIES 375 For example. are limited to x < 1. and (l/V2^)exp(-a: 2 /2).. 315(1 . . and w(y) is the complex error function with y = —Q/(y/2nuioT](7s)Asymptotically. and 0.
Dashed lines outside the threshold curve have growth rates —lmQ/(V2 ln2uoT]as) = 0.2.2. where F is a form factor that depends on the distribution function. J.G. A.1. and 0. the total longitudinal energy drop from impedance. They correspond to -Ima/(V2 In2 W0T)OS) = -0.0. Right: The threshold V vs U' parameters for various beam distributions. z-± < **PWW eI n where / = FQIQ. and -0.e. Ruggiero and V.0. per unit frequency spread n\r]\\/2Tras for mode number n should be less than the total energy spread y/2^P2Eas of the beam. For a Gaussian beam. Pellegrini.M.-0. Keil and W.5 respectively. F = 1. and for a tri-elliptical distribution with ^o(^) = 8(1 .5. This Boussard conjecture has been well tested in the Intersecting Storage Ring (ISR).376 CHAPTER 3.94 [3]. Since the microwave growth rate is usually fast. the threshold condition can be obtained from the local peak current of the beam.-0. Dashed lines inside the threshold curve are stable. is the bunching factor. i. 554 (1980).3. and the the wavelength of the coherent wave is usually small compared with the bunch length.3.G. See e. e/0|Z|||. SYNCHROTRON MOTION Figure 3. CERN ISR TH/68-33 (1968). Schnell. Vaccaro.74 73 E. the Keil-Schnell criterion can be applied to the bunched beam by replacing the average current Io by the peak current /. and -FB = 27r/v/27rcr^.34: Left: The solid line shows the parameters V vs U' for a Gaussian beam distribution at a zero growth rate.-0. . Proc. This is the Keil-Schnell criterion.4. Wang and C. 74Since the growth rate of the microwave instability is normally very fast.1.g.4.0. CERN-ISR-TH-RF/69-48 (July 1969). F « 0. 11th HEACC. p.73 The Keil-Schnell criterion states that if the beam is stable.X2)3/2/3TT.
For a pure inductance impedance. Landau damping for microwave single bunch instability vanishes because of a small synchrotron frequency spread. Microwave instability below transition may arise from the real impedance.75 Landau damping plays an important role in damping collective instability. (3.j(Z^:SC/n) ohms was used to study the growth rate around the transition energy for RHIC.372) 1 = 1O\\ \ 3 ( a 0 0 q w . determination of microwave instability needs careful evaluation of the dispersion integral. Since the beam distribution function is nonadiabatic in the transition energy region. we assume a Gaussian beam model with the threshold impedance determined by the peak current.g. and the distribution function is therefore given by (see Sec. instability exists only below transition.374) shows that the growth rate near the transition energy is nearly equal to the growth rate without Landau damping. Furthermore. This is easy to understand: 75see e.348). Wang.374).Y. Nucl. (3. LONGITUDINAL COLLECTIVE INSTABILITIES D. T h e p e a k c u r r e n t is f (3. We assume a model of collective microwave instability such that the longitudinal modes are nearly decoupled and thus the coherent growth rate can be obtained by solving the dispersion relation Eq. The dispersion integral can be integrated to obtain the coherent mode frequency given by _ 1 3e/0 (Zy/n) 7 r V 3 ^ A t G > ( } ~3 2**/*FEv where JG = 2[l+jy/nyw(y)]. . The Keil-Schnell criterion is not applicable in this region. e. Sci. A ^ (6. we can find the eigenvalue of the growth rate Im (fi(t)) by solving Eq. The solution of Eq. (3.616) where Io is the average current and A is the rms phase-space area of the beam. 2323 (1985).VII. y = nujori %/6a M . 1) <jro(<5) = J^ie-3°ui\ V 7 T w h e r e a^ is given by E q .177). space-charge impedance. Because of a large space-charge impedance. IEEE Trans.g. Lee and J.M.a2^) irass = la y ^ 7 . (3. For a given broadband impedance model with constant Z\\/n. The region of collective instability can be estimated by using the Keil-Schnell criterion. The impedance model Z\\/n = 5 . IV. . Microwave instability near transition energy 377 Near the transition energy. The peak current is located at the center of the bunch A<j> = 0. For a pure capacitive impedance. the growth rate appears to be larger above the transition energy. NS-32. instability occurs when 7 > 7 T . S.
378
CHAPTER 3. SYNCHROTRON MOTION
at 7 = 7 T , the frequency spread of the beam becomes zero, and Landau damping vanishes. Fortunately, the growth rate is also small at 7 ss 7 T . The total growth factor across the transition energy region can be estimated by
G = exp j/(-Imfi) un8table d*j .
(3.375)
The total growth factor is a function of the scaling variable \Z\\/n\Nb/A. Note that the growth factor is much smaller if the initial phase-space area is increased. Phasespace dilution below transition energy has become a useful strategy in accelerating high intensity proton beams through transition energy. The CERN PS and the AGS employ this method for high intensity beam acceleration. Bunched beam dilution can be achieved either by using a high frequency cavity as noise source or by mismatched injection at the beginning of the cycle. The distribution function model Eq. (3.372) does not take into account nonlinear synchrotron motion near the transition energy. For a complete account of microwave instability, numerical simulation is an important tool near transition energy.76 A possible cure for microwave instability is to pass through transition energy fast with a transition energy jump. Furthermore, blow-up of phase-space area before transition energy crossing can also alleviate the microwave growth rate. We have discussed microwave instabilities induced by a broadband impedance. In fact, it can also be generated by a narrowband impedance. Longitudinal bunch shapes in the KEK proton synchrotron (PS) were measured by a fast bunch-monitor system, which showed the rapid growth of the microwave instability at the frequency of 1 GHz and significant beam loss just after transition energy (see Fig. 3.35).77 Temporal evolution of the microwave instability is explained with a proton-klystron model. The narrowband impedance of the BPM system causes micro-bunching in the beam that further induces wakefield. The beam-cavity interaction produces the rapid growth of the microwave instability. This effect is particularly important near the transition energy, where the frequency spread of the beam vanishes, and the Landau damping mechanism disappears. E. Microwave instability and bunch lengthening When the current is above the microwave instability threshold, the instability can cause micro-bunching. The energy spread of the beam will increase until the stability condition is satisfied. For proton or hadron accelerators, the final momentum spread of the beam may be larger than that threshold value caused by decoherence of the synchrotron motion.
76 W.W. Lee and L.C. Teng, Proc. 8th Int. Conf. on High Energy Accelerators, CERN, p. 327 (1971); J. Wei and S.Y. Lee, Part. Accel. 28, 77-82 (1990); S.Y. Lee and J. Wei, Proc. EPAC, p. 764 (1989); J. McLachlan, private communications on ESME Program. 77 K. Takayama et al., Phys. Rev. Lett, 78, 871 (1997).
VII. LONGITUDINAL COLLECTIVE INSTABILITIES
379
Figure 3.35: The longitudinal beam profiles observed at KEK PS revealing microwave bunching in the tail of the bunch. The bottom figure shows the longitudinal bunch profile before the transition energy, the middle figure at 1 ms after the transition energy, and the top figure at 2 ms after the transition energy. The microwave instability occurs near the transition energy for lack of Landau damping. The instability was found to be driven by a narrowband impedance caused by the BPM system. [Courtesy of K. Takayama, KEK]
For electron storage rings, the final momentum spread is equal to the microwave instability threshold due to synchrotron radiation damping. Using the Keil-SchnellBoussard condition of Eq. (3.371), we find
where vs is the synchrotron tune. Note that the bunch length depends only on the parameter f = {IQ\T]\IV1P2E) provided that the impedance does not depend on the bunch length. Chao and Gareyte showed that the bunch lengths of many electron storage rings scaled as a,~e1/(2+a)(3.377) This is called Chao-Gareyte scaling law. For a broadband impedance, we have a = 1. The scaling law is not applicable if the impedance depends on the beam current and bunch length. F. Microwave instability induced by narrowband resonances At low energy, the longitudinal space charge potential, shown as the first term in Eq. (3.357), can be large for high intensity beam bunch. It requires a costly large rf cavity potential to keep beam particles bunched inside the rf bucket. In particular,
380
CHAPTER 3. SYNCHROTRON MOTION
if it requires a beam gap for a clean extraction, and for minimizing the effect of the electron-cloud instability. The longitudinal space charge potential can be compensated by the inductive impedance shown in the second term of Eq. (3.357). We consider a cavity with ferrite ring filling a pillbox. The inductance is
L "~l^ l n iV (3'378)
where fj,' is the real part of the ferrite permittivity, i?i and R2 are the inner and outer radii of the ferrite rings, and I is the length of the pillbox cavity. The inductive inserts carried out at PSR experiment employs coaxial pillbox cavity with 30 ferrite rings each with width 2.54 cm, 12.7 cm inner diameter (id), and 20.3 cm outer diameter (od). The Proton Storage Ring (PSR) at Los Alamos National Laboratory compresses high intensity proton beam from the 800 MeV linac into a bunch of the order of 250 ns. The parameters for PSR are C = 90.2 m, 7 T = 3.1, vx = 3.2, vz = 2.2, vs = 0.00042, and /o = 2.8 MHz. To cancel the space charge impedance at 800 MeV for PSR at the harmonic h = 1, one requires about 3 pillbox cavities. The experimental test for this experiment was indeed successful.78 Unfortunately, the beam also encounters collective microwave beam instability at high intensity. The left plot of Fig. 3.36 shows the initial bunched coasting beam, and the right plot shows the microbunching of the beam under the action of three ferrite inserts.
Figure 3.36: The longitudinal beam profiles observed at PSR the bunched coasting beam in the presence of inductive inserts, where three 1-m long ferrite ring cavities were installed in the PSR ring. [Courtesy of R. Macek, LANL] The microwave instability is induced by a narrowband impedance with Q w 1 at the center frequency of / res « 27/o.79 Although the inductive inserts can be used
78M.A. Plum, et at, Phys. Rev. Special Topics, Accelerators 79see C. Beltran, Ph.D. thesis, Indiana University (2003).
and Beams, 2, 064201 (1999).
EXERCISE 3.7
381
to cancel the space charge impedance, the pillbox cavity can generate a narrowband impedance to cause microwave instability of the beam at higher harmonics. In order to alleviate this problem, it is necessary to broaden the narrowband impedance by either choosing different design geometries for different ferrite inserts, or by heating the ferrite so that the imaginary part (//') of the permittivity is larger at the cavity resonance frequency. At PSR, the cavities was heated to 125-150° C, so that the beam is below the microwave instability threshold.
Exercise 3.7
1. In synchrotrons, beam bunches are filled with a gap for ion-clearing, abort, extraction kicker rise time, etc. Show that the frequency spectra observed from a BPM for short bunches filled with a gap have a diffraction-pattern-like structure. Specifically, find the frequency spectra for 10 buckets filled with 9 equal intensity short bunches. The revolution frequency is assumed to be 1 MHz. 2. Show that the impedance of Eq. (3.363) has two poles in the upper half of the u plane, and find their loci. Use the inverse Fourier transformation to show that the wake function of the RLC resonator circuit is
W = nT^-"rt/2Q c o s ^ - ^ ^ s i n J , |
where £>r = o)r^/l — 1/4Q2. 3. The parameters of the SLC damping ring are E = 1.15 GeV, vx = 8.2, uz = 3.2, a c = 0.0147, -yex<z = 15 7 mm-mrad, aAp/p = 7.1 x 10" 4 , Vrf = 800 kV, C = 35.270 T m, h = 84, /,f = 714 MHz, p = 2.0372 m, and the energy loss per revolution is f0 = 93.1 keV. If the threshold of bunch lengthening is JVB = 1.5 x 1010, use the Keil-Schnell formula to estimate the impedance of the SLC damping ring.80 4. We assume that the growth rate of microwave instability in a quasi-isochronous electron storage ring can be obtained from Eq. (3.365). For electron beams, synchrotron motion is also damped because of the energy dependence of synchrotron radiation energy loss. The damping rate is given by TS = 2ETo/JsUo, where E is the energy of the particle, TQ is the revolution period, the damping partition Js ~ 2, UQ = C1E4/p, C 7 = 8.85 x 10~5 m/(GeV 3 ), and p is the bending radius. Assuming that the growth rate is equal to the damping rate at equilibrium, find the tolerable impedance as a function of the machine parameters. Discuss an example of an electron storage ring at E = 2 GeV. 5. Consider a pillbox-like cavity with length I (see Sec. VII.4). The cavity is filled with ferrite rings with inner and outer radii a and 6 respectively. Show that the longitudinal
80G.E. Fisher et al, Proc. 12th HEACC, p. 37 (1983); L. Rivkin, et al, Proc. 1988 EPAC, p. 634 (1988); see also P. Krejcik, et a!., Proc. 199S PAC, p. 3240 (1993). The authors of the last paper observed sawtooth instability at the threshold current JV B =3x 1010.
382
impedance for TMoio mode is 81 Z1=.ZS_ I
J2na\
CHAPTER 3. SYNCHROTRON MOTION
U'-jy," H^jk^H^jkcb) er
H[l\kca)42)(kcb)
- H^(kcb)H^(kca) - H^(kcb)H[2\kca)'
where Hm are Hankel functions which represent incoming and outgoing waves, ZQ = 377fi is the impedance of free space. kc = ui^/JIe = kJer{^i — j ^ " ) , k = ^ = wy'/ioeo in vacuum, tr is the relative permittivity and fx and n are the real and complex parts of the relative complex permeability.
81The general formula to calculate the shunt impedance is AV = —IZ\\ — —Esi, where Es is the longitudinal electric field, I is the total length, and / is obtained by Ampere's law: / = f Hdl =
VIII. INTRODUCTION TO LINEAR ACCELERATORS
383
VIII
Introduction to Linear Accelerators
By definition, any accelerator that accelerates charged particles in a straight line is a linear accelerator (linac).82 Linacs includes induction linacs; electrostatic accelerators such as the Cockcroft-Walton, Van de Graaff and Tandem; radio-frequency quadrupole (RFQ) linacs; drift-tube linacs (DTL); coupled cavity linacs (CCL); coupled cavity drift-tube linacs (CCDTL); high-energy electron linacs, etc. Modern linacs, almost exclusively, use rf cavities for particle acceleration in a straight line. For linacs, important research topics include the design of high gradient acceleration cavities, control of wakefields, rf power sources, rf superconductivity, and the beam dynamics of high brightness beams. Linacs evolved through the development of high power rf sources, rf engineering, superconductivity, ingenious designs for various accelerating structures, high brightness electron sources, and a better understanding of high intensity beam dynamics. Since electrons emit synchrotron radiation in synchrotron storage rings, high energy e+e~ colliders with energies larger than 200 GeV per beam can be attained only by high energy linacs. Current work on high energy linear colliders is divided into two camps, one using superconducting cavities and the other using conventional copper cavities. In conventional cavity design, the choice of rf frequency varies from S band to millimeter wavelength at 30 GHz in the two beam acceleration scheme. Research activity in this line is lively, as indicated by bi-annual linac, and annual linear collider conferences. Since the beam in a linac is adiabatically damped, an intense electron beam bunch from a high brightness source will provide a small emittance at high energy. The linac has also been considered as a candidate for generating coherent synchrotron light. Many interesting applications will be available if high brilliance photon beam experiments, such as LCLS, SASE, etc., are successful. This section provides an introduction to a highly technical and evolving branch of accelerator physics. In Sec. VIII. 1 we review some historical milestones. In Sec. VIII.2 we discuss fundamental properties of rf cavities. In Sec. VIII.3 we present the general properties of electromagnetic fields in accelerating cavity structures. In Sec. VIII.4 we address longitudinal particle dynamics and in Sec. VIII.5, transverse particle dynamics. Since the field is evolving, many advanced school lectures are available.
VIII. 1 Historical Milestones
In 1924 G. Ising published a first theoretical paper on the acceleration of ions by applying a time varying electric field to an array of drift tubes via transmission lines; subsequently, in 1928 R. Wideroe used a 1 MHz, 25 kV rf source to accelerate potas82 See G.A. Loew and R. Talman, AIP Conf. Proc. 105, 1 (1982); J. Le Duff, CERN 85-19, p. 144 (1985).
384
CHAPTER 3. SYNCHROTRON MOTION
sium ions up to 50 keV.83 The optimal choice of the distance between acceleration gaps is d = p\/2 = Pc/2f, (3.379) where d is the distance between drift tube gaps, pc is the velocity of the particle, and A and / are the wavelength and frequency of the rf wave. A Wideroe structure is shown in Fig. 3.37. Note that the drift tube distance could be minimized by using a high frequency rf source. In 1931-34 E.O. Lawrence, D. Sloan et al, at U.C. Berkeley, built a Wiederoe type linac to accelerate Hg ions to 1.26 MeV using an rf frequency of about 7 MHz.84 At the same time (1931-1935) K. Kingdon at the General Electric Company and L. Snoddy at the University of Virginia, and others, accelerated electrons from 28 keV to 2.5 MeV.
Figure 3.37: Top: Wideroe type linac structure. Bottom: Alvarez type structure. An Alvarez cavity has more than 50 cells. Here /3c is the speed of the accelerating particle, and X =
2TTC/CJ is the rf wave-
length.
To minimize the length of the drift region, which does not provide particle acceleration, a higher frequency rf source is desirable. For example, the velocity of a 1 MeV proton is v = Pc — 4.6 x 10~2c, and the length of drift space in a half cycle at rf frequency /rf = 7 MHz is \vf^1 « l m . As the energy increases, the drift length becomes too long. The solution is to use a higher frequency system, which became available from radar research during WWII. In 1937 the Varian brothers invented the klystron at Stanford. Similarly, high power magnetrons were developed in Great Britain.85
8 3 G. Ising, Arkiv fur Matematik o. Fisik 18, 1 (1924); R. Wideroe, Archiv fur Electrotechnik 2 1 , 387 (1928). 84 D.H. Sloan and E.O. Lawrence, Phys. Rev. 32, 2021 (1931); D.H. Sloan and W.M. Coate, Phys. Rev. 46, 539 (1934). 8 5 The power source of present day household microwave ovens is the magnetron.
VIII. INTRODUCTION TO LINEAR ACCELERATORS
385
However, the accelerator is almost capacitive at high frequency, and it radiates a large amount of power P = IV, where V is the accelerating voltage, / = OJCV is the displacement current, C is the capacitance between drift tubes, and w is the angular frequency. The solution is to enclose the gap between the drift tubes in a cavity that holds the the electromagnetic energy in the form of a magnetic field by introducing an inductive load to the system. To attain a high gradient, the cavity must be designed such that the resonant frequency is equal to the frequency of the accelerating field. A cavity is a structure in which electromagnetic energy can be resonantly stored. An acceleration cavity is a structure in which the longitudinal electric field can be stored at the gap for particle acceleration. A cavity or a series of cavities can be fed by an rf source, as shown in Fig. 3.38.
Figure 3.38: Left: Schematic of a single gap cavity fed by an rf source. The rf currents are indicated by j on the cavity wall. Middle: A two-gap cavity operating at vr-mode, where the electricfieldsat two gaps have opposite polarity. Right: A two-gap cavity operating at 0-mode, where the electricfieldsat all gaps have the same polarity. In 0-mode (or 27r-mode) operation, the rf currents on the common wall cancel, and the wall becomes unnecessary. The Alvarez structure shown in Fig. 3.37 operates at 0-mode. When two or more cavity gaps are adjacent to each other, the cavity can be operated at 7r-mode or 0-mode, as shown in Fig. 3.38. In 0-mode, the resulting current is zero at the common wall so that the common wall is useless. Thus a group of drift tubes can be placed in a single resonant tank, where the field has the same phase in all gaps.86 Such a structure (Fig. 3.37) was invented by L. Alvarez in 1945.87 In 1945-47 L. Alvarez, W.K.H. Panofsky, et al, built a 32 MeV, 200 MHz proton drift tube linac (DTL). Drift tubes in the Alvarez structure are in one large cylindrical tank and powered at the same phase. The distances between the drift tubes, d = /3A,88 are arranged so that the particles, when they are in the decelerating phase, are shielded
Alvarez, Phys. Rev. 70, 799 (1946). 88It appears that the distance between drift tubes for an Alvarez linac is twice that of a Wideroe linac, and thus less efficient. However, the use of a high frequency rf system in a resonance-cavity more than compensates the requirement of a longer distance between drift tubes.
86This
87L.
is the TMOio mode to be discussed in Sec. VIII.3.
386
CHAPTER 3. SYNCHROTRON MOTION
from the fields. In 1945 E.M. McMillan and V.I. Veksler discovered the phase focusing principle, and in 1952 J. Blewett invented electric quadrupoles for transverse focusing based on the alternating gradient focusing principle. These discoveries solved the 3D beam stability problem, at least for low intensity beams. Since then, Alvarez linacs has commonly been used to accelerate protons and ions up to 50-200 MeV kinetic energy. In the ultra relativistic regime with /3 —> 1, cavities designed for high frequency operation are usually used to achieve a high accelerating field.89 At high frequencies, the klystron, invented in 1937, becomes a powerful rf power source. In 1947-48 W. Hansen et al., at Stanford, built the MARK-I disk loaded linac yielding 4.5 MeV electrons in a 9 ft structure powered by a 0.75 MW, 2.856 GHz magnetron.90 On September 9, 1967, the linac at Stanford Linear Accelerator Center (SLAC) accelerated electrons to energies of 20 GeV. In 1973 P. Wilson, D. Farkas, and H. Hogg, at SLAC, invented the rf energy compression scheme SLED (SLAC Energy Development) that provided the rf source for the SLAC linac to reach 30 GeV. In 1990's, SLAC has achieved 50 GeV in the 3 km linac. Another important idea in high energy particle acceleration is acceleration by traveling waves.91 The standing wave cavity in a resonant structure can be decomposed into two traveling waves: one that travels in synchronism with the particle, and the backward wave that has no net effect on the particle. Thus the shunt impedance of a traveling wave structure is twice that of a standing wave structure except at the phase advances 0 or TT. TO regain the factor of two in the shunt impedance for standing wave operation, E. Knapp and D. Nagle invented the side coupled cavity in 1964.92 In 1972 E. Knapp et al. successfully operated the 800 MHz side coupled cavity linac (CCL) to produce 800 MeV energy at Los Alamos. In 1994 the last three tanks of the DTL linac at Fermilab were replaced by CCL to upgrade its proton energy to 400 MeV. Above j3 > 0.3, CCL has been widely used for proton beam acceleration. A combination of CCL with DTL produces the CCDTL structure suitable for high gradient proton acceleration. For the acceleration of ions, the Alvarez linac is efficient for /3 > 0.04. The acceleration of low energy protons and ions relies on DC accelerators such as the
89The linacs designed for relativistic particles are usually called high-/) linacs even though the maximum f) is 1. 90E.L. Ginzton, W.W. Hanson and W.R. Kennedy, Rev. Sci. lustrum. 19, 89 (1948); W.W. Hansen et al, Rev. Sci. lustrum. 26, 134 (1955). 91J.W. Beams at the University of Virginia in 1934 experimented with a traveling-wave accelerator for electrons using transmission lines of different lengths attached to a linear array of tubular electrodes and fed with potential surges generated by a capacitor-spark gap circuit, similar to the system proposed by Ising. Burst of electrons were occasionally accelerated to 1.3 MeV. See J.W. Beams and L.B. Snoddy, Phys. Rev. 44, 784 (1933); J.W. Beams and H. Trotter, Jr., Phys. Rev., 45,849 (1934). 92 E. Knapp et al., Proc. 1966 linac Con}., p. 83 (1966).
VIII. INTRODUCTION TO LINEAR ACCELERATORS
387
Cockcroft-Walton or Van de Graaff. In 1970 I. Kapchinskij and V. Teplyakov at ITEP Moscow invented the radio-frequency quadrupole (RFQ) accelerator. In 1980 R. Stokes et al. at Los Alamos succeeded in building an RFQ to accelerate protons to 3 MeV. Today RFQ is commonly used to accelerate protons and ions for injection into linacs or synchrotrons. Since the first experiment on a superconducting linear accelerator at SLAC in 1965, the superconducting (SC) cavity has become a major branch of accelerator physics research. In the 1970's, many SC post linear accelerators were constructed for the study of heavy ion collisions in nuclear physics.93 Recently, more than 180 m of superconducting cavities have been installed in CEBAF for the 4 GeV continuous electron beams used in nuclear physics research. More than 400 m of SC cavities at about 7 MV/m were installed in LEP energy upgrade, and reached 3.6 GV rf voltage for the operation of 104.5 GeV per beam in 2000.94 The TESLA project had also successfully achieved an acceleration gradient of 35 MV/m.
VIII.2
Fundamental Properties of Accelerating Structures
Fundamental properties of all accelerating structures are the transit time factor, shunt impedance, and Q-value. These quantities are discussed below. A. Transit time factor We consider a standing wave accelerating gap, e.g. the Alvarez structure, and assume that the electric field in the gap is independent of the longitudinal coordinate s. If £ is the maximum electric field at the acceleration gap, the accelerating field is Es=£ cos Lot. (3.380)
The total energy gain in traversing the accelerating gap is
AE = ejii£
cos f
ds = e£gTtr = eV0,
Ttr = ^
^
,
(3.381)
where Vo = £gTtv is the effective voltage of the gap, T tr is the transit time factor, A = 2TTC/CJ is the rf wavelength, and wg/PA is the rf phase shift across the gap. If the gap length of a standing wave structure is equal to the drift tube length, i.e. g = /3A/2, the transit time factor is T tr = sin(7r/2)/(7r/2) = 0.637. This means that only 63% of the rf voltage is used for particle acceleration. To improve the efficiency, the gap length g should be reduced. However, a small g can lead to sparking at the gap. Since
93See e.g., H. Piel, CERN 87-03, p. 376 (1987); CERN 89-04, p. 149 (1994), and references therein. The geometries of these low energy SC cavities are essentially the drift tube type operating at A/4 or A/2 modes. 94 P. Brown et al, Proceedings of PAC2001, p. 1059 (IEEE, 2001).
388
CHAPTER 3. SYNCHROTRON MOTION
there is relatively little gain for g < /JA/4, the gap g is designed to optimize linac performance. The overall transit time factor for standing wave structures in DTL is about 0.8. It is worth pointing out that the transit time factor of Eq. (3.381) is valid only for the standing wave structure. The transit time factor for particle acceleration by a guided wave differs from that of Eq. (3.381). An example is illustrated in Exercise 3.8.7. B. Shunt impedance Neglecting power loss to the transmission line and reflections between the source and the cavity, electromagnetic energy is consumed in the cavity wall and beam acceleration. The shunt impedance for an rf cavity is defined as Rsh = V02/Pd, (3.382)
where V is the effective acceleration voltage, and P<j is the dissipated power. For a o multi-cell cavity structure, it is also convenient to define the shunt impedance per unit length rSh as
r.-^-^•^cav
-Td/^cav
«
£--*,
US rsb
(3.383,
where £ is the effective longitudinal electric field that includes the transit time factor, and dP<i/ds is the fraction of input power loss per unit length in the wall. The power per unit length needed to maintain an accelerating field £ is P^/L = £2/rs^ and the accelerating gradient for low beam intensity is £ = yrShPd/£cav For a 200 MHz proton linac, we normally have rsj, ~ 15 — 50 Mfi/m, depending on the transit time factors. For an electron linac at 3 GHz, rsh « 100 Mfi/m. For high frequency cavities, the shunt impedance is generally proportional to a;1/2 (see Exercise 3.8.4). A high shunt impedance with low surface fields is an important guideline in rf cavity design. For example, using a 50 MW high peak power pulsed klystron, the accelerating gradient of a 3 GHz cavity can be as high as 70 MV/m. The working SLC S-band accelerating structure delivers about 20 MV/m.95 C. The quality factor Q The quality factor is defined by Q = oj\Vst/Pd, and thus we obtain dWJdt = -Pd = -LJWJQ,
95 P.
(3.384)
Raimondi, et al, Proceedings of the EPAC2000, (EPAC, 2000).
97 Alternatively. INTRODUCTION TO LINEAR ACCELERATORS 389 where Wst is the maximum stored energy. However.VIII. we define the stored energy per unit length as Wst = Wst/Lcav.387) where Lcav is the length of the cavity structure and v% is the velocity of the energy flow.tw = -kcav/^g.^ ' or Q = uwst -dPjdS- .. (3. For a traveling wave structure.385) where QL is the loaded Q-factor that includes the resistance of the power source.387). <F. the time for the field to decay to 1/e of its initial value is called the filling time of a standing wave cavity. (3. . (3386) The filling time for a traveling wave structure is96 *F. (3. VIII. it will quickly pass through the wave propagation region unless a wiggler field is employed to bend back the particle velocity vector. welds.3 Particle Acceleration by EM Waves Charged particles gain or lose energy when the velocity is parallel to the electric field. a wave guide designed to provide electric 96We will show that the velocity of the energy flow is equal to the group velocity. 97This scheme includes inverse free electron laser acceleration and inverse Cerenkov acceleration. In general. it can gain energy.385) is twice that of the traveling wave in Eq. On the other hand.SW = 2QL/o. (3. the Q-factor of an accelerating structure is independent of whether it operates in standing wave or traveling wave modes. A useful quantity is the ratio Rsh/Q'Q ^ ' Q u(Wst/Lcm) uwst(6-6m) which depends only on the cavity geometry and is independent of the wall material. A particle traveling in the same direction as the plane electromagnetic (EM) wave will not gain energy because the electric field is perpendicular to the particle velocity. if a particle moves along a path that is not parallel to the direction of an EM wave. and the power loss per unit length becomes dPd ~di = wwst . vt = Pd/wst. etc. For standing wave operation. The conventional definition of standing wave filling time in Eq.
the EM fields can be described by the traveling wave component in Eq. R. R. 1979 Part. We will discuss the choice of standing wave vs traveling wave operation.7. (r. The phase velocity of the EM waves can be slowed down by capacitive or inductive loading. (3. Hr = 0. In general. Miller. A. SYNCHROTRON MOTION field along the particle trajectory at a phase velocity equal to the particle velocity is the basic design principle of rf cavities. s is the longitudinal coordinate. = 0 listed in Table V. (3. . where ZQ = y/zoAo is the vacuum impedance. EM waves in a cylindrical wave guide First we consider the propagation of EM waves in a cylindrical wave guide. Early.k2T . (3. Its high duty factor can be used to accelerate long pulsed beams such as protons.390 CHAPTER 3. Con}.H. These waves are classified into transverse magnetic (TM) or transverse electric (TE) modes.A. 3701 (IEEE.w t l. VI. and k2 = {u/cf . In this section we study the properties of electromagnetic waves in cavities. at the pipe radius r = b.390) The propagation modes are determined by the boundary condition for Es = E$ = 0 . Bane. The standing wave can also accelerate oppositely charged beams traveling in opposite directions.A. On the other hand. traveling in the +s direction. rf cavities for particle acceleration can be operated in standing wave or traveling wave modes. Proc.H.282) in Sec. Kmn = jmn/b. and K.98 Standing wave cavities operating at steady state are usually used in synchrotrons and storage rings for beam acceleration or energy compensation of synchrotron radiation energy loss.391) where j m n are zeros of the Bessel functions Jm(jmn) pendix B Sec. employing high power pulsed rf sources. and continuous wave (CW) electron beams in the Continuous Electron Beam Accelerator Facility (CEBAF). <j>) is the cylindrical coordinate. i.e. V.. k is the propagation wave number in the +s direction. p. SLAC-PUB-3935 (1988). 1979). The EM fields of the lowest frequency TMoi mode. Hs = 0. Ace.2 in Ap- 98See G. are Es = Kr EoMkTr)e-X'a-ut\ (3. Since there is no ends for the cylindrical wave guide.389) Er = j^£oJi(fc r r)e-* s . Miller. the effect of shunt impedance. a traveling wave structure can attain a very high gradient for the acceleration of an intense electron beam pulse. R. Loew.1 (see Appendix B Sec. V). and the coupled cavity linac.8.L. see also Exercise 3. E# = 0.
However. We define wc = krc = 2.282)]. it is not useful for particle acceleration. the particle can not be synchronized with the EM wave during acceleration.405c/6. it becomes the transverse TEM wave. travels at a phase velocity of . Right: Dispersion curve (w/c)2 = fc2 + (2. At high frequency. (3. The phase velocity u/k for a wave without cavity load is always greater than the velocity of light. This mode is a free propagation mode along the longitudinal s direction. However. where kT -> 0. (LJC/LO)2]1/2 (3. 1 radial-node at the boundary of the cylinder [see Eq.405/6)2.211/2 > c.p ) 2 l c [ \OJ J \ V2 . the phase velocity approaches c. i. C. shown in Fig. The wave number of the TMOi wave and the corresponding phase velocity vp become k = ^\l.392) k [1- Unattenuated wave propagation at ui < uoc is not possible. the wave travels forward and backward with a very large phase velocity. the longitudinal component of the EM wave vanishes. Thus the frequency of the TMoi mode is LJ/C = \Jk2 + (2. = £ = . 3. Since the phase velocity propagates faster than the speed of light. The subscript 01 stands for m = 0 in ^-variation. INTRODUCTION TO LINEAR ACCELERATORS 391 Figure 3.39.VIII. v. At high frequencies. ET k ' ET ckZ0 Zo' J B.389) represents an infinitely long pulse of EM waves in the cylindrical wave guide. v .e. the electromagnetic field is transverse. The phase of the plane wave. the phase velocity approaches the speed of light.405/b)2 for the TMoi wave. . At low frequency. Phase velocity and group velocity Equation (3. ks — cut.39: Left: Schematic of a cylindrical cavity.
From Fig. we obtain E(t) = A(t . The power of the TM wave is P=\te[ 2 Js ErH.395) *-F-S| • wo < 3397 > Using Eq.395) into Eq. (3. (3. (3. and the amplitude function of the EM pulse propagates at the "group velocity" {ui-Ljo) = ko + k'(uj-uJo). v* = w = tc2=v*- (-0) 3 40 . (3.tte+wo41 dedw.39 we see that the group velocity is zero at k = 0. SYNCHROTRON MOTION vp = ds/dt = uj/k. In reality. we expand the dispersion wave number around Wo: k{u) = k(u0) + — aw Substituting Eq. we have to discuss a short pulse formed by a group of EM waves.399) where Wm is the magnetic energy. The velocity of the energy flow is Thus the velocity of energy flow is equal to the group velocity. (3.390) has been included.392 CHAPTER 3.396) Note that the phase of the pulse propagates at a "phase velocity" of vp = uo/ko. Since the Maxwell equation is linear. For a quasi-monochromatic wave at the angular frequency CJ0. the group velocity is equal to the velocity of energy flow in the wave guide.dS = \El^~ 2 CZQK^ JO f J!(krr)2nrdr. (3-394) where the dispersion of the wave number of Eq. (3. The propagation of the pulse inside a wave guide becomes E(t.394). (3.k's) ejiuot-kas\ (3.393) where A(t) is the amplitude with a short time duration. we obtain vg = kc2/uj. and the total energy per unit length stored is W = 2Wm = \E20J^-2 fQ J?(krr)2irrdr. s) = A{t)e^at-k^ = 7^fJ AiOeJ^-^-^+^dtdu. For a quasi-monochromatic pulse at frequency w0 in free space. In fact. (3.392) for single-mode wave propagation. s) = ^ff A(Oe J ' [llrt -* (w) . the electric field can be represented by E(t. or vpvg = c2. the pulse can be decomposed in linear superposition of Fourier series.398) where H^ is the complex conjugate of H^. 3.
•••.40: Left: Schematic of a cylindrical cavity.402) .282) in the closed cylindrical pillbox cavity is reproduced as follows: ( Es = Ck2r Jm{kTr) cosm<j>cos ks. The dispersion relation is UJ/C = ^Jk? + k2. p = 0 . 1 .VIII. Similarly. Figure 3. u is the angular frequency.40}. there are also TE modes where the longitudinal electric field is zero. where the longitudinal magnetic field is zero for TM modes. 2 . the TM mode solution of Eq. Right: Dispersion curve (w/c)2 = (pir/d)2 + (2. With proper design of pillbox geometry. We first discuss the standing wave solution of a closed pillbox cavity without beam holes. where d is the length of the pillbox. Here we discuss the standing wave solution of Maxwell's equation for a "closed pillbox cavity. The effect of a chain of cylindrical cells on the propagation of EM waves is discussed in the next section. ET = -CkK E. and kd is the phase advance of the EM wave in the cavity cell. The cylinder has a beam hole for the passage of particle beams (Fig. we obtain k d = PTT. and kr and k are wave numbers of the radial and longitudinal modes. ( Hs = 0. [ # 0 = -jCu>eokr J'm{krr) cos mcj) cos ks. Using the boundary conditions that Er = 0 and E$ = 0 at s = 0 and d." and the effect of beam holes.405/6)2 for TMoip resonance waves (marked as circles) for a closed cylindrical pillbox without beam holes. TM modes in a cylindrical pillbox cavity 393 Now we consider a cylindrical pillbox cavity. (3. J Hr = -jC^^I T Jm{krr) sin m0 cos ks. the phase velocity of the TMoio mode can be slowed to the particle speed for beam acceleration. we obtain kr. With a time dependent factor e7'"'*.p = Cnk -Jm(krr) sinm<f>sinks.mnb = jmn.401) 1 J'm{krr) cos m0 sin ks. 3. INTRODUCTION TO LINEAR ACCELERATORS C. (3. Using the boundary conditions Es = 0 and E^ = 0 at the pipe radius r = b. where both ends of the cylinder are nearly closed. (3.
027 mm. inner diameter of 26 = 83. The solid lines in Fig. we have Es = EoJo{krr).99 mm.793 mm. The circles in Fig. 021. SYNCHROTRON MOTION where b is the inner radius of the cylinder. At / = 2. and the phase velocity vp is equal to c. phase advance of 2TT/3. Medical Electron Accelerators. 1993). To lower the phase velocity. all mode frequencies become horizontal lines. disk diameter of 2a = 26. When the beam hole is completely closed. B^^j^-Jx{kTr). where the effective d parameter is reduced for a single cell structure. the phase shift per cell is about 120°. V).856 GHz. and d = 34.24 mm.842 mm.9 shows parametric dependence of a SLAC-like pillbox cavity at / = 2. Karzmark. When the beam hole radius decreases. it provides a continuous TM mode frequency as a function of wave number k." Because of the coupling between adjacent pillbox-cavities. The details of the TMOio mode are shown in the right plot.19. TMolo. The wall thickness chosen was 6. 020. 3. Increasing the size of the beam hole decreases the coupling capacitance and increases the TMOio mode frequency. 3. 011. 3.404) where LO/C = 2. 3. the discrete mode frequencies become a continuous function of the phase advance kd. Nunan. (McGraw-Hill.41. The EM wave modes can be calculated by finite element or finite difference EM codes with a periodic boundary (resonance) condition and a prescribed phase advance kd across the cavity gap.461 — 81. See also C. the circles in the left plot of Fig. D. Thus the resonance frequency w for the TM mnp mode is For the lowest mode.22 .394 CHAPTER 3.05 m. and the phase-velocity is effectively lowered.41 are the dispersion curves of frequency / vs phase shift kd for TMOnp modes of a SLAC-like pillbox cavity with a = 18 mm.405/6. (3. Xraig S. beam hole radius a and cylinder radius 6 are tailored to provide matched phase advance kd and phase velocity cj/k for the structure. New York. length of the structure of L = 3. and Eiji Tanabe.J. 030 for a closed cylindrical pillbox are shown as circles in the left plot of Fig. Table 3.41. Analytic solution of Maxwell's equations for an actual cavity geometry is difficult. and disk thickness of 5.2 (Appendix B Sec. . The dashed lines show the world line vp = c. the mode frequencies become discrete points. More importantly. Note that the shunt impedance per unit length is maximum at a phase advance " T h e calculation was done by Dr. The wall thickness slightly influences the mode frequencies of TM On i modes.40 (right) show the discrete mode frequencies of TMOIO and TM 0U on the dispersion curve. and j m n are zeros of the Bessel functions JmUmn) = 0 listed in Table V.856 GHz. Li using MAFIA in 2D monopole mode. Both these modes have phase velocities greater than c.856 GHz. The frequencies of the TM modes 010. The actual SLAC structure is a constant gradient structure with frequency of / = 2. b = 43 mm.
2 42. / vs kd. and thus we have b « 2.000 26. Alvarez structure The Alvarez linac cavity resembles the TMOio standing wave mode (see Table 3.405c/w.853 1. The phase advance per cell at a given frequency is mainly determined by the cell length.857 1.475 17.37 41. for TMoip modes for a pillbox cavity with a = 18 mm.5107 7713 29.9: Parametric dependence of the SLAC cavity geometry T(mm) I d (mm) I kd (deg) | / (GHz) I R& (10Bfl) I Q I rsh (10sfVnTr 42.99 120 2.580 39. Since fi increases .685 34.485 | 180 [ 2.8579 0.24 90 2.466 | 17646 | 46. and d = 34.56 41.79 41.10).290 [ 52. 6 = 43 mm. D.14 14848 54. Circles show the TMonp mode frequencies for a closed pillbox cavity.404) is independent of s.415 46.VIII. Right: Dispersion curve of TMoio mode. such as rods and slugs inside the cavity.98 of about 135°. Table 3.99 mm.2 10947 45. The dashed lines show the world line vp = c.653 160 2.416 16507 51.73 41.36 135 2.857 | 2.616 105 2.874 13700 53. are designed to obtain a proper resonance frequency for the TMOio mode. The tank radius and other coupling structures.857 2.805 30.559 12413 50. (3.41: Left: Dispersion curves. The total length is designed to have a distance /3\ between two adjacent drift tubes (cells).92 41. where fie is the speed of the accelerating particles.495 60 2.854 1. INTRODUCTION TO LINEAR ACCELERATORS 395 Figure 3. The resulting electric field of Eq.857 2.
vp w c. Loaded cavity cells can be joined together to form a cavity module. what happens to the EM wave in a chain of cavity cells? If the wave guide is loaded with wave reflecting structures such as iris. and the CEBAF cavity. The phase velocity must be brought to the level of the particle velocity. the phase advance kd = n. Figure 3. the SLAC cavity.42 (top). We observed in Sec. The dispersion relation in this case resembles that in Fig. b . When the a. At these frequencies. At some frequencies the reflected waves from successive irises are exactly in phase so that the irises force a standing wave pattern. The reflected waves for a band of frequencies interfere destructively so that there is no radial field at the irises. this gives rise to a minor perturbation in the propagating wave. frequency / vs phase advance kd of the loaded SLAC-like pillbox cavity. as an example.60 Fermilab (cavity2) 45 1902 59 2J)_ CEBAF SC cavity 1497 7. where .42 shows a slow wave structure. nosecone. D that the propagating wave in an unloaded cylindrical wave guide has phase velocity vp > c. the propagating EM waves can be reflected by obstruction disks.E-field is in the direction of beam momentum. unattenuated propagation is impossible. Opening a beam hole at the center of the cavity is equivalent to a capacitive loading for attaining continuous bands of resonance frequencies.e. For particle beam acceleration. or washers. SYNCHROTRON MOTION along the line. we consider the TM guided wave. The size of the beam hole determines the degree of coupling and the phase shift from one cavity to the next. i. Table 3.66 10.10 shows some properties of an Alvarez linac. Such a chain of loaded wave guides can be used to slow the phase velocity of EM waves. so that the EM wave becomes a standing wave and the group velocity again becomes zero.41 shows.. 3.396 CHAPTER 3.0 EiftA Fermilab (cavity 1) 47 744 55 1. the distance between drift tubes increases as well.39. etc.5 [ « 100 [ 20_ E. Figure 3. 5 5-10 SLAC linac | 2856 | 4. A simple method of reducing the phase velocity is to load the structure with disks. The question is. Loaded wave guide chain and the space harmonics In previous subsections.e.10: Some parameters of basic cylindrical cavity cells Machine I / (MHz) I b (cm) I d (cm) I JVcen I £ (MV/m) Alvarez linac 201. we find that the dispersion curve of a closed cylindrical pillbox cavity resembles that of a cylindrical wave guide except that there are infinite numbers of discrete resonance frequencies. i. 3. Since the irises play no role in wave propagation.2 | 3. shown in Fig. Table 3.25 57.
e.s).410) . 0. 4>. <l>. Es(r." These space harmonics are shown in Fig. the phase change from cavity to cavity along the accelerator gives an overall phase velocity that is equal to the particle velocity. where d is the period of the wave guide.J>. We note further that as kod — 0 or TT. parameters of the disk radii are tailored correctly.<t>. the electromagnetic field can be expanded in Fourier series (or Floquet series).408) (3. H^(r.q (3.42.s.406) With the Floquet theorem for the periodic wave guide: Es{r.409) = iEr£o. s). The solid line branches correspond to forward traveling waves and the dashed line branches are associated with backward traveling waves.<j>)e-^s'd = e^1 £ q=—oo g=—oo £s»)e-^s." andfeois the propagation factor of the "fundamental space harmonic. Bottom: Dispersion curve (tu/c) vs k. The field components of the lowest TMOn mode with cylindrical symmetry become Es = ^EOgJo(kriqr)e-^s-wt\ &r (3.t) = e-Xk°s-^ where g Es. 3. (q = integer) is the propagation wave number for the gth "space harmonic. #.VIII.S + d) = H^(r.t) = e-^s-^Es(r. q Kr. The EM wave of an infinitely long disk loaded wave guide is Es(r.407) 27rg kq = ko + —.s + d) = E3{r. <j>. s. etc. a.cj>. s).q V* = J^E^EoMkr^e-^-^.(3. 7r).405) (3.Ji(^/)e"3hs-ul1.^s). The phase velocity ui/k with a cavity load is equal to the speed of light at a specific point of the dispersion curve.q{r. shown as the intersection of the dashed diagonal line and the solid dispersion curve.t) = e'^-^H^r. The q = 0 space harmonic corresponds to kd £ (—IT.(r. (3.42: Top: Schematic of a chain of cylindrical cavities. forward and backward traveling > branches coincide and they will contribute to enhance the electric field.<f>. 37T). and the q = 1 space harmonic to kd £ (TT. i. INTRODUCTION TO LINEAR ACCELERATORS 397 Figure 3. •^0 q &r.
Each point corresponds to the propagation factor kq. At kod = ir. they must have zero slope at the lower frequency u>o/c. it represents a traveling wave or the maximum of a standing wave. and these curves must join.e. This indicates that the electric field of the qth space harmonic is independent of the transverse position. T h e statement t h a t t h e electric field is independent of transverse position is valid only near t h e center axis of loaded wave-guide structures. The extreme of the pass band is kod = 7T./c)2 . The arrows indicate the maximum electric field directions. i. or the propagation band.1 < cos kod < 1. (3. = V'- 100 One may wonder how t o reconcile t h e fact t h a t t h e tangential electric field component Es must be zero at r = b. .412) K —n. there are infinite numbers of crossings between the horizontal line and the dispersion curve. which has an identical slope in the u>/c vs k curve. where kod = T (see also T Fig.413) where the group velocity is zero. At a given frequency w.100 The dispersion curve of a periodic loaded wave-guide structure (or slow wave structure) is a typical Brillouin-like diagram shown in Fig. 3. Higher order space harmonics have no effect on a beam because they have very different phase velocity. 3. the cavity has lowest rf loss.A*.42. The lower plot shows a similar snapshot for kd = 0. The upper plot shows the snapshot of an electromagnetic wave. The electric field at a snapshot is shown schematically in Fig. where the branches with solid lines correspond to forward traveling wave. the phase velocity is vpq = -r = -.41).. The lengths of kd — n. an identical group velocity: doj duj (3-415) ^ = dk~g = Ik. The range of frequencies [u>o.414) If we draw a horizontal line in the dispersion curve within the pass band of the frequency. 3. Because the dispersion curve is a simple translation of 2ir/d.101 making this a favorable mode of operation for accelerator modules. p>* (3. These crossings are separated into space harmonics. 2TT/3. At an instant of time. SYNCHROTRON MOTION k\ = (o. kq ko + 2irq/d ' Note that kr>q = 0 and Jo(kr^r) = 1 for up>9 = c. TT/2. and at the upper frequency uv/c. 27r/3 and n cavities. ww] is called the pass band.398 with CHAPTER 3. and TT/2 cavities are also shown. (3. and the branches with dashed dots are backward traveling wave.411) (3. The condition for wave propagation is .43. 101 The rf loss is proportional t o \H$\2 on the cavity wall. where kod = 0.
i.e. and •K phase shift structures. The resulting shunt impedance is half of that in traveling wave operation. SUPERFISH. The actual electromagnetic fields must satisfy the periodic boundary conditions.27r/3. the resonance frequency of the electric coupled cavity is102 w m = u)0[l + «(1 . Since Jo(kTfir) = 1.417) where W is the resonance frequency without beam hole coupling. or krfi = 0 for the fundamental space harmonic [see Eq. Note that only half of the kd = TT/2 mode has longitudinal electric field in the standing wave mode. (3.cos k0md)]l/2 . This implies that the transverse force on the particle vanishes as well (see Sec. (3.VIII. For magnetically coupled cavity. (3. and 3D MAFIA.416) In the coupled RLC circuit model. The resonance frequency can be more accurately calculated from powerful finite difference. n/2. the resonance frequency is given by o o = > U[1 + K ( 1 . Bottom: Snapshot at the maximum electric field configuration across each cell for kd = 0. There are N + 1 resonances located at kOmd = miT/N {m = 0. VIII. 2TT/3.2.1. • • •. the energy gain of a charged particle is independent of its transverse position.C O S M ) ] 1 / 2 • . or finite element. and K is the couo pling coefficient. The operating condition vp = c is equivalent to kg = u>/c. 102See Exercise 3. N). and -K are shown. The snapshot represents the field pattern of a traveling wave guide or the maximum field pattern of a standing wave. INTRODUCTION TO LINEAR ACCELERATORS 399 Figure 3.8.411)].5).6. programs such as 2D URMEL. The phase advances kd = 7r/2. the longitudinal electric field of the fundamental space harmonic is independent of the radial position within the radius of the iris.43: Top: Snapshot of a sinusoidal wave. A module made of N cells resembles a chain of N weakly coupled oscillators. The size and the length of cavity cells are also tailored to actual rf sources for optimization. LALA.
because only half of the cavity cells are used for particle acceleration. The filling time of a standing wave structure is a few times the cavity filling time 2QL/W. For example. with drift tubes used to shield the electric field at the decelerating phase. The high-/5 linac can also be operated as a traveling wave guide. a wave guide accelerator. These field free cells are coupled to the main accelerating cavity in the high magnetic field region. e. SYNCHROTRON MOTION F.10). 3.44). e. Standing wave cavities are usually used to accelerate CW beams. the shunt impedance in kd = TT/2 mode operation is reduced by a factor of 2.3. if a cavity has 50 cells.45. In a storage ring. Standing wave operation of a module made of many cells may have a serious problem of many nearby resonances. a standing wave can be used to accelerate beams of oppositely charged particles moving in opposite directions.400 CHAPTER 3. Nagle in 1964. However. Similarly. There are two ways to operate high-/? cavities: standing wave or traveling wave. where the phase velocity is equal to the particle velocity. these empty cells can be shortened or moved outside. where dui/dk has its highest value. This problem can be minimized if the standing wave operates at the kd = n/2 condition. the resulting shunt impedance is 1/2 of that of a traveling wave structure except for the phase advance kd = 0 or n (see Fig.A 49 48 ^ = 7r'50?r'507r'--Since du/dk — 0 for a standing wave at fed = 0 or n.g. Since every other cavity cell has no electric fields in kd — TT/2 standing wave operation. the CEBAF rf cavity at the Jefferson Laboratory (see Table 3. This led to the invention of the coupled cavity linac (CCL) by E. can effectively accelerate particles in its entire length. The idea is schematically shown in Fig. it can have standing waves at I. where QL is the loaded Q-factor. and coupled cavity linacs We have shown that the Alvarez linac operates at the standing wave TMOio mode. and long pulse beams. The CCL cavities operate at TT/2 mode. A wave guide accelerator is usually more effective if the particle velocity is high. Knapp and D. traveling wave.g. to allow time to build up its electric field strength for beam acceleration. There are divided . A small shift of rf frequency will lead to a different standing wave mode. these resonances are located in a very narrow range of frequency. On the other hand. in the proton linacs and storage rings. The effective acceleration gradient is reduced by the transit time factor and the time the particle spends inside the drift tube. The electric field pattern of the main accelerating cavity cells looks like that of a w-mode cavity. 3. where field free cells are located outside the main cavity cells. the forward traveling wave component of a standing wave can accelerate particles. Such a design regains the other half of the shunt impedance and provides very efficient proton beam acceleration for /? > 0. Standing wave.
The filling time for a traveling wave guide is Lcav/vg.VIII.45: A schematic drawing of the 7r/2 phase shift cavity structure (top). Since only the forward traveling wave can accelerate the beam. Note that the particle riding on top of the rightgoing wave that has the phase velocity equal to the particle velocity will receive energy gain Figure 3.10 lists the properties of SLAC linac cavity. a traveling wave cavity can provide a high acceleration gradient for intense electron beams. that is a constant gradient structure operating at a phase advance of 2TT/3. the shunt impedance is 1/2 of that of the traveling wave structure except for kd = 0 and 7 standing wave modes. a standing wave (left) can be decomposed into forward and backward traveling waves (right). With a high peak power rf source. and moved outside to become a coupled cavity structure (bottom). .44: In general. Typical group velocity is about 0.8. where two T neighboring space harmonics contribute to regain the factor of two in the shunt impedance. into "constant gradient" and "constant impedance" structures (see Exercise 3. Table 3. The accelerating cavities of a constant impedance structure are identical and the power attenuation along the linac is held constant. where the field free regions are shortened (middle). INTRODUCTION TO LINEAR ACCELERATORS 401 Figure 3.05c. where Lcav is the length of a cavity and vg is the group velocity. On the other hand. the geometry of accelerating cavities of a constant gradient structure are tapered to maintain a constant accelerating field along the linac.8).
and W be the corresponding physical quantities for a non-synchronous particle. Since the momentum compaction in a linac is zero. and let t. These HOMs. or beam blow up) instabilities. (3. These instabilities are called BBU (beam break up. 9 (1965). A long range wake can affect trailing bunches. the beam in a linac is always below transition energy. and energy of a synchronous particle. Its threshold current can be increased by a quadrupole focusing system. When a beam is accelerated in cavities. AW//3%E is the fractional momentum spread. It also depends strongly on the misalignment of accelerating structure and rf noise.T. and —l/ja is the equivalent phase slip factor.103 The BBU is a transverse instability.R. 104See Jarvis. NS-112. This equation is in fact identical to Eq.21). The change of the phase coordinate is where v = ds/dt and vs = ds/dts are the velocities of a particle and a synchronous particle. Such efforts are instrumental for future linear colliders operating at high frequencies. p. higher order modes (HOMs) can be equally important in cavity design. G. and the subscript s is used for physical quantities associated with a synchronous particle. SYNCHROTRON MOTION So far we have discussed only the fundamental mode of a cavity. where u)/fisc is equivalent to the harmonic number per unit length. Nucl. AW = W-WS. Seeman. (3.104 VIII. HOMs CHAPTER 3. J. and a short range wake can cause a bunch tail to break up.419) where the coordinate s is chosen to coincide with the proper rf phase coordinate. [15]. IEEE Trans. We define the synchrotron phase space coordinates as At = t-ta.418) The accelerating electric field is £ = £0 sin u>t = £Q sin{ips + Atp). 255 in Ref. Let ts. particularly TMnp-like modes. Arp = tp-ipa=uj{t-ta). can affect the threshold current of a linac.C. In reality. (3. Efforts are being made to design or invent new cavity geometries with damped HOMs or detuned and damped HOMs. Crowley-Milling. and M. Sci. ip.4 Longitudinal Particle Dynamics in a Linac Phase focusing of charged particles by a sinusoidal rf wave is the essential core of longitudinal stability in a linac. . Saxon. Operation of the SLAC linac provides valuable information on transverse instability of intense linac beams.402 G. 103T. it also generates long range and short range wakefields. ^>s and Ws be the time. observed first in 1957. rf phase.
105Note that the convention of the rf phase used in the linac community differs from that of the storage ring community by a phase of 7r/2.„. the concept of synchrotron tune is not necessary.424) Since fcsyn ~ 1/I/T 5 . 403 (3. „ • (3. the longitudinal phase space will form islands as discussed in Sec. the wave number of synchrotron motion becomes very small for high energy electrons. In contrast to synchrotrons. i.sink's] « e£ 0 cos V>s A^>. (3. Parametric synchrotron resonances can occur if mvsyn = I is satisfied.422) Hereafter. . the Hamiltonian contour is not a constant of motion. FODO focusing systems. The beam will get the maximum acceleration and a minimum energy spread. if there is a quasiperiodic external focusing structures such as periodic solenoidal focusing systems. However. shown in Fig.e. etc. The linearized synchrotron equation of motion is simple harmonic. where m and i are integers. Tori of phase space ellipses form a golf-club-like shape. the synchrotron tune can be defined as the i/syn = ksynL/(2n). f ^ = -k%nAW. where L is the length of the periodic focusing system. This section will show that all captured particles ride on top of the rf wave. — = e£0 [sin(V>8 + Aip) . III. The beam moves rigidly in high energy electron linacs. Thus the synchronous phase angle is normally chosen as (f>s = | . 3. the linac usually do not have repetitive periodic structures. A. Near a parametric synchrotron resonance.2. The capture condition in an electron linac with vp = c Since (3S% changes rapidly in the first few sections of electron linac.VIII. electron bunches are riding on top of the crest of the rf wave.423) where the wave number of the synchrotron motion is ksyn = \ le£0u)cosips . we use the rf phase convention of the storage ring community. In this textbook.421) (3. INTRODUCTION TO LINEAR ACCELERATORS The energy gain from rf accelerating electric fields is105 dAW — . or periodic doublet focusing systems. as The Hamiltonian for the synchrotron motion becomes H = " JAW)2 2mcr)pJ7J + e£0 [cos(^s + AV>) + A^sinVs]. /3S and 7S are replaced by /? and 7 for simplicity..
428) ^ = _^sinV. and we have used /?2 = 1 and the relation tan (C/2) = ((1 . Let the electric field and the gap .> TT/2. which is usually about 80-150 kV.sin2C.^ .0)/(l + /?)]1/2 = 7 . Letting /3 = v/c. the electric field seen by the electron is fosin^. i. all particles within —IT/2 < fa < IT/2 will be captured into the region IT >fa.429) where the indices 1 and 2 specify the injection and the captured condition respectively.5. me dt Using the chain rule dtp/dt = (dtp/dQidC^/dt). we obtain (-2) 347 (3. (3. If the factor Ylnj = 1. A prebuncher is usually used to prebunch the electrons from a source.Since the phase velocity and the particle velocity are different. ^""sM^H-* Substituting f3 = cos£.A will be captured into the range TT/2 < fa < TT/2 + A 2 /2 (A <C 1).-cos^ = — t a n . For example.cos C)/(l + cos C))1/2 = [(1 .425) Z7T where A = 2nc/ui is the rf wavelength. particles distributed within the range A > fa > .5° in the capture process.\jl2 .429). Eq. we obtain The particle gains energy through the electric field. If Yinj = 1.1The capture condition. We assume a thermionic gun with a DC gun voltage Vo. we can integrate the equation of motion to obtain cosV. particles within an initial phase —TT/3 < fa < TT/3 will be captured inside the phase region IT >fa>> 27r/3. the path length difference between the EM wave and the particle in time interval dt is de={c. In particular.404 CHAPTER 3. favors a linac with a higher acceleration gradient £Q. and we use the fact that dl/\ = dip/2n. what happens to the injected electrons with velocities less than c? Let V be the phase angle between the wave and the particle.e. all injected beam with phase length 20° will be compressed to a beam with a phase length 3.— ( ^ — J =-Yiai. (3.v)dt = ^-dfa (3. The capture efficiency and energy spread of the electron beam can be optimized by a prebuncher. SYNCHROTRON MOTION In an electron linac operating at a phase velocity equal to c. Assuming constant gradient acceleration. which can be thermionic or rf gun.
Synchrotron motion in proton linacs Since the speed of protons in linacs is not highly relativistic. Until an equilibrium state is reached.13 %.11: Properties of rf bucket in conjugate phase space variables 1 (i>. Table 3.8. Electrons that arrive earlier are slowed and that arrive late are sped up. the faster electrons catch up the slower ones. (3. At a drift distance away from the prebuncher. at the same time.e. B. Thus electrons are prebunched into a smaller phase extension to be captured by the buncher and the main linac (see Exercise 3. the energies of individual beam bunches may vary.^) Bucket Area Te ("*'ff «*>)^ aM Bucket Height | 2 (rnc^feS^ Y{A) I hM) 16 ( ^ ) V ' ab(^s) | 2 (fa^go)1/2 y ( ^ ) ' .11 lists bucket area and bucket height for longitudinal motion in proton linacs (see also Table 3.2 for comparison). where a^ips) and F(V's) are running bucket factors shown in Eqs.1 rad will have an energy spread of about 0.48) and (3. individual adjustment of each klystron phase can be used to make a bunch with phase length A ride on top of the rf crest. etc. C. the wakefield induced by the beam travels along at the group velocity. INTRODUCTION TO LINEAR ACCELERATORS 405 width of the prebuncher be £ sin(wi) and g. All captured high energy electrons can ride on top of the crest of the rf wave in order to gain maximum energy from the rf electric field.VIII. Energy spread of the beam In a multi-section linac. The longitudinal particle motion follows a torus of the Hamiltonian flow of Eq. Other effects that can affect the beam energy are beam loading. The final energy spread of the beam becomes This means that a beam with a phase spread of 0.52). The synchrotron motion in ion linac is adiabatic. (3.422). i.1. The rf phase region for stable particle motion can be obtained from ipu and n — ips identical to those in the second the third columns of Table 3. ips = §. A train of beam bunches extracts energy from the linac structure and. wakefields. the synchronous phase angle ips can not be chosen as | . Thus the injection match is important in minimizing the final energy spread of the beam.9). Table 3.
46: A schematic drawing of electric field lines between electrodes of acceleration cavities. The screen produces a focusing force.10 respectively.. (-3) 34 1 (3.433) / H0 V e ^ocos% =\ [AZ(— . Q . Aifr) = <*!*>.However. the field at the exit end increases with time so that the defocussing effect due to the diverging field lines is larger than the focusing effect at the entrance end of the cavity gap. the Hamiltonian becomes H = . E.406 CHAPTER 3.The bunch length in r-coordinate is given by aT = a^jw. we find aAW ~ (w5 0 ) 1/4 (/?7) 3/4 .d w ( ^ ) 2 . -4 rms = TTCTAW/LJCA^. for a constant phase space area Ams. In small bunch approximation. where ^. p[H(AW/u>. the fractional momentum spread will decrease when the beam energy is increased: g ^ = V^r(m3C^7 J • (3-435^ Examples for beam properties in the Fermilab DTL linac and SNS linacs are available in Exercises 3. SYNCHROTRON MOTION The equilibrium beam distribution must be a function of the Hamiltonian. For rf accelerators.434) . i._. Note that the converging field lines contribute to a focusing effect in electrostatic accelerators. Figure 3. cosips) V /. Note that. . Aip)].8.8.\e£oCOS^ {^?p ( — .e.O. and aT ~ (wf o )" 1/4 (/37)" 3/4 . r m s is the rms phase space area in (eVs).e.432) A Gaussian beam distribution with small bunch area becomes where Ho is related to the thermal energy of the beam and the rms energy spread and bunch width are given by /ffomcSffV VAW/u.^. i.3 and 3. = \j ~3 aAt = \ —z fj^s fmc3/3373e£0cosAY/4 =V^ T { tf ) V 7 ym^p^^eto T ' (3 . ..4 (3... Lawrence placed a screen at the end of the cavity gap to straighten the electric field line. but unfortunately it also causes nuclear and Coulomb scattering.
!. Assuming a zero defocussing force. (3. (3.389). (3. we obtain dj-ymf) _ dt ~ euie0 sin ips 2/3-y2c T' ( • ) For a relativistic particle with 7 > 1 . the drift tubes of an Alvarez linac or the irises of a high-/? linac. The transverse force on particle motion is dt = ~eEr . Using Eq.VIII. INTRODUCTION TO LINEAR ACCELERATORS 407 VIII. Er = ~—£ocos > ij>. B& = —^£o cos tp.. e. phase stability requires TT/2 > ips > 0 (below transition energy).437) For a synchronous particle with v = vp. In reality quadrupoles are needed to focus the beam to achieve good transmission efficiency and emittance control in a linac. Thus the defocussing force experienced by the particle at the exit end of the gap is stronger than the focusing force at the entrance of the gap. (3.evB^ = — — (1 .46 shows the electric field lines between electrodes in an acceleration gap./ ) cos rp. if no other external force acts on the particle. where 7' = d'y/ds. the EM field of TMOio mode is Es = £0sin V .44I) Thus the orbit displacement increases only logarithmically with distance along a linac (Loreritz contraction). andfieldstrength increases with time during the passage of a particle. . the constant field strength gives rise to a global focusing effect because the particle at the end of the gap has more energy so that the defocussing force is weaker.440) (3.g.438) becomes Here we obtain 7— = constant = 70a. In an electrostatic accelerator. Eq. For rf linear accelerators. This has been exploited in the design of DC accelerators such as the Van de Graaff or Cockcroft-Walton accelerators.. the transverse defocussing force becomes negligible because the transverse electric force and the magnetic force cancel each other.436) where tp — (uit — w / ds/vp). 5 Transverse Beam Dynamics in a Linac Figure 3. ds Assuming 7 = 70 + j's.. we obtain X-Xo=(7±\nl)x< dx (3.
the transverse force is — * r J (3. the concept of betatron tune is not necessary. 562 in Ref.V ± £ s • LO cc (3. Lee. A mismatched linac will produce quadrupole mode oscillations along the linac structure. where I is also an integer. (3. 5706 (1994).107 106see e. one can define the betatron tune per period as vy = kyL/2n. and £ are integers. It should be designed from a known initial or desired betatron amplitude function and matched through the linac. Since there is no repetitive focusing elements.445) Thus the transverse force on a charged particle is related to the transverse dependence of the longitudinal electric field. This is the basic driving mechanism of synchro-betatron coupling resonances. E49. ^z{t.s) + Kz{s)z{t. Betatron resonances may occur when the condition mvx + nvz — £ is satisfied. Ruth. S. [15].s) = 0.g. Since there is no apparent periodic structure. many linacs employ periodic focusing systems.s)=O. it vanishes if the longitudinal electric field is independent of the transverse positions. p. However. In smooth approximation..443) where y is used to represent either x or z. VxfdsFn = -^Jd8F±.s) + Kx{s)x{t.106 Since TE modes have zero longitudinal electric field. Rev. SYNCHROTRON MOTION Transverse particle motion in the presence of quadrupole elements is identical to that of betatron motion.s) + k2y{s)y{t.s) = 0. Furthermore. In this case. Wakefleld and beam break up instabilities Applying the Panofsky-Wenzel theorem [24]. n. where L is the length of a period. 107See R. The design of cavities that minimize long range wakefields is an important task in NLC research. and ky is the wave number. where m. The linear betatron equation of motion is given by ^x{t.442) where Kx(s) and Kz{s) are focusing functions. the betatron motion in linac is an initial value problem. synchrobetatron resonances may occur when the condition mvx + nvz + li/syn = £ is satisfied.444) F±= ds Vx-Fjl = . These HOMs are also called wakefields. . the linear betatron motion can be described by —y{t. Phys.408 CHAPTER 3.Y. (3. its effect on the transverse motion vanishes as well. Thus we are most concerned with HOMs of the TM waves.
449) k2 — kx \ K2 / hi In t h e limit of equal focusing strength. p(t) is the density of particle distribution. 409 (3.>fci. s) = -j±rr ]t dtp(t)W±(i . x(t. as in Eq. INTRODUCTION TO LINEAR ACCELERATORS In the presence of a wakefield. (3. The motion of the trailing particle due to betatron oscillation of the leading particle becomes Gxi I k\ \ ki „ x2 = j-Xi sin k2s + -s T~2 sin Kis — — sin K2S .t)x(t.VIII. This is the essence of BBU instability. k(t.441) [3].447) X l = Gxu ( 3. If the beam bunch is subdivided into many macro-particles. s is the longitudinal coordinate along the accelerator. one would observe nonlinear growth for trailing particles. and kx and k2 are betatron wave numbers for these two macro-particles. They travel at the speed of light c. Each macro-particle represents half of the bunch charge. We divide an intense bunch into two macro-particles separated by a distance I = 2<rz. [3].we have / x2 —¥ xi sin kis + xi I Afc C \ — ) s cos kis. In the limit Ak — 0.108 108 Including beam acceleration. Detailed properties of the wake function and its relation to the impedance and the transverse force can be found in Ref. (3. and W±(t' — t) is the transverse wake function. If.4 4 8 ) ^^M where eN/2 is the charge of the leading macro-particle. i. s) + k\t. k2 .450) > where Ak = k2 — k\. 4 + k2X2 = (3. the trailing particle can be resonantly excited. s) is the transverse coordinate of the particle. The equation of motion in the smoothed focusing approximation is x'[ + k\xx = 0. s). The amplitude grows linearly with s. the equation of motion is [3] -^x(t. s)x(t. W±(£) is the wake function evaluated at the position of the trailing particle. the leading particle begins to perform betatron oscillation with X\ = Xisinkis. for some reason. .446) where t describes the longitudinal position of a particle. s) is the betatron wave number (also called the focusing function). the trailing macro-particle can be resonantly excited. x\ and x2 are transverse displacements. We will examine its implications on particle motion in a simple macro-particle model.e. the amplitude will grow logarithmically with energy (distance). (3.
Smirnov. Exercise 3. Lceii is the length of the drift tube cell. (3. In an Alvarez linac. AEn are the synchrotron phase space coordinates at the nth cell. Balakin. The SLC linac uses the latter method by accelerating the bunch behind the rf crest early in the linac. the energy spread is equivalent to a spread in focusing strength.11 are kd = n and 2TT/3 respectively. Show that the peak rf magnetic flux density on the inner surface of a pillbox cylindrical cavity in TMoio mode is B^K^-S or % [T] « 50 x 10"4 £ [MV/m].447) provides a good approximation for the description of particle motion in a linac. to restore the energy spread at the end of the linac. This method can also be used to provide BNS damping. SYNCHROTRON MOTION An interesting and effective method to alleviate the beam break up instabilities is BNS damping.8 1. (3. where Zo = fioc is the impedance of the vacuum. The bunch will perform rigid coherent betatron oscillations without altering its shape. where ipn. Show that the phase shifts per cell for the CEBAF and SLAC linac cavities listed in Table 3.451) can be achieved either by applying rf quadrupole field across the bunch length or by lowering the energy of trailing particles. A. It is also worth pointing out that the smooth focusing approximation of Eq.109 If the betatron wave number for the trailing particle is higher than that for the leading particle by the linear growth term in Eq. 109V. Proc.421) can be expressed as mapping equations: AEn+i = AEn + eVcosips Aif>n+i.450) vanishes. Note that BNS damping depends on the beam current. 12th HEACC. 2. and then ahead of the rf crest downstream. . Novokhatsky. p. 119 (1983).410 CHAPTER 3. and V. 3. (3. This means that the dipole kick due to the wakefield is exactly canceled by the extra focusing force. The BNS damping of Eq. and eV is the energy gain in this cell. the longitudinal equations of motion (3. Since the average focusing function is related to the energy spread by the chromaticity and the chromaticity Cx ~ — 1 for FODO cells.420) and (3.
See G. Estimate the total synchrotron phase advance in a cavity.H. .EXERCISE 3.0/6. Since the conductivity is proportional to the mean free path £.H. we can derive the amplitude function for synchrotron motion similar to that for betatron motion. There is little advantage to operating copper cavities at very low temperature.8 411 (a) Using the Courant-Snyder formalism. Soc. In a resonance circuit. 110In the limit that the mean free path I of conduction electrons is much larger than the skin depth (Sskini the surface resistance becomes Rs = (8/9)(\/3iilu}2£/16w<7)1^3.45 0. where Rs = l/o^skin is the surface resistance. Fermilab Alvarez linac Cavity Number Proton energy in (MeV) Proton energy out (MeV) Cavity length (m) Cell length (cm) (first/last) Average field gradient (MV/m) (first/last) Average gap field (MV/m) (first/last) Transit time factor (first/last) Number of cells 4. Pd = \ [ R*\H\2dS = ^ f ^ 2 Js 4 Js / \H\2dS.04/21.64/0. and is proportional to u2/3. wL R stored energy energy dissipation per period' 1 075 10.E.42 37.2/40.aui is the skin depth. £ av = V/Lce\\ is the average acceleration field. (b) Using the table below. calculate the synchrotron phase advance per cell for the first and last cells of cavities 1 and 2. and tps is the synchronous phase.8 2. and a is the conductivity.0 10.110 <5s](jn = y/2//j.30 7. Show that the synchrotron phase advance per cell is $ s y n = 2 arcsin ( ^ j .42 7. where the synchronous phase is chosen to be cos^ s = 1/2. the resulting surface resistance is independent of the mean free path.81 59 The energy stored in the cavity volume is The power loss in the wall is obtained from the wall current. A195. 336 (1984).86/0.54 19. The total energy loss in one period becomes AWd = 2JLPd = w !^Hi / 2 Js ]Hl2dS.8 1.81 55 2 10. 2 2 g where E — "fine2 is the beam energy.02 22. Reuter and E. A is the rf wave length. Proc.44 6. Q is expressed as 2&LI2 \RI2 where w = {LC)~ll2. Sonderheimer.45 0.60/2.62/7. Roy.
defined by ve = P/Wst. The average power flowing through a transverse cross-section of a wave guide is p = I [ E± x H±dS where only transverse components of the field contribute. For TM mode. Identical resonator LC circuits are coupled with disk or washer loading by parallel capacitors 2CP shown in the figure below.405.412 CHAPTER 3. Since <5skin ~ w~1/2.m = \ I \H±\2dS = ^ ^ Thus the total energy per unit length is j \£L\2dS Wst = Wst. (b) Show that the shunt impedance is D _ Zpd2 ^sh _ r 2u}fi . (a) Show that the energy flow. the diameter of the cavity will also be smaller. Us is the surface resistivity. (b) Verify that vg = du/d/3 = ve.856 GHz. ffI = ^°V £l- 7 A P - P-2Z-J^dS X [ k \ f 2WC The energy stored in the magnetic field is Wst. Note here that the shunt impedance behaves like rsh ~ wxl2.n 41 R^nb(b + d)J2{krb) Q ~ n{krb)2J2(krb) ~ ' "^ where kTb = 2. At higher frequencies.m + WKJ* = 2W st . The Q-factor depends essentially on geometry of the cavity. SYNCHROTRON MOTION (a) Using the identity /06 Jf(krr)2nrdr = Trb2J2(krb). the shunt impedance is more favorable.QC « 377f2. 6. m . Find the Q-value for the SLAC copper cavity at / = 2. . we find Q ~ w + 1 / 2 . however. show that the quality factor for a pillbox cavity at TMOio mode is = 2jv\H\2dV SSKafs\H\2dS = _d b_ = 2A05Z0 5skind + b 21^(1+ b/d)' where 6 and d are the radius and length of a cavity cell. is ve = fic/k. and ZQ = 1/fj. 5. which may limit the beam aperture.
(b) Show that the solution of the above equation is . where u>o = l/^/LCs is the natural frequency without coupling at kd = 0. Show that the frequency is u2 = WQ [1 + «(1 . r. Show that the dispersion curve of a magnetically coupled cavity is uil = J1 [1 + «(1 . — Q 1 9 • • • /t — u. these resonators are uncoupled.EXERCISE 3.2 cos(fcd) in + t n _! = 0.8 413 In the limit of large C p . The model describes only the qualitative narrowband properties of a loaded wave guide. (e) Cavities can also be magnetically coupled.cos kd)]. there are higher frequency modes. which give rise to another passband (see Fig. a small beam hole in a pillbox cavity corresponds to Cp > C s . 3.f . and k as the wave number. and K = C s /Cp is the coupling constant between neighboring cavities. Draw the dispersion curve of u vs k. ^.w 2 C p L. show that i n + i . (c) Show that the condition for an unattenuated traveling wave is LUQ < ui < uin. The magnetically coupled-cavity chain can be modeled by replacing 2CP in the LC circuit with £ p / 2 .cos kd)]. In a realistic cavity.. where ^="°( i+2 §) 1/2 ^°( i+ S) is the resonance frequency at phase advance kd — TT. or equivalently. We identify fed as the phase advance per cell. (d) Find k such that the phase velocity vp = c.-n _ p±j[nkd+xo] t — e . C cos(fcd) = 1 4. i. m (a) Applying Kirchoff's law. . m T h e equivalent circuit does not imply that a coupled resonator accurately represents a disk loaded structure. which corresponds to a pillbox without holes.41).
ceii> where J?sh. (a) For a particle traveling at velocity v. • • • . and K = Lp/L is the coupling constant between neighboring cavities. SYNCHROTRON MOTION where wo = 1/-\ZLCS is the natural frequency without coupling at kd = 0. We define the parameter a as 01 ~ 1 dPd 2P d ds ' 112The above calculation for voltage gain in the cavity structure is not applicable for an standing wave structure with kd = 0 and it. where two space harmonics contribute to the electric field so that £s = 2£0 cos ks cosut.1. For an rf structure composed of N cells.is equal to the particle velocity v. (b) For a sinusoidal electric field. i. and the shunt impedance is /2sh = NRa.e. 7. and to is the frequency. Nd] is the longitudinal coordinate.N. Using the definition of shunt impedance. d is the cell length of one period. This means that the voltage gain in the rf structure is AV = Nd£o. show that the shunt impedance of a standing wave rf structure is Thus the shunt impedance for a standing wave structure is equal to 1/2 that of an equivalent traveling wave structure. A constant impedance structure has a uniform multi-cell structure so that the impedance is constant and the power decays exponentially along the structure.408). the electric field of a standing wave rf cavity structure that consists of N cells is £s = £Q cos ks cos ut. where kd is the rf phase advance per cell. show that the total voltage gain in passing through the cavity is MT = -NdE f»fr(*-("/"))M* | sm(k + (U/v))Ndl 2 °[ (k-(ui/v))Nd {k + {u/v))Nd r m = 0. The resonance condition is kd = mir/N. Show that the energy gain is maximum when the phase velocityfc/o. A constant gradient structure is tapered so that the longitudinal electric field is kept constant.. There are two types of traveling wave structures. the power is Pd = N\£0d\2/2RshMl.414 CHAPTER 3. the energy gain of a standing wave structure is only 1/2 that of an equivalent traveling wave structure.ceii is t n e shunt impedance per cell for the traveling wave. (3. the power consumed in one cell is |M 2 /2iU. where s € [0. Show that the maximum voltage gain of the standing wave is (AV) m a x = Nd£o/2.cell- . Discuss the differences between the electric and magnetic coupled cavities. 112 8. Using Eq. k is the wave number.
386)] W g = u.2 T ))(Q(l-e. Calculate the longitudinal bucket and bunch areas in (eVs). The group velocity is equal to the velocity of energy flow. what is the longitudinal brightness of the beam in number of particles per (eVs)? . At a drift distance away from the prebuncher. Compare the rms bunch length in (ns) and in (m) with the rms transverse beam size at exit points of linacs.2 T ). the faster electrons catch up the slower electrons.e . Thus electrons are prebunched into a smaller phase extension to be captured by the buncher and the main linac. Jo (a) In a constant impedance structure. Each microbunch has about A^B = 8. show that the energy gain is AE(L) = eL(2r sh P o a) 1/21 ~ %" .383)] S2 = . show that Pd = P 0 ( l . The following table lists linac and beam parameters. (b) Assuming that rsh and Q are nearly constant in a constant gradient structure. DTL.findthe drift distance as a function of the Vo and Vi. Let the electricfieldand the gap width of the prebuncher be £ sin(wt) and g. CCL. RFQ.EXERCISE 3. where T = Jo a(s)ds. 9. as The total energy gain for an electron in a linac of length L is AE = e f £ds. We assume a thermionic gun with a DC gun voltage Vo.L(l-J(l-e. Electrons that arrive earlier are slowed and that arrive late are sped up. which is usually about 80-150 kV.£ ( l .8 415 the electricfieldis related to the shunt impedance per unit length by [see Eq. Show that the group velocity of a constant gradient structure is [see Eq. (3. An accumulator compresses the 1 ms linac pulse into a 695 ns high intensity beam pulse with 250 ns beam gap. Assuming a small prebuncher gap with Vi = £g -C Vo. Discuss the efficiency of prebunching as a function of relevant parameters. The design of the 2 MW spallation neutron source uses a chain of linacs composed of ion source.08 x 1014 particles per pulse at 60 Hz repetition rate.2 T ) ) . A prebuncher is usually used to prebunch the electrons from a source. (3. 10. which can be thermionic or rf gun.r s h ^ = 2a rsh Pd(s).70 x 108 protons. and SCL to accelerate 2.f\ and the energy gain is AE = e£L = e^P0rshL{l-e-2T). Cth where Po is the power at the input point.
SYNCHROTRON MOTION 1 RFQ 1 DTL L (m) length of the structure 3.8 185 185 1001.37 To. (MeV) •^bucket (eVs) at injection energy Ams (eVs) <yT (ns) I CCL I SRFL 55.5 0.45 ~5".2/0.065 2.12 206.5 402.108 e L (7r-mm-mrad) emittance at exit point _ 0.5 KEext (MeV) 2.2 | - | - | 10.6 86.3" .416 CHAPTER 3.33 fcsyn (m .0 " £0T (MV/m) KE inj (MeV) 0.812 805 805 60-62° 20° 3.1/5.60 0.5 86.21 "O0092~ " g A n.723 38.5 / r f (MHz) %j)s (differ from linac convention by -n/2) 60° 45-65° 3.1 ) /3x/z at exit (m) | 0.8 E|[ (7T-MeV-deg) emittance at exit point 0.7 402.
516 (1963). 52 (1948). The radiation is plane polarized on the plane of the electron's orbit. radiate electromagnetic energy. defined as T = ^hcjc/E. J. Rev. where UJP = c/p is the cyclotron frequency for electron moving at the speed of light.2 • Quantum mechanical correction becomes important only when the critical energy of the radiated photon. 40. Phys. 75. Proc. Acad. Rev. 102.Chapter 4 Physics of Electron Storage Rings Accelerated charged particles. 417 .L. • The radiation spans a continuous spectrum. 70. Schwinger. 74.P. Appl. Nat. Madden and K. Tomboulin and P. ibid. See Phys. 798 (1946). Phys. 10. 2D. 3F.3 applications of this radiation were contemplated. The power spectrum produced by a high energy electron extends to a critical frequency wc = 373wc/2. 810 (1947). its foundation was laid by J. Lett. in particular. and elliptically polarized outside this plane. As far back in 1898. Elder et al. 829 (1947). 18. Rev. The beamstrahlung parameter. Lienard derived an expression for electromagnetic radiation in a circular orbit. is a measure of the importance of quantum mechanical effects. Phys. is comparable to the electron beam energy. where 7 is the relativistic energy factor. E = •ymc2. 380 (1965).4 The 1J. Modern synchrotron radiation theory was formulated by many physicists. Appl. 4R.H. 36. particularly electrons in a circular orbit. This occurs when the electron energy reaches mc1(mcplfi)ll2 « 106 GeV. Phys. 1423 (1956). Codling at the National Bureau of Standards were the first to apply synchrotron radiation to the study of atomic physics. J. 71. Hartman experimentally verified that electrons at high energy (70 MeV then) could emit extreme ultraviolet (XUV) photons. Schwinger. Some of his many important results are summarized below:1 • The angular distribution of synchrotron radiation is sharply peaked in the direction of the electron's velocity vector within an angular width of I/7. Shortly after the first observation of synchrotron radiation at the General Electric 70 MeV synchrotron in 1947. 1912 (1949).R.. Phys. Rev. Sci. hioc = ^hcy3/p. 132 (1954).
and pM = (po.^ . etc. 1 dE . Today. p is the local radius of curvature.\=^\(dv\2 Zmc \dr dr J _l(dE\2} c2 \ dr j J ' . Applications of synchrotron radiation include surface physics.. condensed matter physics. i=l-\v\»-^ where w is the angular cyclotron frequency. For an isomagnetic ring with constant field strength in all dipoles.3 = \ 4. 1. The relativistic generalization of Larmor's formula (obtained by Lienard in 1898) is 2^ (dp. The radiation power arising from circular motion is (4.2 we will show that the power radiated from a circular orbit of a highly relativistic charged particle is much higher than that from a linear accelerator.e. i.418 CHAPTER 4.3) * = &**-&**-%*?• ("> where F± = u\p\ = evB is the transverse force.840 x If)"14 m/(GeV)3 for muons (4.P) is the 4-momentum vector. . medical research. biochemistry. dp. and . advanced manufacturing processes. was commissioned in 1968.846 x 10"5 m/(GeV)3 for electrons C7 = ^ 7 . A.5) 3 {me) [ ? 7 g 3 x 10_i8 m / ( G e V ) 3 for p r o tons. Basic properties of synchrotron radiation from electrons According to Larmor's theorem. K 3mc [{dr) ' ' where the proper-time element dr = dt/j.42. (F) _ _ _ _ _ . PHYSICS OF ELECTRON STORAGE RINGS first dedicated synchrotron radiation source. (4. dp . nearly a hundred light sources are distributed in almost all continents. The energy radiated from the particle with nominal energy EQ in one revolution is where R is the average radius. Tantalus at the University of Wisconsin. the instantaneous radiated power from an accelerated electron is p=J_?eW 47re0 3c 3 =2ro_(dp 3mc \dt &\ dt) ' X ' ' where v is the acceleration rate and r$ — e2/4ireomc2 is the classical radius of the electron.7) .. the energy loss per revolution and the average radiation power become Uo-C^EJp. v = /3c is the speed of the particle. C 8. In Sec.
Jx « 1 is a damping partition number. and the beam is compensated on average by the longitudinal electric field. whose applications include e+e~ colliders for nuclear and particle physics. and electron storage rings for generating synchrotron light and free electron lasers for research in condensed matter physics. II and III we will show that the natural emittance of an electron beam is enat = TCq^263/ Jx. A third generation light source employs high brightness electron beams and insertion devices such as wigglers or undulators to optimize photon brilliance.83 x 10~13 m. A second generation light source corresponds to a storage ring dedicated to synchrotron light production.1%bandwidth].1% of bandwidth). Using long undulators in long straight sections of a collider ring. Therefore a high brilliance photon source demands a high brightness electron beam with small electron beam emittances. one can obtain a wide frequency span tunable high brilliance monochromatic photon source.5 Synchrotron radiation sources are generally classified into generations. Murphy at BNL provides a list of beam properties of synchrotron light sources. and 0 is the bending angle of one half period. Neglecting the optical diffraction. or about ten order higher than the brilliance of X-ray tubes. mostly about 1020 photons/[s(mm-mrad)20. Furthermore. where the lattice design is optimized to achieve minimum emittance for high brightness beam operation. medicine and material applications.bnl. Some examples are SPEAR at SLAC. Using the synchrotron radiation generated from the storage rings. the product of the solid angle and the spot size dtldS is proportional to the product of electron beam emittances exez. and CHESS in CESR at Cornell University. A synchrotron radiation handbook edited by J. The factor T can be optimized in different lattice designs. In Sees. B.ELECTRON STORAGE RINGS 419 where To = 2nR//3c is the orbital revolution period. the longitudinal motion is damped.html . Table 4 lists some machine 5http://www. a beam with short bunch length can also be important in time resolved experiments. This natural damping produces high brightness electron beams. a few first generation light sources can provide photon beam brilliance equal to that of third generation light sources. Because the power of synchrotron radiation is proportional to E4/p2. biology. A first generation light source parasitically utilizes synchrotron radiation in an electron storage ring built mainly for high energy physics research. which is about five to six orders higher than that generated in dipoles.nsls.gov/AccPhys/hlights/dbook/Dbook. where Cq = 3.Menu. Synchrotron radiation sources The brilliance of the photon beam is defined as B = d4N dtdQdS(dX/X) ^ in units of photons/(s mm2-mrad2 0.
18 70. J. and LEP for CERN large electron-positron collider. we note that the emittances of third generation facilities.1 9 55 7 1.374 14.064 0.8 ez [nm] | 35 [8 [ 3.3 9. e+e~ colliders The development of electron and positron storage rings was driven by the needs of particle and nuclear physics research.3 8.1 5.36 35.s ] 3.4 2199. .1 0.0082 ^ [xlO. Fourth Generation Light Sources. are much smaller than those of their collider counterparts. 4.0 6.6 165.96 499.034 0.43 ex [nm] 450 240 64 48 51 8 4.4 768.1 8.2 35.8 h 160 1281 3492 3492 31320 1296 328 frf [MHz] 199. Colliders Light Sources BEPC I CESR 1 LER(e+) 1 HER(e~) I LEP ~APS I ALS E [GeV] 2.2 38.8 9.. APS and ALS. The widely discussed "fourth" generation light source is dedicated to the coherent production of X-rays and free electron lasers at a brilliance at least a few orders higher than that produced in third generation light sources.38 32. Proc.2 3.8 476 476 352.5 499.4 9. The reason is that colliders are optimized to attain a maximum luminosity given by 6M. Some of their properties are listed in Table 4.3 2199.2 14.3 26658.28 vx vz 6. where BEPC stands for Beijing electron positron collider.65 va 0.9 24. Proc.6 7.4 ] 4.96 4.006 0. ed.7 78.18 24. ESRF report (1996).35 60 30.2 352.4 3.2 6 3.9 1104 196.86 | 1.8 9.22 14. SSRL 92/02 (1992).. Fourth Generation Light Sources.51 | 0.016 0. CESR for Cornell electron storage ring.1: Properties of some electron storage rings.3 a[xlO"4] C [m] 240.28 25. PHYSICS OF ELECTRON STORAGE RINGS parameters of the advanced light source (ALS) at LBNL and the advanced photon source (APS) at ANL. eds.93 [ 0.5 6.5 7.6 Table 4.866 2.1 ^ ° [ x l o " 4 e V .48 C. Since 1960.420 CHAPTER 4. Cornacchia and H.18 p [m] 10.08 [ 0. LER and HER for low energy ring and high energy ring of the SLAC B-factory.01 400 152 14.0 3096. Laclare.085 0.5 5.28 76. many e+e~ colliders have served as important research tools for the particle physics.L. From Table 4. Winick.0522 0.
and f0 is the revolution frequency. The linear beam-beam tune shift of hadron colliders is independent of fi*. and eN is the normalized emittance.05 . Results of beam experiments at e+e" colliders show that the beam-beam tune shift is limited by £2 ~ 0. The resulting betatron tune shift is called the linear beam-beam tune shift7 ?z± — 7. /collision = /o-B is the beam encountering rate. the beambeam tune shift for hadron colliders is £ = Nro/iwje = Nro/AireN. r.1UJ 2njaz(ax + az) 2iryax(ax + az) where ro is the classical radius of electrons.10. If the luminosity of a machine is optimized. 7 . Anaxcrz is the cross section area at the interaction point (IP) of colliding beams. particles experience a strong Coulomb force of the opposite beam. ^4. we have £z ~S> £x.are the numbers of particles per bunch. (4. The electric and magnetic forces of the beam-beam interaction are coherently additive. \i __ sx± — NTrop: -. B is the number of bunches.ELECTRON STORAGE RINGS 421 where N+ and N.11) The luminosity expressed in terms of the tune shift parameter with N+ = 7V_ becomes *~ ~ 7Tj2 crxaz c2 2 fl*2~^z /collisionr0 Pz {^••i-^l Note that the luminosity is proportional to axaz. . /3* x are the values of the betatron function at the IP. Because of the nature of the Coulomb interaction. This design constraint for e+e~ colliders differ substantially from that for synchrotron light sources. in order to minimize the beam-beam tune shift the emittance of the beam can not be too small. then. 7Because the horizontal and vertical emittances of hadron beams are normally equal.0. Since ax ^> az for electron beams (see Sec. the beam-beam interaction for particles with a small betatron amplitude is characterized by a quadrupole-like force. During the crossing of the e + and e" beams. the beam-beam interaction for particles at a large betatron amplitude is highly nonlinear. . However. II). and 7 is the Lorentz relativistic factor. where r0 is the classical radius of the particle. _ NTrop.
The motion of the electron is specified by x'(t') with df' t=t. with .1: Schematic drawing of The retarded sealer and vector potentials (4-potential) due to a moving point charge are where J^(f .14) is needed to ensure the retarded condition. Here t' is the retarded time. With the identity /F6(f(t'))dt' = F/\df/dt\. (4.1).422 CHAPTER 4. A(x. will arrive at the observer at time c where R{t') = \x — x'\.t) = — K ' 4ire0c KR ^ . Figure 4. The electromagnetic signal. the scalar and vector potentials become $(£. and t is the observer time. ret (4-!5) . 4. The delta function in Eq.+m (413) _ a _ d^ the coordinates of synchrotron radiation emitted from a moving charge. emitted by the electron at time t' and traveling on a straight path. PHYSICS OF ELECTRON STORAGE RINGS I Fields of a Moving Charged Particle Let x'(t') be the position of an electron at time t' and let x be the position of the observer with R(t) = x — x'(t') (Fig. The unit vector along the line joining the point of emission and the observation point P is h = R{t')/R(t').t') = ec^6(x' — f(t')) is the current density of the point charge with Pfi = (P/c> l)i a n d r{t') is the orbiting path of the charge particle. t' is called the retarded time or the emitter time.*) = — — v ' 4ne0KRret .
t) = -^\^fM 47Te0 [7 2 « 3 i? 2 J 47T£oC [K3R +-J_[^x((n-^)x^)l J ret . defined as the energy passing through a unit area per unit time at the observer location. Using the relations we obtain the electric field as E{x. we obtain > E = JL. (4. (4. Using the identity V — V i ? ^ = n ^ .24) .I.dA/dt and B = V x A. is the Poynting vector S = —[E xB} = — \E\2n. J [*6V [ R2 Ane0 = e + 2-t) + ^ cRW + 2-t)]*> c c " ~^1 J ret \ L h i 1 d (4 17^ = _ £ _ [£^_ft 47re o c L 1 _d_£iinl CKdi' KR J ret f418l K Ki? 2 ' ' Since the time derivative of the vector n is equal to the ratio of the vector Vj_ to R dn _ n x (n x /?) _ (n CM" R ~ we obtain -P)h-0 R ' [ B(f rt _ _ ^ f ^ + A A J_ _ 1 ± A] 1 ' ' ~ 4ne0 « 2 i? 2 CK dt' K ^ c/c df KR \ J ret (A 20) ' [ I 47T€OC \ K2i?2 L\ CK dt' KR / J ret ' Note that the magnetic field is in fact related to the electric field by B = ( 1 / c ) h x E. The electric and magnetic fields are E = —V$ . Thus it suffices to calculate only the electric radiation field.23) The flux. FIELDS OF A MOVING CHARGED PARTICLE 423 where h = R/R = VR. a feature common to all electromagnetic radiation in free space.
which is proportional to 1/i?2. Hoc 16n2e0c 16nzeoc3 (4. dt .424 CHAPTER 4. which is proportional to 1/i?. The second term. n x /3 = |/3| sin0. This field can be transformed into an electrostatic electric field by performing a Lorentz transformation into a frame in which the charge is at rest.25) Note here that the electric field in Eq.e. Both E and B radiation fields are transverse to n and are proportional to 1/R.28) where 9 is the angle between vectors n and /?.27) L J ret and the Poynting's vector (energy flux) is 5 = — E xB = —\Ea\2h. The first term.1 Non-relativistic Reduction When the velocity of the particle is small. i. (4. the radiation field becomes ^ T ^c . PHYSICS OF ELECTRON STORAGE RINGS The total power radiated by the particle is g = (*•$)*£ = «rf|J|'. is a static field pointing away from the charge at time t. is the radiation field. dt' -. (4. Integration over all angles gives the same total radiated power as Larmor's formula: P= J_^ (41 l 47re0 3 c 3 ' ) ' 1. The total energy from this term is zero.M ^ l\ . dQ. 47re \ R 0 (4-26) (4.2 Radiation Field for Particles at Relativistic Velocities For particles at relativistic velocity.23) is composed of two terms. the Poynting's vector becomes ^•^^fei^^-^^^LThe total energy of radiation during the time between 7\ and T2 is W= /•T2+IR2/C) „ (2) 49 (4.^ — |n x (ft x /?)|2 = —f-^tfstfe. related to the acceleration of the charged particle. Mo Moc Thus the power radiated per unit solid angle is ^ = — \Ea\2R2 = .30) M+(Hl/c) (S-n)dt= rtl=T2 •/f=Ti (S-h)—d£. 1.
The integrated power is then T><^ fdP^ l 2e27S2 e2 fdpA2 .35 ) The rms of the angular distribution is also (0 2 ) 1 / 2 = 1/7. The first arises from the denominator with K = 1 — h-fi. The resulting wavelength of the observed radiation is shortened or. At relativistic energies. we have P = (1. the time interval of the electromagnetic radiation dt' of the electron appears to the observer squeezed into a much shorter time interval because a relativistic electron follows very closely behind the photons it emitted at an earlier time. where n = dt/dt' is the ratio of the observer's time to the electron's radiation time. the radiation from a relativistic particle is sharply peaked at the forward angle within an angular cone of 6 « I/7. j3 is parallel to 0.32 ) (433) where 6 is the angle between the radiation direction n and the velocity vector 0. y 'dt' v ' 4TT rjrnc\n*{{n-P)*h\ (l-n-Pf m ) There are two important relativistic effects on the the electromagnetic radiation.^)1/2«l . The angular distribution of the electromagnetic radiation is dPjt') _ remcv2 sin2 0 ( ' dn ~ 4TT ( l . When the observer is in the direction of the electron's velocity vector within an angle of 1/7.I. The second relativistic effect is the squeeze of the observer's time: dt — ndt' w dt'/j2.^ and K * (6* + l/ 7 2) ' • ( 4 . Since the angular distribution is proportional to 1/K 5 . Thus photons emitted at later times follow closely behind those emitted earlier. FIELDS OF A MOVING CHARGED PARTICLE Thus the power radiated per unit solid angle in retarded time is 425 ™£L = tf (S-ft)* = *#{§ • n) = dQ.1) -»• Y( 4 . Note that the instantaneous radiation power is proportional to 1/K 5 . The maximum of the angular distribution is located at e maj( = cos"1 \jg(y/l + 15/?2 . equivalently. Example 1: linac In a linear accelerator.^ c o s 6 ) 5 ' where 9 is the angle between n and /3. the energy of the photon is enhanced. Therefore it appears to the observer that the time is squeezed.
The total radiated power is obtained by integrating the power over the solid angle.2 shows the coordinate system. (4. AE is the energy gain per unit length. we find that the radiation from circular motion is at least a factor of 272 larger than that from longitudinal acceleration. / P x'(t') ^\^ ^v \ . PHYSICS OF ELECTRON STORAGE RINGS dpL 3.e. Therefore the radiation is also confined to a cone of angular width of (0 2 ) 1 / 2 ~ 1/7. Example 2: Radiation from circular motion When the charged particle is executing circular motion due to a transverse magnetic field. A. Ip ' " Figure 4.' .' . where dn B2r2 JpL = jmv = Jmti— = 299. Typically. at p Comparing Eq. i. The power per unit solid angle is then rfP _ dfl ~ e2v2 1 T sin2 0 cos3 $ 1 167r 2 e o c 3 (l-/?cos0) 3 [ ~ 7 2 ( l .426 where CHAPTER 4. -> ''' ] / ' 0/ N ' .2: The coordinate system for synchrotron radiation from the circular motion of a charged particle. (4.79/3B[T] [MeV/m].36). Figure 4.38) with Eq. and p is the bending radius. / 1 ./ 3 c o s 0 ) 2 J e2i.E/As is about 20 MeV/m in the SLAC linac.37) where v = /32c2/p. and 25 to 100 MV/m in future linear colliders.2 6 1 [ 47 2 6 2 cos 2 $] 7 ~ 2 ^ ? ( l + 7 2 6 2 ) 3 [ ~ (1 + 7202)2 J ' (4 . /? is perpendicular to /?.
and the charged particle illuminates the observer for a time interval cdt' « p0 rm s = p/7. the Fourier component has the property G(-LO) = G"(io). FIELDS OF A MOVING CHARGED PARTICLE 427 1. To the observer. i.I. Since the negative frequency is folded back to the positive frequency. however.26).= \G(t)\2. ^. (4. In this case. (4. Since the radiation from the parallel component has been shown to be I/7 2 smaller than that from the perpendicular component. %=r^ .3 Frequency and Angular Distribution The synchrotron radiation from an accelerated charged particle consists of contributions from the components of acceleration parallel and perpendicular to the velocity. The power radiated per unit solid angle is given by Eq. In other words. (Parseval's theorem). _.28). it can be neglected. we can define the energy radiation per unit solid angle per frequency interval as 8 The critical frequency is defined later to be uic = Z^UJP/2. 1 rOO —-= dil J-00 \G(t)\2dt = — \G{uj)\2du>.41) we obtain the total energy radiated per unit solid angle as (flilf TOO _. Using the Fourier transform G(w) = j G(t)ejutdt.31) has an angular width (6 2 ) 1 / 2 ~ 1/7. (4. the acceleration v± is related to the radius of curvature p by v± = v2/p « <? j p. the corresponding time interval At of the radiation is at 72 7dc Thus the frequency spectrum spans a broad continuous spectrum up to the critical frequency wc of order8 At ~ %At>« V = -r• W ~ c zb ~ 7 'p = j 3 u j p - (439) To obtain the frequency and angular distribution of the synchrotron radiation. the radiation emitted by a charged particle in an arbitrary extremely relativistic motion is about the same as that emitted by a particle moving instantaneously along the arc of a circular path. The angular distribution given by Eq. G(t) = (—Y' 2 [REU (4. we should study the time dependence of the angular distribution discussed in the last section.e. Git) = ^J G{w)e-jutdt.42) ZTT J—oo Since the function G(t) is real.40) with electric field E given by Eq. (4. (4.
(4. (4. the amplitude of the frequency distribution becomes 0/7T d eo c •'-oo K = M*T5— )V2 r ft x (ft x 327r J eo c •'-oo fief-V-Wde.44) The Fourier amplitude G(w) is 327r3e0c .49) The corresponding intensity spectrum becomes .[ d3xf(x. (4. i.e. The radiation amplitude is a linear combination of contributions from each charge.)| 2 = 2\G(OJ)\2.)|2 + |G(-u./ X ^ ^ ^ W . we have R=\xf{t')\ tzx-hr{t'). PHYSICS OF ELECTRON STORAGE RINGS ^ = |G(o.46) where x is the distance from the origin to the observer. where i? = \x — r(t')\ is the distance between the observer and the electron. (4./-oo «3 (4. Apart from a constant phase factor.47) where we use integration by parts and the relation K* dt< K ' l j We now consider a group of charged particles ej. ^ e " ^ " ^ . With the observer far away from the source.45) = (W^)1/2 / " " X ( ( " . e/^-jam-r7c _^ £ ej^-^^0 -> .428 with CHAPTER 4.
I. Let (4-51) n — (cos 6 sin $. (4.3." . FIELDS OF A MOVING CHARGED PARTICLE 429 L Z ^ \ a Figure 4. 0). 4. cos 9 cos $. Since the range of the t' integration is of the order of At? ~ p/cry. all horizontal angles are equivalent.^sin. r(t') / A. sinujpt'. (4. /3 = /3(sinujpt'./cose) « | [ ( 1 + e v +1^' 3 ] [i + o(l)] = f£(z + ^ 3 ) + . 0). The vector nx(nxJ3) can be decomposed into h x (n x /?) = p [—ey sin u>pt' + ej_ cos u>pt' sin 0j . where the trajectory is f(t') — p(l — cosojpt'.47) can be expanded as W(f - ±f) = W(f . (4. which is nearly perpendicular to the orbit plane. The radiation is beamed in a narrow cone in the forward direction of the velocity vector.52) where ey is the polarization vector along the plane of circular motion in the outward x direction and e± — h x ey is the orthogonal polarization vector.3: Coordinate system for circular trajectory of electrons. sin 6) be the direction of photon emission. cosujpt'. Because the particle is moving on a circular path.53) . the exponent of Eq.3 shows the coordinate system of a particle moving along a circular orbit. Figure 4. Frequency spectrum of synchrotron radiation The radiation emitted by an extremely relativistic particle subject to arbitrary acceleration arises mainly from the instantaneous motion of the particle along a circular path. The short pulse of radiation resembles a searchlight sweeping across the observer. as shown in Fig. where up = Pc/p is the cyclotron frequency and /3c = df(t')/dt' is the velocity vector. and it is sufficient to calculate the energy flux for the case $ = 0.
(4. the energy and angular distribution function of synchrotron radiation becomes where the amplitudes are Thus the energy radiated per unit frequency interval per unit solid angle becomes EK = T e S ^ O 2 ' 1 + X'f K K ) + TTX^w®] • (461> where the first term in the brackets arises from the polarization vector on the plane of the orbiting electron and the second from the polarization perpendicular to the orbital plane. . 18. 74. (4-56) J™ cos [ ^ (x + ^ 3 ) ] dx = ^ * i / 3 ( f l (4-57) for the modified Bessel function.53) are of the same order of magnitude. 829 (1947). 810 (1947). the radiation is purely plane polarized. 7 1 .9 On the orbital plane.R. The critical frequency wc has indeed the characteristic behavior of Eq. 9 F.55) Note that both terms in the expansion of Eq. With the identity jQ°° xsin [ ^ (x + \x3)] dx = ^ # 2 / 3 ( 0 . (4. The angular distribution has been verified experimentally. PHYSICS OF ELECTRON STORAGE RINGS and e=£-(i+xrj. Phys. Appl. Phys. Elder et al. J. (4. the radiation is elliptically polarized.39). Rev. where X = 0. Away from the orbital plane.430 where CHAPTER 4. 52 (1948). *-!•**-?£.
dfi = @=0 T^i—TH2{—). Figure 4. Thus the energy spectrum at 0 = 0 increases with frequency as 2.5 y=u/uc . and then drops to zero exponentially as e~u/"c above critical frequency. reaches a maximum near uic. We find w dl 3e2 2 where IF. 4.0 I ' 1 1 ' I 1 ' 1 1 I I ' ' 1 I 1 1 1 1 I 1 1 1 1 0 0. uc = Tiuc.64 with y = tu/u>c (Fig.I.91(W/UJC)2/3 for u> <C uic. Thus the synchrotron radiation is confined by W^O 1 ' 8 - (4-63) The synchrotron radiation spans a continuous spectrum up to UJC.5 1 1.4: The functions i?2(y) " and S(y) for synchrotron radiation 0. ^(I^P or if? »i.5 2 2. FIELDS OF A MOVING CHARGED PARTICLE B. 167T3£0C LJC 4. (4-62) we find that the radiation is negligible for £ > 1.where u = hco. Asymptotic property of the radiation Using the asymptotic relation of the Bessel functions 431 ^MvO^.4). The radiation at large angles is mostly low frequency. . Angular distribution in the orbital plane In the particle orbital plane with 0 = 0. C. • / ^ ^ ^ \ ^ H2(y) : o 5 -V I • / / ^^\ ---^^^^^ly) ^~^^^_^^ _ are shown as functions of y = u/uc. High frequency synchrotron light is confined in an angular cone 1/7. the radiation contains only the parallel polarization.
while the second term is the perpendicular component. we find that the parallel polarization carries seven times as much energy as does the perpendicular polarization. (4. . CC (4. Integrating over all angles. Phys. Rev. f .4. Angular distribution for the integrated energy spectrum When the energy flux is integrated over all frequency (see Section 6.67) where S(y) = 1 9-^-yf~K5/3(y')dy'. the flux is ^ dn 0=0 = <-i 4 7) m 3e2 Sm w LeJ 16n3e0hc w 2KuJ = 1. . we obtain the energy flux10 /H = ^7-/>V3(#^7^).J. (4. 1912 (1949). The instantaneous power spectrum becomes /» = ^-I(u) = ^S(-). This result was obtained by Lienard in 1898. The total instantaneous radiation power becomes r°° AP2 FA 2-Kp Jo 367re0p 2n pl where C 7 = 8.576 in Ref. 4.432 CHAPTER 4.85 x 10~5 meter/(GeV)3. 2iTp Ulc U)c PJ = ~ /Mdw =-p— 7 wc = ^ P % . [26]). 75.68) also shown in Fig. £°S{y) = l.61) over the entire angular range. we obtain f°° dP 7e2 -fue ( 5X2 \ Jo dudn ~ 967T£0C (1 + X2)5/2 ^ + 7(1 + X2)) ^' where the first term corresponds to the polarization vector parallel to the orbital plane. (4-69) (4. (4-72) v ywc' [s mr 2 0.33xl013£02^]^2(-) 0 l J 1 0 J. Frequency spectrum of radiated energy flux Integrating Eq. the photon flux density is i-[3i i ^^(S) 1 ' i + ^N'« + T^i«4 In the forward direction Q = 0. E.PtT5.1% bandwidth] ' ' Schwinger. PHYSICS OF ELECTRON STORAGE RINGS D.70) Since the energy of the photon is fuv.
.4. (4. . o The total number of photons emitted per second. du un(u)du = I(u))dw = I(LO)— n or (4.46xlO"£b[GeV]/[A]Gi(-) f uc where roo 2n [s mr2 0. we obtain dT r / 1 %/3e2 5u UJ r°° = 2.J (^) (4. is . The critical photon energy is uc [keV] = hu>c = 0. Thus the radiation due to the bending magnets has a smooth spectral distribution with a broad maximum at the critical frequency uc. 1. where h is Planck's constant. (4. Integrating Eq.665 E%[GeV] B[T] (4.I FIELDS OF A MOVING CHARGED PARTICLE 433 which peaks at y = 1 or u = u)c. we find that the spectral flux vanishes as (w/wc)2^3 for u <C u>c and as e~ulWc for w ~S> u>c.e.62).73) Using the asymptotic properties of the modified Bessel functions of Eq. Let n(u)du be the number of photons per unit time emitted in the frequency interval dui = du/h at frequency u). .1% bandwidth] ^T\ -.76) n{u)=^FO=^§ Uc where Uc O7T Uc r.4 Quantum Fluctuation Electromagnetic radiation is emitted in quanta of energy u — tuu.61) over the vertical angle 0 . 5ac 7 uc 2v^p (4. i.^Gx{y) is shown in Figure 4. Kv3{y)dy> Jujuc (-7 4?) (4.r f°° . we define 4wc as the upper limit for useful photon frequency from bending magnet radiation.78) F(y) = -S(y).79) V . Following the traditional convention. A/".75) Gi(y) = y Jy Ks/3(y')dy' The function S{y) . y Jo rFiy)^1-^. Vo W3P7 8 N= n(u)du=— •!-= —-rJ-.
Uo [MeV] 0.4.86 119. In Table 1.50 1. 19.0 77.0 x 10 9 uc [keV] 2.4 lists synchrotron radiation properties of some storage rings. E is t h e beam energy.5 55 7000 p [m] 10.434 CHAPTER 4.2 240.01 3096. (4.9 C [m] T o [//s] 0. r s and T± are radiation damping times of the longitudinal and transverse phase spaces (to be discussed in Sec.= -^Lory.3 1104 196.2 9 7 1.8 19.4 2199.00 0.6 165 38. 38.37 9. C is the circumference.66 89.0 x 10 9 7 i [ms] 18.28 7. PHYSICS OF ELECTRON STORAGE RINGS where a = e2/4Tre0hc is the fine structure constant.80 2. Note that the number of photons emitted per revolution is typically a few hundred t o a few thousand. II).34 3.80) Table 1. the quantum fluctuation varies as the seventh power of the energy. (482) Cu-^=. = ^u2c. 0.56 7.81) (u2) = ±fo°°u2n(«)du or N{u)-CuucP7-^—{mc2)3—.34 7.4 18.91 0.68 0. 89.52 5.45 0.35 60 30. Table 4.2 3096.7 8. p is t h e bending radius. UQ is the energy loss per revolution. 38. 16.040 | 285 | 777 I 415 | 1166 | 907 | 194 | 7125 |[ 494" iV7 The moments of energy distribution become {u) = 1 r°° 8 AfJo un(u)du = ^7=uc. 9.20 1. and JV7 is t h e average number of photons emitted per revolution. The average number of photons emitted per revolution becomes N7 = N2-K.78 19.3 2199.83) At a fixed bending radius. c Vo (4.9 26658. 2.30 3. .4 768. 155.2: Properties of some high energy storage rings I B E P C I CESR I L E R I H E R I A P S I ALS I LEP I LHC I E [GeV] 2.96 4. 4. (4. To is t h e revolution period.11 2 6 1 .00060 T|| [ms] 8.97 2.2 6 3.8 8. uc is t h e critical photon energy.8 26658. 1.
99. (4. .85 x 10" 5 [(£ [GeV]) 4 /p[m]] [GeV] ~\ 7.38). where K5/3 energy. Plot the angular distribution of synchrotron radiation shown in Eq..37) for /3 = 0.5 and P = 0. show that the number of primarily photons per unit energy interval in one revolution is ~r = -^—2 / K5/3(y)<iy. and show that the integrated power is given by Eq. (4. Find the angle of the maximum angular distribution. Express it in terms of the constants m. (4.5 T at 55 GeV beam energy? What will the energy loss per revolution be at 100 GeV? With the present LEP dipole magnets.0 0.5 5453 908 E [GeV] p [m] 7 uc [keV] Up [keV] N7 5. 3. A particle of mass m and charge e moves in a plane perpendicular to a uniform. 4. and show that the integrated power is given by Eq.28 0. the magnetic flux density in a LEP dipole is B = 592.99.78 x 1Q-6[(E [TeV]) 4 /p[m]] [GeV] <IN 9v^c/0 r ° „ . and 0 is the critical pnoton for electrons. _ 4nrpmp<?y4 ~ ~ 3p is the energy loss per revolution. uc = 3hcj3/2p _ f 8.5 G. (4. Find the maximum angular distribution of synchrotron radiation. 2.059 3246 123 10. At 55 GeV. du 8n ulc Ju/uc is the Bessel function of order 5/3. which will desorb the surface molecules.2 165 107632 17612 119 9.e.67).1 1.69). Show that the total number of primary photons in one revolution is given by 15V3Uo Verify 7V7 of the machines in the table below. Using Eq.5 and p = 0.96 13699 19.78 261495 3518 7136 Tl68 rings I APS 7 38. The synchrotron radiation generated by the circulating beam will liberate photo electrons from the chamber walls. (4.1 435 Exercise 4. The photon yields depend on the photon energy and the chamber wall material.61) for /? = 0.j and B. What happens if you design the LEP with a magnetic flux density of 0. static magnetic induction B.EXERCISE 4. (a) Calculate the total energy radiated per unit time.3 3530 1429 567 Electron storage LEP I HER(B) 55 9 3096. (b) Find the path of the electron.2 53289 21316 8526 3. Plot the angular distribution of synchrotron radiation shown in Eq. at what energy will the beam lose all its energy in one revolution? . for protons. Proton storage rings VLHC I SSC I LHC 50000 20000 8000 15000 10108 3096.
7 1 [A]. 11 The resulting pressure increase is given by kS ds where i) is the molecular desorption yield (molecules/photon). Grobner. (b) Show that the total number of photons per unit time (s) is given by M = 4 .60 x 1 0 1 6 4 1 * [ A ] as pl [photons/ml. 8. integrate the intensity over all angles. 5 is the pumping speed (liter/s).Od x 10 (^m])2 [W\.72) and (4. 454 in Ref.b. p. Verify Eqs. 1 4 x l O 1 7 . . (4. (4. and prove that the parallel polarization carries seven times as much energy as that of the perpendicular polarization. (a) For an accelerator with an average current / [A]. In designing a high energy collider. Verify Eq. 7.74). [15]. PHYSICS OF ELECTRON STORAGE RINGS 6. and k = 3. Show that the total number of photons per unit length in the dipole magnet is given by11 ^— = 6. .2 x 1019 (molecules/torr-liter) at room temperature. show that the total synchrotron radiation power is given by F. See 0. you need to take into account the problems associated with gas desorption due to synchrotron radiations.66).436 CHAPTER 4.
On the other hand. and R is the average radius of a storage ring. 3% of its total energy. The energy loss per revolution at 100 GeV is 2. and methods of manipulating the damping partition number. The balance between damping and excitation provides natural emittance or equilibrium beam size.096 km) will lose 0. p (4. electrons lose energy in a cone with an angle about 1/7 of their instantaneous velocity vector. In this section we discuss damping time. and C1 = 8. beam emittances. there is a damping-fluctuation partition between the longitudinal and transverse radial planes.84)] and the average beam energy is compensated by longitudinal electric field. The energy of circulating electrons is compensated by rf cavities with longitudinal electric field.e. Since higher energy electrons lose more energy than lower energy electrons [see Eq. (4. The vertical emittance is determined by the residual vertical dispersion function and linear betatron coupling. The damping (e-folding) time is generally equal to the time it takes for the beam to lose all of its energy.II. The balance between quantum fluctuation and phase-space damping provides natural momentum spread of the beam.18 GeV per turn.85 x 10~5 m/(GeV) 3 is given by Eq. p is the local radius of curvature. For example. an electron at 50 GeV in the LEP at CERN (p = 3.5). . i. (4. The total energy radiated in one revolution becomes Uo — ——f 2TT -r J p1 (= C 7 —for isomagnetic rings). synchrotron radiation is a quantum process. The photon emission is discrete and random. The longitudinal and transverse motions are coupled through the dispersion function. This mechanism provides transverse phase-space damping. damping partition. Furthermore. there is radiation damping (cooling) in the longitudinal phase space.6) Therefore the average radiation power for an isomagnetic ring is (P)-~cCjEA (4 7) where To = /3C/2TTJ? is the revolution period. RADIATION DAMPING AND EXCITATION 437 II Radiation Damping and Excitation The instantaneous power radiated by a relativistic electron at energy E is *-£7-•£*>**• < 484 ' where B is the magnetic field strength. and the quantum process causes diffusion and excitation. and gain energy through rf cavities in the longitudinal direction.9 GeV. quantum fluctuation.
C is the accelerator circumference. l3Here. V(T) = Vos\n(f> = Vosinwrf(r + r s ). (4. For simplicity. where the energy gain in the rf cavity is to compensate the energy loss in synchrotron radiation. EJ AF (4. if the field is linear with respect to displacement. since a particle having nonzero betatron amplitude moves through different regions of magnetic field. W =% . (4. and the difference in arrival time is13 (~* A p AT = ac Thus the time derivative of the r coordinate is C Ej — = acT0—. its rate of synchrotron radiation may differ from that of an electron with zero betatron amplitude.e. we obtain12 U(E) =U0 + WAE.89) where the rf frequency is wrf = hu>0 = h2ir/T0.85) at.438 CHAPTER 4. AE) be the longitudinal phase-space coordinates of a particle with energy deviation AE from the synchronous energy. Let (C(T + TS).90) 12In fact. (4. First. we assume a sinusoidal rf voltage wave in the cavity and expand the rf voltage around the synchronous phase angle (f>s = hu>oTs. i. w0 is the revolution frequency. and let (CTS. where a c is the momentum compaction factor. the phase slip factor is rj = ac — 1/y2 fa ac for high energy electrons. and h is the harmonic number. Now we consider the case of a storage mode without net acceleration. Thus the net energy change is d(AE) dt eV(T)-U(E)^ TQ where the radiation energy loss per revolution is U(E) = UQ + WAE. Thus Eq. However. PHYSICS OF ELECTRON STORAGE RINGS II. we consider the longitudinal equation of motion in the presence of energy dissipation. The path length difference between these two particles is AC = acC^r. the electron loses energy U(E) by radiation. E=E0 where Eo is the synchronous energy. and gains energy eV(r) from the rf system. We will show later that the coefficient W determines the damping rate of synchrotron motion.86) !=*¥• = During one revolution. [/rf = eV(r) = Uo + eVr (4.84) around synchronous energy. We assume that all particles travel at the speed of light. 0) be those of a synchronous particle. 1 Damping of Synchrotron Motion Expanding the synchrotron radiation power of Eq. the radiation power averaged over a betatron cycle is independent of betatron amplitude. . (4.85) does not depend explicitly on betatron amplitude.
V'= wrfVo cos(a>rfTs). Since the damping rate is normally small. Particle motion is damped toward the center of the bucket.96) . where W <*E = ^ . OE -C WS. (4-91) Thus.95) Figure 4. c J p c J p h/Q (4. aceV ws2 = .87).dt = fp^ds > i J J as = .4 lists the longitudinal and transverse damping times T\\ = 1/CXE and T± of some storage rings. i. (4.lp. we have dJ^l = at Jo Combining with Eq.5: A schematic drawing of damped synchrotron motion.00).e. the solution can be expressed as r{t) = Ae~aEt cos(wst . we need to evaluate W. (4. 1/e damping time is 103 — 104 revolutions. .92) g + 2« B f+u<?r = 0. (4.II. Typically. Table 1. in small amplitude approximation. Since the radiation energy loss per revolution is £ ™ = fP.94) This is the equation of a damped harmonic oscillator with synchrotron frequency ws and damping coefficient a^.93) (4. we obtain heVr-WAE). RADIATION DAMPING AND EXCITATION 439 Figure 4.il + -)ds = .5 illustrates damped synchrotron motion.^ — . (4./P 7 (l + ~^)ds. The damping partition To evaluate the damping rate. with i/o = eVosin(wrfrs).
and we have used cdt/ds = (1+rr/p). p is the radius of curvature.D1 d^ cf \ Eo BoEodx Eo p j E Thus the damping coefficient becomes a £ = 2 ^ ^ ^ = ^ 2 + 2?) - (4'100) Here T> is the damping partition number.440 CHAPTER 4. Since we are interested in the dependence of total radiation energy on the off-energy coordinate and (xp) = 0. . 14The transverse displacement x is the sum of betatron displacement and off-momentum closed orbit. 14 the derivative of radiation energy with respect to particle energy w=e±mifi&+°z\ dE c J [dE p E JE Using F 7 ~ E2B2 of Eq. The damping partition number V is a property of lattice configuration. (4. we obtain dE and rft/rad = Eo „.84). we replace x by D(AE/E0). . /dipole (4. ds. cC/o/1 7 VP Bdx)]Eo = [/f (?+ »<•>)*] [/?]"'' V = ~<fD(s)(-2+2K(s)) Z7!" J ("01) where A'(s) = Bi/Bp is the quadrupole gradient function with Bi = dB/dx. PHYSICS OF ELECTRON STORAGE RINGS where D is the dispersion function. For an isomagnetic ring.97) Eo Bo dE Eo Bo dE dx Eo Bo Eo dx' K > If f ^ 2^rjDd5 P.102) \P The integral is to be evaluated only in dipoles.4.
RADIATION DAMPING AND EXCITATION Example 1: Damping partition for separate function accelerators 441 For an isomagnetic ring with separate function magnets. The damping coefficient for separate function machines becomes The damping time constant.II. Figure 4.105) and OE « 2(Py)/E. Example 2: Damping partition for combined function accelerators For an isomagnetic combined function accelerator. to be discussed in the next section. The momentum change resulting from recoil of synchrotron .103) ' y where QC is the momentum compaction factor.6: Schematic drawing of the damping of vertical betatron motion due to synchrotron radiation. we find (see Exercise 4. The synchrotron motion is highly damped at the expense of horizontal betatron excitation. where K(s) = 0 in dipoles.2. II. is nearly equal to the time it takes for the electron to radiate away its total energy.2 Damping of Betatron Motion A. which is the inverse of OE. Since normally ac -C 1 in synchrotrons. This process damps the vertical betatron oscillation to a very small value.1) £> = 2 . Transverse (vertical) betatron motion A relativistic electron emits synchrotron radiation primarily along its direction of motion within an angle I/7. V -C 1 for separate function machines. The energy loss through synchrotron radiation along the particle trajectory with an opening angle of I/7 is replenished in the rf cavity along the longitudinal direction.^ (4. V=^l^ds = ^ 2K pJ p p (4.
442 CHAPTER 4. Now the energy gain from rf accelerating force is on the average parallel to the designed orbit (see Fig. where vertical betatron coordinate z is plotted as a function of longitudinal coordinate s. dipole magnets. PHYSICS OF ELECTRON STORAGE RINGS radiation is exactly opposite to the direction of particle motion. p E The corresponding change of amplitude A in one revolution becomes ASA = (fz'Az1) = -<(/?z') 2 )^. Since the betatron motion is sinusoidal. the betatron amplitude is unchanged except for a small increment in effective focusing force.109) = -z'^. z' = ?±. Eq.e. and the energy of each photon is small in comparison with particle energy. such that 5P is parallel and opposite to P with \c5P\ = u. and (3 is betatron function.106) where A is betatron amplitude.6). <j> is betatron phase.108) is valid because the momentum direction of photon emitted is along the trajectory of betatron motion.6 illustrates betatron motion with synchrotron radiation. we have ({pz<?) = A2/2. however. (4. . (4. where electrons emit synchrotron radiation.. are distributed in a storage ring but rf cavities are usually located in a small section of a storage ring. Let p±_ be the component of momentum p that is perpendicular to the designed orbit. (4. When an electron loses an amount of energy u by radiation. The betatron phase-space coordinates are z = Acos(f>.) averages over betatron oscillations in one revolution. and The time variation of the amplitude function is then I^ Adt 15Although = !M = _J^ To A 2ET0' ( 4H2) K ' . it does not change z either. i. z' = . Figure 4. (4. A2 = z2 + {pz'f. the momentum vector P changes by 5P. the new slope is decreased by the increment of longitudinal momentum at cavity locations where the change of z' is15 ^^-7(1-7'-y(1-7»Az' = -z'^ (4"108) (4.107) P When an accelerating field is applied to the electron.. 4.s i n 0 . Since the radiation loss changes neither slope nor position of the trajectory.110) where (.. and UQ is synchrotron radiation energy per revolution.
Since phase-space coordinates are not changed by any finite impulse.116). The radiation damping arises from the combination of energy loss in the direction of betatron orbit and energy gain in the longitudinal direction from rf systems. The horizontal displacement from the reference orbit is A ri An x = xp + xe. xe is the off-energy closed orbit. (4.108).114) The resulting change of betatron amplitude can be obtained from the betatron phase average along an accelerator. xe = D{s)—.118) 16Here we use SI = (1 + x/p)ds with x = xp. + %) %-ds. where xp is the betatron displacement. . Horizontal betatron motion The horizontal motion of an electron is complicated by the off-momentum closed orbit. as shown in Eq.116) into Eq. x' = x'g + x'e. x'e = D'{s)—. and D(s) is the dispersion function. = Aco84>. x'0 = -~sm<j>. Substituting the energy loss u in an element length S£ with (4. The off-momentum closed orbit does not contribute to the change in betatron amplitude. When the energy of an electron is changed by an amount u due to photon emission. (4.II. (4. We consider betatron motion with Xf. 6x'0 = -5x'e = -D'(s)^. A2 = x} + (px'p)2.115) the change in betatron amplitude becomes ASA = xpSxf. y B dx p J eh (4. + px'pSx'p = -(Dip + P2D'x'e)^.7. 4. because we are interested in the effect on betatron motion. B. the off-energy closed orbit xe changes by an amount 5xe = D(s) (u/E) shown schematically in Fig.16 we obtain the change in betatron amplitude as ASA = xeD (l + l^-xf. (4. the resulting betatron amplitude is Sip = -Sxe = -D(a)l. RADIATION DAMPING AND EXCITATION and the damping coefficient is 443 The radiation loss alone does not result in betatron phase-space damping.
119) is positive. there is an increase in horizontal betatron amplitude due to synchrotron radiation.101). . az = Jza0. This resembles the random walk problem.121) as = ( l . (4. and the electron energy is changed by u.122) 17This is easy to understand. i. The fractional betatron amplitude increment in one turn becomes where V is the damping partition number given in Eq. we observe that the right side of Eq. and the resulting betatron amplitude will increase with time. At a location marked by a vertical dashed line. we obtain the net horizontal amplitude change per revolution as The damping (rate) coefficient becomes (4. which perturbs the betatron motion. Emission of a photon excites betatron motion of the electron. and thus the off energy closed orbit is shifted by Sxe.7: Schematic illustration.e. aE = JEOC0. where (xp) = 0 and (zjj) = \A^.^ . In particular. (4. the electron emits a photon.444 CHAPTER 4.101). A small and not so important effect is a stronger focusing field for betatron motion.112).. (4. Here we have neglected all terms linear in x'p. We are now looking for the time average over the betatron phase. radiation damping coefficients for the three degrees of freedom in a bunch are ax .2 > ) . Sands [27]. of quantum excitation of horizontal betatron motion arising from photon emission at a location with nonzero dispersion functions. (4.17 Including the phasespace damping due to rf acceleration given by Eq. where the damping partition V is given by Eq.JxaQ. In summary. PHYSICS OF ELECTRON STORAGE RINGS Figure 4. Hot. after M. (4. because their average over the betatron phase is zero.
Wiggler magnets.II. XE = T0/TE.124) provided that all fields acting on the particle are predetermined and are not influenced by the motion of electrons. . Note that the damping time. Rev. for a fixed B-field.{P. Some typical damping times for electron storage rings are listed in Table 1. depend on radiation energy Uo per turn. insertion devices.) = ~ jjT0 °' Tz _ 2E _ AirRp _ IE ~ J.123) (4.3 Damping Rate Adjustment The damping partition and damping times are determined by the lattice design. The corresponding damping decrements are defined as K=T0/TX. Phys. Increase U to increase damping rate (damping wiggler) Phase-space damping rates.JXE3 MP-. which consist of strings of dipole magnets 18 K. longitudinal and transverse dampers powered by amplifiers sensing beam displacement.7. (4. = 1-2?. The corresponding damping time constants are _ Tx = 2E _ 4-irRp _ 2E CC. and the damping partition numbers . such as undulators and wigglers.JzE* ~~JjT00' 2E AnRp _ IE ~ JE(P7) ~ CCJJEE* ~ JETTO °' TE where To is the revolution period. is inversely proportional to the cubic power of energy and. apart from damping partition numbers.) ~ cC. I l l . for constant p. We discuss below some techniques for damping rate adjustment. Robinson. \Z=T0/TZ. induced current in rf cavity. wakefields. 373 (1958).125) The damping rate of an individual particle or a portion of a bunch can be modified if additional forces are introduced that depend on the details of particle motion. II. A. and electron and stochastic cooling devices. Some examples are image current on vacuum chamber wall. However. can be used to adjust beam characteristic parameters.4. is inversely proportional to the square of energy. satisfy the Robinson theorem18 52Ji = Jx + Jz + JE = 4 or JX + JE = 3 Jz = h JE = 2 + V 445 (4. RADIATION DAMPING AND EXCITATION where a0 = (P7}/2E.
Stability of the electron beam can be achieved only by having a positive damping partition number. PHYSICS OF ELECTRON STORAGE RINGS with alternate polarities excited so that the net deflection is zero. The growth time at 3.126) and the damping rate is enhanced by a w = —~ = a0 + awiggier(4. used combined function isomagnetic magnets.130) Since xco < 0.z% (4. K > 0 for a focusing quadrupole. Thus the energy oscillations are strongly damped (JE ~ 4) and the horizontal oscillations become anti-damped (Jx ss — 1). At the CERN PS. If the rf frequency is increased without changing the dipole field. The resulting energy loss per revolution becomes Uy. which is much shorter than the cycle time of 1. the 33 GeV AGS at BNL. (4.5 GeV as part of the LEP injection chain.129) The actual closed orbit can be expressed as x = xc0 + x$.128) where K = (\/Bp)(dBz/dx) is the focusing function. The potential for betatron motion in a quadrupole is V0 = \K(S) (X2 . p = 70 m).446 CHAPTER 4. etc. (4. and xp is the betatron coordinate. (l/Bz)(dBz/dx) < 0.1). (4.2 s. = Uo + C/wjggier. which can be facilitated by decreasing the orbit radius R.e.2. The potential for betatron motion becomes V0 = l-K{s) (xj -z2 + 2x0xco + x\o)..127) The damping time is shortened by a factor of (1 + f/wiggier/^o)"1B. the 28 GeV PS at CERN. This is similar to the effect of a . i. where xco < 0 is a new closed orbit relative to the center of a quadrupole. The reason for the change in damping partition due to orbit radius variation is as follows. the effective dipole field xC0K(s) in a quadrupole and the quadrupole field have opposite signs. and K < 0 for a defocussing quadrupole. and the change of radius ARis ±L = _ ^ = -aA.6 to 3. can be used to increase the radiation energy and thus enhance damping rate.5 GeV is about 76 ms {Jx « —1. the mean radius will move inward. horizontal emittance is an important issue. where V « 2 (see Exercise 4. Change V to repartition the partition number Many early synchrotrons. in facilitating the acceleration of e+/e~ from 0. such as the 8 GeV synchrotron (DESY) in Hamburg.
32) The CERN PS lattice is composed of Ncen = 50 nearly identical combined function FODO cells with a mean radius of R = 100 m.II. RADIATION DAMPING AND EXCITATION 447 Robinson wiggler. Hubner.117) we get the change in damping partition due to closed orbit variation (see Exercise 4. (4. where Bp is the momentum rigidity.101) or Eq. From K. with loss of useful aperture. Robinson wiggler Without a Robinson wiggler. (4. 4. 4. The resulting change of damping partition is (see also Exercise 4.8 shows Jx. and the fractional off-momentum shift <5S = —Af/acf. JE vs AR for the CERN PS.8).118) as A ( T ) =^?/^2JD2^S' (-3) 411 where we have used xco = D5S.8: The variation of the damping partition number of the CERN PS with the strength of the Robinson wiggler.2.117) gives the additional change of betatron amplitude in Eq. 7T / \Bp) TT2R (4.133) Figure 4.9) f={2JK^ds)[f^S]~l. C. it is preferable to change the damping partition number by using the Robinson wiggler. AV = ^<fD(^P\2dsARK^fAR. (4. p. (4. and this limits the dynamical aperture of circulating beams. a fairly large change in AR is needed to attain Jx = 1. If the gradient and dipole field of each magnet satisfy Kp < 0. The effective dipole field arising from the closed orbit in a quadrupole is given by BpK {xp + xco). changing the damping partition requires a large shift of the mean orbiting radius (Fig. as shown in Fig. Thus. CERN 85-19. Without the Robinson wiggler. which consists of gradient dipoles.9. discussed below. Using Eq. Substituting the contribution of quantum excitation from the quadrupole into Eq.2. (4. the damping . 226 (1985). Figure 4. The combined effect is that the damping partition V will get a negative contribution from these quadrupoles.9).
<0 partition of Eq. and Lw are respectively the bending radius. Bw. The change of damping partition is AV = m^^l(l + ^lY\ 2irplJ (4.101) can be made negative. The resulting line density of beam bunches is likewise reduced to prevent collective instabilities. A string of four identical magnet blocks having zero net dipole and quadrupole fields will not produce global orbit and tune distortion in the machine. and the length of each wiggler.448 CHAPTER 4. which enhances damping of horizontal emittance and reduces damping in energy oscillation. The emission time is short and thus the synchrotron radiation can be considered as instantaneous. the wiggler field strength. When a photon is emitted. its derivative. PHYSICS OF ELECTRON STORAGE RINGS Figure 4. where gradient dipoles with B^ are used to change the damping partition number. the wiggler contributes a negative term to the damping partition of Eq. dB^/dx. p is the radius of curvature.4 Radiation Excitation and Equilibrium Energy Spread Electromagnetic radiation is emitted in quanta of discrete energy. Since this time is very short compared with the revolution period and the periods of synchrotron and betatron oscillations. and p is the bending radius of ring magnets and (D) the average dispersion function in wiggler locations. (4. In a semi-classical picture.9: Schematic drawing of a Robinson wiggler.101). II. The Robinson wiggler has been successfully employed in the CERN PS to produce Jx « 3. This can be verified as follows. (4. Since these magnets have Kwpw < 0. quantum emission can be considered instantaneous.34) ' Bw dx 2npl \ where p w . the time during which a quantum is emitted is about P® c P C7 6 B[Tesla] 12 where 7 is the relativistic Lorentz factor. and B is the magnetic flux density. . the electron energy makes a small discontinuous jump.
t0). . the probable change in amplitude will be 5A2 = (A2 . where p = (n) is the average rate per second. <-5"> = -W Z = Vtt' (4139) 19The probability of an electron emitting n photons per second is given by a Poisson distribution f(n) = p"e~p/ni.I I RADIATION DAMPING AND EXCITATION 449 Another important feature of synchrotron radiation is that emission times of individual quanta are statistically independent. . is AE = Aoejul!!{t-to\ (4. The amplitude of oscillation will grow until the rates of quantum excitation and radiation damping are on the average balanced.. (t > h). Now if the energy is suddenly decreased by an amount u at instant t\ via quantum emission. Since the time t\ is unpredictable. Since the energy of each photon [keV] is a very small fraction of electron energy.dA2. The cumulative effect of many such small disturbances introduces diffusion similar to random noise. Effects of quantum excitation When a quantum of energy tiu is emitted.138) . expressed in complex representation. the emission of successive quanta is a purely random process. P{n) = (l/V3ip>-("-") 2 /2p. (4. In the limit of large p.136) where (. The impulse disturbance sets up a small energy oscillation. A.137) The quantum emission has changed the amplitude of synchrotron oscillation. i.e. In the absence of any disturbance and damping. d(A2) . (4.135) where ^o is the amplitude of synchrotron motion. which satisfies Poisson distribution. The variance of Poisson distribution a2 is equal to p.2A0ucosuj%{tl .)t stands for time average. the energy of the electron is suddenly decreased by an amount hui. the energy deviation AE from the synchronous energy. The growth is limited by damping. The cumulative effect of many such random disturbances causes energy oscillation to grow (as in a random walk). the amplitude growth rate becomes . the energy oscillation of the particle becomes AE = Aoeju"{t-to) where A2 = A2 + u2.ue^1'^ = A^^-^. . The damping process depends only on the average rate of energy loss.A2)t = u2. Poisson distribution approaches Gaussian distribution.19 Discontinuous quantized photon emission disturbs electron orbits. Qualitatively. where JV" is the rate of photon emission. (4. whereas the quantum excitation fluctuates about its average rate. and LJS is the synchrotron frequency.
the damping time of A2 is TE/2. the radiated power is P" deseed orbit = ^ ^ = ( V ^ V - ( 4 M 5 ) 20For an order of magnitude estimation. PHYSICS OF ELECTRON STORAGE RINGS B.142) This shows that the amplitude growth rate depends on mean energy loss (u2) of electrons.450 CHAPTER 4. and is proportional to 7 2 . and TE « E/Py to obtain an rms energy oscillation amplitude of UE O \/Efujc ~ 7 2 . (4./tu^c. The equation for the synchrotron amplitude thus becomes ^>=-2<^+A^.141) To attain a better calculation on the equilibrium beam momentum spread. We define the mean square energy fluctuation rate GE as GE = Win2}}. = ^ fN{u2)ds. it is reasonable to average the excitation rate by averaging M{u2) over one revolution around the accelerator. Let n(u)du be the photon density at energy between u and u + du.144) On the design orbit. Equilibrium rms energy spread Since damping time of the amplitude A is TE = I/as. The mean square equilibrium energy width becomes O\ = \GETE.140) where the stationary state solution is (A2) = | JVU2TE . at TE (4. (4. we use u RJ hu)c. which depends on electron energy E and local radius of curvature p. A qualitative estimation of the rms beam energy spread for sinusoidal energy oscillation is20 a B2 = ^ = j ^ « V (4.143) where the subscript s indicates an average over the ring. M a P-. (4. The amplitude growth rate due to quantum fluctuation becomes ^1LZ=/ dt Jo u2n{u)du = N(u2). The energy fluctuation is roughly the C geometric mean of electron energy and critical photon energy. the quantum fluctuation should be obtained from the sum of the entire frequency spectrum because the photon spectrum of synchrotron radiation is continuous. and damping time TE and synchrotron period l/ws are much longer than revolution period To. as shown in Eq. .94). (4. JV = Jo n{u)du. Since the radius of curvature may vary widely along the ring.
J \pw\2 + ^)(l +:|i)~\ (4.147) where Cu = 55/24^3. such as undulators and wigglers. RADIATION DAMPING AND EXCITATION Equation (4. . can change the rms energy spread of Eq. the bunch length is shorter with higher rf voltage. GE = and 3-Cuhcj3-^-}(l/p3) The fractional energy spread is then -i-Sw= (4148) (4I49) ' <f>° = ^ ( 1 / A J5 n_ = x io_13 m where c » 9 4mc 32\/3wc For an isomagnetic ring.Using Eq. |p w | < p.152) are radiation integrals for ring dipoles and wigglers respectively. the bunch length is also affected by wakefields [3]. ^ F \±/ h> / (4.144). Adjustment of rms momentum spread Insertion devices. C. the rms energy spread will normally be increased by insertion devices.83) then gives 451 N{u>) designed orbit = tchclt+jjL. we obtain (f)2 = i ^ ! i r c ^ h. yJJEp[m\ ^4-151) Note that the energy spread is independent of the rf voltage. and the resulting phase-space area is smaller.143) and TE = 2E/JE(P1). J |pw|3 I2w = / -. i.146) we obtain (4.—r^ds. Two competing effects determine the equilibrium energy spread. Because the magnetic field of insertion devices is usually larger than that of ring dipoles. (4.II. the resulting equilibrium energy spread becomes 4 = 4o(l where ^3 = / r-^ds. which will increase quantum fluctuation GESince the damping time is also shortened. Insertion devices increase radiation power. J \p\3 I2 = / r-rrds. In many electron storage rings. (4. For a bunch with a given momentum spread. J |p|2 I3w = / -—prds. 4&VSJEE or JEP f-(°-62xl0"6)7^rit.e. with lower rf voltage phase-space area is larger.
1 (4. 2ir/u>s. Beam distribution function in momentum The energy deviation AE at any instant t is a result of contributions from the emission of quanta at an earlier time £. (4. and \J2-KOE V27T(TE The bunch length in time is Or = -^-VE.155) the corresponding longitudinal time component relative to a synchronous particle [see Eq.. which are positive and negative with equal probability. Normally the damping time is much longer than the synchrotron period.6 = Eu>&r/ac). For a particle executing synchrotron motion with AE(t) = Acos(ust . and the distribution function becomes g{A) = N^e'A2l2^ °E = N24e-All°i. Since the normalized phase-space ellipse is a circle.U)]. the sum at any time t consists of a large number of individual terms. The central limit theorem (see Appendix A) implies that the distribution function of energy amplitude is Gaussian: 9{AE) = ^ = ^ e .158) which depends on the rf voltage. (4.153) where Ui is the energy of a quantum emitted at time U.87)] is T=^sm(ujst-x).159) . We define the invariant amplitude A2 = AE2 + 92.X). The normalized phase-space coordinates are (AE. (4.{t . and £*'s are randomly distributed. °A (4. (4. the Gaussian distribution of a beam bunch is tf (A£. (4. We can write AE{t) = £ Uie-aE{t-u) cos [u. 6 ) = NB^(AE)^(0). \J2-KOE (4. and x is an arbitrary phase factor.154) where OE is the rms standard deviation. Since the typical value of AE(t) is much larger than the energy of each photon. PHYSICS OF ELECTRON STORAGE RINGS D./VB is the number of particles in a bunch.156) where ac is the momentum compaction factor.452 CHAPTER 4.A £ 2 / 2 ^ .157) where .
i. The rate of change of betatron amplitude (emittance) is obtained by replacing u2 with M(u2) and averaging over the accelerator. i. Using the variable W = A2 with dW = 2AdA. RADIATION DAMPING AND EXCITATION 453 where a\ = {A2} = 2a2E. it is not invariant in regions with dipoles. (4." 7 W . U = j ^ [D2 + (PXD' .f z W . we obtain OX Tx The equilibrium rms width becomes (a2) = -TxGx 21The and o*xfil = ^ & (a2).+ A ^ + {PxD' .e.fD)'] (J)». we get the probability distribution function as h(W) = tf J L e . (4. the resulting amplitude growth becomes 8(a2) = %{^-f. (4.112). Sx'g = -D' {u/Eo).161) The resulting change in the Courant-Snyder invariant is 6* = I [DH + {pxD> . Adding the damping term of Eq. Averaging betatron coordinates xp. the %-function is invariant.x'p. 5 Radial Bunch Width and Distribution Function Emission of discrete quanta in synchrotron radiation also excites random betatron motion.165) emittance growth in a transport line is de/dt = -^ $* N(u2)Hds.162) where the ^-function depends on the lattice design. with (W) = 2<r|. In an accelerator straight section.II. .\p'xDf] . (4.21 where (•••)„ stands for an average over a complete revolution. where there is no dipole.160) II. The emission of a quantum of energy u results in a change of betatron coordinates.e. Sxp = -D {u/E0). (4. where /3X and f}'x are the horizontal betatron amplitude function and its derivative with respect to longitudinal coordinate s.f x)] | .
unn r i 3(P.e. (4. The emittance of Eq. PHYSICS OF ELECTRON STORAGE RINGS Using Eq. Since the betatron and synchrotron frequencies differ substantially. Comparing with the energy width for the isomagnetic ring. Gaussian quadrature can be applied to obtain o-x = 4 » . where 6 is the dipole angle of a half cell. the natural emittance of an electron storage ring is proportional to j263. i.g is the average "H-function in dipoles. the result can be simplified to ex—— -Uq Px — -Jxy . where Cq = 3.7 i r — e x p j .168) Since the H-function is proportional to LO2 ~ pO3. + * « (4-172) .166) is also called the natural emittance. (4. the normalized natural emittance of an electron storage ring increases with energy.' « ^ ' ' ' } . we find & ~ Jx \ E ) • ( ' The horizontal distribution function The distribution function for particles experiencing uncorrelated random forces with zero average in a simple harmonic potential well is Gaussian: *(*/0 = . For isomagnetic storage rings.Lb7) where (H)ma.)(n/\p\3) E2{1/p2) 3Cqcro75(-H/\p3\) 3(p2)(1/p2) . The normalized emittance is proportional to 7303.)- ' V2naxffx np {-'» + ' ^ .454 CHAPTER 4. (4.83 x 1CT13 m is given by Eq. (4.150). we obtain _ 3 r f ox--. (4.83) for M{u2). Unless the orbital angle of each dipole is inversely proportional to 7.m> Now the total radial spread has contributions from both betatron and energy oscillations. (H)mag = ^~f Z7T/9 Jdipole nds. the distribution in phase-space coordinates follows the Courant-Snyder invariant ^•^. { 2axfjx ) <.^ } - (4-17°) Since the betatron oscillation period is much shorter than the damping time.
When the electron emits a photon at a nonzero angle with respect to its direction of motion. Recalling that obtain = GE.179) . z axes respectively. Let ex and ez be the horizontal and vertical emittances with e* + e z =enat. r7 a*= % II. Emittance in the presence of linear coupling Sometimes it is desirable to introduce intentional horizontal and vertical betatron coupling. we obtain o\ = TZGZ/3Z ^ rEGE ~ Jrf&or ezHCq(fi)m/P. Including both damping and quantum fluctuation. Thus the vertical oscillation is energy independent and is less than the radial oscillation by a factor of I/7 2 . Emission of a single photon with energy u gives rise to an average change of invariant betatron emittance 8{a2z) = (u/E0)292zfiz. The transverse angular kicks on phase-space coordinates become 5x = 0.— + ~17\ • (4'173) Vertical Beam W i d t h Synchrotron radiation is emitted in the forward direction within a cone of angular width I/7. we o\ Using Jz = 1. we neglect it. The vertical beam size is damped almost to zero. We consider only the effect of random kick on vertical betatron motion. Consider the emission of a photon with momentum u/c at angle 91 from the electron direction of motion. •&0 Sz = 0. -fro (4.6 2 Px(s){n)™s . The transverse kick is then equal to 97u/c. we obtain 455 . (4. Sz' = ^9Z.II. RADIATION DAMPING AND EXCITATION For an isomagnetic ring. 6x' = ±9X. D2(s)] —J. 9Z are projections of 9y onto x.161). where we expect 97 < 1/7. JE{Pl) ^U'] (4-178) °l*CM/p which is very small. it experiences a small transverse impulse. the equilibrium beam width is 0% = -ATZGZ$Z. When the coupling is introduced.174) where 9X. (4.175) (N(v?9ppz)s _ (N(u2){9l)pz)s G* = (Af{u2))(pz) * 7 2£2 & ^ ^ > (JV(M 2 )) (4-176) where we have used the fact that (Of) ~ I/7 2 . the quantum excitation is shared up to an equal division. Since Sx' is small compared with that of Eq. (4.
PHYSICS OF ELECTRON STORAGE RINGS where the natural emittance enat is Eq.166). the Gaussian distribution. Here (5) is the spin polarization. Since the aperture of an accelerator is limited by accelerator components such as vacuum chambers. injection or extraction kickers. etc. K £z — Z ! 1+ K ^natj (4. which has an infinitely long tail. .180) where the coupling coefficient K is (see Exercise 4.8). beam position monitors.7.74). A. can produce a radial displacement as large as the aperture. 11 = J D/pds ac = II/2TTR Uo = 1. 11. (4. Table 4.7 Radiation Integrals To summarize the properties of electron beams. V = hll2 6nRax = recy3{I2-h). Quantum lifetime Even when the aperture is large.8 Beam Lifetime We have used a Gaussian distribution function for the electron beam distribution function. QirRaE = recj3{2I2 + h) ex = Crfh/(I2-h) 11. The horizontal and vertical emittances can be redistributed with appropriate linear betatron coupling £x = Z . is only an ideal representation when the aperture is much larger than the rms width of the beam so that particle loss is small. electrons. we list radiation integrals in the left column of Table II. 1 1+K Enat. JE = 2 + Ii/I2. and the corresponding physical quantities in the right column.404 E4 [GeV] 72 [GeV] 12 = J i/p^ds h = I l/\p\3ds ha = I 1/P3ds I4 = f(D/p)(l/p2 + 2K)ds K=(l/Bp)(dBz/dx) h = fH/\p\3dS (aE/Ef = Cg72I3/(2I2 + 74) (S) = PsTha/h Jx = l-h/I2.3: Radiation integrals and their effects on properties of electrons.456 CHAPTER 4. and PST = —8/5-\/3 is the Sokolov-Ternov radiative polarization limit. which suffer sufficient energy fluctuation through quantum emission.2.. 6nRaz = recj3{I2 .
it is most likely to return to the main body of the distribution because of faster damping at large amplitude.m. The loss rate becomes N dt rq K ' where rq is the quantum lifetime. Rev. We discuss quantum lifetime for radial and longitudinal motion below.II. To estimate beam lifetime. i. we set up a diffusion equation for h(W). f=M. an equal flux of electron passes inward and outward through Wo. Bai et al.e. See M. Phys.187) 22 Note that the formula is valid only in a weakly damping system. is small. 3493 (1997). RADIATION DAMPING AND EXCITATION 457 If the chance of an electron being lost at the aperture limit. • * = — • ( 4 1 8 4 ) The flux inward through Wo due to damping is = 2NWoh(Wo) W at w0 TX In a stationary state. Thus the quantum lifetime is 22 r rq = f^.e. TX(W) (4. with Wo > (W). so that the probability for the electron to have W > Wo is small.186) where Wo has been replaced by W. We assume an equilibrium distribution without aperture limit and consider an electron at amplitude Wo. within its damping time. then the loss probability per unit time is the same for all electrons.. W (4. 1 ^ =-I. .183) h(W) = ~e-w^w\ where W = a2.181) Radial oscillation We consider radial betatron oscillation x = acoswpt. w at = ™wm TX = Nw_e-*. dW 2W . Once the electron gets into the tail region (W > Wo) of the distribution. (4. (4.182) Quantum excitation and radiation damping produce an equilibrium distribution given by {W) = 2al. E 5 5 . (4. i.
0S) and (0. the rms beam velocity spreads in the beam moving frame satisfy the characteristic property (W/2 » <(4)2)1/2 * <(ApbM.^ [ c o s <j> .)2>1/2. (j> = her. (0. The Hamiltonian of synchrotron motion is H ( 5 .0.190) (4.t = Ha/(H)).^T-JE— ooirna uc \~ cos & + (£ . (4-194) where xp and z@ are betatron coordinates. <j>) = -huac62 + ^ .193) Thus the momentum deviation in the rest frame of the beam is reduced by the relativistic factor 7. (4.c o s 0 s + (^-0s)sin^]. the aperture is limited by rf voltage and bucket area. From Eq.188) where 5 = Ap/p = AE/E. Because of synchrotron radiation damping and quantum fluctuation in the horizontal plane. the deviation of the momentum Apb of a particle from that of the synchronous particle.187). (4. If the nonlinear term in the momentum compaction factor is negligible and the synchrotron tune differs substantially from zero. the Hamiltonian is invariant. and h is the harmonic number. (4.7r-^) = ^ ^ [ .cj>s) sin & ] .192) B.458 CHAPTER 4. L 2 J (4. T The Hamiltonian has two fixed points.cos </>B + (<j>.189) The stable rf phase angle (f>s is determined by the energy loss due to synchrotron radiation with eVosin^s = Uo = C7E4/p. is related to the momentum deviation in the laboratory frame Ap by Aph = Ap/-y. PHYSICS OF ELECTRON STORAGE RINGS Synchrotron oscillations For synchrotron motion. and £ . (4. z'p = dzp/ds are the slopes of the horizontal and vertical betatron oscillations.191) where the average value of the Hamiltonian is (H) = hojoac{aE/E)2. The value of the Hamiltonian at the separatrix is Hsx = J ff(O.) sin J . which has zero momentum. T — <j>8). the quantum lifetime is T = * 1P (. Touschek lifetime In the beam moving frame. and p0 is the momentum of a . (4. x'p — dxp/ds.
10. 407 (1963).195) v = 2px/m.10.init = (-px. ^_Ji_M dn~ (v/cyUnH 3_i sin20j' . 4. In the spherical coordinate system.23 The Touschek effect has been found to be important in many low emittance synchrotron radiation facilities. The velocity difference between two particles in the CMS is (4. . Rev. Particle loss resulting from large angle Coulomb scattering gives rise to the Touschek lifetime. Since the transverse radial momentum component of the orbiting particle is much larger than the transverse vertical and longitudinal components. as shown in Fig.. and z base vectors. This process was first pointed out by Touschek et al. which becomes a limiting factor for high brightness electron storage rings. where the momenta are expressed in the x. s.0).10. With the geometry shown in Fig. We consider the Coulomb scattering of two particles in their center of mass system (CMS) with momentum pi.197) Bernardini et at.scatt = (px sin x cos (p. we assume that the initial particle momenta of scattering particles are only in the horizontal direction. and z as orthonormal curvilinear coordinate system. px cos x. which transfers horizontal momentum into longitudinal momentum in the center of mass frame of scattering particles. Let x De the angle between the momentum Pi. and let (p be the angle between the i-axis and the projection of the momentum of the scattered particle onto the x-z plane. Since the transverse horizontal momentum spread of the beam is much larger than the momentum spread of the beam in the longitudinal plane. RADIATION DAMPING AND EXCITATION 459 synchronous particle.10: The schematic geometry of Touschek scattering. the momentum of a scattered particle is Pi. s. 0. (4. 23C. 4. Lett.II. px sin x sin ip). 0.scatt of a scattered particle and the s-axis. 4196) ( ' where r^ is the classical electron radius. Phys. large angle Coulomb scattering can transfer the radial momentum to the longitudinal plane and cause beam loss. in the Frascati e+e" storage ring (AdA). the differential cross-section is given by the Moller formula. We use x.init = (px. Figure 4.0) and p2.
201) (4. and the Touschek loss rate becomes24 dN N2 1 —r. we get the integrals of the vertical and longitudinal planes as (2V^7TCTZ)~1 and (2y/7rcrs)~1 respectively. N is the total number of particles in the bunch. i.198) and the momentum transfer to the longitudinal plane in the CMS is APcms=px\cosx\(4. (4. Thus the total cross-section leading to particle loss in the CMS is crT = ~ / . x'2)5(x1 . (4.= 2 — dt 7 vK(jzas J r / va p(xj. the Touschek loss rate becomes where the factor I/7 2 takes into account the Lorentz transformation of aTv from the CMS to the laboratory frame. the vertical and longitudinal planes can be integrated easily.The scattering angle 9 is related to x a n d <fi by cos 6 = sin x cos ip. and v = dx/dt. |cosx|>— (4.scatt.l + m^l. (4. In the laboratory frame.460 CHAPTER 4. where dV is the volume element.e. and the factor 2 indicates that two particles are lost in each Touschek scattering.202) where n is the density of the beam bunch. da y27r Sm X X J\cosx\>£p/lPx Arl rcos-1 (Ap/jpx) I" V 4 3 1 (v/c)i Jo Jo [(l-sin 2 xcos 2 (/j) 2 ~ 1 . . The number of particles lost by Touschek scattering in the CMS becomes dN = 2aTNndx. PHYSICS OF ELECTRON STORAGE RINGS where the momentum of the other scattered particle is — pi.199) Now we assume that the scattered particles will be lost if the scattered longitudinal momentum is larger than the momentum aperture.x2)dxidx'1dx2dx'2. Thus the loss rate in the CMS is dN/dt = 2 / aTvn2dV.200) where Ap = {2vs/hac)Y(<ps) is the rf bucket height.s i n 2 x c o s 2 ^ J = T ^ f ^ . ndx is the target thickness. Since Touschek scattering takes place only in the horizontal plane.204) 24 Using the Gaussian distribution.x'1)p(x2.
and + £>(£) = >/£/0°° ( ( ^ j 5 [u + \tlnt \t ln^w + $} e " ? ) e""du- (4207 ) The Touschek loss rate is inversely proportional to the 3D volume axazas. RADIATION DAMPING AND EXCITATION 461 where az and as are respectively the rms bunch height and bunch length. p is the bending radius. (4. Figure 4. and the function 6(xi .7 r C. and 6 is the orbital bending angle in one half period. [see Eq. Using the Gaussian density function P{X>X>) = 2Sf 6XP ["2^ (^ + {PxX' " f " )2 )] ' ldN_ Ndt~ Nrlc HrncV W*axv.204) to obtain ' where £ = (Ap/ja^)2 = (pxAp/j2mcax)2. (4. the parameter £ is ^W^WJE (4208) where J7L and . ax = With typical parameters Ap w ap.<x. ap/p = JCqj/^/JEp s/(5xex. (4. the betatron amplitude function can change appreciably.x2) indicates that the scattering process takes place in a short range between two particles.11.211) Sec. as shown in Fig.II. D(£) is a function varying slowly with the parameter £ in accelerator applications. and ex = .7 2 0 3 /^ [see Eq. In a low emittance storage ring. The actual Touschek scattering rate should be averaged over the entire . Ill]. 4.11: The Touschek integral D{0 ofEq. In this parameter region. { Ap ) m h [ (4205) we easily integrate the integral of Eq.7k are the damping partition numbers. aPx = jmccrx//3x.207). (4. T is the lattice dependent factor.149)]. Thus the typical £ parameter for Touschek scattering is about 10~3 to 1.
0 is the bending angle of a half FODO cell. p. i. See. (a) In thin-lens approximation. It can be affected by linear coupling. the rf voltage increases with energy with Vrf oc 7 2 . Exercise 4. and show that the damping partition number for the separate function FODO cell lattice is l-fsin 2 (c&/2) R62 sin 2 ($/2) p ' where R is the average radius of the ring. LeDufF. peak intensity. (c) Use the midpoint rule to evaluate the integral of the damping partition T>. Show that the damping partition number is V = 2 — (acR/p) for an isomagnetic combined function lattice. p is the bending radius of the dipole.e. The damping partition number of DBA lattices is independent of the betatron tunes. 25Actual calculation of Touschek lifetime should include the effect of the dispersion function. p is the bending radius of the dipoles. we can approximate (-D(f)) = 1/6 to obtain 48 7 W^/Ag\ 3 Nrlc \ p ) x ' The Touschek lifetime is a complicated function of machine parameters. i.2 1. etc.25 The beam current in many high brightness synchrotron radiation light sources is limited by the Touschek lifetime. and at a fixed energy the Touschek lifetime is proportional to Vrf because as oc VrJ1'2.g. show that the damping partition number for an isomagnetic combined function accelerator made of N FODO cells is given by g -o R( 2- V p \2Nsm{$/2)J ' where R is the mean radius of the accelerator. Touschek lifetime calculation is also available in MAD [19]. (b) Show that the damping partition for a separated function double bend achromat with sector dipoles (discussed in Exercise 2. CERN 89-01. PHYSICS OF ELECTRON STORAGE RINGS ring. 114 (1989).462 CHAPTER 4. .4. and 6 is the bending angle of a half DBA cell.. and $ is the phase advance per cell. and V = (acR/p) for an isomagnetic separate function lattice with sector magnets. If we choose Ap « Wap. J. rf parameters.16) is / sin(^/iV)\ ^ 62 V (*/N) ) ~ J' where N is the number of DBA cells for the entire lattice.e. and $ is the phase advance per cell. e. we obtain rT oc 7 6 . -T=\N^)s 1 / 1 dN\ = ^fN-dTds- 1 / 1 dNJ (4 ' 209) Since D(£) a a slowly varying function.
where the radiation integrals are ha. (a) For a separate function isomagnetic machine with sector dipoles. 6.2lf GerV ' For a SPEAR-like ring. (b) Show that the contribution from the edge angles of a non-sector type magnet to the integral 7 4b is26 tan<$i tanc52 where 5\ and 62 are entrance and exit angles of the beam. NS-20. and M.063 MeV and F(ip' : 3685) = 0. D is the dispersion function.2 463 2. on Nucl. 26 R. find the energy spread at the J/tp and tp' energies. 5.149). we learn that the beam energy spread can reduce the effective reaction rate. and K = is the quadrupole gradient function. Sands. 900 . Verify Eq.. The rms beam energy spread is given by Eq. = f -gds. What is the constraint of the IR design such that the total center of mass energies for all electron-positron pairs are identical? Discuss possible difficulties. Discuss your result. From the previous problem.215 MeV. J p h= -jrfs.)lh. the production rate is reduced by a factor of T/OE4. IEEE Trans. Show that the vertical emittance resulting from residual vertical dispersion is given c z _ C .2(?WIP| 3 ) ~ ql J*WP2) ' where Hz = ±-[Dl + (pzD'z + azDz)2]. Lee. The damping partition number V for energy spread and natural emittance is given by V = (74a + 1hx. T(J/ip : 3100) = 0. D\ and £>2 are values of the dispersion function at the entrance and exit of the dipole with £>2 = (1 . P.EXERCISE 4. Sci. show that /4a = 2nacR/p2 and /4b = 0. Morton. For example. 3. (4.cos 9)p + Di cos d + [pD[ + Dx tan 5{j sinfl. Helm. where a c is the momentum compaction factor and R is the average radius of the synchrotron. The beam energy spread of a collider should be of the order of the width of the resonance in the energy region of interest. Note that. when the energy spread is large. J p2(l/Bp)dBz/dx Here p is the bending radius. J ps I4h = i K—ds.[MeV] = 1. Show that . (4.J.H. with p = 12 m and JE ~ 2. Now imagine that you want to design an interaction region (IR) such that the higher energy electrons will collide with lower energy positrons or vice versa.L.. Pz (1973).162) for the change of betatron amplitude in photon emission. M.
(a) The synchrotron radiation power is P = ^2^E2B2. ^ . ~C{e2 .118) show that the average rate of betatron amplitude diffusion per revolution is <£ = <¥4(E0 + sef ID \^ A vncti J y + B'X^+2B'B0 + p 2B.J J *. as shown in Eq. 7. where ax. (4.e 0 ).ax(ex . the horizontal action of each particle can interchange with its vertical action.^ + (ax + az + 2C)-^. and the magnetic field can be expanded as B = Bo+B'xco+B'xp. (4.ex) azez.+ [axaz + C{ax + az)]ex = ax{az + C)e0. show that the emittance can be expressed by Eq. Use the following model to find emittances of electron storage rings. the average radius and the beam energy are changed by AR/R = -Af/fo and Eo + 8e. Using Eq. Use the following steps to derive the expression for AV/AR. . eo is the natural emittance. The equation of motion for emittance of an electron storage ring near a linear coupling resonance is —j^ = -£• = -C(ex . while the total action of the particle is conserved.363). and C is the linear coupling constant. Z7T If the rf frequency is altered.464 CHAPTER 4. Make a realistic estimate of the magnitude of the vertical emittance arising from the residual vertical dispersion.e2) . Find the equilibrium emittances. Near a betatron coupling resonance. The damping partition T> can be decreased by moving the particle orbit inward. az are damping rates. PHYSICS OF ELECTRON STORAGE RINGS fiz and az are vertical betatron amplitude functions.+ {az + az + 2 C ) ^ + [axaz + C(ax + az))ez = axCe0. (a) Show that the equations of motion for horizontal and vertical emittances are . (2.180) where the K parameter is given by C 8. (b) For ax = az = a. and Dz and D'z are the residual vertical dispersion function and its derivative with respective to s.
Show also that the quadrature horizontal beam size of the electron beam is cr2 = 3pC?72/[n(3 .8 for the CERN PS. show that ASS ~ sin 2 ($/2) " TT^B Aii- and A P (d) Compare your estimation with that in Fig.2 (b) Show that the change in damping partition with respect to xc0 is 465 £-('/o"*)(/>r A P „ 8JVc2ell Axco ~ TT2E ' For an isomagnetic FODO cell combined function machine.2n)/(l .An)}. where p is the bending radius. In fact. .n.n) 3 / 2 .n). Show that (H) = p/(l . ^ = n/(l . and e^ = Cq^2/n^/l . 9.n).4.5) with focusing index 0 < n < 1. 4.n). J £ = (3 4n)/(l . where 1 A / _ 1 Afl ac /o a c il is the fractional momentum deviation from the momentum at frequency foShow that the variation of the damping partition with respect to Ss is For a FODO cell combined function lattice. V = (1 . (c) The above analysis assumes that xco = AR. Consider a weak focusing synchrotron (Exercise 2. xco = D5S.EXERCISE 4. show that where iVcen is the number of FODO cells.
72^3. . Because synchrotron light sources from electron storage rings are tunable.1% of bandwidth). and the %-function is given by Eq. See e. electronic processing. Jx « 1 is the damping partition number.. Wiedemann. H. (4. cell biology. the {'H) and the resulting natural emittance obey the scaling laws: {•H)/Jx = fuuicep03 and e. the synchrotron light fans out to an angle equal to the bending angle of the dipole magnet. etc. where 7 is the Lorentz factor. they have been widely applied in basic research areas such as atomic. it would be useful to understand the limit of achievable emittance in order to determine the optimal solution for a given lattice.258). The synchrotron light spectrum is continuous up to a critical frequency of 3c73 Beyond the critical photon frequency. Computer codes such as MAD [19] or SYNCH [20] can be used to optimize {%). Horizontally. molecular. chemistry. (2. Thus a small electron beam emittance is desirable for a high brightness synchrotron radiation storage ring. Low emittance Ring Design.466 CHAPTER 4.g.27 The amplitudes of the betatron and synchrotron oscillations are determined by the equilibrium between the quantum excitation due to the emission of photons and the damping due to the rf acceleration field used to compensate the energy loss of the synchrotron radiation. where the product of the solid angle and the spot size dSldS is proportional to the product of electron beam emittances exez. Since % ~ L62 = p93. The brilliance of a photon beam is defined as dtdfldS(d\/X) y ' in units of photons/(s-mm2-mrad2 0. the power of the synchrotron light decreases exponentially e1. microbiology. 390 in Ref. For an isomagnetic ring.— LJ/UJC). and solid state physics. the horizontal emittance reduces to ex = Cqj2{H)dipole/JxP. condensed matter. The horizontal (natural) emittance is where Cq = 3.atticeC. However. (4.167) The objective of low emittance optics is to minimize {H) in dipoles. [14] (1988).211) 27Many review articles have been published on this subject. PHYSICS OF ELECTRON STORAGE RINGS III Emittance in Electron Storage Rings The synchrotron light emitted from a dipole spans vertically an rms angle of 1/7 around the beam trajectory at the point of emission.83 x 10 13 m. p. = ^. p is the bending radius.
III. three-bend achromat. Morton. In this section. and each application has its special design characteristics. Sommer. Figure 4. Internal report DCI/NI/20/81 (1981). The low emittance point of a FODO cell lattice at 8 GeV corresponds to that of a PEP low emittance lattice [G. Lee and P.212) depends essentially only on the lattice design factor ^lattice for electron storage rings at constant 7$. the lattice of a high energy collider is usually composed of arcs with many FODO cells and low 0 insertions for high energy particle detectors. four-bend (QBA). Popular arrangements include the double-bend achromat. D.J. etc.12: Normalized emittances of some electron storage rings plotted vs the designed beam energies. and 6 is the total dipole bending angle in a bend-section. 461 (1987)].28 Sci. For example. Figure 4. IEEE PAC. three-bend (TBA). Electron storage rings have many different applications. lattices for synchrotron radiation sources are usually arranged such that many insertion devices can be installed to enhance coherent radiation while attaining minimum emittance for the beam. M. we review the basic beam characteristics of these lattices.^) 3 (4.. . Note that the emittances tend to be larger for machines with FODO cell lattices than for those with DBA or TBA lattices.K. in Proc. On the other hand. etc.l Emittance of Synchrotron Radiation Lattices Storage ring lattices are designed to attain desirable electron beam properties. p. M. Possible lattice design includes FODO cells.L. The resulting normalized emittance 6n = jex = ^atticeC. Nucl. Internal 28G. IEEE Trans. NS-20. The function of arcs is to transport beams in a complete revolution. BNL-50522 (1976). III. the double-bend achromat (DBA) or Chasman-Green lattice. and n-bend achromats (nBA). Brown et al. Green.12 compiles normalized emittances of some electron storage rings. Potaux. H. 900 (1973). R. EMITTANCE IN ELECTRON STORAGE RINGS 467 where the scaling factor lattice depends on the design of the storage ring lattices. Helm.
Nucl. PEP-303 (1979). ' Z. 2. Y.29 A FODO cell is usually configured as { | Q F B Q D B | Q F } . Note that the dispersion invariant does not change appreciably within the FODO cell. Some high energy colliders have been converted into synchrotron light sources in parasitic operation mode. p. 33 (1980).3.1). $. the optical function is /3 ' = S 2L . Wiedemann. we can approximate {H) in the dipole by averaging %F and Ha (see also Exercise 4. Ropert. SLAC Tech Note. 1 ~ S „ m 2 ) $. Wiedemann. Here we discuss the characteristics of FODO cell lattices.[28]. D° = L9 . 30 R. Argonne Report (1985). Helm and H. ^(1 + Sm . the change of the dispersion functions of a FODO cell is shown in Fig. where Q F and QD are focusing and defocussing quadrupoles and B is a dipole magnet (see Chap. 2.. Kamiya and M. The %-function is therefore % is invariant outside the dipole region. as shown in the left plot of Fig. / is the focal length.30 ^ 1 . PHYSICS OF ELECTRON STORAGE RINGS A. 158 in Ref. 9 is the bending angle in a half cell. Kihara. Instr. A. In thin lens approximation. L. and $ is the phase advance per cell. 4. Methods 172. H. . $ 2f=Sm2' < J > and D' = L6 . 2 9 H. Since the dispersion invariant does not vary much from QF to QD.468 CHAPTER 4. Wiedemann. Teng. II). where two invariant circles of radii JH^ and J'HT at the defocussing and focusing quadrupole locations are joined at dipole locations to form a small loop.13. $. D/\/]3). ^1-2**2)> 1 where L is the half cell length of a FODO cell. Report ESRP-IRM-71/84 (1984). LS-17. Sec. FODO cell lattice FODO cells have been widely used as building blocks for high energy colliders and storage rings. 3 cos($/2) r(l + isin($/2)) 2 ~ 2 P sin 3 ($/2) [ (l + sin($/2)) (l-|sin($/2))2l (1 . 2)' o ^ =^ 2L . ri?f(1+28in2)' 1 . KEK 83-16 (1983).34. In normalized coordinates ((aD + PD')/y/P. The ratio 'HF/'HD is typically less than 1.sin($/2)) J " ' W + [ The coefficient T of the FODO cell becomes FODO _ l-fsin2($/2) " sin 3 ($/2)cos($/2) Jx • [ b) report DCI/NI/30/81 (1981).
III.14). The coefficient decreases rapidly with phase advance of the FODO cell. or a pair of doublets.13: Left: the ratio HT/HD. Wrulich.31 The resulting emittance of the FODO cell dominated lattice is (4.3 at <j> « 140°. Nonlinear magnetic fields can become critical in determining the dynamical aperture. undulators. chromatic sextupoles are 31A. The {O QF 0 } section may consist of a single focusing quadrupole. 22. Part. where we assume Jx = 1. and rf cavities. At this phase advance. Since the dispersion function is nonzero only in this section. The right plot of Fig.217) W o = ^PODOC^3B.13 shows the coefficient T as a function of phase advance per cell. 4. We note that the factor T has a minimum of about 1. or triplets with reflection symmetry for dispersion matching. right: T with Jx = 1 for a FODO cell lattice. The betatron function matching [00] section can be made of doublets or triplets for attaining optical properties suitable for insertion devices such as wigglers. . Double-bend achromat (Chasman-Green lattice) The simplest Chasman-Green lattice is made of two dipole magnets with a focusing quadrupole between them to form an achromatic cell (see Exercise 2. the chromaticity and the sextupole strength needed for chromaticity correction are large.5. Accel. 257 (1987). EMITTANCE IN ELECTRON STORAGE RINGS 469 Figure 4. A possible configuration is [00] B {O Q F 0 } B [00].
/30.2a o psin 0(1 . PHYSICS OF ELECTRON STORAGE RINGS also located in this section. (4. (4.| B + | c } . (4.40 sin 6 + 5 sin 20)/65.| F ) di + | A . B(6) = (6 .221) where L and 6 = L/p are the length and bending angle of dipole(s) in a half DBA cell. E-+1. E{9) = 2 ( 1 .14) H((t>) = Ho + 2(a0D0 + /3oD'o) sin c/> . and a 0 .^pe3B(9) + f/e'ae). D'=(l °-)sm<P + D'0cos<j) .3sin20)/(403). With the normalized scaling parameters do = §' d'0 = ^' P° = T' ^ = ^oL' "o = a o.470 CHAPTER 4. In general.220) where Ho = 7O-DQ + 2aoDoD'o + PQD'O2. B->1.cos (j>) + D o cos <t> + pD'o sin (f>.cos9)/92. we need Do = 0 and D'Q — 0 to attain the achromatic condition. The evolution of the "H-function in a dipole is (see Exercise 2.cos <j>). p is the bending radius. (4.8 cos 0 + 2 cos 20)/0\ C{0) = (306 . we get (H) = (aoDo + PoDo)e2E(6)-±(joDo +^e2A(e) + aoD'o)P62F(e) .2(j0D0 + a0D'0)p{l . F{6)=6{6-sme)/63. we find A->1. (4. and A{9) = (66 . and Do and D'o are respectively the values of the dispersion function and its derivative at s = 0. C-»l.223) .219) where <f> = s/p is the bend angle at a distance s from the entrance of the dipole.222) the averaged %-function becomes (H) = pe3\^od2 + 2aodod'o + hd'o2+(&aE-1^F^do + {PoE .218) (4. In the small angle limit.cos (j>)2 . Averaging the %-function in the dipole. For the Chasman-Green lattice. and 7Q are the Courant-Snyder parameters at s = 0.4. the dispersion function inside the dipole is D = p(l .cos 0) +Po sin2 4> + 7op2(l . F->1.
3. and the corresponding betatron amplitude functions are 6C . In zero gradient approximation.227) In thin lens approximation.224) With the identity /3o7o = (1 + &l). (4. decreases slightly with increasing 9 because of the horizontal focusing of the bending radius. a longer dipole magnet will give a smaller emittance. i. shown in Fig. (4. the minimum of (H) becomes (see Exercise 4. (H) = H(6)/4.15B2. and we obtain 1 (^)MEDBA = J y f f ^ Wlth 3 ^MEDBA = 3 SMEDBA = ^ ^ Q ^ ' g^" The corresponding minimum emittance is ^MEDBA = FMEDBACql2e3. since %(</>) ~ 4>3. The dispersion action %{&) outside the dipole is an important parameter in determining the aperture requirement. and the ^-function at the end of the dipole is (4.e. y/TEB .228) where FUEDBA = 1/(4^15 Jx). the horizontal betatron phase advance across a dipole for the MEDBA lattice is 156. In small angle approximation. i.e. 2. we get the average ^-function as <W)=pfl»[|A-|B + | c ] .7°.37. EMITTANCE IN ELECTRON STORAGE RINGS Bl. we find H — 0 at s = 0. For a minimum emittance (ME) DBA lattice. 8V5A Po = 7TEd' °° = — • 7o = W (4226) The factor G = y/16AC — 15B2.4) MMBDBA = •jjfiPP' (4225) where G = y/16AC . Minimum emittance DBA lattice 471 Applying the achromatic condition with d0 — d'0 = 0. the average of % is 1/4 of its maximum value. the dispersion "H-function for a MEDBA lattice at the end of a dipole is W) = -LPe>. and the phase advance in the dispersion .III.
The resulting emittance is smaller than the corresponding FODO cell lattice by a factor of 20 to 30.472 CHAPTER 4. PHYSICS OF ELECTRON STORAGE RINGS matching section is 122° (see Exercise 4. Since the phase advance is large. B2. Thus each MEDBA module will contribute about 1. The ELETTRA lattice has 12 superperiods. Examples of low emittance DBA lattices Many high brilliance synchrotron radiation light sources employ low emittance DBA lattice for the storage ring. The total phase advance of each ELETTRA DBAperiod is about 429°. which does not include the phase advance of the zero dispersion betatron function matching section for the insertion devices. and the low emittance DBA lattice of APS at Argonne for 7 GeV electron storage ring (right).9439 to increase damping partition number Jx (see Sec.4). III.38 £ MEDBA and — « 3.14 shows the lattice functions of a nearly minimum emittance DBA lattice ELETTRA at Trieste in Italy for 2 GeV electron storage ring (left). while the corresponding phase advance of APS DBA-period is about 319°. The ELETTRA lattice employs defocussing combined-function dipole with q = y |J5i|/Bp£dipoie = 0.l.14: The low emittance lattice functions for a superperiod of ELETTRA (left) and APS (right).64.2 unit to the horizontal betatron tune. £ MEDBA Figure 4. and the APS lattice has 40 superperiods. Figure 4.D).4°. the chromatic properties of lattices should be carefully corrected. The resulting horizontal emittances of these lattices are respectively « 1.3. . Thus the minimum phase advance for the MEDBA module is 435.
EMITTANCE IN ELECTRON STORAGE RINGS B3. The optical functions that minimize (71).229 ) where /?* is the value of the betatron amplitude function at the symmetry point of the dispersion free straight section.min = ^ .230) (H) = ^P0 3 (faA . Minimum ('H)-function lattice Without the achromat constraint. each module of an accelerator lattice has only one dipole. First. this minimum emittance condition can not be reached. . Triplet DBA lattice 473 A variant of the DBA lattice is the triplet DBA.37.e.232 ) where G = ylQAC . 2 (4. Because there is no quadrupole in the [00] section of the DBA. are symmetric with respect to the center of the dipole. C. Note also that G is nearly equal to G. C = -C . the minimization procedure for (H) can be achieved through the following steps.30). 32See Exercise 4. 2.6.221). (4. The minimum emittance is32 WPI« = ^jCrf9*' ( 4 .231) With the relation /?o7o = 1 + QQ> *he minimum emittance is (^)ME = J^P03' ( 4 . 6 and d'aMa = -\E.3E2. where we find that the stability condition is incompatible with the achromat condition. i.3. the lattice is very simple (see the lower plot of Fig. From Eq.a0B + ^ c ) with A = 4A. and I is the length of the dipole. 4 4 (4. B = W2EF. Therefore. (H) can be minimized by finding the optimal dispersion functions with ddQ ' dd'o ' where we obtain rfo.-F2.III. 2. where a quadrupole triplet is used in the dispersion matching section for the achromat condition.15B2 is also shown in Fig. the dispersion and betatron amplitude functions.
The required minimum betatron amplitude function is B* = -B* It is interesting that the minimum B value for a ME lattice is actually larger than that of a MEDBA lattice.230) and (4. we obtain H(0) = H(9) = ~^=p93 [WE2 . ao = —• V15B .235) In small bending angle approximation. The horizontal beam width is given by the quadrature of the betatron beam width and the momentum beam width. we find that the minimum {%) without achromatic constraint is smaller by a factor of 3 than that with the achromat condition.236) where S2 = {aE/E)2 = Cqj2/pJE is the equilibrium energy spread in the beam.e.3. 7o = ^ - 2VTEA (4233) The waist of the optimal betatron amplitude function for minimum ("H) is located at the center of the dipole. s*ME = Lji. It is appropriate to define the dispersion emittance as ed = lx{D5)2 .232). (4. and the dispersion matching section is 133.234) where J-ME = l/(12\/l5Ji). The brilliance of the photon beam from an undulator depends essentially on the electron beam width.B'X(D5)(D'5) + BX(D'6)2 = U(0)62. PHYSICS OF ELECTRON STORAGE RINGS From Eq.4° (see Exercise 4.233) for the ME condition. Using Eq.79. the corresponding minimum betatron amplitude function at the waist is 8^B = L/\/60. In small angle approximation with ^ C l . The values of the dispersion H-function on both sides of the dipole are important in determining the beam size in the straight sections. i. Each minimum emittance module with a single dipole would contribute a horizontal betatron tune of 0.474 CHAPTER 4. Thus the horizontal betatron tune of this minimum emittance single dipole module is 284.^-BEF + ^-AF2} G"1.5). 3vl5 I 2 2 J (4. To attain the minimum emittance. e^ is invariant in the . (4.4°. The minimum condition corresponds to = PO = 8C 7EG> . we have 7i(9) = j^pO3 = 4("H)ME. where insertion devices such as undulators are located. the betatron phase advance across the dipole is 151°. which is equal to |H(0)| MEDBA . Because the "H-function is invariant in the straight section. (4. where A -» 1. The attainable emittance is eME = ?u*Cql2e\ (4. and C -> 1. B ->• 1.
the divergence of the ME lattice is smaller than that of the MEDBA lattice. To simplify the design of a DBA in a synchrotron storage ring.33 D. A defocussing combined function dipole has the advantage of having minimum /3X inside the dipole.°^B(q) + ^ C ( 9 ) ] . the dispersion function is D(s) = .240) .7 ^ ( c o s h JlK^s p\Kx\ and {%) is (H) = p63 ^A(q) . J i « 1 or J"£ « 2 ^ . a0 = a 0 .40 sinh q + 5 sinh 2q -5 • 33 The brilliance of a photon beam is inversely proportional to the phase space areas of the electron beam in the straight section.III. we obtain 475 For a separated function lattice. /30 = /30/L. The dispersion function in a combined function dipole satisfies D" + KXD = . P where Kx = (1/p2) + B\/Bp < 0 is the effective defocussing strength function and p is the bending radius of the dipole. combined function dipoles have often been used.(4.1).239) where q = ^J\KX\L.236). (4. (4. 7o = loL.a S a m m photon brilliance by minimizing the betatron beam-emittance.5. Thus designing the lattice may be slightly easier. and thus the resulting photon brilliance is proportional t 0 l/V£MEeiot. (4.i> i-e. Although the total electron rms beam size in the straight section for a MEDBA lattice is the same as that of an equivalent ME lattice. EMITTANCE IN ELECTRON STORAGE RINGS straight section. e. The total effective emittance for a bi-Gaussian distribution becomes 1 Crffi tA?W\ The decrease in betatron beam size in minimum emittance lattice is accompanied by an equal amount of increment in the dispersion beam size. CW = _ 6 . . in the Elettra at Trieste and in the UVU and X-ray rings in NSLS (see Exercise 2.235) into Eq. JE « 2. . Substituting 7/(0) of Eq. Minimizing emittance in a combined function DBA We have discussed the minimum {H) only for sector dipoles.22).8 cosh q + 2 cosh 2q 30q . and 3(sinh 2q-2q) ..g. For a DBA.
PHYSICS OF ELECTRON STORAGE RINGS Figure 4. The TBA is a combination of DBA lattices with a single dipole cell at the center. the emittance can be reduced accordingly. R is the average radius. The emittance factor V16AC .476 CHAPTER 4. we use small angle approximation. a combined function DBA gives rise to a larger (Ti). etc. Three-bend achromat Now we are ready to discuss the minimum emittance for three-bend achromat (TBA) lattices. To simplify our discussion.15S2 for a combined function DBA lattice is shown as a function of quadrupole strength q = y/Kt. the Pohang Light Source (PLS). (4. However.15: The factor V16AC . we find y/16AC — 15B2 > 1. there is another factor that can change the emittance of the machine. the Taiwan Light Source (TLS).15 shows %/16/lC — 155 2 vs quadrupole strength. and p is the bending radius.e. Note that the combined function DBA can achieve lower emittance due to damping partition. Depending on the focusing strength. which is good provided that the bending angle for each dipole is less than 30°. The damping partition number for radial betatron motion is given by ~g-^. which have been used in synchrotron radiation sources such as the Advanced Light Source (ALS) at LBNL. i. S i n h g ^ = l+ 2 E.15B2/JX is also shown. The normalized dispersion coordinates for the minimum emittance DBA and minimum emittance single dipole lattices are given respectively by VP O Pi . Thus the minimum of (%) is _ V16AC-15B* {n)mtn 3 ~ WE p • Figure 4. As expected.241) 3 P where ac is the momentum compaction factor.
Thus we have proved a theorem stating that an isomagnetic TBA with equal length dipoles can not be matched to attain the advertised minimum emittance.235). 1940 (1996). where the normalized dispersion functions are transformed by coordinate rotation.76°. For an isomagnetic storage ring. The necessary condition for finite angle can be obtained by equating Eq. Optical matching between the MEDBA module and the ME single dipole module is accomplished with quadrupoles. l^. see S. and the matching condition of Eq. the center dipole for the TBA should be longer by a factor of 3 1//3 than the outer dipoles in order to achieve dispersion matching. (4. . (4. where p\ and Lx are the bending radius and length of the DBA dipoles.247) requires Li = 2>1I3L\ for isomagnetic storage rings. (4. or pi = \/3p2 for storage rings with equal length dipoles.e. we can prove the following trivial theorem: The emittance of the matched minimum TBA (QBA. {X**\-{ 1p / ^ rME ' V cos # sin $ W XMEDBA \ V-sin$ cos<3?MP /' b111 * LOb ^ / V -"MEDBA / . In this case. provided the middle dipole is longer by a factor of 3 1 / 3 than the outer dipoles. ' V*-™) at the dispersive ends of the dipoles in the MEDBA lattice. i. Phys. The formula for the attainable minimum emittance is identical to that for the MEDBA. Lee.Y.I l l EMITTANCE IN ELECTRON STORAGE RINGS 477 aD + PD' ^MEDBA^ " 7 8(15)V4 LT pi . E 54.227) and Eq. or nBA) lattice is = ^ A (4 248) 4V15JX' [ ' where 9\ is the bending angle of the outer dipoles.zwj where $ is the betatron phase advance. and . Rev. e METBA 34This necessary condition is valid in small angle approximation. The necessary condition for achieving dispersion phase space matching is34 §=4- (-4) 4 27 The phase advance is $ = 127. and pi and L2 are those of the ME dipoles. = D V2(15)V^f 3 T 4V2(15)V4 (4244) oD + 0D'= V^ Lf p2 ' [*-Mb) at the entrance and exit locations of the dipole in the ME lattice.
0E=<'Eo[1 ( PW. will not affect the dispersion function outside the wiggler.20).4.2 Insertion Devices A. I40. For insertion devices in zero dispersion regions. Sec. Damping wigglers have been successfully employed in LEP to enhance the momentum spread for Landau damping of coherent instability and.-20) : I ( Pyr 6) I S 3LW 4LW 0 L_ : • 2LW Figure 4.(2LW . the wiggler. /4W. 9 = 0 W = £ w / p w is the bending angle of each dipole. where /?w = p/eBy. Since the rectangular magnet wiggler is an achromat (see Exercise 2. I30.e.4.478 CHAPTER 4. B) I (-p w .r ) . /3 W . Lw < s < 2LW ' . PHYSICS OF ELECTRON STORAGE RINGS III. is the bending radius. located in a zero dispersion straight section.s) 2 ]/2p w ' V [S)~ I -[2L W . Effects of insertion devices on the emittances The rm beam emittance ex and the fractional energy spread (JE/E in electron storage rings can be expressed in terms of the radiation integrals (listed in Table II.s ] / P w . Depending on the radiation integrals. and L w is the length of each wiggler dipole.16: A schematic drawing of a section of a vertical field wiggler. II. a n d ^50 a r e radiation integrals of bending dipoles. However. this wiggler can generate it's own dispersion (see Exercise 2.251) \ 730 / V JI20 + ho / where I20. and 75w are radiation integrals of wigglers. to reduce the horizontal emittance. at the same time. i.7): Since insertion devices also contribute to the radiation integrals. (4. the emittance can be reduced while the momentum spread can be enhanced. Example 1: Ideal vertical field wiggler in zero dispersion sections We consider a simple ideal vertical field wiggler (Fig.20). the emittance and the energy spread can increase or decrease.7. 4. V J50 / V •'20 ~~ -"40 / (4-25°) + -j—l[l+OT . and 72w.16). U{S) ~ \ -[2L2W . the natural emittance and the energy spread of the beam can be expressed as *-= ^ ( 1 + 7=) (1 + 7=^7=)"'.
/PW The contribution of each wiggler period to I2 is Now. EMITTANCE IN ELECTRON STORAGE RINGS 479 Now we assume that the wiggler magnet is located in a region with ax = — \@'x ~ 0 with high jjx. (4. the ideal dipole field discussed in the last section does not exist. wigglers or undulators.z cos £ws. (4. in synchrotron radiation storage rings can greatly enhance the brilliance and wavelength of the radiation.e.e.254) where we have used J5o = 2TT('H)O/P2 and 720 = 2TT/P for the isomagnetic storage ring with /40 « 0 and 74w « 0. ~ 4X' z " 47 y 7 ~ 4.III.257) . < 4252) (4. Since we normally have fix S> p w . cosh ky. (4. i.w = fxo (1 + o 2P/o/\ ^ w & e t ) (1 + —iVwOw . we assume that there are Nw wiggler periods in an isomagnetic storage ring.255) This condition is usually satisfied.rp2 ' (4256) where LWitotai = 4ArwZ/w is the total length of the undulator. Bz = By. thus there is no net focusing in the horizontal plane. P2 Pv. Since all magnetic fields must satisfy Maxwell's equation. Example 2: Effects of undulators and wigglers with sinusoidal fields Insertion devices. Thus the condition for e xw < eI>0 is —p^P^l < I- (4.250) becomes e*. The focal length of the vertical betatron motion and the tune shift resulting from the rectangular wiggler dipole are respectively f ~ 4 ^ e .253) where each dipole contributes an equal amount to this radiation integral with 6 W = LY. and (/3Z) is the average betatron amplitude function in the wiggler region. and the radiation integral 75w is approximately *-5*f-^ /2w = 4 % = — 6 W . We consider a simple model of a nonlinear undulator with a planer vertical modulation field. the H-function can be approximated by 7i « f3xD'2. i. It is worth pointing out that the edge defocussing in the rectangular vertical field magnets cancels the dipole focusing gradient of 1/p2. adding wiggler magnets in regions where the dispersion function is zero will generally reduce the beam emittance. The emittance of Eq.
.z sin KWS.480 CHAPTER 4. or ^|| = w X /^^i=(^l-^-^sin 2 fcw S j / i K2 \l/2 !_lJ^l4^£ 27 2 = 1_i±M_^sin2. The velocity vector of an electron in the planar undulator is /? = /3j_x + P\\s. x'co = (3j_ = ——sinfc w s = —^sinfe w s. .l/j2. and the corresponding horizontal and longitudinal magnetic fields are Bx = 0 and Bs = —Bw sinh kwz sin k^s. (4.258) Thus the Hamiltonian of particle motion is H=-(px-r cosh Kzsin A:ws)2 + .e.rms = Kw/V^ is called the rms undulator parameter. • x. For Kw < 1. The quantity ii'w. the device is called an undulator. The corresponding vector potential is Ax =--~-Bw cosh k^z sin ky.w = Kw/fiy « Kw/y with the wiggler parameter defined as Kv = ^ ^ = 0.cosfc w s).w is the wiggler wave number.s.261) „ Pw "'w ^ w w ^ where we use l/pwA.p 2 . As = 0. (4. The transverse electron angular divergence inside the undulator or wiggler is equal to Kw/j. Az = 0.z cos kws.263) Note that the magnitude of the longitudinal velocity oscillates at two times the undulator wave number. it is called a wiggler. Pw 2fc w Pw T h e nonlinear magnetic field can b e neglected if t h e vertical b e t a t r o n motion is small with kwz <C 1. T h e horizontal closed orbit becomes 1 TC 1 xco = — — ( 1 . z H 5 — = — sinh ky.4 lists wiggler parameters of the some insertion devices for third generation light sources. (4-260) sin^/cwssmh2fcwz px . PHYSICS OF ELECTRON STORAGE RINGS where A. the wiggler period is Aw = 2ir/kw.259) where p w is the bending radius of the wiggler field. The equation of motion is x" = — cosh ky. .934 B^ [T] Aw [cm].wS.262) Table 4. (4. 27 2 472 (4. where 02=01 + tf\=l. (4. for ivTw ^> 1. i. P \ . i?wpw = vle with particle momentum p.
The nonlinear field in the wiggler can also affect the dynamical aperture.5 5 14 1. Thus the nonlinear wiggler is achromatic if kwLw is an integer multiple of 2n.4: parameters of some undulators and wigglers Machine 1 E [GeV] I B [T] I Aw [cm] 1 L [m] I K~ ALS 1.65 I 2.III.2 |5 | 1.5 1.267) . it also produces vertical ^-function modulation.2 Photon Factory 2.63 5. Since the vertical field undulator introduces a vertical quadrupole field error. The emittance and the energy spread become 4 = 4 o (1 + ^ ( ^ ) 3 ) (4.18 14 .5 5.2 481 APS I7 I 0.5 ESRF 6 1.25 TO 2 12~~ . where Lw is the wiggler length. For an off-momentum particle.63 2. EMITTANCE IN ELECTRON STORAGE RINGS Table 4.8 9.2 5.96 65 1.5 6 3. The betatron tunes should avoid all low order nonlinear resonances. Pvf "• D'= i-sin/c w s.45 6 3.3~ The vertical closed orbit is not affected by the vertical field undulator. the vertical field generates average vertical focusing strength and vertical betatron tune shift given by sin2fcws 1 _ 1 r j3z{s)ds _ (/?2)Lw.7 Elettra 2 5 30 3 140 1.5 10 3. However.5 6.totai where Lv is the total length of the undulator.15 9 4. Pw "* (4.264) where we have assumed Do — D'o = 0. the vertical field undulator also gives rise to a dispersion function in the insertion region: D= l— (l-cosifcws). The radiation integrals of the sinusoidal wiggler are 2 = lt' 3 = ^' /w /w '-"d^2(l + T ^ ( ^ ) 2 ) ~l • (25 46) where we have approximated ~H ss fixDa.
4. .n ~ 7?r mm-mrad at 6 GeV. .482 CHAPTER 4. The transverse equation of motion for electrons traveling at nearly the speed of light in the longitudinal direction inside the wiggler is dp n ymc— = Pcs x B.s) + p\\s. (4.= (z cos A. (4. (4. we need to include higher order nonlinear terms in the magnetic field. i.e. The n can be written as (see Fig. (4. to a normalized natural emittance of enat. .272) 35 In order for the ideal helical magnetic field to obey the Maxwell's equation. dt or -* By. the helical undulator does not produce a large tune shift in linear approximation. —. PHYSICS OF ELECTRON STORAGE RINGS Since the dispersion generated by an undulator is usually small. z) are unit vectors of the curvilinear coordinate system for the transverse radial.268) Aw Aw where (£. longitudinal.s + zsmky.17). T^l h = </>x + ipz+(l- ]-02)s (4.269): = ^-(xsmujy. we neglect all higher order terms in the following discussions.ZCOSLOJ) + put's. „ .t' . Note that the magnitude of the transverse velocity vector is /3j_ = -R"W/T with P* = ^ + ^ = 1-1. . (4.ws — x sin rews). J. = By.269) where the wiggler parameter Ky.271) c ww7 where t' is the reference frame of the moving electrons. «!_!+*£. as -ymc P = —-(xcosky. fl. {x cos dp eBy. at the expense of rms momentum spread. the emittance can be reduced by wigglers located in zero dispersion regions. .262). and transverse vertical directions. Example 3: Ideal helical undulators or wigglers We next consider a helical wiggler with magnetic field35 — • 2TT S 2TT S hisin——). Installation of damping wigglers in PEP had once been proposed to reduce the emittance by a factor of 10. is defined in Eq. The displacement vector of the electron in the wiggler is obtained by integrating Eq. Let the observer be located at one end of the wiggler.270) Unlike the planer undulator. For linear betatron motion. fC (4.
irw Thus wL corresponds to the characteristic frequency of the device in the observer's frame. The integrand of the radiation integral in Eq. with (j>2 4.17: Coherent addition of radiation from electrons in wigglers or undulators. (4. i.273) can be transformed to I + Kl + W ^ t . EMITTANCE IN ELECTRON STORAGE RINGS 483 Figure 4. (4.e.273) Let £ = wwt'. i. t t =t h-f(t') i + Kl + —^ = ^ c 2Y 12e\. (4. ^ w smw w i H cos uwt. ww7 ww7 (4. where these angles are of the order of -. The actual frequency should be obtained by solving Eq. (4. Eq. Longitudinal coherence gives rise to resonance condition of single frequency of diffraction like structure. When 4> and ip are not zero. The radiation integral of Eq.277) 6 = "' ( s ^ ) V 2 fl ([^ . (4. we expect to have higher harmonic in the spectrum.e.III. (4.I + Kl + W^+I + Kl + W"**- (4274) It is apparent to see that the periodic motion of the electron in the wiggler are transformed to the observer at a frequency boosted by the factor shown in Eq.ip2 = 92.276) for t' as a function of t. *>>• = I + K ( 27^w + 1 ^ - ( 4 .13) of the classical radiation formula is given by — K K n x (n x ft) = x[(j) cos wwt'] + z[<t> sin w w i']. The observer's time t is related to the electron's time t' via the retarded condition.274).3.1 C 0 S ^ + [4>~ Tsin^)e"^- .274) as wLt = £ — puiL sin f + quih cos ^.275 > We can rewrite Eq. Let us use the notation wL for the laser frequency.47) becomes (4. 1 <j>K» . with 27y.
Aw and Kw.282) Due to the coherent interference nature.J j w 5 q i (as)'"*•*""•*>Lk*-Tcosfl* +[(f> -sinf]£J x e x p j .L((?) = no.278) where JVW is the number of the wiggler period. \ L = £L(I + KI) (4.. PHYSICS OF ELECTRON STORAGE RINGS Now considering the periodic structure of the wiggler. we obtain s . Since a wiggler magnet may have a .279) and the spectral coherent factor S(u)/u>h) is sharply peaked at integer harmonics of 5(W/Wj Wn .e.« »£. the spectra are similar to those of synchrotron radiation from dipoles. the frequency spectrum is discrete.p s i n £ +gcos£) 1 df. (4'280) = na.[iV w sin^J ~ == [ ^ W ( W . or AiW- ^ p ^ • (4-281) Thus the photon energy can be adjusted by tunning the electron energy. The spectral distribution of the diffraction pattern has a full width half maximum at the n-th harmonic: ^ « -4. B.484 CHAPTER 4. The frequency spectrum will also be broadened by the momentum spread of the electron beam. Synchrotron radiation has a continuous spectrum up to the critical frequency wC]W = 373c/2pw.1(1+ ^)A w [cm] ei-M0)-(1 + ia)Aw[cm]. (4. Summary on characteristics of radiation from undulators and wigglers If the wiggler parameter is large.^ ) K j ' * . Kw > 1. the apparent angular frequency wL(0) at the forward direction is CJL(0)=LUL(9 = 0) = Y^CUW. i. The maximum power is proportional to N%. L (0) ( l + ^ ^ J The corresponding photon energy at the fundamental frequency is _0 1 95^[GeVL .j — ( £ . or the wiggler parameters. The photon flux is proportional to the number of electron due to incoherent nature. 13. (4.
.III. Figure 4.3 ^ 2 . during the time that the electron travels one undulator period. [A]. The emittance of the photon beam is equal to A/4TT. The angular aperture and the source radius of the radiation are (6 2 ) 1 / 2 = ^JX/NWXW and y/XN^X^/A-K.18: Schematic drawing of the sinusoidal orbit of an electron in a planar undulator and the electromagnetic wave (vertical bars).. Figure 4. the synchrotron radiation spectrum generated in a wiggler is shifted upward in frequency. The pulse length of a photon from a short electron bunch is A i = iVwAw_A^cos0%A^Ai /3|. The fractional bandwidth is then Au/uji = l/(2iVw). J = w = ' ^ (n = 1.18 shows schematically the sinusoidal electron orbit and electromagnetic radiation (vertical bars).B [ T ] ^ [m\ 18. .6 B [ T ) £2 [GeV] r .2. If Kw < 1. Such a wiggler is also called a wavelength shifter. where the electron (circle) lags behind the electromagnetic wave by one wave length in traversing one undulator period for n = l.e. (4-283) where 0 is the observation angle with respect to the undulator plane. The resonance condition for constructive interference is achieved when the path length difference between the photon and electron. EMITTANCE IN ELECTRON STORAGE RINGS 485 stronger magnetic field..c c c } Thus the frequency bandwidth is *»-s=]Sr=5b (4-285) where Ai is the wavelength of the fundamental radiation. is an integer multiple of the electromagnetic wavelength.1 Aw [cm] nE2 [GeV] ( 2 w 7 _2 2 . Optical resonance cavities have been used to enhance the radiation 3 6 The critical wavelength from a regular dipole is _ i-Kmc _ 0. _ Ac. the radiation from each undulator period can coherently add up to give rise to a series of spectral lines given by36 " l+i^+72e2 2^f 13.dipole . i. The resonance condition is achieved when the electron travels one undulator period. Aw//?|| .).A w c o s 0 = n\n. it lags behind the electromagnetic wave by one full wave length for the n = 1 mode.007135 f .
1708 in Ref. minimizing the natural emittance in an accelerator lattice and minimizing the vertical emittance by correcting the linear coupling will provide higher beam brightness. See. beam lifetime. III. Torre. efforts are being made in many laboratories to demonstrate the self-amplified spontaneous emission (SASE) principle. and to achieve single pass X-ray FELs such as the Linac Coherent Light Source (LCLS) at SLAC and DESY. emittance preservation in linacs. This is a subject of active research in thefieldof accelerator physics and technology.486 CHAPTER 4. If we neglect the effects of the residual vertical dispersion function. p.Thus.H. the vertical emittance is determined mainly by the residual vertical dispersion and the linear betatron coupling. G. and S. The beam brightness is proportional to NB/CXCZ.E. etc. longitudinal bunch compression. beam brightness limitation. CERN 89-03 (1989). where NB is number of electrons per bunch. The natural emittance of an electron storage ring obeys the scaling law £nat = TCql26\ where 9 is the total bending angle of dipoles in a half-cell.l. CERN 90-03 p. Giannessi. R. to produce an infrared FEL. l 2&?] I l/(4Vl5Jx) | 2/3'/{3LJx) [ (5 + 3 cos $)/[2(l . 254 (1990). Deacon et al. Pantell. Dopanfilis. the vertical emittance is arrived from the linear betatron coupling with ex + ez = enat.. 38. this field has been very active. and C l/(12\/l5X) T = (4.A. Madey's group [D. TBA or nBA . with many regular workshops and conferences. Since then. e. [12]. III. Low emittance lattices and the dynamical aperture In Sec. which can be attained by high quality linacs with high brightness rf-gun electron sources or by high brightness storage rings. G. Rev. and precise undulators. L. PHYSICS OF ELECTRON STORAGE RINGS called the free electron laser (FEL). beam intensity limitation. A.cos $) sin $ Jx] for ME triplet DBA for FODO cell lattice.g. Some of these issues are the emittance. dynamical aperture. At the same time. Dattoli. Torre.286) for ME lattice for MEDBA. we discuss only the physics issues relevant to high brightness electron storage rings.. A. ' ' 3 7 The free electron laser was realized in 1977 by J. Lett. Dattoli and A. 892 (1977)]. . Here. Phys.37 With the progress in small emittance beam sources from photocathode rf guns.3 Beam Physics of High Brightness Storage Rings High brilliance photon beams are generally produced by the synchrotron radiation of high brightness electron beams. we have studied methods of attaining a small natural emittance.
ground motion. the high brightness lights emitted from undulators in many synchrotron light sources can reach the diffraction limit. where k = 2TT/A is the wave number in the longitudinal direction. Thus DBA. L is the length of the dipole. To maximize beam brightness for synchrotron radiation with insertion devices. Figure 4. Thus we find axax> = crx{akx/k) > l/{2k) = A/(4TT). where ax and <Jkx are the rms beam width and the rms value of the conjugate wave number. the electron beam emittance that reaches the diffraction limit is ediff > ~(4.. The equality is satisfied for a Gaussian wave packet.^ .288) where the subscript r stands for radiation.19 shows the required bending angle per half period as a function of beam energy and the corresponding critical wavelength.. and other error sources. With emittance given by Eq. the required emittance is about tag « 10~ n m. lattices with zero dispersion-function straight sections are favorable. (4. Since a strong focusing machine is much more sensitive to the dipole and quadrupole errors. Note that the dipole angles of all existing synchrotron light sources are above the diffraction limit (solid line). High energy linacs may be the only way to reach such a small emittance. the diffraction limit condition is TC 7203 .289) 4TT For hard X-ray at energies 10 keV. and $ is the phase advance of a FODO cell. The correction of large chromaticities in these lattices requires strong chromaticity sextupoles. . This means that the synchrotron radiation at the critical frequency emitted from these light sources can not reach the diffraction limit However.287). or nBA lattices are often used in the design of synchrotron radiation sources (see Sec. B. The circle symbols are bending angles per dipole for existing synchrotron light sources.— (4 290) ql ~373 ~ 3 B 7 2 ' [ ' where the critical wavelength of synchrotron radiation is used.III. EMITTANCE IN ELECTRON STORAGE RINGS 487 Here ft* is the betatron amplitude function at the symmetry point of the dispersion free straight section. Diffraction limit Since the phase space area of a photon beam with wavelength A is38 Ax r A4 = AzrAz'r = ara'T > A/4TT = ephoton. the lifetime and brightness of the beam can be considerably reduced by power supply ripple. TBA. Low emittance lattices require strong focusing optics. Dynamical aperture can be limited by strong nonlinear resonances and systematic chromatic stopbands. which is difficult to attain in electron storage rings. Multiple-families of sextupoles are needed to correct geometric and chromatic aberrations. III. 3 8 The conjugate phase space coordinates of a wave packet obey the uncertainty relation axokx > i .l). (4.
many high brightness storage rings employ positrons with full energy injection. and it is usually alleviated by increasing the beam energy. The beam gas scattering processes include elastic and inelastic scattering with electrons and nuclei of the gases. The quantum lifetime can be controlled by the rf cavity voltage. the corresponding bunch length will be decreased and the peak beam current may be limited by collective beam instabilities. which results in emittance dilution. The beam emittances in low energy storage rings are usually determined by the intrabeam scattering.8). Beam lifetime Since high energy photons can desorb gases in a vacuum chamber. The Touschek scattering discussed in Sec. p. E. C. and B = 1 T.19: The solid line shows the required bending angle vs energy for a synchrotron light source. II. n = 3. This is particularly important for high-charge density lowenergy beams (< 1 GeV). Two high energy machines with small bending angles are PEP and PETRA.22 x 1022P [torr] m~3 is the density of the gas. Circles show the bending angle of each dipole for existing synchrotron light sources. ion trapping. The Touschek lifetime depends on a high power of 7.39 bremsstrahlung. etc. and /3c is the speed of the particle. ionization.488 CHAPTER 4. An effect associated with beam gas scattering is the multiple small angle Coulomb scattering. vacuum pressure is particularly important to beam lifetime in synchrotron radiation sources (see Exercise 4. 39See e. with parameters T = l/A\/TE.1. The beam-gas scattering lifetime is T = ~ ^ ? =a ? n g WC' ('9) 42 1 where <7tot is the total cross-section.. Because of these problems. Another solution is to increase the rf voltage. If the longitudinal momentum of the scattered particle is outside the rf bucket. where the emittance of the electron beam is equal to the emittance of the photon beam at the critical wavelength. Weihreter. CERN 90-03.8 arises from the Coulomb scattering that transfers transverse horizontal momentum into longitudinal momentum. However. The dashed line shows the corresponding critical wavelength. Jx = 1.g. PHYSICS OF ELECTRON STORAGE RINGS Figure 4. 427 (1990) for analysis on the vacuum requirement for compact synchrotron radiation sources. the particle will be lost. The small angle multiple Coulomb scattering between beam particles within a bunch is called the intrabeam scattering. .
(b) Show that the dispersion invariant at the center of the dipole is U= sin 3 (*/2)L(*/2) (1 " I Sln2 f + h Sln4 f >" . Those issues have been discussed in Chapters 2 and 3. de-Qing HOMs of rf cavities. (a) Using thin lens approximation. VII. Dividing the dipole into two pieces. we can express the dispersion function transfer matrix of the half dipole by (I MkB=[0 VO \L 1 0 L9/S\ 9/2 .EXERCISE 4. In storage ring. The turbulent bunch lengthening or microwave instability leads to increase in bunch length and momentum spread (see Sec. enlarging the tune spread with Landau cavities. These operational issues should be addressed in the operation of a storage ring facility.3 489 D. 3). Collective beam instabilities Collective instabilities are important to high intensity electron beams. 1 / where L and 9 are the length and the bending angle of the dipole in the half cell. lifetime degradation. ground motion. Chap. The broadband impedance can be reduced by minimizing the discontinuities in the vacuum chamber. there are high-Q components such as the rf cavities. show that the dispersion function at the center of the dipole is £0(1 . etc.3 1.1 sin2 f ) ' sin 2 ($/2) 9__ sin($/2)' where $ is the phase advance per cell and L is the half cell length. fluctuation. The results are emittances dilution. stability of the beam orbit is also an important issue. etc. The transverse microwave instability has usually a larger threshold provided that the chromaticities are properly corrected. un-shielded beam position monitors. Exercise 4. The single beam instabilities are usually driven by broadband impedance. Power supply ripple. Methods to combat these collective instabilities are minimizing the impedance by careful design of vacuum chamber. and active feedback systems to damp the collective motion [3]. Besides collective beam instabilities. mechanical vibration. and/or human activities can perturb the beam. These accelerator components can lead to coupled bunch oscillations. intensity limitation.4.
(4. Plot T vs the phase advance of the FODO cell.3 n ~™%6) fa .4. the dispersion transfer matrix is cos <p / M = I — (l/p)sin(f \ 0 p sin (p p (1 — cos tp) \ cosy s'm(p I 0 1 / where p is the bending radius. 2/3 ~ ) sin2(9. PHYSICS OF ELECTRON STORAGE RINGS (c) Applying Simpson's rule. show that the average of the %-function in the dipole is (H) = Ho + 2(a0DQ + poD'o)(1 p. 2. In a zero gradient dipole.g. e. (a) Using the relation .2( 7 0 A. show that the average of H-function is ' [ sin 3 ($/2) cos($/2) j ' X P The number in brackets is the T factor of Eq. . and <p = s/p is the beam bending angle along the dipole. + a o ^ ) p ( l sin2i9. where £ is the length of the dipole.211). Express the {H) in Eq.11. cos26^ 2sini9 where 9 is the bending angle of dipoles in a half cell and p is the bending radius. 6 < 60°. For a double-bend achromat (DBA).221). show that average H in a dipole is <«>-*•(!-?•£). Using Exercise 2. (4. using the small bending angle approximation.490 CHAPTER 4.0070 = 1 + ajji show that the minimum of {H) is WMEDBA = ^7J^ 3 with /— 6 8VT5 (b) Show that the minimum (H) occurs at s* = | l with B* 3 P 4%/60 (c) Show that the horizontal betatron phase advance across the dipole for a MEDBA lattice is (tan" 1 VT5 + tan" 1 5\/l5/3). 3. where the numerator depends slightly on the dipole configuration.
show that the dispersion and the betatron amplitude functions inside the dipole are where s = 0 corresponds to the entrance edge of the dipole. i is the length of the dipole magnet. Verify the following properties of a minimum emittance (ME) lattice. 5. and /?* = £/V60. in small angle approximation. Extend your result to find a formula for the condition necessary for a matched MEnBA without using small angle approximation. This configuration appears in the SOR Ring in Tokyo and the ACO Ring in Orsay. (c) Evaluate (aD + /3D')/y/]3^ and D/i/fi^ at the exit point of the dipole magnet for the ME condition and show that Use the result to show that the phase advance of the matching section is 2tan" 1 (9/\/l5). 6. A variant of the double bend achromat is to replace the focusing quadrupole by a triplet. (a) When (%) is minimized. A minimum emittance n-bend achromat (MEnBA) module is composed of n — 2 ME modules inside a MEDBA module. The configuration of the basic cell is40 40Because there is no quadrupole in the straight section. Show that the necessary condition for matching ME modules to the MEDBA module in small angle approximation is ^ ME = 3 P - MEDBA Thus an isomagnetic nBA can achieve optical matching for the minimum emittance only if the middle dipole is longer than the outer dipole by a factor of 3 1 / 3 . such a configuration has the advantage of a very compact storage ring for synchrotron light with dispersion free insertions.3 491 (d) Evaluate (aD + fiD')/y/fi^ and D/yffix at the exit points of the dipole magnet for the MEDBA condition and show that Use the result to show that the phase advance of the dispersion function matching section of the MEDBA is 2tan" 1 (7/\/l5)4.EXERCISE 4. where combined function dipoles are used. (b) Show that the horizontal betatron phase advance in the dipole is 2tan~x vT5. Find the minimum emittance. .
(b) Show that the betatron phase advance of the dispersion matching section for a minimum emittance triplet DBA is i/> = 2 arctan / At 2 + 9 f . jo. Plot the betatron phase advance of the matching section and 3* 11 as a function of £. The method is applicable to a sector dipole or a rectangular dipole.492 • CHAPTER 4. and I. ('H)min = pO ~~op~i where 2Li is the length of the zero dispersion straight section.240) should be used./3o. and £ is the length of the dipole. (c) What happens to (H) and the natural emittance if the dipole is replaced by a combined function magnet? (d) Study the linear stability of the triplet DBA lattice. the emittance will be altered by insertion devices that alter the betatron amplitude function. and D'o into Eq.221). B • Here 2L\ is the length of the zero dispersion straight section. 13 \ Ue + u+iJ . the betatron amplitude function inside the dipole is P-P + p where s = 0 corresponds to the entrance of the dipole. is the length of the dipole. 41The formula can be obtained by substituting ao. PHYSICS OF ELECTRON STORAGE RINGS • \ \ Triplet DBA • • n B QF u QD n QF • . (a) In small angle approximation. Eq.Do. (4. show that the average of the 7i function in the dipole is 41 <«>^'[!+Mti+T^)]. Show that the minimum emittance of the triplet DBA is 20* where I VP U 20' Since the emittance is proportional to the betatron amplitude function at the insertion region. (4. Since the mid-point of the straight section is the symmetry point for the lattice function. For a combined function dipole. where £ = L\jl. .
q = \fKL is the defocussing strength of the dipole. where 9 = L/p is the bending angle of the dipole. where A = 4A . and Dg and D'g are respectively the values of the dispersion function and its derivative at s = 0. with (H) = ±p930oA-aoB + ^C).^ 5 _ sinh^(cosh^. A) = Po/L. First.3 7.Do co s n 0.EXERCISE 4. . d'Oimin = --E{q).I) 2 . Aw .. and _ 2(coshg-l) _ 6(sinhg-g) _ 3sinh2g-6g ^3 .min = -F{q). and ao.— ^ 3 .40 sinh q + 5 sinh 2q B(q) = _ .1) + . 70 = 7o£.2EF. C(q) = -5 .8 cosh g + 2 cosh 2<j 3Qq . ^w -2 ' Fw . pvK where K = -Bi/Bp— 1/p2 is the defocussing strength with B\ = dBz/dx. L is the length of the dipole. B = 3B . 6 .1). and 70 are the Courant-Snyder parameters at s = 0. d'Q = D'Q/0. the normalized betatron amplitude functions are So = «o. (b) The minimization procedure can be achieved through the following steps. pK vK 493 7=)sinh</i + . <f> = \fKs is the betatron phase. . C=\C\F2.3E2./3o. (H) can be minimized byfindingthe optimal dispersion functions with d(H) dd0 Ul d{U) _ dd'o U' Show that the solution is do. s = 0 corresponds to the entrance of the dipole. The dispersion function in the combined function dipole is 0 D = ^— (cosh < .~(jodo + aod'o)F(q) 4A(9)-^)+^)]. where HQ = 7oi?o + 2otoDoD'0 + PQD'Q.Do cosh <j> + ~^D'O sinh<j>. The evolution of the 'H-function in a dipole is D' = {DQVK-\ 2 2 %{<t>) = •Ho + —7={aoDa+/3oD'0)sinh4>-—(y0D0 + a0D'0){cosh<l>-l) pVK pK+ ^ s i n h 2 ^ + A ( c 0 S h ^ . (a) Averaging the 'H-function in the dipole. show that (H) = Ho + p93 [(5orfo + M)E(q) . do = DQ/L9.
= -G-' \/l5S ^ = —G— . (d) Show that the value of the dispersion "H-function at both ends of the dipole for the ME condition is W(0) = U{q) = ^=^p03{6CE2 .1 . (a) Using the parameters of PEP with Jx = 1. If the phase advance is tuned to 98° per FODO cell.0159 m. What happens if similar undulators fill 30 straight sections? . (f) Discuss the effect of damping partition number for the combined function ME lattice. To decrease the emittance further.6 x 10"6iVw + 2.9 2 J . The PEP is a high energy e+e~ collider at SLAC with circumference C = 2200 m and bending radius p = 165 m. Using the parameter listed in Table 4.4 for the APS.1 and the undulator parameters listed in Table 4. wigglers installed at zero dispersion locations. (b) Using a typical set of the wiggler parameters: ~B^ (T) I Aw (cm) I K I 6 W (mr) I p w (m) L26 I 12 I 14.15B2 with s A=VTf6' 8(5 Q° . 2 ^ 1 Plot G vs q. show that the minimum (H) is where G = Vl6AC . (e) Show that the damping partition number Jx for horizontal motion is Jx = l . where (/3X) « 16 m.ac+ -2 (coshq . can be used to decrease the damping time with a minimum increase in quantum diffusion integrals.12 I 1.20 | 25 show that the emittance is ex ' w ~ £a:0 l 1 + 1.^BEF + ^AF2}. PHYSICS OF ELECTRON STORAGE RINGS (c) Using the relation /3o7o = 1 + <*o. the natural emittance of the electron beam is e nat = 5.1 nm at 6 GeV.494 CHAPTER 4. where (H)o is the average of the ^-function for the storage ring without wigglers. show that ('H)o = 0.. How many wiggler periods are needed to reduce the emittance by a factor of 10? What is the total length of the wigglers? What will be the momentum spread of the beam? 9. 8.5xl0-3iVw' where JVW is the number of wiggler periods. estimate the effect of an undulator in the zero dispersion straight section (/3X « 10 m) on the emittance and momentum spread of the beam.
639333739. 2TTS „ / 27rs\l D =— a.#S/E TWISS STOP (a) In thin lens approximation.S3.80 Q3: QUAD.L6. ANGLE=PI/40. E2=PI/80.78365 L8: DRIFT. 7 = 800.35 L9: DRIFT.22365 L6: DRIFT.0 cm. Study the attached lattice d a t a file for the APS and answer the questions below.S4.5. L=0.06. L=.Q3. Aw = 8. 2TT7 L Aw V Aw J\ Find the ratio of the transverse beam sizes arising from the dispersion function and betatron motion. TITLE.L8.L1.L7.S1 . L=. L=0. s i n \-z [1 .-HSECTOR) USE.Q2. L=.1400.1. "APS STORAGE RING LATTICE '96 VERSION" L0: DRIFT. L=. SUPER=40 PRINT.Q5. L=. .22365 L6: DRIFT.83365 L5: DRIFT. El=PI/80.L3.— ..66 L2: DRIFT.1194. K2= 15.L9) SECTOR: LINE=(HSECTOR.50 Q2: QUAD. Show that the dispersion function generated by a helical wiggler in a dispersion-free straight section is „ #wAw L . L=. L=.60 SI: SEXTUPOLE. L=3.3 495 10.17365 L3: DRIFT. Kl=-. L=0. as = 0. and eN = 2ir mm-mrad. L=.1 %. L=. L=0. Ql: QUAD.2527 S4: SEXTUPOLE. HARM0N=1296 HSECTOR: LINE=(LO. SECTOR. L=0.& S2.17365 M: SBEND.Q4.80954550.780136057. where we assume Kv = 5.M.2527 S3: SEXTUPOLE. L=O.36 LI: DRIFT. K2=-16. L=0.. Px — Pz = 10 m.22365 L7: DRIFT. what are the quadrupole strengths of Q4 and Q5 for the achromat condition? Do the data in the MAD input file agree with your thin-lens approximation calculation? (b) What is the purpose of sextupoles pairs (SI and S2) and (S3 and S4)? (c) What is the absolute minimum emittance of this lattice at 7 GeV? Compare your result with the emittance listed in Table 4.2527 S2: SEXTUPOLE. L=.c o s . Kl= .EXERCISE 4.5150.50 Q5: QUAD. L=0. L=3. Kl=-.5O Q4: QUAD.L2. 11.L5. V0LT=9. Kl= .4700.17365 L4: DRIFT.2527 RF: RFCAVITY. L=0.Q1. Kl=-.L4. K2= 5. K2=-11.45435995.41158941.
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Lett. (4. and collective beam instabilities. 497 . In Chapter 4.M. where a collective instability induces microbunching in electron beam for coherent laser action in a long undulator. laser-particle interaction. The stimulated radiation can generate high power coherent radiation from infrared to Xray. Phys. we discussed incoherent spontaneous synchrotron radiation of each individual electron in dipole or wiggler fields. Rev.W. 36 (1976) 717.1 The idea has been extended to vacuum ultra-violet (VUV) and X-ray production by a process called Self-Amplified Spontaneous Emission (SASE). The radiation is incoherent. Robinson and P. 13. the radiated electromagnetic wave plays no role on the motion of electrons. 42 (1971) 1906. there is no correlation between electromagneticwaves radiated by any two electrons. Elias et al.e. 2J. beam cooling. 914 in Ref. [10]. R. Sprangle. The spectral coherence of synchrotron radiation in undulators (see Eq. the effects of space-charge force.J.280)) is attained through the electromagnetic-wave interference radiated by a single electron. I would like to provide introduction to the following two topics: free electron laser and beam beam interaction. The power or intensity of the radiation is proportional to the number of electrons in a bunch. Beside the quantum fluctuation and energy dissipation. p. The first section in this chapter addresses physics of beam-laser interaction and the free electron laser. advanced nonlinear beam dynamics. radiation damping and quantum fluctuation in electron storage rings. The efficiency of these radiation is only a few percent. a Review of FELs. Appl. see also C. 12. Phys. There are many textbooks and workshop proceedings on these advanced topics [10. and synchrotron radiation. However. 11. For high power operation.L. Madey J. i. impedance and collective beam instabilities. nonlinear beam dynamics. Nevertheless.. 14]. beam-beam interaction. linac.Chapter 5 Special Topics in Beam Physics In preceding chapters. it is necessary to induce laser oscillation in a laser cavity consisting of undulator and mirrors. we have focused on particle dynamics of betatron and synchrotron motions. this introductory textbook does not address advanced topics including free-electron laser.
The radiation is generated by coherent transition from population-inverted states to a low-lying state of a lasing medium made of atomic or molecular systems. For a complete updated list.1.ucsb. The force is highly non-linear. in SLAC-R-521 Chapter 4.08.2 Figure 5. Its wavelengths are tunable from millimeter to visible and potentially ultraviolet to x-ray. Herefco= 2TT/A. and is a slowly varying function of * and s. coherent. the beam-beam interaction becomes an important topic in accelerator physics because it plays a major role in limiting the luminosities of all high energy colliders. high power radiation. UQ = 2TTC/X.html.edu/www/vLfel. Since 1960's. <j>o is an arbitrary initial phase of the EM wave. When two beams collide. see the World Wide Web Virtual Library of the Free Electron Laser at http://sbfel3.268)3 is E = Eo\xsm(koS — tx>ot + <f>a) + zeos(fcos — u>ot + 4>o)}.1 summarizes the existing laboratories with free electron laser research facilities. I Free Electron Laser (FEL) Lasers are coherent and high power light (radiation) sources. the limit of beam-beam interaction in colliders is found to be about 0. For simplicity.1. The circularly polarized plane electromagnetic (EM) wave. In the presence of the electromagnetic fields. we limit our discussion to ID FEL-theory. Since the advance of the collider concept. 3For 2See . (4. The space charge force between two countertraveling bunches produces large impulse on each other. Its optical properties possess characteristic of conventional lasers: high spatial coherence and near the diffraction limit. see Exercise 5. FEL Physics. accelerator scientists devote great efforts in developing colliders. and t is the time coordinate. SPECIAL TOPICS IN BEAM PHYSICS The center of mass energy available in fixed target experiments is limited by the kinematic transformation. and induces noises in the detector area. degrades beam lifetime and beam stability.1) propagating along the wiggler axis s. When the beam-beam potential is coupled with the betatron and synchrotron motion in particle accelerators. A free electron laser employs relativistic electron-beams in undulators to generate tunable.02 to 0. where the amplitude Eo is independent of the transverse coordinates. the equation of motion for e. s is the longitudinal distance.498 CHAPTER 5. produced by the relativistic electron-beam in a helical wiggler magnetic field Bw of Eq.g. it perturbs the beam distribution. (2001) and reference there in. physicists and engineers have discovered many techniques to minimize the effects of beam-beam interaction. where two counter-traveling beams are made to collide at the interaction points. Methods of finding a larger tolerable beam-beam interaction is needed to enhance luminosities in high energy colliders. radiation from planar undulator. Depending on the beam-damping time. B = -s x E (5.
I FREE ELECTRON LASER (FEL) 499 Figure 5.2 2 * oi"6 ^Gev»2 **w«' is satisfied. + Nation- (5. Since the space charge force (see Exercise 2. 5-sm^.3 (K + ko)s .269). the importance effect of the EM fields is that it can cause the electron beam to microbunch itself for producing possible coherent radiation.2) is proportional to I/7 2 . where P 7 is the instantaneous radiation power in the wiggler and r0 is the electron classical-radius.c. Besides these operational facilities listed in the graph. or i =— = as where 4>= . dj eEa(3i_ . electron is ^=eE + ec0x{B + Bw) + FS. The effect of the electromagnetic waves E and B on the electron orbit is small.2) where fie is the speed of the electron. The wavelengths of these projects are of the order of 0.1: A compilation of the existing FEL laboratories with associated FEL wavelength. we neglect the radiation force. we obtain the energy-exchange between the electron and electromagnetic wave: mc 2 7 = — eE • j3c. The effect of radiation reaction force is also small.u>ot + (/>o- (5. it is negligible at the energy of our consideration. High gain Xray FEL projects. provided that the condition: ^ r = J t = s ^ = 4 .3. Using Eqs. there are about 10 FEL development centers in universities and National Laboratories. the electron orbital motion is essentially determined by the wiggler magnetic field.1) and (4. and Fradiation are respectively the space charge force and the radiation reaction force.C.e. jmc1 5. However. The energy exchange is maximum when the stationary . (5. i. are not listed on this graph. — sin0 = mcz eEoKv.1 nm. Hereafter.4) The wiggler field provides electron trajectory while the EM-fields interact with electrons for energy exchange. such as the LCLS and XFEL to be completed around 2007. Fs.
500 CHAPTER 5. and thus the resonance condition should be replaced by the rms undulator parameter KWiTms = K^j\pi. <j>) are conjugate phase-space coordinates and the longitudinal coordinate s is the independent variable. uj0At = wo( j^~^f) = 2TT.3) and (5.e. (4. The corresponding Hamiltonian is . the longitudinal velocity vector is given by Eq.1.263). or for resonance photon wavelength at a given beam energy 7. See also Exercise 5.4 The resonance condition can also be expressed as .I Small Signal Regime In a small radiation loss regime.270) is used. the electron energy is near the resonance energy. = _eEoK^ f = 2U (5.11) 4For a planar undulator.18. (5.7) can be derived from the Hamiltonian tf = A:w(l + ^ ) 7 . graphically represented in Fig. 4.^ ^ c o s ^ (5-7) (5-8) where (7. (4. i. I. The equation of motion for the phase angle cj> becomes 4>' = M i . The energy exchange between the electron and the external electromagneticfieldscan be obtained by solving Hamiltonian's equation.4>) are conjugate phase-space coordinates and longitudinal distance s serves as the independent "time" coordinate.10) m& 7^ where (r\. When this resonance condition is satisfied.1. _ Aw(l + ^ ) _ l + Kl '^6d» + 4-s)]"4-*^H 7r2A ' °r K~K^f~- (55) (5-6) for resonance electron beam energy at a given photon wavelength A. electrons lag behind the EM wave by one wavelen
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Hypothyroidism
Hypothyroidism
(Science: endocrinology) a deficiency of thyroid activity.
in adults, it is most common in women and is characterised by decrease in basal metabolic rate, tiredness and lethargy, sensitivity to cold and menstrual disturbances. If untreated, it progresses to full blown myxoedema.
in infants, severe hypothyroidism leads to cretinism.
in juveniles, the manifestations are intermediate, with less severe mental and developmental retardation and only mild symptoms of the adult form.
When due to pituitary deficiency of thyrotropin secretion it is called secondary hypothyroidism.
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Understanding the Life Cycle of Threads in Java: A Comprehensive Guide with Code Examples
As a Java developer, it‘s crucial to have a solid grasp of how threads work and the various states they go through during their life cycle. Threads allow you to write efficient, responsive applications by enabling concurrent execution of tasks. By understanding the life cycle of threads, you can effectively manage their creation, execution, and termination.
In this comprehensive guide, we‘ll dive deep into the life cycle of threads in Java. We‘ll explore each state in detail, provide code examples to illustrate how threads transition between states, and discuss best practices for managing threads in your Java applications. Whether you‘re a beginner or an experienced Java developer, this guide will give you the knowledge you need to master thread management.
What are Threads in Java?
Before we dive into the life cycle, let‘s first define what threads are in the context of Java. A thread is a lightweight unit of execution that represents an independent path of control within a program. Each thread has its own call stack and local variables, allowing it to execute concurrently with other threads.
The main advantage of using threads is that they enable parallelism and efficient utilization of system resources. By dividing a program into multiple threads, you can perform multiple tasks simultaneously, such as handling user input, updating the UI, and performing background processing. This results in more responsive and performant applications.
Java provides built-in support for creating and managing threads through the Thread class and the Runnable interface. You can create a new thread by either extending the Thread class or by implementing the Runnable interface.
The Life Cycle of a Java Thread
Now that we understand what threads are, let‘s explore the different states that a thread goes through during its life cycle. The life cycle of a Java thread consists of the following states:
1. New
2. Runnable
3. Running
4. Blocked/Waiting
5. Terminated
Let‘s take a closer look at each state and see how threads transition between them.
1. New State
When a thread is created using the new keyword, it enters the New state. At this point, the thread has not yet started executing and is not eligible for CPU time. It remains in the New state until the start() method is called on it.
Here‘s an example of creating a new thread:
public class MyThread extends Thread {
@Override
public void run() {
System.out.println("Thread is running");
}
}
MyThread thread = new MyThread();
In this example, we create a new thread by extending the Thread class and overriding the run() method. The thread is in the New state until we call the start() method.
2. Runnable State
Once the start() method is called on a thread, it enters the Runnable state. In this state, the thread is eligible to run and is waiting for CPU time to be allocated to it by the thread scheduler. The thread scheduler determines which thread gets to run based on factors such as thread priority and available system resources.
Here‘s an example of starting a thread:
MyThread thread = new MyThread();
thread.start();
By calling the start() method, we transition the thread from the New state to the Runnable state. The thread is now ready to be executed by the CPU.
3. Running State
When the thread scheduler allocates CPU time to a thread in the Runnable state, it enters the Running state. In this state, the thread is actively executing its task by running the code inside its run() method.
A thread can transition from the Running state back to the Runnable state in the following scenarios:
• The thread voluntarily yields control of the CPU by calling the yield() method.
• The thread is preempted by the thread scheduler to allow other threads to run.
• The thread is blocked or enters the waiting state.
Here‘s an example of a thread in the Running state:
public class MyThread extends Thread {
@Override
public void run() {
System.out.println("Thread is running");
// Perform some task
}
}
MyThread thread = new MyThread();
thread.start();
Once the start() method is called, the thread enters the Running state and executes the code inside the run() method.
4. Blocked/Waiting State
A thread can enter the Blocked or Waiting state in several situations:
• When a thread is waiting for a lock to be acquired, it enters the Blocked state. This happens when the thread tries to enter a synchronized block or method, but the lock is currently held by another thread.
• When a thread calls the wait() method on an object, it enters the Waiting state. The thread remains in this state until another thread calls the notify() or notifyAll() method on the same object.
• When a thread calls the sleep() or join() method, it enters the Timed Waiting state. The thread remains in this state for a specified period of time or until the joined thread completes its execution.
Here‘s an example of a thread entering the Blocked state:
public class SharedResource {
public synchronized void doSomething() {
// Critical section
}
}
public class MyThread extends Thread {
private SharedResource resource;
public MyThread(SharedResource resource) {
this.resource = resource;
}
@Override
public void run() {
resource.doSomething();
}
}
SharedResource resource = new SharedResource();
MyThread thread1 = new MyThread(resource);
MyThread thread2 = new MyThread(resource);
thread1.start();
thread2.start();
In this example, if thread1 acquires the lock on the SharedResource object and enters the doSomething() method, thread2 will be blocked until thread1 releases the lock.
5. Terminated State
A thread enters the Terminated state when it completes its execution or is explicitly terminated using the stop() method (which is deprecated and should be avoided). Once a thread is terminated, it cannot be restarted.
Here‘s an example of a thread entering the Terminated state:
public class MyThread extends Thread {
@Override
public void run() {
System.out.println("Thread is running");
// Perform some task
System.out.println("Thread is terminated");
}
}
MyThread thread = new MyThread();
thread.start();
When the run() method completes execution, the thread automatically enters the Terminated state.
Special Case States
In addition to the main states, there are a few special case states that a thread can enter:
• Timed Waiting: A thread enters this state when it calls the sleep() or wait() method with a specified timeout. The thread remains in this state until the specified time elapses or it is interrupted.
• Parked: A thread can be parked using the LockSupport.park() method. It remains parked until it is unparked using the LockSupport.unpark() method or interrupted.
• Interrupted: A thread can be interrupted by calling the interrupt() method on it. This sets the interrupted status of the thread, which can be checked using the isInterrupted() method.
Here‘s an example of a thread entering the Timed Waiting state:
public class MyThread extends Thread {
@Override
public void run() {
try {
Thread.sleep(5000); // Sleep for 5 seconds
System.out.println("Thread resumed");
} catch (InterruptedException e) {
System.out.println("Thread interrupted");
}
}
}
MyThread thread = new MyThread();
thread.start();
In this example, the thread enters the Timed Waiting state for 5 seconds when it calls the sleep() method. After the specified time elapses, the thread resumes execution.
Best Practices for Managing Threads
When working with threads in Java, there are some best practices to keep in mind:
1. Avoid using the stop() method to terminate threads, as it can lead to resource leaks and inconsistent state. Instead, use a flag or condition variable to signal the thread to stop gracefully.
2. Be cautious when using the suspend(), resume(), and destroy() methods, as they are deprecated and can cause deadlocks and other synchronization issues.
3. Use synchronization mechanisms like synchronized blocks and methods to ensure thread safety when accessing shared resources.
4. Be mindful of the potential for deadlocks when multiple threads are waiting for each other to release locks.
5. Use higher-level concurrency utilities like ExecutorService, CountDownLatch, and CyclicBarrier to manage threads and coordinate their execution.
6. Properly handle interruptions by checking the interrupted status and responding appropriately.
Debugging Thread Issues
When working with threads, you may encounter various issues such as deadlocks, race conditions, and thread starvation. Here are some tips for debugging thread-related issues:
1. Use logging statements to track the execution flow and identify potential issues.
2. Utilize thread dumps to gather information about the state of threads and identify deadlocks.
3. Use the jconsole or jvisualvm tools to monitor thread activity and detect resource contention.
4. Employ synchronization mechanisms correctly and avoid nested locks to prevent deadlocks.
5. Test your code thoroughly with different thread configurations and scenarios to identify race conditions and synchronization issues.
Conclusion
Understanding the life cycle of threads in Java is essential for writing efficient and robust concurrent applications. By knowing the different states a thread can be in and how it transitions between them, you can effectively manage thread creation, execution, and termination.
Remember to follow best practices when working with threads, such as avoiding deprecated methods, using synchronization appropriately, and handling interruptions gracefully. When debugging thread-related issues, utilize logging, thread dumps, and monitoring tools to identify and resolve problems.
With a solid grasp of the Java thread life cycle and best practices, you‘ll be well-equipped to write high-performance, concurrent applications that make the most of system resources.
Happy threading!
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There are numerous organizations within the academic, federal, and commercial sectors conducting large scale advanced research in the field of sustainable energy. This research spans several areas of focus across the sustainable energy spectrum. Most of the research is targeted at improving efficiency and increasing overall energy yields.[94] Multiple federally supported research organizations have focused on sustainable energy in recent years. Two of the most prominent of these labs are Sandia National Laboratories and the National Renewable Energy Laboratory (NREL), both of which are funded by the United States Department of Energy and supported by various corporate partners.[95] Sandia has a total budget of $2.4 billion [96] while NREL has a budget of $375 million.[97]
A report by the United States Geological Survey estimated the projected materials requirement in order to fulfill the US commitment to supplying 20% of its electricity from wind power by 2030. They did not address requirements for small turbines or offshore turbines since those were not widely deployed in 2008, when the study was created. They found that there are increases in common materials such as cast iron, steel and concrete that represent 2–3% of the material consumption in 2008. Between 110,000 and 115,000 metric tons of fiber glass would be required annually, equivalent to 14% of consumption in 2008. They did not see a high increase in demand for rare metals compared to available supply, however rare metals that are also being used for other technologies such as batteries which are increasing its global demand need to be taken into account. Land, whbich might not be considered a material, is an important resource in deploying wind technologies. Reaching the 2030 goal would require 50,000 square kilometers of onshore land area and 11,000 square kilometers of offshore. This is not considered a problem in the US due to its vast area and the ability to use land for farming and grazing. A greater limitation for the technology would be the variability and transmission infrastructure to areas of higher demand.[54]
As competition in the wind market increases, companies are seeking ways to draw greater efficiency from their designs. One of the predominant ways wind turbines have gained performance is by increasing rotor diameters, and thus blade length. Retrofitting current turbines with larger blades mitigates the need and risks associated with a system-level redesign. As the size of the blade increases, its tendency to deflect also increases. Thus, from a materials perspective, the stiffness-to-weight is of major importance. As the blades need to function over a 100 million load cycles over a period of 20–25 years, the fatigue life of the blade materials is also of utmost importance. By incorporating carbon fiber into parts of existing blade systems, manufacturers may increase the length of the blades without increasing their overall weight. For instance, the spar cap, a structural element of a turbine blade, commonly experiences high tensile loading, making it an ideal candidate to utilize the enhanced tensile properties of carbon fiber in comparison to glass fiber.[47] Higher stiffness and lower density translates to thinner, lighter blades offering equivalent performance. In a 10 (MW) turbine—which will become more common in offshore systems by 2021—blades may reach over 100 m in length and weigh up to 50 metric tons when fabricated out of glass fiber. A switch to carbon fiber in the structural spar of the blade yields weight savings of 20 to 30 percent, or approximately 15 metric tons.[48]
Wind-to-rotor efficiency (including rotor blade friction and drag) are among the factors impacting the final price of wind power.[16] Further inefficiencies, such as gearbox losses, generator and converter losses, reduce the power delivered by a wind turbine. To protect components from undue wear, extracted power is held constant above the rated operating speed as theoretical power increases at the cube of wind speed, further reducing theoretical efficiency. In 2001, commercial utility-connected turbines deliver 75% to 80% of the Betz limit of power extractable from the wind, at rated operating speed.[17][18][needs update]
2010 was a record year for green energy investments. According to a report from Bloomberg New Energy Finance, nearly US $243 billion was invested in wind farms, solar power, electric cars, and other alternative technologies worldwide, representing a 30 percent increase from 2009 and nearly five times the money invested in 2004. China had $51.1 billion investment in clean energy projects in 2010, by far the largest figure for any country.[155]
Several groups in various sectors are conducting research on Jatropha curcas, a poisonous shrub-like tree that produces seeds considered by many to be a viable source of biofuels feedstock oil.[117] Much of this research focuses on improving the overall per acre oil yield of Jatropha through advancements in genetics, soil science, and horticultural practices. SG Biofuels, a San Diego-based Jatropha developer, has used molecular breeding and biotechnology to produce elite hybrid seeds of Jatropha that show significant yield improvements over first generation varieties.[118] The Center for Sustainable Energy Farming (CfSEF) is a Los Angeles-based non-profit research organization dedicated to Jatropha research in the areas of plant science, agronomy, and horticulture. Successful exploration of these disciplines is projected to increase Jatropha farm production yields by 200-300% in the next ten years.[119]
The array of a photovoltaic power system, or PV system, produces direct current (DC) power which fluctuates with the sunlight's intensity. For practical use this usually requires conversion to certain desired voltages or alternating current (AC), through the use of inverters.[4] Multiple solar cells are connected inside modules. Modules are wired together to form arrays, then tied to an inverter, which produces power at the desired voltage, and for AC, the desired frequency/phase.[4]
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Wohl Degradation
Delve into the fascinating world of Organic Chemistry with a thorough exploration of the Wohl Degradation. This comprehensive resource unravels complicated chemical processes, from understanding the Wohl Degradation mechanism to its practical applications and significance in Biochemistry. You'll gain an in-depth knowledge of this reaction, its history, and its role in glucose metabolism. Benefit from case studies like the degradation of glucose and the involvement of dehydration in the mechanism. As well as definitions and the meaning of technical terms within this scientific field.
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Delve into the fascinating world of Organic Chemistry with a thorough exploration of the Wohl Degradation. This comprehensive resource unravels complicated chemical processes, from understanding the Wohl Degradation mechanism to its practical applications and significance in Biochemistry. You'll gain an in-depth knowledge of this reaction, its history, and its role in glucose metabolism. Benefit from case studies like the degradation of glucose and the involvement of dehydration in the mechanism. As well as definitions and the meaning of technical terms within this scientific field.
Understanding the Wohl Degradation in Organic Chemistry
The Wohl Degradation is a notable reaction employed in organic chemistry, specifically used for the transformation of sugars into smaller fragments. This reaction was pioneered by Alfred Wohl, hence its nomenclature. Be aware that the comprehension of this reaction is valuable to your proficiency in the sphere of organic chemistry.
The Wohl Degradation technique is essentially used when a sugar molecule needs to be broken down into smaller pieces, through the conversion of a sugar into an aldehyde and a ketone.
Exploring the Wohl Degradation mechanism process
Now let's delve deeper into the Wohl Degradation process, to understand how it works. A detailed study of the process will help you gain a better understanding of not only this specific degradation technique but also organic chemistry reactions in general.
Consider Fructose, a hexose sugar. When it undergoes the Wohl Degradation process, the outcome will be a four-carbon product (tetrose) like Erythrose, in addition to a two-carbon fragment (dihydroxyacetone).
Events leading up to the Wohl Degradation reaction
To begin with, the sugar molecule is turned into a glycosylamine through the reaction with hydroxylamine. This is then followed by the formation of an osazone via a series of steps.
• Conversion of sugar into glycosylamine, which is performed by reacting the sugar with hydroxylamine (\( NH2OH \)).
• Followed by the transformation of glycosylamine into an osazone.
• Subsequently, the osazone is heated with an acid to induce degradation.
• Finally, the process results in the formation of an aldehyde or ketone, alongside a new, smaller sugar.
Enumerating the steps involved in the mechanism process
The key component of the Wohl Degradation process is the series of steps leading to the transformation from glycosylamine to osazone.
Step 1: The sugar is converted into a glycosylamine by reacting the sugar with hydroxylamine (NH2OH).
Step 2: The glycosylamine is transformed into an osazone.
Step 3: The osazone, upon heating with acid, breaks down - this is the key Wohl Degradation event.
Step 4: This process results in the formation of an aldehyde or ketone, alongside a new, smaller sugar.
Fascinatingly, the Wohl Degradation process doesn't just randomly split the sugar molecule. It neatly splits the sugar into an aldehyde or ketone and a smaller sugar, which is exactly half the size of the original sugar. This is because of the symmetrical nature of many sugar molecules. This careful, calculated degradation is what makes this reaction so useful in organic chemistry.
Wohl Degradation Mechanism Dehydration: A Closer Look
In the realm of organic chemistry, and more specifically within the Wohl Degradation reaction, dehydration plays a huge part in the execution of the process. It is the dehydration step that propels the forward motion of the reaction, allowing for the degradation to take place.
Identifying the role of dehydration in the Wohl Degradation mechanism
This pivotal role of dehydration, in terms of its function within the Wohl Degradation mechanism, is chiefly accentuated during the transformation of osazone. The vital part to understand here is that the process hinges on expedient and accurate dehydration reactions.
Think of it like this: without the dehydration step, the Wohl Degradation reaction would fail to reach completion. The water molecules, still held within the osazone, act as a barrier to degradation. But with them removed by dehydration, the reaction is able to progress.
To understand this further, let's explore what happens before and after dehydration occurs. Prior to dehydration, the sugar molecule is converted into a hydrazone. This is achieved by nucleophilic addition of the amine to the carbonyl, followed by protonation of nitrogen and loss of water. The hydrazone then cyclizes to a furanose-like structure which is subsequently transformed into an osazone.
Dehydration in this context refers to the removal of water from the structure, transforming it into an osazone.
Following the removal of water, the osazone structure is then ready to be split apart in the actual degradation step of the Wohl Degradation reaction. Here, the dehydration role is implying that water (a usual by-product of many chemical reactions) isn’t produced, but is instead actively removed from the reacting molecule to allow the reaction to progress.
How does dehydration facilitate the Wohl Degradation reaction?
In essence, the dehydration within the Wohl Degradation reaction acts as a preparatory step for the actual degradation, or breakdown, of the molecule. By removing the constituent water molecules through dehydration, the chemical rigidity of the osazone is lowered, permitting the further series of reactions that follow.
• Step 1: The dehydration process begins with an osazone molecule. In the case of a sugar, osazone is the result of a reaction with phenylhydrazine.
• Step 2: The dehydration of osazone can be induced by heating with acetic acid, facilitating the removal of water molecules from the osazone.
• Step 3: Two phenylhydrazine molecules are liberated in the process, leaving behind a dehydrated portion of the original sugar molecule.
• Step 4: This dehydrated product is more reactive, allowing for the subsequent degradation mechanism to occur more readily.
This illustrates just how vital hydration levels are to organic chemistry reactions, such as the Wohl Degradation. By following the specific dehydration step, you are enabling the osazone to become primed for the degradation process, making it less stable and thus more reactive. Without this crucial dehydration step, the Wohl Degradation reaction wouldn't be able to proceed as effectively or at all.
Importantly, the specific details of these steps can vary slightly depending on the specific sugar molecule which is being degraded. However, the general principles regarding dehydration remain the same.
Practical Examples of Wohl Degradation
In the context of learning chemistry, practical examples can go a long way in cementing the understanding of a given concept. This holds true for the Wohl Degradation as well, an organic chemistry reaction that is not only theoretical but also has extensive practical applications. To understand this better, let's explore some practical examples in the world of organic chemistry.
Wohl Degradation examples in Organic Chemistry
The Wohl Degradation reaction is frequently used in organic chemistry to break down larger sugar molecules into smaller ones. This happens through a series of steps that result in an aldehyde or a ketone, as well as a smaller sugar molecule.
For instance, if a six-carbon sugar (hexose) such as glucose or fructose undergoes Wohl Degradation, it will yield a four-carbon product (tetrose) and a two-carbon fragment (dihydroxyacetone). This is a fine practical instance of how the Wohl Degradation technique is applied in the chemical degradation of sugars.
Two vital steps involved in this process include the concerted cyclization of the sugar molecule while it is in its phenylhydrazone form, and the subsequent elimination of the phenylhydrazine molecules. Both of these steps involve the dehydration of the sugar molecule, which is a prerequisite for the degradation process to occur.
Cyclization, based on its name, involves the formation of a cyclic (ring) structure within the sugar molecule, which sets up the scene for the elimination of the phenylhydrazine molecules. Elimination, as the term suggests, is the process where unneeded molecules (phenylhydrazine in this case) are eliminated from the structure.
Observing Wohl Degradation in an aldohexose situation
Using an aldohexose as a practical example, such as glucose or mannose, can demonstrate the Wohl Degradation reaction very efficiently.
To start off, the hexose sugar gets transformed into a glycosylamine via the reaction with hydroxylamine to create the osazone. This osazone then undergoes a series of reactions which include dehydration and a ring closure mechanism that transforms it into a furanose-like structure. This furanose-like structure, when heated with acid, then undergoes degradation via a concerted mechanism that ends up forming two smaller sugar fragments.
The final degradation process can be simplified as such:
Hexose \( \rightarrow \) Glycosylamine \( \rightarrow \) Furanose-like structure \( \rightarrow \) Tetrose + Dihydroxyacetone
Case study: Wohl Degradation of glucose
Delving even deeper, we can delve into the case study of Wohl Degradation of glucose. As a hexose sugar, glucose proves to be the perfect candidate for this reaction process.
Initially, glucose pairs with phenylhydrazine in order to form the phenylglycosazone. This then undergoes cyclical dehydration to form a cyclic structure, which prepares it for the subsequent degradation.
Following the cyclical dehydration, the structure undergoes a phenylhydrazone elimination, leading to the degradation of the glucose into a tetrose and dihydroxyacetone.
In essence, the Wohl Degradation of glucose can be summarized as:
Glucose \( \rightarrow \) Glucosazone \( \rightarrow \) Cyclic Dehydration \( \rightarrow \) Phenylhydrazone elimination \( \rightarrow \) Tetrose + Dihydroxyacetone
The Wohl Degradation reaction provides an elegant and precise way to degrade larger sugar molecules, like glucose, into smaller fragments, allowing organic chemists to study and harness the properties of these smaller sugar structures. As you can see, this reaction is key to molecule degradation in organic chemistry, and understanding it opens doors to understanding more complex organic reactions.
Interpreting the Wohl Degradation Meaning
As you delve into the natural sciences, particularly organic chemistry, you will come across a multitude of complex terminologies. 'Wohl Degradation' is one such term. Like all scientific terms, 'Wohl Degradation' has a precise and specific meaning. However, understanding this term is more than just a taxonomy exercise. At its core, the meaning of 'Wohl Degradation' revolves around the process which it refers to, a distinct reaction mechanism used extensively in the field.
Breaking down the technical terms: Wohl Degradation definition
In order to understand the full significance and interpretation of the Wohl Degradation, it's instrumental to first comprehend its technical definition. In simplest terms, the Wohl Degradation is an organic chemistry process that leads to the degradation of a specific type of molecules known as sugars, or more accurately pentoses and hexoses - five and six carbon sugars respectively.
The term 'degradation' here denotes a reduction or breakdown of these sugar molecules into smaller fragments. Importantly, the Wohl Degradation is characterised by a unique series of reaction steps which include cyclisation, dehydration steps and a concerted degradation step.
Naturally, these terms may be confusing without further elaboration, so let's break these down:
• Cyclisation – a reaction step that involves the formation of a ring or cyclic compound from a linear molecule. This is often an enzymatic process, which brings two distant atoms within reach to create a new bond.
• Dehydration – a chemical reaction that involves the removal of water (H2O) from a molecule. The outcome of this process is a more condensed molecule that is primed for further chemical interaction.
• Concerted degradation – a multi-step process leading to the cleavage of the molecule leading to the production of smaller fragments via a sequence of actions that take place together, or 'in concert'.
Together, these complex steps comprise the Wohl Degradation, named after the chemist who discovered this sequence of reactions, and collectively they function to facilitate the controlled degradation of sugar molecules.
History and origin of the Wohl Degradation term
The genesis of the Wohl Degradation term is inevitably tied with the legacy of Wilhelm Rudolph Wohl. Born in the late 19th century, Wohl was a German chemist who is most recognised for his contributions to carbohydrate chemistry, particularly his discovery of the degradation process that now bears his name.
The degradation of sugar molecules explored by Wohl was a relatively new field of study in the early 1900s. His work marked a significant milestone in the understanding of sugar chemistry leading to advancements in the structural elucidation of carbohydrates.
First publicised around 1912, the degradation technique revealed an elegant method of converting an aldohexose into an aldotriose, and ketohexose into a diketotriose. It was subsequently named the Wohl Degradation in honour of Wohl's groundbreaking discovery.
One of Wohl's most famous experimentations involved breaking down glucose, a six-carbon sugar, into glyceraldehyde, a three-carbon sugar using phenylhydrazine. It is here where he first observed chemical dehydration playing a key role in the reaction sequence. This study became a primary model for what we now know as the Wohl Degradation.
A historical understanding of the Wohl Degradation term helps to contextualise this reaction within the broader field of organic chemistry. Wohl's discovery sparked new research avenues and has since been a cornerstone in the study of carbohydrates and the biochemistry of sugar metabolism.
In summary, understanding the in-depth meaning of terms such as 'Wohl Degradation' goes beyond the mere definition. It can embolden a more profound comprehension of the reactions involved, give a nod of respect to the pioneers of the field, and appreciate the transformation that has taken place within the scientific landscape over the past century.
Application of the Wohl Degradation in Biochemistry
The landscape of biochemistry is studded with numerous crucial natural processes, and the application of the Wohl Degradation is a facet that cannot be overlooked. In essence, the Wohl Degradation has paved the way for the in-depth investigation of saccharides, particularly aldohexoses, and has answered several questions related to the structural elucidation of carbohydrates.
Understanding the Significance of Wohl Degradation of Aldohexose
Among the myriad chemical reactions in biochemistry, the Wohl Degradation holds a notable place due to its significant function in the breakdown of aldohexose. When applied to an aldohexose, a six-carbon sugar with an aldehyde functional group, Wohl Degradation facilitates a reduction process converting this sugar into an aldotetrose, a four-carbon sugar.
To grasp the importance of this conversion, it's pivotal to first understand the role of aldohexoses. These sugars provide an essential energy source for living organisms. Being soluble, they can be easily transported through the body. A perfect example illustrating this application is glucose, a primary aldohexose, which is the main energy source for cells.
The process of converting an aldohexose into a smaller sugar molecule, an aldotetrose, via the Wohl Degradation consists of several critical steps:
• The aldohexose reacts with phenylhydrazine to form a phenylhydrazone compound.
• Subsequent treatment with phenylhydrazine leads to the formation of glycosazone via a series of reactions.
• The glycosazone facilitates cyclization resulting in a three-membered ring structure.
• Further treatment leads to a phenylhydrazone elimination which degrades the original aldohexose into a smaller sugar molecule.
A common biochemical application of this reaction lies in the structural determination of unknown sugars. By performing Wohl Degradation on an unknown sample, you can identify smaller derivatives, which are typically easier to analyse. This can provide valuable clues to the identity of the original larger sugar molecule.
Role and Importance of Wohl Degradation in Glucose Metabolism
The biochemistry of glucose metabolism is the cornerstone of energy production in cells, and strikes as a representative example to illustrate the application of Wohl Degradation. During metabolism, glucose - an aldohexose - is broken down, primarily through a process called glycolysis. However, the complexity of the glycolytic pathway can be simplified if we introduce Wohl Degradation.
Indeed, Wohl Degradation reaction can be employed to mimic glycolysis. In general terms, glucose metabolism involves similar steps to Wohl Degradation, such as the formation of an intermediate compound that is later broken down into smaller derivative products. Even more, these smaller products are often substrates for comprehending metabolic pathways.
Let's break down this process:
• First, glucose reacts with phenylhydrazine to produce a phenylhydrazone compound, phenylglucosazone.
• Then, the phenylglucosazone undergoes cyclization and dehydration, forming a three-membered ring structure.
• Upon further treatment, phenylhydrazone is eliminated.
• This results in the Wohl Degradation of glucose, yielding erythrose and dihydroxyacetone.
In essence, the reactions depicted in the Wohl Degradation could be considered as a greatly simplified version of glucose metabolism. Here, glucose degradation yields two three-carbon compounds that are applicable in further metabolic pathways. This process, resembling a step in glycolysis, underscores the importance of the Wohl Degradation and its application in biochemistry.
The breaking down of glucose aids in the understanding of more complex biochemical pathways. It offers insights into the inner workings of carbohydrate metabolism and bioenergetic processes, making it fundamental to the comprehension of overall glucose metabolism.
In a broader perspective, this reaction technique uncovers the intricacies of the sugar molecule degradation process, and facilitates human understanding of how cells derive energy from glucose. This also prompts the further exploration of complicated processes such as glycolysis, thereby enriching the span of biochemistry, and illuminating how it's crucial to life and human understanding of it.
Wohl Degradation - Key takeaways
• The Wohl Degradation process carefully splits a sugar molecule into an aldehyde or ketone and a smaller sugar, due to the symmetrical nature of many sugar molecules.
• In the Wohl Degradation reaction, dehydration is a key step, particularly during the transformation of osazone, and it propels the forward motion of the reaction.
• Dehydration removes water molecules from the osazone, a key step for the Wohl Degradation reaction to progress and reach completion.
• The Wohl Degradation reaction has been practically applied in organic chemistry to break down larger sugar molecules into smaller ones, for instance, a six-carbon sugar (hexose) like glucose or fructose can be degraded into a smaller four-carbon product (tetrose) and a two-carbon fragment (dihydroxyacetone).
• The Wohl Degradation is named after Wilhelm Rudolph Wohl, who discovered this sequence of reactions. It's primarily applied in the controlled degradation of sugar molecules, particularly the carbohydrates, pentoses and hexoses (five and six carbon sugars respectively).
Frequently Asked Questions about Wohl Degradation
The Wohl degradation mechanism is a chemical process used to shorten the carbon chain of sugars. It involves the oxidation of an aldose to an aldonic acid, followed by a reduction and then an elimination reaction, producing an aldose with one less carbon atom.
An example of Wohl Degradation is the breakdown of maltose into two glucose molecules. This reaction involves steps of acetylation, reduction, and hydrolysis to degrade the disaccharide into monosaccharides.
Wohl degradation is a popular method in carbohydrate chemistry used for structural transformations of sugars. It involves the conversion of an aldose into an aldonic acid, followed by decarboxylation and reduction to produce an aldose with one less carbon atom.
Wohl's method of degradation is a chemical process used to breakdown carbohydrates into simpler forms. It involves halogenation followed by reduction, effectively shortening the carbohydrate chain. It's named after German chemist and researcher, Alfred Walter Wohl.
In the Wohl Degradation process, an aldose is first converted into an acyclic diketone through oxidative cleavage. This diketone then undergoes a series of chemical reactions resulting in the formation of a ketose, a type of carbohydrate, that has one less carbon atom than the original aldose.
Final Wohl Degradation Quiz
Wohl Degradation Quiz - Teste dein Wissen
Question
What is the Wohl Degradation in organic chemistry?
Show answer
Answer
The Wohl Degradation is a reaction employed in organic chemistry for the conversion of sugars into smaller fragments, specifically through turning a sugar into an aldehyde and a ketone.
Show question
Question
How does the Wohl Degradation process work?
Show answer
Answer
The Wohl Degradation process starts by transforming a sugar molecule into a glycosylamine through reacting with hydroxylamine. Then, it is turned into an osazone, which, when heated with acid, breaks down to form an aldehyde or ketone along with a new, smaller sugar.
Show question
Question
What is the initial step in the Wohl Degradation process?
Show answer
Answer
The initial step in the Wohl Degradation process is the conversion of a sugar molecule into a glycosylamine by reacting with hydroxylamine.
Show question
Question
What is the role of dehydration in the Wohl Degradation mechanism?
Show answer
Answer
Dehydration in the Wohl Degradation mechanism removes water molecules from the osazone structure, making it less stable and thus more reactive. This allows the degradation process to occur more readily.
Show question
Question
What happens during the dehydration process in the Wohl Degradation mechanism?
Show answer
Answer
In the case of a sugar molecule, the dehydration process begins with an osazone molecule, produced by reaction with phenylhydrazine. Dehydration is induced by heating with acetic acid, facilitating the removal of water molecules from the osazone. This results in a more reactive, dehydrated product.
Show question
Question
How does hydration level impact the Wohl Degradation reaction?
Show answer
Answer
The hydration level significantly impacts the Wohl Degradation reaction. Without dehydration, the osazone structure remains stable, acting as a barrier to degradation. Therefore, dehydration is essential to make the osazone structure less stable and more reactive, allowing for the degradation process.
Show question
Question
What is the Wohl Degradation reaction used for in organic chemistry?
Show answer
Answer
The Wohl Degradation reaction is used in organic chemistry to break down larger sugar molecules into smaller ones, yielding an aldehyde or ketone and a smaller sugar molecule.
Show question
Question
What are the two vital steps involved in the Wohl Degradation process?
Show answer
Answer
The two vital steps in the Wohl Degradation process are the concerted cyclization of the sugar molecule and the subsequent elimination of the phenylhydrazine molecules. Both steps involve dehydration of the sugar molecule.
Show question
Question
How can the degradation of an aldohexose like glucose be summarised using the Wohl Degradation process?
Show answer
Answer
The Wohl Degradation of glucose can be summarised as: Glucose to Glucosazone, cyclical dehydration, Phenylhydrazone elimination, and finally yielding a tetrose and dihydroxyacetone.
Show question
Question
What does the term "Wohl Degradation" refer to in organic chemistry?
Show answer
Answer
Wohl Degradation is an organic chemistry process that leads to the degradation of a specific type of molecules known as sugars, comprising a unique series of reaction steps including cyclisation, dehydration and a concerted degradation step.
Show question
Question
What does the term 'cyclisation' mean in the context of Wohl Degradation?
Show answer
Answer
'Cyclisation' is a reaction step in Wohl Degradation that involves the formation of a ring or cyclic compound from a linear molecule.
Show question
Question
Who is the Wohl Degradation named after and what is his contribution in the field of organic chemistry?
Show answer
Answer
The Wohl Degradation is named after Wilhelm Rudolph Wohl, a German chemist recognised for his contributions to carbohydrate chemistry, particularly his discovery of the sugar degradation process that bears his name.
Show question
Question
What is the Wohl Degradation in the context of biochemistry?
Show answer
Answer
The Wohl Degradation in biochemistry is a process that facilitates the breakdown of aldohexose, a six-carbon sugar with an aldehyde functional group, into an aldotetrose, a four-carbon sugar. This reaction enables in-depth investigation and structural elucidation of carbohydrates.
Show question
Question
What is the importance of the Wohl Degradation in the investigation of saccharides?
Show answer
Answer
The Wohl Degradation enables the conversion of larger sugar molecules into smaller ones, which are easier to analyse. This allows for the structural determination of unknown sugars, providing valuable clues to the identity of the original larger sugar molecule.
Show question
Question
How is the Wohl Degradation process involved in glucose metabolism?
Show answer
Answer
The Wohl Degradation can be employed to mimic glycolysis, a pathway in glucose metabolism. The degradation of glucose yields two three-carbon molecules, which are applicable in further metabolic pathways. It simplifies the understanding of complex biochemical pathways.
Show question
Test your knowledge with multiple choice flashcards
What is the Wohl Degradation in organic chemistry?
How does the Wohl Degradation process work?
What is the initial step in the Wohl Degradation process?
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Flashcards in Wohl Degradation15
Start learning
What is the Wohl Degradation in organic chemistry?
The Wohl Degradation is a reaction employed in organic chemistry for the conversion of sugars into smaller fragments, specifically through turning a sugar into an aldehyde and a ketone.
How does the Wohl Degradation process work?
The Wohl Degradation process starts by transforming a sugar molecule into a glycosylamine through reacting with hydroxylamine. Then, it is turned into an osazone, which, when heated with acid, breaks down to form an aldehyde or ketone along with a new, smaller sugar.
What is the initial step in the Wohl Degradation process?
The initial step in the Wohl Degradation process is the conversion of a sugar molecule into a glycosylamine by reacting with hydroxylamine.
What is the role of dehydration in the Wohl Degradation mechanism?
Dehydration in the Wohl Degradation mechanism removes water molecules from the osazone structure, making it less stable and thus more reactive. This allows the degradation process to occur more readily.
What happens during the dehydration process in the Wohl Degradation mechanism?
In the case of a sugar molecule, the dehydration process begins with an osazone molecule, produced by reaction with phenylhydrazine. Dehydration is induced by heating with acetic acid, facilitating the removal of water molecules from the osazone. This results in a more reactive, dehydrated product.
How does hydration level impact the Wohl Degradation reaction?
The hydration level significantly impacts the Wohl Degradation reaction. Without dehydration, the osazone structure remains stable, acting as a barrier to degradation. Therefore, dehydration is essential to make the osazone structure less stable and more reactive, allowing for the degradation process.
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API with NestJS #101. Managing sensitive data using the AWS Secrets Manager
AWS NestJS
This entry is part 101 of 146 in the API with NestJS
When managing the architecture of our system, we often deal with sensitive data. It’s our job to ensure they don’t fall into the wrong hands. An excellent example of confidential information is the database password and the Json Web Token secret key. In this article, we explore how we can use the AWS Secrets Manager to increase the security of our NestJS application.
Defining environment variables
It’s tough to hide a piece of information when it is included in the source code of our application.
database.module.ts
With the above approach, everyone with access to our code has full access to our database. This is a significant security issue that might lead to compromising our database. This might be especially apparent if we write open-source software, but it is not limited to it. For example, we might have teammates we trust enough to provide them with the code, but we wouldn’t want them tinkering with the production database.
Besides that, our testing environment will surely use a different database than the production environment. When we include environment-specific details in our code, we don’t have a straightforward way of reusing our code across different environments.
We can solve the above problems by externalizing specific values in the form of environment variables. The NestJS application we’ve created during this series has a bunch of them. A good example is the password of our PostgreSQL database. A good way of introducing an environment variable is to add it to our . When doing that, we force NestJS to check if all necessary environment variables are provided. If we forget to provide a specific variable we marked as required, the application won’t start.
app.module.ts
Using environment variables, we can now improve our and avoid hardcoding sensitive information in our code.
database.module.ts
Providing the values for the environment variables
When we develop and run our application locally on our machine, we can provide the values for our environment variables by creating a dedicated file called .
It is a good practice to avoid commiting the file to the repository.
.env
Since our application runs in Docker, we need to point it to the file containing our environment variables.
docker-compose.yml
If you want to know more about running NestJS using Docker, check out API with NestJS #91. Dockerizing a NestJS API with Docker Compose
Environment variables values in ECS
In one of the previous parts of this series, we learned how to deploy our NestJS application using Amazon Elastic Container Service. One of the important parts of it was providing the environment variables for our application running in the cluster.
So far, we’ve been doing that by putting the values directly into the task definition.
Unfortunately, this has some downsides. First, we need to acknowledge that in real-life scenarios, a web application is managed by a whole team of people. Each person might have their own AWS account and access to certain parts of our configuration.
With the above approach, everyone who can access our task definition has access to all of our environment variables, including sensitive data. To deal with this issue, we can use the AWS Secrets Manager.
Introducing AWS Secrets Manager
With AWS Secrets Manager, we can control access to sensitive information, such as database credentials and private keys. We can also rotate them by configuring AWS to change the passwords automatically.
Integration with RDS
Let’s start by opening the Secrets Manager dashboard and going to the secrets page. When we do that, we notice that we might already have some secrets defined.
This is because Relational Database Service (RDS) is integrated with the Secrets Manager. When we created our database, AWS stored our credentials in the Secrets Manager.
If you want to read morea bout using the Relational Database Service, check out API with NestJS #93. Deploying a NestJS app with Amazon ECS and RDS
When we click on the name of our secret, we can access all of the associated values.
To be able to refer to the secret values in the Elastic Container Service, we need the Amazon Resource Name (ARN) of our secret. You can find it at the top of the page.
Creating new secrets
Besides the database, we also have other sensitive information in our environment variables, such as the JWT secret key. Therefore, let’s create a new secret to hold it.
To do that, we need to click the “Store a new secret” button on the secrets page. Then, we must choose the right secret type and define key/value pairs. In the case of our JWT token, the only thing we want to store for now is the secret key.
We also need to provide our secret with a name.
Allowing the service to use our secrets
By default, our services can’t access any of our secrets. To allow that, we need to create an IAM policy with the correct permissions.
Make sure to put the correct resource names in the resources part of the above interface. You can find the Amazon Resource Name (ARN) of each secret on its page in the Secrets Manager.
We also need to give a name to our policy.
By default, AWS uses the IAM role when executing our ECS tasks.
We can create our custom IAM role containing the permissions in the and our new policy. When doing that, it’s important to select the correct use case.
We also need to provide our new role with the required permissions.
Thanks to doing the above, we can choose our new role when defining the task definition. If we do that, our service will be able to use the secrets we’ve created.
Using the secret values
The last step is to modify our task definition to use the values from the secrets manager.
To use a value from the secrets manager, we must choose ValueFrom as the value type. The most crucial thing is using the right resource name as values in our environment variables.
The resource name consists of the following parts: . By adding at the end of the resource name, we can refer to one of the keys of our secret, for example, .
Therefore, in our case, we use the following values:
Summary
In this article, we’ve increased the security of our architecture. To do that, we stored sensitive data necessary for our NestJS application to run in the AWS Secrets Manager. While doing that, we also had to create additional policies and roles so that our service would have access to the secrets. Thanks to all of the above, we’ve improved our workflow and learned more about AWS.
Series Navigation<< API with NestJS #100. The HTTPS protocol with Route 53 and AWS Certificate ManagerAPI with NestJS #102. Writing unit tests with Prisma >>
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Should the next standby power target be 0-watt?
Should the next standby power target be 0-watt?
TitleShould the next standby power target be 0-watt?
Publication TypeReport
Year of Publication2017
AuthorsAlan K Meier, Hans-Paul Siderius
Date Published06/2017
Abstract
The standby power use of appliances continues to consume large amounts of electricity. Considerable success has been made in reducing each device's use, but these savings have been offset by a huge increase in the number of products using standby power and new power requirements for maintaining network connections. Current strategies to reduce standby have limitations and may not be most appropriate for emerging energy consumption trends. A new strategy for further reductions in standby, the "Standzero" option, encourages electrical products to be designed to operate for short periods without relying on mains-supplied electricity. Energy savings are achieved through enhanced efficiency and by harvesting ambient energy. A sensitivity analysis suggests many appliances could be designed to operate for at least an hour without relying on mains power and, in some cases, may be able to operate indefinitely at zero watts until activated.
LBNL Report Number
LBNL-2001019
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Gas-Phase Oxidation of Reactive Organometallic Ions Anuj Joshi Sofia Donnecke Ori Granot Dongju Shin Scott Collins Irina Paci J Scott McIndoe 10.26434/chemrxiv.11909397.v1 https://chemrxiv.org/articles/Gas-Phase_Oxidation_of_Reactive_Organometallic_Ions/11909397 Analysis of highly reactive compounds at very low concentration in solution using electrospray ionization mass spectrometry requires the use of exhaustively purified solvents. It has generally been assumed that desolvation gas purity needs to be similarly high, and so most chemists working in this space have relied upon high purity gas. However, the increasingly competitiveness of nitrogen generators, which provide gas purity levels that vary inversely with flow rate, prompted an investigation of the effect of gas-phase oxygen on the speciation of ions. For moderately oxygen sensitive species such as phosphines, no gas-phase oxidation was observed. Even the most reactive species studied, the reduced titanium complex [Cp<sub>2</sub>Ti(NCMe)<sub>2</sub>]<sup>+</sup>[ZnCl<sub>3</sub>]<sup>–</sup> and the olefin polymerization precatalyst [Cp<sub>2</sub>Zr(µ-Me)<sub>2</sub>AlMe<sub>2</sub>]<sup>+</sup> [B(C<sub>6</sub>F<sub>5</sub>)<sub>4</sub>]<sup>–</sup>, only exhibited detectable oxidation when they were rendered coordinatively unsaturated through in-source fragmentation. Computational chemistry allowed us to find the most plausible pathways for the observed chemistry in the absence of observed intermediates. The results provide insight into the gas-phase oxidation of reactive species and should assure experimentalists that evidence of significant oxidation is likely a solution rather than a gas-phase process, even when relatively low-purity nitrogen is used for desolvation. 2020-02-28 13:13:52 Oxidation Mass spectrometry electrospray ionization decomposition oxygen olefin polymerization
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Discrete Differential Geometry Assignment 0
DDG Week2 Writing Assignment
2.1
Show that VE+F=1V - E + F = 1 for any polygonal disk.
For a simple n-sided polygon with n vertices, n edges, and 1 face, the equation above holds. When conncting another n-sided polygon to form a disk, the polygon can connect to the existing disk by merging mm edges. This will generate nmn-m edges, n(m+1)n-(m+1) vertices, and 1 new face.
V=V+nm+1E=E+nmF=F+1VE+F=VE+F+(n(m+1))(nm)+1=VE+F=1V' = V + n - m + 1 \\ E' = E + n - m \\ F' = F + 1 \\ V' - E' + F' = V - E + F + (n-(m+1)) - (n-m) + 1 = V - E + F = 1
The last equality stems from our inductive assumption.
2.2
In a platonic solid, there are FF m, n-gons meeting at VV vertices.
(F/m)(nF/2)+F=2(F/m) - (nF/2) + F = 2
2.8
Cl(S)
St(S)
Lk(S)
2.9
2.10
2.11
A0 = [
[1,1,0,0,0],
[1,0,1,0,0],
[1,0,0,1,0],
[1,0,0,0,1],
[0,1,0,0,1],
[0,1,1,0,0],
[0,0,1,1,0],
[0,0,0,1,1]
]
A1 = [
[1,0,0,1,1,0,0,0],
[1,1,0,0,0,1,0,0],
[0,1,1,0,0,0,1,0],
[0,0,1,1,0,0,0,1]
]
Coding
Code is somewhere, I haven't decided where to put it. The screenshots below should verify that the c++ code is working to solve the exercises.
All tests green: 11 tests passed
Star star
Link Link
Closure closure
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Common Pipistrelle
Common PipistrelleCommon Pipistrelle
Latin name: Pipistrellus pipistrellus
The Common Pipistrelle is the most commonly encountered bat in Aldernery and is recognisable from it’s broad wings and jerky flight pattern. Often hunts in fixed paths over grasslands, woodland and urban areas. Most active for two hours before and after dawn and dusk respectively. During this period a single individual is capable of consuming over 3000 insects which they eat on the wing.
Echolocation Frequency: Strongest at 45 kHz.
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Anthropocene Epoch, unofficial interval of geologic time, making up the third worldwide division of the Quaternary Period (2.6 million years ago to the present), spanning the period from the second half of the 18th century to the present. It is characterized as the time in which the collective activities of human beings (Homo sapiens) began to substantially alter Earth’s surface, atmosphere, oceans, and systems of nutrient cycling. A growing group of scientists argue that the Anthropocene Epoch should follow the Holocene Epoch (a formal interval of geologic time that spans the most recent 11,700 years). The name Anthropocene is derived from Greek and means the “recent age of man.”
Although American biologist Eugene Stoermer coined the term in the late 1980s, Dutch chemist and Nobelist Paul Crutzen is largely credited with bringing public attention to it at a conference in 2000, as well as in a newsletter printed the same year. In 2008 British geologist Jan Zalasiewicz and his colleagues put forth the first proposal to adopt the Anthropocene Epoch as a formal geological interval.
The scale of human activity
Changes in rock strata and the makeup of the fossils they contain are used to mark the boundaries between formal intervals of geologic time. Throughout Earth’s history, periods of upheaval characterized by mass extinctions, changes in sea level and ocean chemistry, and relatively rapid changes in prevailing climate patterns are captured in the layers of rock. Often these periods mark the end of one interval and the beginning of another. The formalization of the Anthropocene hinges on whether the effects of humans on Earth are substantial enough to eventually appear in rock strata. Most scientists agree that the collective influence of humans was small before the dawn of the Industrial Revolution during the middle of the 18th century; however, advancements in technology occurring since then have made it possible for humans to undertake widespread, systematic changes that affect several facets of the Earth system.
At present, human beings have a profound influence over Earth’s surface, atmosphere, oceans, and biogeochemical nutrient cycling. By 2005, humans had converted nearly two-fifths of Earth’s land area for agriculture. (Cultivated land accounted for one-tenth of the land surface, whereas roughly three-tenths were used for pasture.) An additional one-tenth of Earth’s land area was given over to urban areas by this time. According to some estimates, humans have harvested or controlled roughly one-quarter to one-third of the biomass produced by the world’s terrestrial plants (net primary production) on a yearly basis since the 1990s. Such sweeping control over Earth’s plant production has been attributed in large part to the development of a method of industrial nitrogen fixation called the Haber-Bosch process, which was created in the early 1900s by German chemist Fritz Haber and later refined by German chemist Carl Bosch. The Haber-Bosch process synthesizes ammonia from atmospheric nitrogen and hydrogen under high temperatures and pressures for use in artificial fertilizers and munitions. The industrialization of this process increased the amount of usable nitrogen in the world by 150 percent, which has greatly enhanced crop yields and, along with other technological developments, facilitated the exponential rise in the world’s human population from about 1.6–1.7 billion in 1900 to 6.9 billion by 2010.
As the human population grew, energy use increased, and energy derivation from wood and easily obtained fossil fuels (i.e. petroleum, natural gas, and coal) expanded. Carbon dioxide (CO2) released by cooking fires and other sources during preindustrial times was dwarfed by the amount released by industrial furnaces, boilers, coal-fired power plants, gasoline-powered vehicles, and concrete production during the 20th and early 21st centuries. In the 1950s climate scientists began to track the annual increase in average global carbon dioxide concentrations in the atmosphere, which rose from approximately 316 parts per million by volume (ppmv) in 1959 to 390 ppmv a half century later. Many climatologists contend that the buildup of CO2 in the atmosphere has contributed to a global rise in average surface temperatures of 0.74 °C (1.3 °F) between 1906 and 2005, loss of sea ice in the Arctic Ocean and the breakup of ice shelves along the Antarctic Peninsula, reduction in the size of mountain glaciers, changes in prevailing weather patterns, and more-frequent occurrence of extreme weather events in different parts of the world.
Furthermore, the oceans absorb much of the CO2 released into the atmosphere by human activities, and this absorption has driven the process of ocean acidification. Seawater pH has fallen by 0.1 between about 1750 and 2010, a 30 percent increase in acidity. Marine scientists fear that continued increases in ocean acidity will slow, and possibly cease, the construction of reefs by corals in many parts of the world, dissolve the shells and skeletons of mollusks and corals, and interfere with the metabolic processes of larger marine animals. Since coral reefs are hubs of biodiversity in the oceans, the loss of coral will likely contribute to the demise of multitudes of other marine species either directly, through habitat loss, or indirectly, through changes in marine food chains. Other human-induced changes to the hydrosphere include the damming and diversion of rivers and streams, the rapid extraction of groundwater from freshwater aquifers, and the creation of large oxygen-depleted areas near the mouths of rivers.
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Category:
What is the UV Index?
Article Details
• Written By: Darlene Goodman
• Edited By: Michelle Arevalo
• Last Modified Date: 05 July 2017
• Copyright Protected:
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Many scientists suggest that the sun’s ultraviolet (UV) light waves are harmful to human skin and eye tissue. As a result, the World Health Organization (WHO) created the UV index, a standard system for measuring the amount of UV light that penetrates Earth’s atmosphere. The linear scale is one way for governments and scientists to measure and track ultraviolet light intensity, as well as to warn the public about potential dangers associated with high UV levels.
UV radiation is a specific set of wavelengths on the light spectrum. The waves are shorter than visible light. Often classified as UVA and UVB, these light waves are typically considered more dangerous to the skin and eyes than visible light. The UV index measures the amount of these potentially harmful waves that reaches the lower atmosphere.
There are several factors affecting the ultraviolet light in a given area. First, the position of the sun in the sky is often important. Seasons can affect UV by changing the sun’s distance and angle of light in relation to Earth. Also, UV is often strongest at latitudes closer to the equator.
For the most part, the UV index measures the intensity of light waves at solar noon, or the time of day when the sun is highest in the sky. Sunlight is typically strongest at this time. Solar noon may not be the same as noon on a clock.
Second, atmospheric conditions may affect ultraviolet levels, as well. A thicker atmosphere results in lower radiation, so the UV index level is often different from mountain to valley. Cloud cover may also have an effect, but it does not make a large change, because UV radiation can typically penetrate clouds better than other light wavelengths. In addition, ozone in the high atmosphere may filter harmful UV rays.
Finally, ground reflection may also play a role in the UV index. Snow, water, and sand can reflect UV light. This reflection can intensify the level of ultraviolet rays striking an individual outdoors in these conditions because, not only do they receive UV directly from the sun, but it is reflected back up at them from the ground.
There are several ways to limit an individual’s exposure to UV light. Sunglasses with UVA and UVB filters can protect a person’s eyes from damaging rays. Sunscreens of at least Sun Protection Factor (SPF) 15 are often recommended to protect skin.
Individuals may also wish to seek shade, remain indoors, or at least avoid direct sunlight during peak sunlight hours. Depending on the time zone, this period usually falls between 10:00 a.m. and 4:00 p.m. People may also wish to cover their skin by wearing long sleeves, trousers, and a wide-brimmed hat.
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Besides providing a UV index forecast, a local weather service -- in conjunction with the EPA -- can also issue warnings and advisories if the atmospheric conditions make going outside for extended periods of time potentially harmful.
A UV Alert will be issued if the next day's UV forecast is unusually high for a given area in a given time of year. Essentially it's an opportunity to remind people about sun safety (using sunscreen, covering skin exposed to the sun's rays, wearing sunglasses, etc.) when the risk of sunburn and other damage is especially high.
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Dashboard > OverDrive > Home > Gist
Gist
Added by Ted Husted, last edited by Ted Husted on Nov 04, 2005 (view change)
Labels:
(None)
The OverDrive project is constructing a set of realistic "best practice" applications for the Nexus framework.
Nexus Framework
At a Glance
Presentation Layer ASP.Net Spring.Web Nexus ViewControl Nexus ViewHelper (View)
Application Layer Spring.Net Nexus Catalog Nexus Commands Nexus Contexts (Controller)
Persistence Layer iBATIS.Net DataMaps Database Web Services (Model)
The Nexus framework exposes the application layer to the presentation layer through one or more Helper objects. The Helpers are a facade. A ViewControl base class works with the Helpers to read values from the controls, invoke the application logic, and determine the result. The result may include error messages, a set of values, or a list of values – all ready to display.
By "ready to display", we mean that all the type conversions, text formatting, and localizations have already been done. The UI controls can wrap the text in markup without additional post-processing. Providing ready-to-display text is an essential feature, since Nexus is designed to be used with multiple presentation layers. And, yes, we do consider unit tests to be a presentation layer!
Nexus does not replace presentation frameworks like ASP.NET or Struts. Nexus provides the missing link between presentation frameworks and the business logic that drives your application.
The Nexus back-end is an extended version of the Commons Chain of Responsibility (CoRe). Our Agility product is a port of the original Jakarta Commons CoRe codebase. The Nexus product is our extension to Agility. Nexus adds the features we need to use a Chain of Responsiblity as a business facade.
"Chain of Responsibility pattern"
"Avoid coupling the sender of a request to its receivere by giving more than one object a chance to handle the request. Chain the receiving objects and pass the request along the chain until an object handles it."
Design Patterns by Gamma, Helm, Johnson, and Vlissides (ISBN 0201633612).
The Nexus extensions to Agility feature an advanced Context with attributes common to most applications. There are attributes for storing an Exception, lists of Errors, and lists of generic Messages (Fault, Alerts, and Hints). A convenient "IsNominal" property tells us if there are Alerts or a Fault to display.
Nexus Catalog
The Nexus Catalog makes it easy to retrieve a Context and Command in one call. The caller can pass a Command ID and get back a Context with the Command embedded as an attribute. After filling the Context with values, we can "execute" the Context. The Catalog retrieves the Command, and then passes the Context to its Command. For populating a page, we can also ask for a Command ID, and get back the Context after the Command has executed, in a single call.
Nexus Context
The Context provides a "Criteria" attribute, which is used like a sandbox to store input and output values. The Helpers utilize the Criteria rather than the Context. Other framework Commands automatically convert or format the values between the Criteria and Context.
Request Processing
The Nexus Catalog is extended so that we can execute any given Command as part of a larger Chain of Commands. At runtime, the Catalog creates a Chain and wraps "pre-opt" and "post-op" chains around the instant Command (which could also be a Chain). The pre-op and post-opt chains are configured along with other Commands in the Catalog. In effect, the Catalog creates a "Back Controller" to ensure certain things always happen on each request (Command invocation). The pre-opt Chain, instant Command (or Chain), and post-op Chain work as a request processor.
The standard pre-op Chain converts input, and the standard post-op Chain formats output. But we can also do things like link a logger into the post-op Chain. It's very much like the way Subversion uses pre-commit and post-commit triggers, except that the "commit" is a Command.
Nexus Processors
Pluggable Processors handle the conversion and/or formatting for a kind of field. The "kind" might be a data type, like "integer", or a formatting type, like "telephone number", a combination of both, or even a calculated attribute.
The Processors are linked to a Field Table. Each field that needs special handling can be listed in the Field Table and associated with a Processor. The Field Table and Processors also contain the message templates that are used to create validation errors. For lists returned from a database, a special Processor can iterate over each row of the list, and call the Processor for each column, to create a formatted version in the Critiera.
Input and Output
When listing a Command in the Catalog, we can also list the input and output values the command expects. Input can be specified as "required" or as "related". A Chain automatically aggregates the input field list from its Commands, to insure that all required input is provided. We can also specify a command's output values, and the Chain will consider the output from one command valid input to a subsequent command.
The Helpers use the Command's list of input fields to to automatically read or bind values to the controls. The standard pre-op Chain also uses the field list and Field Table to generate validation errors.
Form Binding and Reading
Most often, you can read or bind an entire form with a single line of code. A complete block, including error checking, may take four or five lines.
Binding
Here is a typical idiom for populating a form:
IViewHelper helper = ExecuteBind(FIND_COMMAND);
bool okay = helper.IsNominal;
if (!okay) Page_Alert = helper;
Reading
Here is a typical idiom for reading input from a form:
IViewHelper helper = ReadExecute(SAVE_COMMAND);
bool okay = helper.IsNominal;
if (!okay) Page_Alert = helper;
For more about Nexus, see the WhitePaper page and PhoneBook application.
OVR-2@OVR-JIRA
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| 0.900579 |
@article{10272/8286, year = {2008}, url = {http://hdl.handle.net/10272/8286}, abstract = {Planktonic foraminifer assemblages from core PRGL1-4 have been studied to reconstruct sea surface temperatures (SST) in the Gulf of Lions during Marine Isotope Stages 6 and 7 based on the modern analog technique. This method consists of a comparison between core and modern sample assemblages assuming that similar planktonic foraminifer assemblages develop under the same ecological conditions and that foraminifer ecological preferences have not changed in time. During stage 6 (glacial) a strong millennial variability is observed in SST, whereas in stage 7 (interglacial) the astronomical forcing controls SST. These features have been already reported in temperature records from other areas out of the Mediterranean Sea, which means that SST in the Gulf of Lions during stages 6 and 7 was influenced by global climate changes. Moreover, some differences exist between paleotemperature records from different areas in the Mediterranean region. In the Gulf of Lions temperature records are more extreme since this area is directly influenced by Mistral and Tramontane winds, which cause important water cooling during cold periods. Furthermore, this study suggests that seasonality in the Gulf of Lions is not influenced by Northern Hemisphere summer insolation}, keywords = {Modern analog technique}, keywords = {Sea surface temperatures}, keywords = {Gulf of Lions}, keywords = {Marine Isotope Stages 6 and 7}, title = {Reconstrucción de paleotemperaturas en el golfo de León durante los estadios isotópicos 6 y 7 utilizando la técnica de los análogos modernos}, title = {Paleotemperature estimates in the Gulf of Lions during Marine Isotope Stages 6 and 7 based on the modern analog technique}, author = {González Mora, Beatriz and Sierro, Francisco Javier and Berné, S.}, }
|
__label__pos
| 0.851441 |
Invasive Species Compendium
Detailed coverage of invasive species threatening livelihoods and the environment worldwide
Abstract
Infection with a trematode parasite differentially alters competitive interactions and antipredator behaviour in native and invasive crayfish.
Abstract
Parasites can have profound effects on host behaviour and species interactions, but the consequences of these impacts are inadequately understood. Three common crayfish in northern Wisconsin and Michigan (native Orconectes virilis, non-native O. propinquus and non-native and invasive O. rusticus) are intermediate hosts for trematode parasites, Microphallus spp. Some species in the genus Microphallus alter host behaviour, increasing their predation risk, but the effects of microphallids on crayfish are unknown. Orconectes propinquus replaces O. virilis in most lakes where they are introduced, and O. rusticus replaces both. These species replacements have major effects on macrophytes, macroinvertebrates and fish. Therefore, differential parasite impacts on crayfish could have community-level effects if competitive outcomes are altered. We examined the shelter affinity of infected and uninfected individuals of all three species in laboratory experiments in the presence and absence of a conspecific. We also observed behaviour during agonistic interactions, and measured boldness by quantifying how quickly crayfish emerged from shelter with a predatory fish present. Infection with Microphallus substantially altered crayfish shelter affinity, shelter competition and boldness, though infection affected each species differently. Infection reduced shelter affinity in O. propinquus and the ability of O. virilis to compete for shelter against uninfected conspecifics. Infected crayfish were bolder in the presence of a predatory fish. Our results suggest that infection with Microphallus alters crayfish behaviour so that all three species are more vulnerable to predation. Orconectes propinquus is likely to suffer the greatest increase in predation when infected, due to a reduced affinity for shelter coupled with increased boldness. In lakes where crayfish species coexist, O. rusticus will probably be less affected by the parasite than either congener. Therefore, crayfish parasites could alter crayfish abundance and species composition in north temperate lakes via behavioural modifications.
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__label__pos
| 0.822099 |
Skip to contentSkip to navigation
Philips Sleep Apnea Device Recall - Click for details
Your Questions — 4 minutes
What types of water should be used in your continuous positive airway pressure (CPAP) device?
Biron
Distilled, demineralized, bottled, ozonated, boiled or tap water? Which ones are recommended, and which ones should be avoided?
CPAP tanks are usually made of materials that tend to degrade when in contact with heated minerals. Therefore, using mineral-free water prevents premature wear and build-up of a whitish residue. That's why only two types of purified water are recommended for CPAP tanks: demineralized water and distilled water.
Distilled water
This water is free of minerals and microorganisms. It is the purest commercially sold water as it contains the fewest organic contaminants.
Demineralized (or deionized) water
This water has been stripped of all its salts and minerals using a process of demineralization (or deionization).
You can find both types of water in grocery stores, pharmacies, and supermarkets. To avoid potential contamination, keep the bottles in a cool, dark place.
Also, there are domestic water demineralization devices, such as reverse osmosis systems. When properly maintained, they make it possible to use tap water in CPAP equipment.
The following types of water are not recommended for everyday use but can be used occasionally for troubleshooting. As they contain minerals (calcium, magnesium, iron, etc.), it is necessary to thoroughly clean the tank after using the device (see the box for cleaning instructions).
Ozonated water
Ozonation is a water sterilization process designed to destroy pathogens. Indeed, ozone is an oxidizing agent and a powerful disinfectant capable of eliminating viruses and bacteria that may have survived the filtration stage.
Boiled water
Contrary to popular belief, using tap water is not harmful to you, as long as it is safe to drink. Still, it can affect the lifespan of your accessories, especially your reservoir.
Tap water
Contrary to popular belief, using tap water is not harmful to you, as long as it is safe to drink. Still, it can affect the lifespan of your accessories, especially your reservoir.
Bottled spring water shares the same properties as tap water.
Note that a water softener does not make tap water usable since it usually removes only calcium and magnesium. Purifiers equipped with a carbon filter (e.g., Brita) reduce the chlorine content without removing minerals and leave impurities that can damage your equipment.
Important reminders
• Never add essential or scented oils to the water as they can damage the device.
• Remember to empty and rinse the tank after each use.
• Every week, wash the tank with warm, soapy water and rinse with plenty of clean water. You can place certain models in the top rack of the dishwasher (check the manufacturer's instructions). Allow it to dry away from direct sunlight or any heat source before reassembling.
• In case of mineral deposits, soak the tank for 10 minutes in a solution of one part white vinegar (5% acetic acid) to two parts water. Empty the solution and rinse thoroughly before letting it dry.
Biron
|
__label__pos
| 0.933816 |
In-depth survey report of carbon monoxide emissions and exposures on express cruisers under various operating conditions
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In-depth survey report of carbon monoxide emissions and exposures on express cruisers under various operating conditions
Filetype[PDF-807.33 KB]
• English
• Details:
• Description:
"Under an interagency agreement with the United States Coast Guard, working in collaboration with an industry consultant, National Institute for Occupational Safety and Health (NIOSH) researchers evaluated carbon monoxide (CO) exposures on ten express cruiser boats from several manufacturers. The evaluated boats were new and included several different models. These boats had gasoline-powered propulsion engines and used gasoline-powered generators to provide electricity for onboard appliances. This study was performed for the U.S. Coast Guard to better understand how CO poisonings may occur on express cruisers, identify the most hazardous conditions, and begin the process of identify controls to prevent/reduce CO exposures. Boats were evaluated while stationary and at multiple speeds, ranging from 5 to 25 miles per hour. CO concentrations were measured by multiple real-time instruments, which were placed at different locations on the boats with overhead, enclosing canopies set at various configurations. Many of the evaluated boats generated hazardous CO concentrations: peak CO concentrations often exceeded 1,100 parts per million (ppm), while average CO concentrations were well over 100 ppm at the stern (rear). Two boats with a combined exhaust system (exhausting at the sides and underwater) had dramatically lower CO concentrations than any of the other evaluated boats (about 40% lower). Based on the results and observations made in this report, the following major findings are summarized below: 1. When the canvas is deployed and boat is underway, CO concentrations exceeded the immediately dangerous to life and health (IDLH) level near the swim platform for many of the evaluated boats. 2. The combination of travel at low speeds into the wind with the canvas fully deployed and no forward hatches, windows or front panels opened maximized the station wagon effect, pulling significant amounts of CO into the cockpit. 3. Different exhaust configurations have a major impact on how CO concentrations are entrained into the cockpit and other occupied areas. Accordingly, boats equipped with underwater exhaust exhibited significantly lower CO concentrations than vessels equipped with other exhaust designs. 4. CO concentrations are typically higher at the stern of the boat and become gradually lower toward the front of the boat. 5. Stationary smoke tests in the engine compartment showed satisfactory sealing of the bulkhead between the engine and adjacent compartments on all boats. Based on the preceding findings, the following recommendations are made to reduce CO concentrations on express cruisers: 1. Boat manufacturers should consider underwater exhaust that will significantly reduce CO concentrations inside the cockpit and other occupied areas compared to surface exhaust. 2. Because of the station wagon effect, some canvas configurations should not be used while boat is moving or propulsion and/or generator engines are running. 3. The possibility of adding force draft blowers into the cabin, creating a positive pressure to minimize potential CO intrusions, should be studied. Auxiliary blowers can be fitted and routed to ventilate the cockpit and swim platform areas in order to minimize negative pressure areas throughout the vessel. 4. Since properly sealed cabin doors directly influenced the CO concentration in the cabin area, door suppliers should be encouraged to develop better sealing methods and designs. 5. Windshield manufacturers should be encouraged to study the possibility of maximizing ventilation of occupied areas by improving the design of the center and side wings of the windshield. 7. Due care should be exercised when designing the powered ventilation system on the engine compartment, locating the air intake on the opposite side of the generator exhaust. Also, potentially moving the intake much farther forward on the vessel would help minimize the intake of CO exhaust into the engine compartment. 8. The development of cleaner burning engines (propulsion and generators) with catalytic converters should continue since they have the potential to greatly reduce CO concentrations to safer levels. 9. The American Boat and Yacht Council (ABYC) should examine their standards and emphasize ventilation problems that can lead to CO intrusions, taking a strong position against surface exhaust designs for propulsion engines." - NIOSHTIC-2 NIOSHTIC no. 20030253 289-11a.pdf
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__label__pos
| 0.707062 |
font size
Polycystic Kidney Disease (cont.)
What is autosomal recessive PKD?
Autosomal recessive PKD is caused by a mutation in the autosomal recessive PKD gene, called PKHD1. Other genes for the disease might exist but have not yet been discovered by scientists. We all carry two copies of every gene. Parents who do not have PKD can have a child with the disease if both parents carry one copy of the abnormal gene and both pass that gene copy to their baby. The chance of the child having autosomal recessive PKD when both parents carry the abnormal gene is 25 percent. If only one parent carries the abnormal gene, the baby cannot get autosomal recessive PKD but could ultimately pass the abnormal gene to his or her children.
The signs of autosomal recessive PKD frequently begin before birth, so it is often called "infantile PKD." Children born with autosomal recessive PKD often, but not always, develop kidney failure before reaching adulthood. Severity of the disease varies. Babies with the worst cases die hours or days after birth due to respiratory difficulties or respiratory failure.
Some people with autosomal recessive PKD do not develop symptoms until later in childhood or even adulthood. Liver scarring occurs in all patients with autosomal recessive PKD and tends to become more of a medical concern with increasing age.
What are the symptoms of autosomal recessive PKD?
Children with autosomal recessive PKD experience high blood pressure, urinary tract infections, and frequent urination. The disease usually affects the liver and spleen, resulting in low blood cell counts, varicose veins, and hemorrhoids. Because kidney function is crucial for early physical development, children with autosomal recessive PKD and decreased kidney function are usually smaller than average size. Recent studies suggest that growth problems may be a primary feature of autosomal recessive PKD.
Medically Reviewed by a Doctor on 3/19/2014
Patient Comments
Viewers share their comments
PKD - Experience Question: Please describe your experience with PKD.
PKD - Autosomal Dominant Symptoms Question: What symptoms did you experience with autosomal dominant PKD?
PKD - Autosomal Dominant Treatment Question: What was the treatment for autosomal dominant PKD?
PKD - Autosomal Recessive Symptoms Question: What symptoms did you or someone you know experience with autosomal recessive PKD?
PKD - Autosomal Recessive Treatment Question: What treatments were effective with your case, or someone you knows case of autosomal recessive PKD?
Source: MedicineNet.com
http://www.medicinenet.com/polycystic_kidney_disease/article.htm
Women's Health
Find out what women really need.
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Search the Medical Dictionary for Health Definitions & Medical Abbreviations
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__label__pos
| 0.724945 |
EPP
地球与行星物理
ISSN 2096-3955
CN 10-1502/P
In situ evidence of resonant interactions between energetic electrons and whistler waves in magnetopause reconnection
Zhi Li, QuanMing Lu, RongSheng Wang, XinLiang Gao, HuaYue Chen
2019, 3(6): 467-473. doi: 10.26464/epp2019048
Keywords: magnetic reconnection, whistler waves, magnetosphere, energetic electrons
Study on electron stochastic motions in the magnetosonic wave field: Test particle simulations
Kai Fan, XinLiang Gao, QuanMing Lu, and Shui Wang
doi: 10.26464/epp2021052
Keywords: magnetosonic waves, electron stochastic motions, bounce resonances, test particle simulations
Supported by Beijing Renhe Information Technology Co. LtdE-mail: [email protected]
|
__label__pos
| 0.899973 |
.htaccessと.htpasswdファイルを使って手動でベーシック認証（Basic Auth）をかける方法
.htaccess
web制作時にサイトを公開するまではベーシック認証をかけておきたいケースがありますよね。レンタルサーバーであれば、レンタルサーバーの管理画面から設定できることもありますが、その他のサーバーを利用している場合には.htaccessと.htpasswdファイルを使って認証をかけるのが一般的です。
やりたいこと
ベーシック認証のポップアップ
ページにアクセスすると、上記のようなベーシック認証（Basic Auth）の画面を表示したいと思います。
ユーザー名とパスワードを設定して、ログインできるようにしていきます。
結論
# 認証タイプをベーシック認証を指定
AuthType BASIC
# 認証の名前
AuthName "Input your ID and Password."
# .htpasswd ファイルまでのフルパス
AuthUserFile "/var/www/html/.htpasswd"
# 全てのユーザーを対象にする
require valid-user
# ユーザー名とパスワードを「:（半角コロン）」で繋げる
user:password
# パスワード例
sinciate:$apr1$kGbO0bjk$g8WnslsTgcEHh84MYqirz/
.htaccessファイルのあるディレクトリ以下のページにベーシック認証が設定されます。
また、パスワードは暗号化することが推奨されています。
パスワードの暗号化について
パスワードは平文でも使用できますが、セキュリティを強化するには暗号化する必要があります。
「ベーシック認証 パスワード 暗号化」などとウェブ検索すると、暗号化してくれるページがいろいろ出てきますが、1つだけご紹介します。
http://www.cityjp.com/cript/crpt.cgi
こちらのサイトで、ユーザー名とパスワードを入れると秒速で.htpasswdファイルに入力する1文を生成してくれます。
ベーシック認証パスワードの暗号化
MD5を使った暗号化が強度が強いので、1番上の1行を使うとよいですね。
三上 龍志｜株式会社シンシエイト
この記事を書いた人
三上 龍志｜株式会社シンシエイト
2005年からWeb制作に従事。システム開発ベンチャーでエンジニアとしてWeb開発、Webコンサルティング会社でマーケター・新規事業開発を経て2015年に当社を創業。顧客の成果に顧客よりも本気になることをテーマに、Webを通じて顧客の事業を加速させるために日々奮闘中。
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マーケティングとWeb制作で
貴社の経営課題を解決します
市場調査や競合調査を始め、企業やサービスの優位性を理解した上で、UI/UX設計やコンテンツマーケティング、SEO・ネット広告を中心としたWebマーケティングを通じて、ビジネスを加速させるご提案をしています。
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__label__pos
| 0.719196 |
LATEST VERSION: 9.5.2 - RELEASE NOTES
Pivotal GemFire® v9.5
SELECT Statement
The SELECT statement allows you to filter data from the collection of object(s) returned by a WHERE search operation. The projection list is either specified as * or as a comma delimited list of expressions.
For *, the interim results of the WHERE clause are returned from the query.
Examples:
Query all objects from the region using *. Returns the Collection of portfolios (The exampleRegion contains Portfolio as values).
SELECT * FROM /exampleRegion
Query secIds from positions. Returns the Collection of secIds from the positions of active portfolios:
SELECT secId FROM /exampleRegion, positions.values TYPE Position
WHERE status = 'active'
Returns a Collection of struct<type: String, positions: map> for the active portfolios. The second field of the struct is a Map ( jav.utils.Map ) object, which contains the positions map as the value:
SELECT "type", positions FROM /exampleRegion
WHERE status = 'active'
Returns a Collection of struct<portfolios: Portfolio, values: Position> for the active portfolios:
SELECT * FROM /exampleRegion, positions.values
TYPE Position WHERE status = 'active'
Returns a Collection of struct<pflo: Portfolio, posn: Position> for the active portfolios:
SELECT * FROM /exampleRegion portfolio, positions positions
TYPE Position WHERE portfolio.status = 'active'
SELECT Statement Results
The result of a SELECT statement is either UNDEFINED or is a Collection that implements the SelectResults interface.
The SelectResults returned from the SELECT statement is either:
1. A collection of objects, returned for these two cases:
• When only one expression is specified by the projection list and that expression is not explicitly specified using the fieldname:expression syntax
• When the SELECT list is * and a single collection is specified in the FROM clause
2. A collection of Structs that contains the objects
When a struct is returned, the name of each field in the struct is determined following this order of preference:
1. If a field is specified explicitly using the fieldname:expression syntax, the fieldname is used.
2. If the SELECT projection list is * and an explicit iterator expression is used in the FROM clause, the iterator variable name is used as the field name.
3. If the field is associated with a region or attribute path, the last attribute name in the path is used.
4. If names cannot be decided based on these rules, arbitrary unique names are generated by the query processor.
DISTINCT
Use the DISTINCT keyword if you want to limit the results set to unique rows. Note that in the current version of GemFire you are no longer required to use the DISTINCT keyword in your SELECT statement.
SELECT DISTINCT * FROM /exampleRegion
Note: If you are using DISTINCT queries, you must implement the equals and hashCode methods for the objects that you query.
LIMIT
You can use the LIMIT keyword at the end of the query string to limit the number of values returned.
For example, this query returns at most 10 values:
SELECT * FROM /exampleRegion LIMIT 10
ORDER BY
You can order your query results in ascending or descending order by using the ORDER BY clause. You must use DISTINCT when you write ORDER BY queries.
SELECT DISTINCT * FROM /exampleRegion WHERE ID < 101 ORDER BY ID
The following query sorts the results in ascending order:
SELECT DISTINCT * FROM /exampleRegion WHERE ID < 101 ORDER BY ID asc
The following query sorts the results in descending order:
SELECT DISTINCT * FROM /exampleRegion WHERE ID < 101 ORDER BY ID desc
Note: If you are using ORDER BY queries, you must implement the equals and hashCode methods for the objects that you query.
Preset Query Functions
GemFire provides several built-in functions for evaluating or filtering data returned from a query. They include the following:
Function Description Example
ELEMENT(expr) Extracts a single element from a collection or array. This function throws a FunctionDomainException if the argument is not a collection or array with exactly one element.
ELEMENT(SELECT DISTINCT *
FROM /exampleRegion
WHERE id = 'XYZ-1').status = 'active'
IS_DEFINED(expr) Returns TRUE if the expression does not evaluate to UNDEFINED. Inequality queries include undefined values in their query results. With the IS_DEFINED function, you can limit results to only those elements with defined values.
IS_DEFINED(SELECT DISTINCT *
FROM /exampleRegion p
WHERE p.status = 'active')
IS_UNDEFINED (expr) Returns TRUE if the expression evaluates to UNDEFINED. With the exception of inequality queries, most queries do not include undefined values in their query results. The IS_UNDEFINED function allows undefined values to be included, so you can identify elements with undefined values.
SELECT DISTINCT *
FROM /exampleRegion p
WHERE IS_UNDEFINED(p.status)
NVL(expr1, expr2) Returns expr2 if expr1 is null. The expressions can be query parameters (bind arguments), path expressions, or literals.
TO_DATE(date_str, format_str) Returns a Java Data class object. The arguments must be String S with date_str representing the date and format_str representing the format used by date_str. The format_str you provide is parsed using java.text.SimpleDateFormat.
COUNT
The COUNT keyword returns the number of results that match the query selection conditions specified in the WHERE clause. Using COUNT allows you to determine the size of a results set. The COUNT statement always returns an integer as its result.
The following queries are example COUNT queries that return region entries:
SELECT COUNT(*) FROM /exampleRegion
SELECT COUNT(*) FROM /exampleRegion WHERE ID > 0
SELECT COUNT(*) FROM /exampleRegion WHERE ID > 0 LIMIT 50
SELECT COUNT(*) FROM /exampleRegion
WHERE ID >0 AND status LIKE 'act%'
SELECT COUNT(*) FROM /exampleRegion
WHERE ID IN SET(1,2,3,4,5)
The following COUNT query returns the total number of StructTypes that match the query’s selection criteria.
SELECT COUNT(*)
FROM /exampleRegion p, p.positions.values pos
WHERE p.ID > 0 AND pos.secId 'IBM'
The following COUNT query uses the DISTINCT keyword and eliminates duplicates from the number of results.
SELECT DISTINCT COUNT(*)
FROM /exampleRegion p, p.positions.values pos
WHERE p.ID > 0 OR p.status = 'active' OR pos.secId
OR pos.secId = 'IBM'
|
__label__pos
| 0.982824 |
"Standard Deviation" Essays and Research Papers
11 - 20 of 500
Standard Deviation and Minimum Order
mean+0.67*SD (standard deviation). According to z table, z equal to 0.67 when probability is 0.75. Therefore, we can calculate quantity for each style include the risk of stock out by using formulate Q*=mean+z*SD. Therefore, we can get the maximum order units for each style in order to avoid stock out. Figure 1 |Style |Price |Average |Standard |2*standard |P=1-8%/(24%+8|Z |Q*=Average+z*SD | | | |forecast |deviation |deviation...
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Probability: Standard Deviation and Pic
9551/SQRT(300),1) = 0.0783 (c) A circuit contains three resistors wired in series. Each is rated at 6 ohms. Suppose, however, that the true resistance of each one is a normally distributed random variable with a mean of 6 ohms and a standard deviation of 0.3 ohm. What is the probability that the combined resistance will exceed 19 ohms? How "precise" would the manufacturing process have to be to make the probability less than 0.005 that the combined resistance of the circuit would exceed 19...
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Random Variable and Standard Deviation
the mean from part b find the standard deviation of the probability distribution. 8. A computer password consists of two letters followed by a five-digit number, none of which can be repeated. After 3 tries the computer locks down and notifies security. a) What is the probability of guessing the correct password on the first try? b) What it the probability of guessing the correct password within three tries? 9. Find the mean and standard deviation of the binomial distribution...
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Standard Deviation and Probability
order is long and uncertain. This time gap is called “lead time.” From past experience, the materials manager notes that the company’s demand for glue during the uncertain lead time is normally distributed with a mean of 187.6 gallons and a standard deviation of 12.4 gallons. The company follows a policy of placing an order when the glue stock falls to a predetermined value called the “reorder point.” Note that if the reorder point is x gallons and the demand during lead time exceeds x gallons...
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Standard Deviation and Gulf View Condominiums
Sales Price | | Days to Sell | | | | | | | | Mean | 474007.5 | Mean | 454222.5 | Mean | 106 | Standard Error | 31194.293 | Standard Error | 30439.72954 | Standard Error | 8.256078 | Median | 437000 | Median | 417500 | Median | 96 | Mode | 975000 | Mode | 305000 | Mode | 85 | Standard Deviation | 197290.03 | Standard Deviation | 192517.7534 | Standard Deviation | 52.21602 | Sample Variance | 3.892E+10 | Sample Variance | 37063085378 | Sample Variance | 2726.513 | Kurtosis...
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Standard Deviation and Double Degree
in analysing the data is determining if outliers exists within the data. The presence of outliers must be evaluated because their existence could distort the data and make it inaccurate. In order to determine if outliers exist the average and standard deviation must be calculated in order to calculate the Z score, which will show, wither or not outliers exist. In this instance to outliers where found present in the data set as all of the data fell within the +3,-3 range, the largest positive outlier...
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Quiz: Standard Deviation and Confidence Interval Estimate
corresponds to a 94% level of confidence. A. 1.88 B. 1.66 C. 1.96 D. 2.33 2. In a sample of 10 randomly selected women, it was found that their mean height was 63.4 inches. Form previous studies, it is assumed that the standard deviation, σ, is 2.4. Construct the 95% confidence interval for the population mean. A. (61.9, 64.9) B. (58.1, 67.3) C. (59.7, 66.5) D. (60.8, 65.4) 3. Suppose a 95% confidence interval for µ turns out to be (120, 310). To make...
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Standard deviation abstract
Standard Deviation Abstract QRB/501 Standard Deviation Abstract Standard Deviations Are Not Perverse Purpose: The purpose of this article is to illustrate how using statistical data, such as standard deviation, can help a cattleman choose the best lot of calf’s at auction. The statistical data used in these decision making processes can also help the cattleman with future analysis of the lots purchased and existing stock. Research Question: How can understanding the standard deviation...
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Biology Homework Stansard Deviation
14, 14, 15, 15, 16. The mean is 14.0mm.What is the best estimate of the standard deviation?
-1mm 5 1000 bananas were collected from a single plantation and weighed.Their masses formed a normal distribution. How many bananas would be expected to be within 2 standard deviations of the mean?
-950 6 In a normal distribution, what percentage of values fall within ±1 standard deviation of the mean and
±2 standard deviations of the mean?
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standard deviation
Standard deviation can be difficult to interpret as a single number on its own. Basically, a small standard deviation means that the values in a statistical data set are close to the mean of the data set, on average, and a large standard deviation means that the values in the data set are farther away from the mean, on average. The standard deviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standard deviation. A small standard deviation can...
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Do Antibiotics Cause Constipation
Introduction :-
Are you taking antibiotics and wondering “Do antibiotics cause constipation?” You are in the right place, We will discuss this in detail
Can Antibiotics Cause Constipation?
Although antibiotics might induce gastrointestinal problems, constipation is not one of them. Antibiotics, on the other hand, might affect the gut microbiota, causing gastrointestinal disorders such as diarrhoea or constipation. The risk of constipation is determined by the antibiotic used and other personal risk factors. If a person develops constipation while taking antibiotics, they should not blame the medication but rather consider boosting their hydration and fibre diet as well as daily exercise to keep their bowels moving. If a person experiences severe symptoms, such as a new or worsening fever, or if their side effects continue to worsen, they should seek medical treatment.
How To Treat Constipation Caused By Antibiotics?
If a person develops constipation while taking antibiotics, they should not blame the medication but rather consider boosting their hydration and fibre diet as well as daily exercise to keep their bowels moving. If the constipation is caused by antibiotics, there are numerous treatments available, including:
Stool softeners, such as docusate sodium, can help soften stools and make them easier to pass.
Laxatives, such as polyethylene glycol (PEG), can help stimulate bowel movements and relieve constipation.
Probiotics: Taking probiotics can help restore the balance of bacteria in the gut and reduce the risk of constipation caused by antibiotics.
Fibre supplements, such as psyllium, can help bulk up stools and encourage regular bowel movements.
Drinking enough fluids, such as water and herbal tea, can help keep stools soft and prevent constipation.
Daily exercise helps bowel movements and prevent constipation.
Before mixing any two medications or beginning any new treatment for antibiotic-induced constipation, consult with your doctor or chemist.
What Is The Time Frame To Recover From Antibiotic Caused Constipation?
There is no specific time frame for recovering from antibiotic-induced constipation. Even after the antibiotics have been removed from a person’s body, the alterations to the stomach that induce constipation may persist. There are two ways antibiotics might cause constipation. The first is by wreaking havoc on gut bacteria, and the second is by depleting the body of critical nutrients that aid in digestion. Antibiotics seldom induce constipation, but they can cause diarrhoea, cramps, and nausea. If constipation is severe, unpleasant, or occurs in conjunction with other gastrointestinal symptoms, a person should consult a doctor.
What Can Increase The Risk Of Constipation?
Some risk factors that may enhance the chance of antibiotic-induced constipation include:
Older people are more likely than younger adults to develop constipation as a result of antibiotics.
The longer a person takes antibiotics, the greater the risk of getting constipation.
Some antibiotics are more likely than others to produce constipation.
People who have pre-existing gastrointestinal issues, such as irritable bowel syndrome (IBS), are more likely to have antibiotic-induced constipation. 1. Poor diet: A diet low in fibre and high in processed foods can increase the risk of antibiotic-induced constipation. Lack of exercise: A sedentary lifestyle can raise the risk of antibiotic-induced constipation.
How Do Antibiotics Affect Gut Bacteria?
Antibiotics can have a substantial impact on the gut microbiome, which is the diverse mix of bacteria that grows in the stomach and aids digestion. Antibiotics can have a number of negative effects on the gut microbiota, including decreased species diversity, changes in metabolic activity, and the selection of antibiotic-resistant organisms, resulting in downstream effects such as antibiotic-associated diarrhoea and recurring difficile infections. Most antibiotics operate by killing or stopping bacteria from developing, but because they can’t tell the difference between good and bad bacteria, they can wreak havoc on the gut’s healthy bacteria. Changes in the gut microbiome can induce a variety of gastrointestinal problems, including infections. diarrhoea.
Are There Any Natural Therapies That Can Treat Constipation Caused By Antibiotics?
While antibiotics almost never cause constipation, there are some natural therapies that can be used to prevent or alleviate constipation while on antibiotics.
Increase your intake of fluids and fibre. Drinking plenty of water and eating high-fibre meals can help keep your intestines flowing and prevent constipation.
Probiotic supplements can help restore the balance of healthy bacteria in the stomach and avoid gastrointestinal problems.
Herbal medicines, such as senna, psyllium, and aloe vera, can be used to ease constipation. However, before using any herbal medicines, consult with a doctor or pharmacist, especially if you are taking antibiotics.
To avoid unwanted alteration of the gut flora, antibiotics should only be used when absolutely essential.
Can Drinking Water Reduce The Risk Of Constipation?
Drinking more water while taking antibiotics can help reduce constipation.
Constipation is frequently caused by dehydration, which makes it difficult to pass a bowel movement. Drinking enough water and being hydrated can therefore help keep the intestines flowing and prevent constipation. Increasing fibre intake and exercising regularly, in addition to drinking more water, can help reduce constipation when taking antibiotics. If constipation is severe, painful, or happens in conjunction with other gastrointestinal symptoms, a person should consult a doctor.
What High-Fibre Foods Can Help Prevent Constipation?
While taking antibiotics, eating high-fibre foods can help prevent constipation. Here are some high-fibre dietary examples:
Beans, lentils, and chickpeas are high in fibre and can aid in digestion.
Almonds, chia seeds, and flaxseeds are high in fibre and can aid in constipation prevention.
Apples, oranges, berries, pears, and figs are high in fibre and can aid in bowel movement.
Broccoli, carrots, peas, and leafy greens like spinach and kale are all high in fibre.
Fibre-rich foods such as brown rice, whole wheat bread, and whole grain pasta can help reduce constipation.
Conclusion
Hence , after going through the blog, you are now able to answer the question “Can antibiotics make you constipated? If you have any questions, let us know in the comments below.
By Caitlyn
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تمام کارهای مربوط به برد مدار چاپی را به ما بسپارید :)
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Practical PCB Layout Tips
Practical PCB Layout Tips
Engineers tend to pay most attention to circuits, the latest components, and code as important parts of an electronics project, but sometimes a critical component of electronics, the PCB layout, is neglected. Poor PCB layout can cause function and reliability problems. This article contains practical PCB layout tips that can help your PCB projects work correctly and reliably.
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?What's a PCB
Overview
One of the key concepts in electronics is the printed circuit board or PCB. It's so fundamental that people often forget to explain what a PCB is. This tutorial will breakdown what makes up a PCB and some of the common terms used in the PCB world.
Blank PCB from the ClockIt Kit
Over the next few pages, we'll discuss the composition of a printed circuit board, cover some terminology, a look at methods of assembly, and discuss briefly the design process behind creating a new PCB.
What's a PCB?
Printed circuit board is the most common name but may also be called "printed wiring boards" or "printed wiring cards". Before the advent of the PCB circuits were constructed through a laborious process of point-to-point wiring. This led to frequent failures at wire junctions and short circuits when wire insulation began to age and crack.
-> Mass of wire wrap
courtesy Wikipedia user Wikinaut <-
A significant advance was the development of wire wrapping, where a small gauge wire is literally wrapped around a post at each connection point, creating a gas-tight connection that is highly durable and easily changeable.
As electronics moved from vacuum tubes and relays to silicon and integrated circuits, the size and cost of electronic components began to decrease. Electronics became more prevalent in consumer goods, and the pressure to reduce the size and manufacturing costs of electronic products drove manufacturers to look for better solutions. Thus was born the PCB.
LilyPad PCB
PCB is an acronym for the printed circuit board. It is a board that has lines and pads that connect various points together. In the picture above, there are traces that electrically connect the various connectors and components to each other. A PCB allows signals and power to be routed between physical devices. Solder is the metal that makes the electrical connections between the surface of the PCB and the electronic components. Being metal, the solder also serves as a strong mechanical adhesive.
Composition
A PCB is sort of like a layer cake or lasagna- there are alternating layers of Continue ...
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Six Things to Consider When Designing Your PCB
Unless your PCB is designed correctly in the first place, you are going to run into issues sooner or later.
Designing a PCB for one of today's products can be very complex, but this aspect of things is often overlooked. Instead, the focus falls upon the more "interesting" aspects of the product, like the FPGAs or MCUs. The fact remains, however, that unless the board is designed correctly in the first place, you are going to run into issues sooner or later.
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The Importance Of IPC Standards For PCB Manufacturing
Technological advances have ensured that Printed Circuit Boards cannot only perform complex functions they can also be produced inexpensively. This is the exact reason why PCBs are an integral part of so many devices. However, the quality of the device is directly proportional to the quality of the PCB used. PCB failure can, therefore, have debilitating consequences wherein entire systems can fail. It is therefore extremely important to stick to some quality measures in the PCB design and manufacturing process.
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Prediction: South-Facing Ivy Growth Smaller than North-Facing Ivy
I predict that the ivy plant that grows on the south side of the wall will be smaller than the north facing ivy. This is because they have less limiting factors affecting them such as availability of light. The south side ivy have no problem with availability of light as the earth is tilted on its axis facing the sun, so this side has more sunlight. The purpose of the leaves to be able to photosynthesise, so I think that the north leaves will be bigger in size as they need a larger surface area in order to photosynthesise at the same rate as the smaller south ivy.
Conclusion:
For the ivy growing on the north side of the wall the results are generally very varied.
The graphs show skewed results, as there is an uneven distribution of growth by the plant, and there is no pattern between the data collected. Between 50-80mm on the north petiole length there are more results, with the highest amount being 6 petioles at 75-80mm in length.
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The north leaf length has the most in the group 30-35mm with 11 petioles in this category. Again there is an uneven distribution, but the numbers seem to decline, as the length gets bigger. The common width is between 40-45mm with 9 and most of the ivy plants having a leaf width of 35-70mm before we see a significant decline at 70-75mm with only one plant.
The ivy growing on the south side has results, which aren't as varied as the north side ivy.
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There is a much more even distribution amongst the plants. This suggests to me that they have less limiting factors acting against them. These limiting factors can affect he rate of photosynthesis in a plant, these factors can be light intensity, carbon dioxide levels and temperature. The equation for photosynthesis is:
Light
CO2 + H2O O2 + C6H12O6
This equation shows that you need the input variables, which in this case is carbon dioxide and water to produce the output variables that are oxygen and glucose. Both light intensity and carbon dioxide levels feature in the equation but temperature doesn't. However photosynthesis is driven by enzymes that work better in warmer conditions, but if the temperature is too hot they become denatured and therefore cannot carry out their function.
The south petiole length of 25-30 mm and 30-35mm have the same number of ivy, which is 11. With 20-25mm and 35-40mm with 9 and 8. This instantly shows a different picture to the north graphs as their results didn't steadily increase and decrease but grew statically and erratically. The south leaf width between 30-35mm there was 14 plants, this was the highest number in the group and the south leaf length had also 14 plants but this time in the 35-40mm group.
If I compare the highest number of results for leaf length, with and petiole for the north and south. I can see that there is a considerable amount of difference in the sizes of the leaf.
North
South
Petiole
75-80mm
6/50
25-30mm 30-35mm
11/50
Width
40-45mm
9/50
30-35mm
14/50
Length
30-35mm
11/50
35-40mm
14/50
Using this table I can see that the south side growing ivy has grown at similar sizes ranging from about 25-40mm. Whilst the north side ivy ranges from 30-80mm which is a 50mm difference on the north side and 15mm on the south side. This tells me that there are more limiting factors affecting the ivy plants on the north side of the wall.
Factors affecting the growth of the ivy on the north side of the wall can be temperature, water and carbon dioxide. All these factors are needed in photosynthesis as shown by the equation. Light
CO2 + H2O O2 + C6H12O6
Plants need to photosynthesise they use the energy for carbohydrates, proteins and fats. If there is an increase or decrease in temperature the enzymes that catalyse this process are denatured. This means that photosynthesis is affected. Also plants need sunlight to photosynthesise so as there is more sunlight on the south side of the wall. I know this as the sun tilts towards the sun like this:
This can show why the petiole lengths are longer as they need to grow longer so that their leaves can reach the sunlight to photosynthesise. This agrees with my prediction as I said that the south side plants would be smaller than the north side plants. This is also proven by the averages of each category shown below in a table.
Averages
North ivy
South ivy
Petiole length
75.1
31.8
Leaf length
46.9
34.66
Leaf width
61.36
31.96
In each category the averages show that the North ivy has a larger petiole length, leaf length and width, as it has had to adapt to its surroundings due to factors affecting it. So this table of results shows that my prediction is correct, as the ivy on the south side of the wall is smaller than the north facing ivy.
Transpiration can also be another limiting factor in this process. Transpiration is the loss of water from a plant. It is caused by evaporation of water from inside the leaves via the stomata. The biggest rate of transpiration occurs in hot, dry and windy conditions. To prevent this from occurring plants have a waxy layer (cuticle) on their leaves, which stops them losing too much water. You will find the plants in hot climates have to adapt by having a thicker layer of wax. This can affect the ivy leave because there will be more water vapour on the south side as temperature is higher, so the air is more saturated causing less transpiration to occur. The north leaves have a large surface area that can aid transpiration but they have long petioles that restrict surface area to make transpiration more difficult, this is an example of a plant adapting to its environment. So the north ivy leaves are more varied than the south as shown by the results, proving my prediction correct.
The results confirm that my prediction is correct. This is due to the earth's tilt on its axis causing the availability of sunlight to be more limited on the north side. This caused the north ivy to grow larger leaves and petioles to deal with the situation, as they would need a bigger surface area to trap the sunlight for the photosynthesis process and longer petioles to reach the sunlight on the south side of the wall. This is shown by the results, which is portrayed by the graphs. In conclusion the petiole lengths, leaf widths and lengths are larger on the north facing ivy wall than the south facing ivy wall, due to the North side being in shadow because of the Earth's axis. Which in turn causes the lengths to be longer and bigger to be able to complete the photosynthesis reaction with the sunlight obtained.
Evaluation
The results show that the generally the sizes of the south ivy are smaller than the north ivy. This is due to the position of the leaves on the wall and what factors have affected their growth. The most important factor that I think caused a difference in these plants is the availability of sunlight due to their position, north or south. This is devised on the fact that my results only show the sizes that the leaves and petioles grew to. If the experiment was done again, then temperature and availability of sunlight could be measured. I would measure sunlight and temperature levels with the use of a solar meter. If the levels recorded were different for example the south receives more sunlight and has a higher temperature this would justify my conclusion. As I said that there are more limiting factors affecting the north ivy plant and sunlight is needed for photosynthesis, and temperature to catalyse the enzymes needed for photosynthesis. This is shown by the results, as the north petiole lengths are longer as they need to grow further to reach the sunlight.
As I myself did not carry out the experiment I have to take into consideration that it was done as a fair test and with the same variables used each time, for example the same ivy plant used to measure leaf length and width and petiole length. From my graphs I can see that there are some anomalous results, the results of the north petiole length for example. The results seem to increase to a peak of 6 between 75-80mm but the next group between 80-85mm there are no results. But this could be to due to inaccurate measurement of the plant or an error in the data collected. Also another reason for anomalous results is genetic difference, which could be due to the limiting factors that have affect the north ivy plants. As the leaves generally have to grow longer and larger to obtain sunlight for photosynthesis, so some of the leaves may grow to excess, likewise they may not even grow to the average size at all because of this.
Also if the leaves were picked randomly from the top or bottom of the plant this would too make a difference as the top leaves would have more sunlight available meaning they would have a smaller surface area. Finally the results given were as whole numbers so there could be a degree of inaccuracy if decimal places were not used. However as my prediction agreed with the results obtained, I would say that the experiment was successful as my hypothesis that the south side ivy plant would be smaller was correct. This enabled me to write a conclusion with the scientific evidence needed to prove my prediction correct. There was enough data given for me to have some good graphs with many different groups and sizes. This too helped to conclude that my hypothesis is correct; as I could determine a ratio, averages and percentages, and also see whether the south plants were smaller than the north plants or vice versa.
To ensure that the measurements recorded were accurate, if I were to do the experiment again, I would increase the sample size from fifty to hundred to get a wider range of results that can prove to be more accurate. Again the averages, ratios and percentages would be recorded to see if they coincided with the prediction. Also I could test the pH of the soil where the ivy plants grow as this too can be a factor that can limit or aid growth, for example if the soil was to acidic or alkaline. I would collect soil samples from each side of the wall and filter them through filter paper into a water beaker. I would then use universal indicator and see what colour the soil changes. I would compare the colours against a pH chart. If they were different then this result would support the conclusion as this could affect the process of photosynthesis.
The colour of the leaves can be recorded against for example a colour chart also the total height can be measured, this can also show the amount of chlorophyll in the plant, which is also needed in photosynthesis. This too can support the conclusion, as I know from my results that the north ivy leaves were bigger in size thus having a larger surface area. The larger surface area could mean that there is more chlorophyll present or the same amount present as the smaller south ivy leaf, if that is the case than genetic variation has occurred and the plant has had to adapt to its surroundings. The total height of the north and south ivy plants can be subtracted to note the difference. Also the location where the plant is growing for example under a tree at the top or bottom of a hill. All these factors can help further the investigation to determine why the dimensions of the north and south ivy plants differ.
Updated: Apr 29, 2023
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Prediction: South-Facing Ivy Growth Smaller than North-Facing Ivy. (2020, Jun 02). Retrieved from https://studymoose.com/ivy-plants-new-essay
Prediction: South-Facing Ivy Growth Smaller than North-Facing Ivy essay
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Example #1
0
public void testLinenoFunctionCall() {
AstNode root = parse("\nfoo.\n" + "bar.\n" + "baz(1);");
ExpressionStatement stmt = (ExpressionStatement) root.getFirstChild();
FunctionCall fc = (FunctionCall) stmt.getExpression();
// Line number should get closest to the actual paren.
assertEquals(3, fc.getLineno());
}
@Override
public AstNode functionCall(AstNode target, Iterable<AstNode> arguments) {
FunctionCall fc = new FunctionCall();
fc.setTarget(target);
if (!Iterables.isEmpty(arguments)) {
fc.setArguments(list(arguments));
}
return fc;
}
Example #3
0
public void testJSDocAttachment4() {
AstRoot root = parse("(function() {/** should not be attached */})()");
assertNotNull(root.getComments());
assertEquals(1, root.getComments().size());
ExpressionStatement st = (ExpressionStatement) root.getFirstChild();
FunctionCall fc = (FunctionCall) st.getExpression();
ParenthesizedExpression pe = (ParenthesizedExpression) fc.getTarget();
assertNull(pe.getJsDoc());
}
Example #4
0
public void testRegexpLocation() {
AstNode root = parse("\nvar path =\n" + " replace(\n" + "/a/g," + "'/');\n");
VariableDeclaration firstVarDecl = (VariableDeclaration) root.getFirstChild();
List<VariableInitializer> vars1 = firstVarDecl.getVariables();
VariableInitializer firstInitializer = vars1.get(0);
Name firstVarName = (Name) firstInitializer.getTarget();
FunctionCall callNode = (FunctionCall) firstInitializer.getInitializer();
AstNode fnName = callNode.getTarget();
List<AstNode> args = callNode.getArguments();
RegExpLiteral regexObject = (RegExpLiteral) args.get(0);
AstNode aString = args.get(1);
assertEquals(1, firstVarDecl.getLineno());
assertEquals(1, firstVarName.getLineno());
assertEquals(2, callNode.getLineno());
assertEquals(2, fnName.getLineno());
assertEquals(3, regexObject.getLineno());
assertEquals(3, aString.getLineno());
}
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Serverside Configuration «Prev Next»
Lesson 7Using the Oracle Net Assistant, part 2
ObjectiveUse the Oracle Net Assistant to choose naming methods.
Net Assistant Choosing Naming Methods
Network - Profile, Service Names, Listeners, Oracle Names Servers
To use the Oracle Net Assistant to choose naming methods, you must first select Profile from the menu hierarchy on the left side of the Assistant interface. The first tab under Profile is the Naming tab, where you can choose the naming methods. By default, Oracle Net will attempt to resolve a service name to a network address using the following three naming methods in the order in which they appear:
1. Local naming (specified in the Oracle Oracle Net Assistant as TNSNAMES)
2. Centralized naming using Oracle Names (specified in the Oracle Oracle Net Assistant as ONAMES)
3. Host naming (specified in the Oracle Oracle Net Assistant as HOSTNAME)
In the next lesson, you will have the opportunity to simulate the procedure for defining trace levels for the listener.
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Focused Ultrasound
Focused Ultrasound
Overview Focused ultrasound is the union of two different innovative technologies: Focused ultrasound– which gives the energy to treat tissue deep in the body precisely and noninvasively. Magnetic resonance or ultrasound imaging— which is used to recognize and target the tissue to be treated, guide and control the treatment in real-time, and confirm the effectiveness…
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fortraveladvicelovers.com
Coral reefs: where are they found, how are they formed and which are the most beautiful
Who I am
Martí Micolau
@martímicolau
EXTERNAL REFERENCES:
SOURCES CONSULTED:
wikipedia.org, lonelyplanet.com
Article rating:
Content warning
Coral reefs are underwater forests rich in living species. Consider that at least 25% of all marine species in the world live in the waters of coral reefs. Also known by the English name Reef, the majority of coral reefs in the world are actually made up of many smaller fractions, connected in a single ecosystem.
Here's everything you need to know about barrier Reef: how it is formed, where it is in the world, and which are the biggest and most beautiful!
Index
1. How the coral reef is formed
2. The most beautiful coral reefs in the world
3. Coral bleaching and death
4. Curiosities about the coral reef
5. User questions and comments
How the coral reef is formed
The spectacular coral reefs they are "built" by the Antozoi, small octopus-shaped organisms that need clear, illuminated and oxygenated waters in order to live. These tiny polyps they gather in colonies called Coralli, and live in symbiosis with unicellular algae called zooxanthellae.
Through the photosynthesis process of these algae, the small organisms constitute a sort of skeleton of calcium carbonate which assumes a protective and support function. Over time, these skeletons merge with each other creating coral structures hard as rock. The structures are called "barriers" when they are separated from the coast by a shallow lagoon. When they are found near the coast, they are called "coral reefs".
The most beautiful coral reefs in the world
As already specified, corals need certain conditions to live, including good lighting, sea temperatures between 20 ° and 30 ° C and high salinity. These conditions unite the areas of the central Pacific and the Australian east coast, not surprisingly, almost all of the existing reefs are concentrated in these areas. On the contrary, the western coasts of the continents are not suitable for developing barriers due to cold currents.
But where are the most beautiful and largest barriers in the world located? Let's find out in the following ranking.
1 - Great Barrier Reef, Australia
La Great Barrier Reef of Australia it is located off the coast of Queensland and is known to be the largest coral reef in the world. It is made up of some 3.000 barrier systems. Just think that it is so big that it is visible from space.
Since 1981 it has been part of the UNESCO World Heritage Site.
2 - Coral reef in the Red Sea, Egypt
The Red Sea coral reef is found off the coasts of Egypt, Israel and Saudi Arabia. Ten percent of the 1.200 species found in this coral reef are unique to this area. This place includes the Blue Hole of Dahab, one of the most popular and dangerous dive sites in the world.
3 - Reef of New Caledonia, New Caledonia
The New Caledonian Barrier Reef, in the South Pacific, is the third longest barrier in the world. More than 1.000 different species - many of which have not yet been classified - live within this coral reef. New Caledonia encloses a 1.500 km circular lagoon and reaches an average depth of 25 meters. In 2008, UNESCO included it among the World Heritage Sites, giving it the name of "Lagoons of New Caledonia".
4 - Mesoamerican Reef, Yucatán, Belize, Guatemala and the islands of the Honduras Bay
The Mesoamerican Reef, located in the Caribbean basin is the largest coral reef in the Atlantic Ocean. The coral reef extends almost 1.126 km, from the Yucatan Peninsula to the Bay of Islands, in Honduras. Over 500 species of fish and 65 types of coral live within this large reef system. It has been a UNESCO World Heritage Site since 1996.
5 - Coral reef of the Maldives Islands, Indian Ocean
The Maldives are the largest coral reef system in the entire Indian Ocean.
The islands that make up the atoll are formed by volcanic eruptions and contain more than 1.300 coral reefs.
6 - Apo Coral Reef, Philippines
The Apo Reef is the largest barrier in the Philippines. This barrier is 800km long and covers 67,877 acres off the coast of Mindoro Island and is surrounded by a mangrove forest. Due to previous problems, in 2007, the Philippine government enacted a ban on reef fishing to help restore and preserve its pristine nature.
7 - Belize barrier reef, Caribbean Sea
The Belize Barrier Reef is a part of the Mesoamerican Reef system. The reef stretches from Ambergris Caye in the north to Cayes Sapodilla in the south.
This coral reef is protected by the UNESCO program, which deals with the world heritage of humanity.
8 - Saya de Malha, Indian Ocean
Saya de Malha Banks in the Indian Ocean is the largest submerged reef in the world. This ridge connects the Seychelles and Mauritius islands to the Mascarene Plateau. Together with its coral reef, the marine habitat facilitates the life of particular species such as turtle and blue whales.
9 - Andros Reef, Bahamas
The coral reef of Andros, in the Bahamas, stretches for approximately 167 km in length. The island is located along the edge of a ocean trench known as the "language of the sea". This means that the barrier extends downwards, this particularity allows it to reach one depth of almost 2 km.
10 - Florida Keys, United States
The Florida Keys Reef system is thethe only coral reef system in North America. This system extends 160km along the southeastern coast of Florida, from Key Biscayne to the Dry Tortugas. The reef is protected as if it were an underwater state park.
Images and videos
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Coral bleaching and death
In recent years, the symbiotic relationship between coral polyps and algae has been altered by rising water temperatures, which in turn are caused by global warming. For reasons not yet fully known, this is leading to the discoloration of corals as well as their progressive death. According to a study conducted by the James Cook University of Australia, over 90% of the Great Barrier Reef has been affected by the bleaching phenomenon.
Other causes of coral death are insane fishing, tourism, ecological imbalances and pollution.
Curiosities about the coral reef
• What is the largest coral reef in the world?
It is the great Australian coral reef, which extends for about 2.300 km
• Is there a coral reef in Sardinia?
There are no coral reefs in the Italian territory. The closest is the Red Sea Reef along the coast of Egypt. Among the main places where you can admire it there is Marsa Alam.
• How long does it take for a coral reef to form? It takes thousands of years and several millions of colonies to form relevant coral structures
Audio Video Coral reefs: where are they found, how are they formed and which are the most beautiful
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Secondary Structure
The term secondary structure refers to the interaction of the hydrogen bond donor and acceptor residues of the repeating peptide unit. The two most important secondary structures of proteins, the alpha helix and the beta sheet, were predicted by the American chemist Linus Pauling in the early 1950s. Pauling and his associates recognized that folding of peptide chains, among other criteria, should preserve the bond angles and planar configuration of the peptide bond, as well as keep atoms from coming together so closely that they repelled each other through van der Waal's interactions. Finally, Pauling predicted that hydrogen bonds must be able to stabilize the folding of the peptide backbone. Two secondary structures, the alpha helix and the beta pleated sheet, fulfill these criteria well (see Figure ). Pauling was correct in his prediction. Most defined secondary structures found in proteins are one or the other type.
Figure 1
Alpha helix. The alpha helix involves regularly spaced H‐bonds between residues along a chain. The amide hydrogen and the carbonyl oxygen of a peptide bond are H‐bond donors and acceptors respectively:
The alpha helix is right‐handed when the chain is followed from the amino to the carboxyl direction. (The helical nomenclature is easily visualized by pointing the thumb of the right hand upwards—this is the amino to carboxyl direction of the helix. The helix then turns in the same direction as the fingers of the right hand curve.) As the helix turns, the carbonyl oxygens of the peptide bond point upwards toward the downward‐facing amide protons, making the hydrogen bond. The R groups of the amino acids point outwards from the helix.
Helices are characterized by the number of residues per turn. In the alpha helix, there is not an integral number of amino acid residues per turn of the helix. There are 3.6 residues per turn in the alpha helix; in other words, the helix will repeat itself every 36 residues, with ten turns of the helix in that interval.
Beta sheet. The beta sheet involves H‐bonding between backbone residues in adjacent chains. In the beta sheet, a single chain forms H‐bonds with its neighboring chains, with the donor (amide) and acceptor (carbonyl) atoms pointing sideways rather than along the chain, as in the alpha helix. Beta sheets can be either parallel, where the chains point in the same direction when represented in the amino‐ to carboxyl‐ terminus, or antiparallel, where the amino‐ to carboxyl‐ directions of the adjacent chains point in the same direction. (See Figure 2 .)
Figure 2
Different amino acids favor the formation of alpha helices, beta pleated sheets, or loops. The primary sequences and secondary structures are known for over 1,000 different proteins. Correlation of these sequences and structures revealed that some amino acids are found more often in alpha helices, beta sheets, or neither. Helix formers include alanine, cysteine, leucine, methionine, glutamic acid, glutamine, histidine, and lysine. Beta formers include valine, isoleucine, phenylalanine, tyrosine, tryptophan, and threonine. Serine, glycine, aspartic acid, asparagine, and proline are found most often in turns.
No relationship is apparent between the chemical nature of the amino acid side chain and the existence of amino acid in one structure or another. For example, Glu and Asp are closely related chemically (and can often be interchanged without affecting a protein's activity), yet the former is likely to be found in helices and the latter in turns. Rationalizing the fact that Gly and Pro are found in turns is somewhat easier. Glycine has only a single hydrogen atom for its side chain. Because of this, a glycine peptide bond is more flexible than those of the other amino acids. This flexibility allows glycine to form turns between secondary structural elements. Conversely, proline, because it contains a secondary amino group, forms rigid peptide bonds that cannot be accommodated in either alpha or beta helices.
Fibrous and globular proteins
The large‐scale characteristics of proteins are consistent with their secondary structures. Proteins can be either fibrous (derived from fibers) or globular (meaning, like a globe). Fibrous proteins are usually important in forming biological structures. For example, collagen forms part of the matrix upon which cells are arranged in animal tissues. The fibrous protein keratin forms structures such as hair and fingernails. The structures of keratin illustrate the importance of secondary structure in giving proteins their overall properties.
Alpha keratin is found in sheep wool. The springy nature of wool is based on its composition of alpha helices that are coiled around and cross‐linked to each other through cystine residues. Chemical reduction of the cystine in keratin to form cysteines breaks the cross‐links. Subsequent oxidation of the cysteines allows new cross‐links to form. This simple chemical reaction sequence is used in beauty shops and home permanent products to restructure the curl of human hair—the reducing agent accounts for the characteristic odor of these products. Beta keratin is found in bird feathers and human fingernails. The more brittle, flat structure of these body parts is determined by beta keratin being composed of beta sheets almost exclusively.
Globular proteins, such as most enzymes, usually consist of a combination of the two secondary structures—with important exceptions. For example, hemoglobin is almost entirely alpha‐helical, and antibodies are composed almost entirely of beta structures. The secondary structures of proteins are often depicted in ribbon diagrams, where the helices and beta sheets of a protein are shown by corkscrews and arrows respectively, as shown in Figure 3 .
Figure 3
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Though it feels like the end to many, the MESA technique is often a new beginning
Though it feels like the end to many, the MESA technique is often a new beginning
MESA is a technique where a urologist obtains sperm from part of the male reproductive tract, such as the epididymis. These collected sperm are used in ICSI. this procedure is used when there are no sperms present in the ejaculation.
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From Many, One
In Stars by Brian Koberlein4 Comments
A single star is a wonder. A million stars is a story.
A star can burn for billions, even trillions of years. With human history spanning mere centuries, how can we possibly understand the lifespan of a star? If we only had the Sun to study, understanding it’s history would be difficult, but we can observe millions of stars, some ancient and some still forming. By looking at these stars as a whole we can piece together the history and evolution of a star. It is similar to taking pictures of a single day on Earth, and using it to piece together the story of how humans are born, live and die.
One of the ways this is done is through a Hertzsprung-Russell (HR) diagram. The brightness of a star is plotted against its color. When we make such a plot, most stars lie along a diagonal line where the bluer the star the brighter it is. Given a large enough sample of stars, we can presume that the ages of stars are randomly distributed. Since most stars lie along this line (known as the main sequence) they must spend most of their lives there. So it’s clear that stars have a long stable period where they burn steadily. Other stars are red, but still quite bright. One would expect red stars to be dimmer than blue stars since they have a lower temperature. In order to be so bright, they must be quite large. These red giants are stars that have swollen up as their cores heat up in a last-ditch effort to continue fusing hydrogen. Some stars are large enough to start fusing helium in this stage. Since helium burns hotter, these stars brighten into blue giants. In the end, however, most stars collapse into white dwarfs when core fusion ends. They become hot but small stars, blue-white in color but quite dim.
While an HR diagram gives us a snapshot of stellar lifetimes, they don’t tell the whole story. Another way to categorize stars is through their spectra. Different elements in a star’s atmosphere absorb particular wavelengths of light. By looking at the pattern of wavelengths absorbed we can determine which elements the star contains. On a basic level can categorize stars by their metallicity. While stars are mainly hydrogen and helium, they contain traces of other elements (which astronomers call metals). The metallicity of a star is by its ratio of iron to helium, known as [Fe/He]. This is expressed on logarithmic scale relative to the ratio of our Sun. So the [Fe/He] of our Sun is zero. Stars with lower metallicity will have negative [Fe/He] values, and ones with higher metallicity have positive values. Since “metals” are formed by fusion in the cores of stars, those stars with higher metallicity must have formed from the remnants of earlier stars. Our Sun is likely a third generation star. One of the things metallicity tells us is that stars toward the center of our galaxy formed earlier than stars in the outer regions. Through millions of stars we not only understand the history of stars but the history of galaxies.
As we continue to gather more data on stars, they continue to tell us a rich collective story.
Comments
1. Great post, as usual.
How varied are the relative abundances of various “metals”, in stars of the same metallicity?
For example, is there much variation when you compare elements produced in stars which do not go supernova (i.e. up to ~Fe), with those which do (up to Pb and Bi)?
1. Author
Metallicity is more of a measure of the components making up the outer layers of a star. Heavier elements beyond iron are produced in the last moments of a star, so you wouldn’t really see them in the atmospheres of stars.
2. If a large enough star begins to fuse iron in the last moments before it goes supernova, does the presence of iron in a later-generation star affect it’s potential lifetime in any way?
I understand that fusing iron is the death knell to a stellar core because it involves a net energy loss to produce it or any elements heavier than it. But, does the presence of iron or heavier elements in the protostellar cloud during the star’s formation limit the lifetime of the star to less than that of one containing a similar mass of hydrogen but with less heavier elements?
Or is it only the process of actual Fe fusion that has any effect, and Fe doping doesn’t ‘poison’ the star to any degree?
1. I think it might make some difference, albeit only a small one.
Even if all the Fe (and Co and Ni) a massive star ‘inherited’ when it formed were to end up in the core, well before the fusion stage before ‘iron fusion’, it’d be a pretty small total amount (even the most ‘metal-rich’ main sequence stars have merely percent levels, combined, of elements other than H and He). So the core collapse might happen somewhat sooner than in really metal-rich stars than really metal-poor ones.
The really big difference is the presence of any metals (astronomer-speak; everything other than H, He, and perhaps Li is a “metal”); Population III stars – those with essentially zero metals – are thought to have quite different properties than the Pop I and Pop II main sequence (MS) stars we see today; H and He – both atoms and ions – have relatively few ‘electronic transition’ energy levels, so radiation transfer is quite different (how the fusion energy generated in the core gets out to the star’s surface; yes, there’s also convection), so stars with much greater masses than the most massive of today’s MS ones can be (relatively) stable … and die in different kinds of supernovae (I think; Brian?).
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Volume 16, Issue 2 (December 1994)
The Effect of Mold Size and Mold Material on Compressive Strength Measurement Using Concrete Cylinders
CODEN: CCAOAD
Format Pages Price
PDF Version 8 $25 ADD TO CART
Abstract
Fourteen concretes were cast, with and without air entrainment and fly ash, using various sizes of cylindrical mold made from different materials. Cylinders were cured in the standard manner and tested for compressive strength, primarily at an age of 28 days. The range of strengths measured was 30 to 50 MPa. Each cast consisted of 60 cylinders, with 12 replicates performed for each of the five types of mold: 150-mm diameter cardboard and plastic molds (notation, C150 and P150); 100-mm diameter plastic molds (P100); and 75-mm cardboard and plastic molds (C75 and P75). The length/diameter ratio of all cylinders was 2.0.
For 150-mm cylinders, the variability of strength measurement, as indicated by the average coefficient of variation (cv), did not depend upon the type of mold material used (cardboard or plastic). For 75-mm cylinders, the same observation was made; cv did not depend upon mold-material.
However, when plastic molds were used, there were significant differences in cv as cylinder size varied. The average cv for P75 cylinders was 4.9 MPa, while it was 3.2 MPa for P150 cylinders; the increase is highly significant. On the other hand, P100 cylinders gave a cv of 3.6 MPa, which is not significantly different than that for P150 cylinders.
For cardboard molds, where 150- and 75-mm cylinders were tested, the cv was not a function of cylinder size.
The effect of mold material and diameter on the magnitude of measured strength was also examined. No distinct trends were observed. Differences in observed strengths due to type and size of mold were influenced by the type of concrete tested and the procedure for initial storage during the first 24 h of curing. However, when all 28-day strengths were considered in paired-t analyses, the following differences were found to be significant: (1) the use of 75-mm molds (C75 and P75) resulted in a higher strength of about 1 MPa, compared to the larger molds (P100, P150, C150); (2) for 150-mm molds, the use of plastic instead of cardboard resulted in a lower strength, about 1 MPa; and (3) the strength obtained from concrete cast into plastic molds increased as the mold size decreased; the difference between P100 and P150 molds was 1 MPa, while the difference between P75 and P150 molds was, on average, 1.6 MPa.
Some of the differences in strength among the mold types was traced to differences in moisture loss during the first 24 h of storage. Cylinders stored under plastic sheet (in accordance with the American Society for Testing and Materials (ASTM) specifications), were about 3 MPa stronger than those stored in sealed plastic bags (in accordance with the Canadian Standards Association (CSA) requirements). Erratum to this paper appears in 17(1).
Author Information:
Day, RL
Professor, University of Calgary, Calgary, Alberta
Stock #: CCA10294J
ISSN: 0149-6123
DOI: 10.1520/CCA10294J
ASTM International is a member of CrossRef.
Author
Title The Effect of Mold Size and Mold Material on Compressive Strength Measurement Using Concrete Cylinders
Symposium , 0000-00-00
Committee C09
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README.md
gruesome - A Groovy port of the QuickCheck unit test framework
HOMEPAGE
http://www.yellosoft.us/quickcheck
INSTALL
$ make install
EXAMPLE
$ make install
$ groovy example.groovy
*** Failed!
[833472555]
+++ OK, passed 100 tests.
+++ OK, passed 100 tests.
LICENSE
FreeBSD
REQUIREMENTS
Optional
DEVELOPMENT
Testing
Ensure the example script works as expected:
$ bundle
$ cucumber
Feature: Run example tests
Scenario: Running example tests # features/run_example_tests.feature:3
Given the program has finished # features/step_definitions/steps.rb:1
Then the output is correct for each test # features/step_definitions/steps.rb:5
1 scenario (1 passed)
2 steps (2 passed)
0m1.167s
Local CI
Guard can automatically run testing when the code changes:
$ bundle
$ guard -G Guardfile-cucumber
...
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Differences Between North- and South-Facing Slopes
Differences Between North- and South-Facing Slopes
••• vovik_mar/iStock/GettyImages
The face a slope presents to the sun – north or south – plays a role in the local climate created on it. This "microclimate" helps determine the types of plants that colonize the slope and influences which animals are drawn to the area seeking their preferred foods and suitable shelter. The basic difference between north- and south-facing slopes – the relative amount and intensity of sunlight they receive – leads to profound ecological differences, similar (but reversed) in the Northern and Southern Hemisphere.
Amount of Sunlight
In the Northern Hemisphere, north-facing slopes in latitudes from about 30 to 55 degrees receive less direct sunlight than south-facing slopes. The lack of direct sunlight throughout the day, whether in winter or summer, results in north-facing slopes being cooler than south-facing slopes. During winter months, portions of north-facing slopes may remain shaded throughout the day due to the low angle of the sun. This causes snow on north-facing slopes to melt slower than on south-facing ones. The scenario is just the opposite for slopes in the Southern Hemisphere, where north-facing slopes receive more sunlight and are consequently warmer. Near the equator, north- and south-facing slopes receive roughly the same amount of sunlight because the sun is almost directly overhead. At the poles, north and south slopes tend to be either shrouded in darkness all winter long, or bathed in sunlight all summer long, with only slight variation between the slopes in spring and fall.
Depth of Soil
Depth of soil on a slope, whether it faces north or south, depends on the steepness of the slope. The steeper the incline, the higher the rate of soil erosion from rain runoff. Soils on steep slopes are primarily made up of rock fragments because pieces of lightweight organic matter, such as leaves, wash away before they can decompose into soil. Slopes that have a gentle incline tend to accumulate a deeper layer of soil. In the Northern Hemisphere, soil on south-facing slopes dries out faster and is warmer than soil on north-facing slopes due to longer exposure to sunlight – the opposite applies in the Southern Hemisphere.
Effect of Rainfall
The amount of rain that falls on a slope and is taken up by existing vegetation is determined by how steep the slope is, rather than whether it faces north or south. Rain runs more quickly off steeper slopes and does not have time to be taken up by plants. Rain falling on less steep inclines stays in the soil longer and is utilized by plants and trees, generally resulting in larger plants and/or colonization of plants with higher hydration needs. Slope aspect can figure into this, however: Vegetation on south-facing slopes in the Northern Hemisphere, for example, has less time to take up water because of the drying effect of the sun.
Effect on Plant Communities
Given the effects of varying solar insolation, plant communities can vary widely between north- and south-facing slopes. In the Northern Hemisphere, warmer south-facing slopes green up sooner in spring, stay greener longer in the fall and tend to be drier than north-facing slopes. Plants that tolerate these hot, dry conditions – which, depending on the region, may be oaks, pines or drought-tolerant shrubs and grasses –grow well on southern slopes in their native range. A few feet away, a cooler, moister north-facing slope with a gradual incline may be dotted with closed mixed-hardwood or conifer forest and shade-tolerant wildflowers. Trees capture indirect sunlight better than low-growing grasses.
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Tundra Characteristics
How Do Mountains Affect Precipitation?
Characteristics of Grassland Biomes
Temperate Woodland & Shrubland Flowers
Types of Swamp Grass
Names of Plants That Live in Grasslands
Native Plants of the Texas Coastal Plains
Texas Geography & Soil Types
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Dont Go!
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Vehicular Power Bus – Hell on Earth?
Connecting a piece of gear to an automotive bus is a terrible thing to do… unless the gear manufacturer understands the perils that await including spikes, surges and dips. Fortunately, military specifications consolidate decades of observations to help us understand the vehicle power bus.
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Bridging ligand
From Wikipedia, the free encyclopedia
Jump to: navigation, search
An example of a μ2 bridging ligand
A bridging ligand is a ligand that connects two or more atoms, usually metal ions.[1] The ligand may be atomic or polyatomic. Virtually all complex organic compounds can serve as bridging ligands, so the term is usually restricted to small ligands such as pseudohalides or to ligands that are specifically designed to link two metals.
In naming a complex wherein a single atom bridges two metals, the bridging ligand is preceded by the Greek character 'mu', μ,[2] with a superscript number denoting the number of metals bound to the bridging ligand. μ2 is often denoted simply as μ. When describing coordination complexes care should be taken not to confuse μ with η ('eta'), which relates to hapticity. Ligands that are not bridging, are called terminal ligands (see figure).
List of bridging inorganic ligands[edit]
Virtually all ligands are known to bridge, with the exception of amines and ammonia.[3] Common inorganic bridging ligands include most of the common anions.
bridging ligand name example
OH hydroxide [Fe2(OH)2(H2O)8]4+, see olation
O2− oxide [Cr2O7]2-, see polyoxometalate
SH hydrosulfido Cp2Mo2(SH)2S2
NH2 amido HgNH2Cl
N3− nitride [Ir3N(SO4)6(H2O)3]4-, see metal nitrido complex
CO carbonyl Fe2(CO)9, see metal carbonyl#Bridging carbonyls
Cl- Chloride Nb2Cl10, see metal halide#Halide ligands
H- Hydride B2H6
CN- Cyanide approx. Fe7(CN)18, see cyanometalate
Many simple organic ligands form strong bridges between metal centers. Many common examples include organic derivatives of the above inorganic ligands (R = alkyl, aryl): OR, SR, NR2, NR2− (imido), PR2 (phosphido, note the ambiguity with the preceding entry), PR2− (phosphinidino), and many more.
Examples[edit]
Bonding[edit]
For doubly bridging (μ2-) ligands, two limiting representation are 4e and 2e bonding interactions. These cases are illustrated in main group chemistry by [Me2Alμ2-Cl]2 and [Me2Al(μ2-Me]2. Complicating this analysis is the possibility of metal-metal bonding. Computational studies suggest that metal-metal bonding is absent in many compounds where the metals are separated by bridging ligands. For example, calculations suggest that Fe2(CO)9 lacks a Fe-Fe bond by virtue of a 3-center, 2-electron bond involving one of three bridging CO ligands.[4]
Representations of two kinds of M-bridging ligand interactions, 3-center, 4 electron bond (left) and 3-center, 2 electron bonding.[4]
{clear left}
Polyfunctional ligands[edit]
Polyfunctional ligands can attach to metals in many ways and thus can bridge metals in diverse ways, including sharing of one atom or using several atoms. Examples of such polyatomic ligands are the oxoanions CO32− and the related Carboxylate, PO43−, and the polyoxometallates. Several organophosphorus ligands have been developed that bridge pairs of metals, a well-known example being Ph2PCH2PPh2.
See also[edit]
References[edit]
1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "bridging ligand".
2. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "µ- (mu)".
3. ^ Werner, H. (2004). "The Way into the Bridge: A New Bonding Mode of Tertiary Phosphanes, Arsanes, and Stibanes". Angew. Chem. Int. Ed. 43 (8): 938–954. doi:10.1002/anie.200300627. PMID 14966876.
4. ^ a b Jennifer C. Green, Malcolm L. H. Green, Gerard Parkin "The occurrence and representation of three-centre two-electron bonds in covalent inorganic compounds" Chem. Commun. 2012, 11481-11503. doi:10.1039/c2cc35304k
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VariablePackage: compilerToCDocOverviewCGDocRelNotesFAQIndexPermutedIndex
Allegro CL version 9.0
Unrevised from 8.2 to 9.0.
8.2 version
peephole-optimize-switch
As with all compiler switch variables, the value of this variable can be t, nil, or a function object that accepts five arguments and returns t or nil. The arguments passed to the function will be the values of the safety, space, speed, debug, and compilation-speed optimization qualities, in that order. nil is equivalent to a function that always returns nil and t to a function that always returns t. When the value is a function and we say t (or true) or nil (or false) in the text below, we mean that the function returns, respectively, t or nil.
When true, the compiler performs peephole optimization. Peephole optimizations include removing redundant instructions, such as a jump to the immediately following location. Initially true when speed is greater than 0.
See compiling.htm for information on the compiler.
Copyright (c) 1998-2012, Franz Inc. Oakland, CA., USA. All rights reserved.
Documentation for Allegro CL version 9.0. This page was not revised from the 8.2 page.
Created 2012.5.30.
ToCDocOverviewCGDocRelNotesFAQIndexPermutedIndex
Allegro CL version 9.0
Unrevised from 8.2 to 9.0.
8.2 version
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NEW DATABASE - 350 MILLION DATASHEETS FROM 8500 MANUFACTURERS
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CD4055BE Texas Instruments CMOS BCD-to-7-Segment LCD Decoder/Driver with Display-Frequency Output 16-PDIP -55 to 125 ri Buy
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two digit 7-segment display with decimal
Catalog Datasheet Results Type PDF Document Tags
Abstract: DRIVE USING THE ST6-REALIZER 1 PRINCIPLE OF OPERATION A 7-Segment Display consists of 7 LED's , components (Figure 3). 3/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 3. 7-segment drive , as per Figure 4. 4/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 4. 7-segment , defined as described in Table 2: Table 2. 7-segment coding in one byte D7 D6 D5 D4 D3 D2 , APPLICATION NOTE 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER by Olivier Rouy ... Original
datasheet
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78.98 Kb
ST62 many seven segment display datasheet abstract
datasheet frame
Abstract: DRIVE USING THE ST6-REALIZER 1 PRINCIPLE OF OPERATION A 7-Segment Display consists of 7 LED's , components (Figure 3). 3/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 3. 7-segment drive , as per Figure 4. 4/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 4. 7-segment , defined as described in Table 2: Table 2. 7-segment coding in one byte D7 D6 D5 D4 D3 D2 , APPLICATION NOTE 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER by Olivier Rouy ... Original
datasheet
6 pages,
116.92 Kb
7-segment LED display common anode 7 digit 7segment display 2 7-segment 7 segments led led 7 segment display Seven-Segment Numeric LED Display 7 SEGMENT DISPLAY COMMON CATHODE "7 Segment Display" common anode 7-segment display common cathode 7 segment display common cathode 7-segment display datasheet abstract
datasheet frame
Abstract: 2-digit, 7-segment type for common-anode LED displays for 2-digit display such as numbers for TV and , 70 CD7211 CD7211 CD7211A CD7211A CD7211M CD7211M CD7211AM CD7211AM Non-multiplexed 4-digit, 7-segment LCD types 3-6 Backplane signal , (°C) of Pins* CA3081 CA3081 CA3082 CA3082 Directly drive 7-segment incandescent and LED displays 7-transistor common , * Operating Ta Range (°C) CA3161 CA3161 BCD-to-7-segment type for common-anode LED displays {in combination with , Sequencer driver Sequentially turns on 1 of 14 characters (2 of 28) when used with two CA3208 CA3208's 35 to 55 ... OCR Scan
datasheet
1 pages,
102.46 Kb
4-DIGIT 7-SEGMENT 7 segment cc CA3082 CA3161 CA3168 CA7211 Common-anode 7-segment LED display POWER SUPPLY WITH 7 SEGMENT DISPLAY CD7211M CD7211AM CD7211A CA3250 CA3251 datasheet abstract
datasheet frame
Abstract: display drive using the ST6-REALIZER Figure 3. 7-segment drive with bitwise lookup table However , 7-segment display drive using the ST6-REALIZER Figure 4. 7-segment drive with bytewise lookup table By , from 0 to 99. 5/8 7-segment display drive using the ST6-REALIZER Figure 5. Multiple digit , AN842 AN842 APPLICATION NOTE 7-segment display drive using the ST6-REALIZER INTRODUCTION Seven , using the ST6-REALIZER is described. June 2008 Rev 2 1/8 7-segment display drive using the ... Original
datasheet
8 pages,
88.69 Kb
led numeric display 7 digit 7-segment 7 segment digital display led 7 segment display 7 segment cathode common cathode 7-segment display 4 digits 7-segment led display common anode 7-segment display common cathode 7 segment display "7 Segment Display" 7 SEGMENT DISPLAY COMMON CATHODE 7 Segment common cathode AN842 AN842 abstract
datasheet frame
Abstract: Series 7-Segment Displays 3-27 3-27 10-Position FNS700 FNS700 DIP Socket Optoelectronic Products General Description The FNS700 FNS700 is a 10-position DIP socket with two rows of five positions each. It is designed for use with all Fairchild 0.362-inch 7-segment LED displays (FND300 FND300 Series). Package Outline I"" .002 , 1-8 LED Bar Graph Display, Digit Hardware Bar Graph Display Device No. Character Height Inches , Data Sheet Page No. FNA12 FNA12 .050 Both Red 12-Element Bar Display None 1.7 200 3-3 Digit Hardware ... OCR Scan
datasheet
2 pages,
28.96 Kb
FNS700 FNA12 LED bar graph FND300 12 element LED bar display 10-POSITION 10 bar led display "LED Bar Graph" datasheet abstract
datasheet frame
Abstract: Driving 7-Segment Gas Discharge Display Tubes with National Semiconductor Drivers Driving 7-Segment Gas , Data AC Coupled From MOS-Output 3 Driving 7-Segment Gas Discharge Display Tubes with National , VOLTAGE CATHODE DECODER DRIVER The DS8880 DS8880 offers 7-segment outputs with high output breakdown voltage of , DS8884A DS8884A decodes four lines of BCD input and drives 7-segment digits of gas-filled displays There are two , INTRODUCTION Circuitry for driving high voltage cold cathode gas discharge 7-segment displays ... Original
datasheet
4 pages,
93.07 Kb
AN-84 BECKMAN C1995 DS7880 DS8880 beckman display DS8884A panaplex II 7 segment decoder TTL 7-Segment Display Driver with Decoder sperry introduction of bcd 7 segment two digit cathode 7-segment decoder gas discharge display sperry DS8880 abstract
datasheet frame
Abstract: select font character 7-segment mode, writing digit data to use font map data with decimal place lit , 7-Segment Displays with the MAX6954 MAX6954 Abstract: This article is how-to guide, intended as a quick learning , versatile display driver, capable of controlling a mix of discrete, 7-segment, 14-segment, and 16-segment , connection scheme for 7-segment digits that is compatible with the MAX6954 MAX6954 multiplex scheme and the built in , illustrates a sixteen digit, 7-segment application circuit for the MAX6954 MAX6954. Table 1. Connection Scheme for ... Original
datasheet
9 pages,
228.78 Kb
7 Segment Displays 7-segment common cathode pin connection MAX6955 MAX6954 digital clock with 7segment APP3210 an3210 common anode 7-segment display 10 pins 4-DIGIT 7-SEGMENT LED DISPLAY common cathode 7-segment display driver common anode 7-segment display driver 7-segment 6 digit clock circuit datasheet abstract
datasheet frame
Abstract: microprocessors or digital systems to an LED display. Included on chip are an 8-byte static display memory, two types of 7-segment decoders, multiplex scan circuitry, and high current digit and segment drivers for , 8-Byte Static Display Memory 7-Segment Hexadecimal and Code B Decoders Output Drive Suitable for LED , ICM7218 ICM7218 Printer Friendly Version 8-Digit LED Microprocessor-Compatible Multiplexed Display , HEXA/CODE B/SHUTDOWN), 4 separate display data input lines, and 3 digit address lines. Display data is ... Original
datasheet
2 pages,
102.35 Kb
ICM7228 ICM7218DIJI 7 SEGMENT DISPLAY ALPHANUMERIC ICM7211AM 4 digit 7 segment LED display common LED hexadecimal display 4 digit 28 segment display 4-DIGIT 7-SEGMENT MULTIPLEX 7 LED DISPLAY TYPES 4 digit 7 segment decoder 4-DIGIT 7-SEGMENT LED DISPLAY MULTIPLEX Common-cathode 7-segment LED display ICM7218 ICM7218AIJI ICM7218 abstract
datasheet frame
Abstract: drain P-channel transistor outputs organized as four 7-segment digits. The devices are available with , devices simplify the task of implementing a cost-effective alphanumeric 7-segment display for , decode true BCD to a 7-segment decimal output. These devices are actually mask-programmable to provide , voltage segment drivers provide four 7-segment digits • Multiplexed BCD input (7235) • High speed processor Interlace (7235M 7235M) • 7-segment hex (0-9, A-F) or Code-B (0-9, dash, E, H, L, P, blank) output ... OCR Scan
datasheet
6 pages,
241.73 Kb
BCD to 7segment decoder common anode ICM7235A 7segment 7-segment display driver ICM7217 ICM7226 POWER SUPPLY WITH 7 SEGMENT DISPLAY 7segment two digit 16 pin NEC Vacuum Fluorescent Display ICM7235 two digit cathode 7-segment decoder BCD to 7segment decoder ICM7235 abstract
datasheet frame
Abstract: 7-segment digits. The devices are available with two input configurations. The basic devices provide four , cost-effective alphanum eric 7-segment display for microprocessor systems, w ithout re quiring extensive ROM or , terminals of a four-digit, 7-segment non-multiplexed vacuum fluorescent display. The outputs are taken from , explicitly in Table 1 Either decoder option will correctly decode true BCD to a 7-segment decimal output. , voltage segment drivers provide tour 7-segment digits · Multiplexed BCD Input (7235) · High speed ... OCR Scan
datasheet
6 pages,
398.6 Kb
ICM7235A driver 7segment binary to 7segment decoder common anode 7235 7 SEGMENT DISPLAY ALPHANUMERIC 4 digit 40 pin IC configuration ICM7234 7235M ICM7234 abstract
datasheet frame
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DISPLAY DRIVE USING THE ST6-REALIZER Figure 3. 7-segment drive with bitwise lookup table However this Rouy 2/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER 1 PRINCIPLE OF OPERATION A 7-Segment Display Don't care g f e d c b a 5/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 4. 7-segment drive Figure 1. Structure of a 7 segment display Table 1. LED's turned on for a given digit X = Don't care 7-segment coding in one byte The resulting byte is then transferred to 7 digout components by using an
www.datasheetarchive.com/files/stmicroelectronics/stonline/books/ascii/docs/4389-v5.htm
STMicroelectronics 11/01/2000 9.19 Kb HTM 4389-v5.htm
7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 3. 7-segment drive with bitwise lookup table ST6-REALIZER 1 PRINCIPLE OF OPERATION A 7-Segment Display consists of 7 LED's arranged in a figure-eight Figure 4. 7-segment drive with bytewise lookup table By using the simulation features of the : Table 2. 7-segment coding in one byte The resulting byte is then transferred to 7 digout components by Application Note ST6 - 7 SEGMENT DISPLAY DRIVE
www.datasheetarchive.com/files/stmicroelectronics/stonline/books/ascii/docs/4389-v4.htm
STMicroelectronics 16/01/2001 9.22 Kb HTM 4389-v4.htm
SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 4. 7-segment drive with bytewise lookup table By using ST6-REALIZER 1 PRINCIPLE OF OPERATION A 7-Segment Display consists of 7 LED's arranged in a figure-eight Figure 3. 7-segment drive with bitwise lookup table However this approach implies the use of 7 lookup table is a byte defined as described in Table 2: Table 2. 7-segment coding in one byte The resulting application, where two digits are used to display values ranging from 0 to 99. 6/6 7 SEGMENT DISPLAY DRIVE
www.datasheetarchive.com/files/stmicroelectronics/stonline/books/ascii/an/4389.htm
STMicroelectronics 17/09/1999 9.31 Kb HTM 4389.htm
(Figure 3). 4/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 3. 7-segment drive with 7-Segment Display consists of 7 LED's arranged in a figure-eight pattern, such that by selectively pow- DRIVE USING THE ST6-REALIZER Figure 4. 7-segment drive with bytewise lookup table By using the the optional Decimal Point character. Figure 1. Structure of a 7 segment display Table 1. LED's as described in Table 2: Table 2. 7-segment coding in one byte The resulting byte is then
www.datasheetarchive.com/files/stmicroelectronics/stonline/books/ascii/docs/4389-v3.htm
STMicroelectronics 25/05/2000 9.19 Kb HTM 4389-v3.htm
7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 3. 7-segment drive with bitwise lookup table ST6-REALIZER 1 PRINCIPLE OF OPERATION A 7-Segment Display consists of 7 LED's arranged in a figure-eight Figure 4. 7-segment drive with bytewise lookup table By using the simulation features of the : Table 2. 7-segment coding in one byte The resulting byte is then transferred to 7 digout components by Application Note ST6 - 7 SEGMENT DISPLAY DRIVE
www.datasheetarchive.com/files/stmicroelectronics/stonline/books/ascii/docs/4389-v1.htm
STMicroelectronics 20/10/2000 9.36 Kb HTM 4389-v1.htm
components (Figure 3). 4/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 3. 7-segment drive with 7-Segment Display consists of 7 LED's arranged in a figure-eight pattern, such that by selectively pow- ST6-REALIZER Figure 4. 7-segment drive with bytewise lookup table By using the simulation features of the the optional Decimal Point character. Figure 1. Structure of a 7 segment display Table 1. LED's turned described in Table 2: Table 2. 7-segment coding in one byte The resulting byte is then transferred to 7
www.datasheetarchive.com/files/stmicroelectronics/stonline/books/ascii/docs/4389.htm
STMicroelectronics 02/04/1999 7.48 Kb HTM 4389.htm
components (Figure 3). 4/6 7 SEGMENT DISPLAY DRIVE USING THE ST6-REALIZER Figure 3. 7-segment drive with 7-Segment Display consists of 7 LED's arranged in a figure-eight pattern, such that by selectively pow- ST6-REALIZER Figure 4. 7-segment drive with bytewise lookup table By using the simulation features of the the optional Decimal Point character. Figure 1. Structure of a 7 segment display Table 1. LED's turned described in Table 2: Table 2. 7-segment coding in one byte The resulting byte is then transferred to 7
www.datasheetarchive.com/files/stmicroelectronics/stonline/books/ascii/docs/4389-v2.htm
STMicroelectronics 14/06/1999 7.44 Kb HTM 4389-v2.htm
or digital systems to an LED display. Included on chip are an 8-byte static display memory, two types of 7-segment decoders, multiplex scan circuitry, and high current digit and segment drivers for Multiplex Scan Circuitry 8-Byte Static Display Memory 7-Segment Hexadecimal and Code B Decoders Output ( WRITE and HEXA/CODE B/ SHUTDOWN ), 4 separate display data input lines, and 3 digit address lines. Display data is written into the internal memory by setting up a digit address and strobing the WRITE
www.datasheetarchive.com/files/intersil/device_pages/device_icm7218.html
Intersil 07/09/2006 22.52 Kb HTML device_icm7218.html
display. Included on chip are an 8-byte static display memory, two types of 7-segment decoders, multiplex Display Memory 7-Segment Hexadecimal and Code B Decoders Output Drive Suitable for LED Displays Voltage Reference ICM7218 ICM7218 ICM7218 ICM7218 8-Digit LED Microprocessor-Compatible Multiplexed Display Decoder separate display data input lines, and 3 digit address lines. Display data is written into the internal and Random Access Versions Decimal Point Drive On Each Digit Related Documentation Application
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Intersil 13/10/2005 22.43 Kb HTML device_icm7218-v1.html
No abstract text available
www.datasheetarchive.com/download/56814331-186286ZD/demosafe.zip (Demo30.asm)
STMicroelectronics 04/02/2000 86.54 Kb ZIP demosafe.zip
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Usability, Customer Experience & Statistics
What's a Z-Score and Why Use it in Usability Testing?
Jeff Sauro • September 17, 2004
The Power of Z
A common statistical way of standardizing data on one scale so a comparison can take place is using a z-score. The z-score is like a common yard stick for all types of data. Each z-score corresponds to a point in a normal distribution and as such is sometimes called a normal deviate since a z-score will describe how much a point deviates from a mean or specification point.
In Six Sigma parlance, z-score and process sigma are used interchangeably and are sometimes called z-equivelents. Strictly speaking, the process sigma and z-equivalents are loosely tied to the statistical z-score. The statistical z-score has very strict definitions derived from the rules of the normal distribution. For most applications in Six Sigma, ignoring some of those constraints is innocuous. In usability testing the benefit of the standardization from process sigmas allow us to meaningfully compare disparate measures like task completion and time on task.
The z-score/process sigma is calculated by subtracting your sample mean from a target data point and dividing by the target standard deviation. This value is a measure of the distance in standard deviations of a sample from the mean and is expressed using the Greek letter σ. If your sample is 3 standard deviations from the spec limit, you would describe your process as 3 sigma. or 3σ
The further away a sample is from the spec limit the higher the z-score and process sigma. A higher process sigma means a less defective process. The term Six Sigma originates from the z-score. 6σ means that six standard deviations lie between the mean of a sample and the nearest specification limit. To visualize the Z-score see the Interactive Graph of the Standard Normal Curve
Each process sigma has two equivalent values which provide a meaningful way to compare data and understand how defective a process is:
1. DPMO: Each expresses the probability of a defect in terms of a defect per million opportunities or DPMO. That is, if a condition were to occur one million times, how many times out of that one million would a defect occur? A process sigma of .5 is equal to 308,000 defects per million opportunities. And a process sigma of 2.5 means that 6,210 out of 1 million times there will be a defect. For a sample that is 6σ, the DPMO is .0.001. Some organizations prefer to think in terms of defects per opportunities instead of the more abstract "standard deviations above the spec limit."
2. Probability of a Defect: The process sigma can also be described in terms of a probability of a defect. A z-score of .5 means there is a 30% probability of encountering a defect. A z-score of .25 means there is a 40% probability of a defect. For a sample that is 6σ, the probability of a defect is .0000001%. Note: Values do not include a 1.5σ shift.
Why use a Process Sigma?
The process sigma is helpful in three ways:
1. It allows you to compare disparate types of data (seconds, which are a continuous measurement with task completion which is binary with errors which are discrete count data)
2. It provides you with a probability of a defect
3. You can meaningfully compare two different products or processes:
1. The process sigma for one release of a software product can be compared to subsequent versions
2. You can compare two different products' process sigmas
3. You can compare one module of the same product with a different module on the same product
4. You can use the properties of the normal distribution to aide in assessing and improving your data set.
About Jeff Sauro
Jeff Sauro is the founding principal of MeasuringU, a company providing statistics and usability consulting to Fortune 1000 companies.
He is the author of over 20 journal articles and 5 books on statistics and the user-experience.
More about Jeff...
Learn More
Related Topics
Z-score
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Posted Comments
There are 13 Comments
January 25, 2013 | Soma wrote:
I want to comparision of two Z score but I have not getting more idea about the comparision of the Z score.
October 3, 2012 | anonymous wrote:
Comprehensive
September 11, 2012 | Khalil wrote:
I am UX (QoE) Scientist and I found this post easy to understand and informative. Thanks
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April 18, 2012 | pramod wrote:
It is very good to find these kind of learning material on web. it makes the learning easier.
October 13, 2011 | anonymous wrote:
The Interactive graph drives the point home like a nail.
February 24, 2011 | PALASH ROY wrote:
Very good thing
November 23, 2010 | Dr. John wrote:
wonderful! simple and comprehensive.
September 11, 2009 | Barbara wrote:
Discuss types of research where using the t statistic may be an appropriate alternative to using a z-score.
August 25, 2008 | DERYA TERZIOGLU wrote:
what is the difference between z score and sigma value?
May 28, 2008 | samar wrote:
that was great job thank you very much
May 28, 2008 | samar wrote:
that was great job thank you very much
March 22, 2008 | Michael wrote:
How do you compare two z scores to see if there is a significant difference between the two? would that be a t-test, or something else?
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Jeff's Books
Customer Analytics for DummiesCustomer Analytics for Dummies
A guidebook for measuring the customer experience
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Quantifying the User Experience 2nd Ed.: Practical Statistics for User ResearchQuantifying the User Experience 2nd Ed.: Practical Statistics for User Research
The most comprehensive statistical resource for UX Professionals
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Excel & R Companion to Quantifying the User ExperienceExcel & R Companion to Quantifying the User Experience
Detailed Steps to Solve over 100 Examples and Exercises in the Excel Calculator and R
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A Practical Guide to the System Usability ScaleA Practical Guide to the System Usability Scale
Background, Benchmarks & Best Practices for the most popular usability questionnaire
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72 Answers to the Most Common Questions about Quantifying the Usability of Websites and Software
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Skip to Content
Medical Term:
anticipation
Pronunciation: an-tis′i-pā′shŭn
Definition:
1. Appearance before the appointed time of a periodic symptom or sign.
2. Progressively earlier age of manifestation of a hereditary disease in successive generations; may be factitious (because of heightened awareness of early signs of the disease or because these signs are more conspicuous in the young) or authentic (because of progressive loss of epistatic and modifier genes by recombination and segregation, or because of expansion of unstable alleles in successive generations).
3. An increase in the severity of a phenotype in successive generations of a family, often associated with an increase in the number of trinucleotide repeats in a causative gene (e.g., fragile X syndrome, myotonic dystrophy, Huntington disease).
(web1)
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Developer
Additional Logging and Tracing Code Samples
These examples illustrate how to include code in your apps for uploading logs and traces for Store apps and .NET applications.
These examples illustrate basic implementations, and as a result the upload process cannot be canceled. To cancel the upload, create a CancellationTokenSource and pass its CancellationToken as the last parameter of the SendAsync method of the netclient's instance. Then you can show a button, for example "Cancel" and then you can signal the CancellationToken. The cancellation of the upload process is then handled by the Networking and Supportability libraries.
Windows Store app uploader example code
class UploadResult : SAP.Supportability.IUploadResult {
public int ResponseStatusCode { get; set; }
public string Hint { get; set; }
}
class SupportabilityUploader : SAP.Supportability.IUploader {
SAP.Net.Http.HttpClient clientRef = null;
Uri serverUri= null;
public SupportabilityUploader(SAP.Net.Http.HttpClient client, Uri host, bool uploadBtx = true) {
if (client == null) throw new ArgumentNullException("client");
if (host == null) throw new ArgumentNullException("host");
this.serverUri = new UriBuilder(host.Scheme, host.Host, host.Port, (uploadBtx ? "btx" : "clientlogs")).Uri;
this.clientRef = client;
}
public IAsyncOperation<SAP.Supportability.IUploadResult> SendAsync(IReadOnlyDictionary<string, string> headers, Windows.Storage.Streams.IInputStream payload) {
return Task.Run<SAP.Supportability.IUploadResult>(async () => {
var result = await this.clientRef.SendAsync(() => {
var request = new HttpRequestMessage(HttpMethod.Post, this.serverUri) {
Content = new StreamContent(payload.AsStreamForRead())
};
foreach (var header in headers) {
request.Content.Headers.TryAddWithoutValidation(header.Key, header.Value);
}
return request;
});
return new UploadResult() { ResponseStatusCode = (int)result.StatusCode,
Hint = await result.Content.ReadAsStringAsync() };
}).AsAsyncOperation();
}
}
Windows .NET application uploader example code (only the method which is different from the Store app)
public async Task<SAP.Supportability.IUploadResult> SendAsync(IReadOnlyDictionary<string, string> headers, System.IO.Stream payload) {
var result = await this.clientRef.SendAsync(() => {
var request = new HttpRequestMessage(HttpMethod.Post, this.serverUri) {
Content = new StreamContent(payload)
};
foreach (var header in headers) {
request.Content.Headers.TryAddWithoutValidation(header.Key, header.Value);
}
return request;
});
return new UploadResult() { ResponseStatusCode = (int)result.StatusCode,
Hint = await result.Content.ReadAsStringAsync() };
}
Log creation upload example code for Store Apps and .Net applications
var logManager = SAP.Supportability.SupportabilityFacade.Instance.ClientLogManager;
logManager.SetLogLevel(SAP.Supportability.Logging.ClientLogLevel.Info);
logManager.SetLogDestination(SAP.Supportability.Logging.ClientLogDestinations.FileSystem | SAP.Supportability.Logging.ClientLogDestinations.Console);
var logger = logManager.GetLogger("testLogger");
logger.LogWarning("sample");
logger.LogError("sample error message");
logger.LogError("sample log 2");
string message = null;
try {
await SAP.Supportability.SupportabilityFacade.Instance.ClientLogManager.UploadClientLogsAsync(new SupportabilityUploader(httpClient, serverUri, false));
} catch (Exception ex) {
var supportabilityException = ex as SAP.Supportability.ISupportabilityException;
message = ex.Message + ((supportabilityException != null) ? ("(" + supportabilityException.UploadResult.ResponseStatusCode + ")") : "");
}
BTX generation and upload example for Store apps and .Net applications
var traceManager = (SAP.Supportability.Tracing.E2ETraceManager)SAP.Supportability.SupportabilityFacade.Instance.E2ETraceManager;
traceManager.ClientHost = "WinDemo-Client";
traceManager.TraceLevel = SAP.Supportability.Tracing.E2ETraceLevel.Low;
var transaction = await traceManager.StartTransactionAsync("NewTransactionWin");
var step = transaction.StartStep();
var request = step.StartRequest();
request.SetRequestLine("GET http://www.test.com HTTP/1.1");
request.SetRequestHeaders(new Dictionary<string, string> {
{"SAP-PASSPORT",request.PassportHttpHeader} ,
{"X-CorrelationID","correlationID0101"}
});
request.SetByteCountSent(100);
request.EndRequest();
step.EndStep();
transaction.EndTransaction();
string message = null;
try {
await SAP.Supportability.SupportabilityFacade.Instance.E2ETraceManager.UploadBtxAsync(new SupportabilityUploader(httpClient, serverUri));
} catch (Exception ex) {
var supportabilityException = ex as SAP.Supportability.ISupportabilityException;
message = ex.Message + ((supportabilityException != null) ? ("(" + supportabilityException.UploadResult.ResponseStatusCode + ")") : "");
}
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Skip to main content
• Illustration by Cyndie C.H. Wooley.
File Size: 640 KB
Comprehensive Ophthalmology
Comparison of sporadic retinoblastoma (A), where 2 independent mutations (“hits”) in the RB1 gene occur in a somatic cell, with hereditary retinoblastoma (B), where a germline mutation is present in every cell and second mutations can arise in multiple cells, leading to multiple tumors.
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--- old/src/share/classes/sun/applet/AppletPanel.java 2013-11-18 10:54:17.000000000 +0400 +++ new/src/share/classes/sun/applet/AppletPanel.java 2013-11-18 10:54:17.000000000 +0400 @@ -794,18 +794,13 @@ doInit = true; } else { // serName is not null; - InputStream is = (InputStream) - java.security.AccessController.doPrivileged( - new java.security.PrivilegedAction() { - public Object run() { - return loader.getResourceAsStream(serName); - } - }); - ObjectInputStream ois = - new AppletObjectInputStream(is, loader); - Object serObject = ois.readObject(); - applet = (Applet) serObject; - doInit = false; // skip over the first init + try (InputStream is = AccessController.doPrivileged( + (PrivilegedAction)() -> loader.getResourceAsStream(serName)); + ObjectInputStream ois = new AppletObjectInputStream(is, loader)) { + + applet = (Applet) ois.readObject(); + doInit = false; // skip over the first init + } } // Determine the JDK level that the applet targets. @@ -1239,20 +1234,13 @@ // append .class final String resourceName = name + ".class"; - InputStream is = null; byte[] classHeader = new byte[8]; - try { - is = (InputStream) java.security.AccessController.doPrivileged( - new java.security.PrivilegedAction() { - public Object run() { - return loader.getResourceAsStream(resourceName); - } - }); + try (InputStream is = AccessController.doPrivileged( + (PrivilegedAction) () -> loader.getResourceAsStream(resourceName))) { // Read the first 8 bytes of the class file int byteRead = is.read(classHeader, 0, 8); - is.close(); // return if the header is not read in entirely // for some reasons.
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Documentation API - CvIT.ca
An application programming interface (API) is a protocol intended to be used as an interface by software components to communicate with each other.
An API may include specifications for routines, data structures, object classes, and variables.
When used in the context of web development, an API is typically defined as a set of Hypertext Transfer Protocol (HTTP) request messages, along with a definition of the structure of response messages, which is usually in an Extensible Markup Language (XML) or JavaScript Object Notation (JSON) format.
While "Web API" is virtually a synonym for web service, the recent trend (so-called Web 2.0) has been moving away from Simple Object Access Protocol (SOAP) based services towards more direct Representational State Transfer (REST) style communications.
Note : API accepts HTML POST data only
All data should be POSTed to [ https://api.cvit.ca ]. A single job array contains the following elements:
title job title, string
short_descriptionjob description summary, string [optional]
long_descriptionjob description long, string
keywordskeywords, string [optional but strongly recommended]
id job id, integer - required only for update, deactivate, activate, delete tasks.
apikeyapi key, string - Provided by CvIT.ca - contact us [ [email protected] ]
taskapi task, string [insert, update, deactivate, activate, delete]
INSERT method example
//php example of making an 'insert' api call
$post_string = 'jobs[0][title]='.urlencode('Technological Architect').'&'.
'jobs[0][short_description]='.urlencode('A short description').'&'.
'jobs[0][long_description]='.urlencode('A long description').'&'.
'jobs[0][keywords]='.urlencode('network, Montreal, architecture').'&'.
'apikey=AbCdEfGhIjKlMnOpQrStUvXz01234567&'. // api key here
'task=insert';
$url = 'https://api.cvit.ca';
$ch = curl_init($url);
curl_setopt($ch, CURLOPT_POST, 1);
curl_setopt($ch, CURLOPT_POSTFIELDS, $post_string);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
$response = curl_exec($ch);
curl_close($ch);
print($response);
?>
UPDATE method example
//php example of making an 'update' api call
$post_string = 'jobs[0][title]='.urlencode('PHP Programmer').'&'.
'jobs[0][short_description]='.urlencode('A short description about the job').'&'.
'jobs[0][long_description]='.urlencode('A long description about the job').'&'.
'jobs[0][keywords]='.urlencode('php, New York, api, architecture').'&'.
'jobs[0][id]='.urlencode('10442').'&'.
'apikey=AbCdEfGhIjKlMnOpQrStUvXz01234567&'. // api key here
'task=update';
$url = 'https://api.cvit.ca';
$ch = curl_init($url);
curl_setopt($ch, CURLOPT_POST, 1);
curl_setopt($ch, CURLOPT_POSTFIELDS, $post_string);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
$response = curl_exec($ch);
curl_close($ch);
print($response);
?>
DEACTIVATE method example
//php example of making a 'deactivate' api call
$post_string = 'jobs[0][id]='.urlencode('10442').'&'.
'apikey=AbCdEfGhIjKlMnOpQrStUvXz01234567&'. // api key here
'task=deactivate';
$url = 'https://api.cvit.ca';
$ch = curl_init($url);
curl_setopt($ch, CURLOPT_POST, 1);
curl_setopt($ch, CURLOPT_POSTFIELDS, $post_string);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
$response = curl_exec($ch);
curl_close($ch);
print($response);
?>
ACTIVATE method example
//php example of making an 'activate' api call
$post_string = 'jobs[0][id]='.urlencode('10442').'&'.
'apikey=AbCdEfGhIjKlMnOpQrStUvXz01234567&'. // api key here
'task=activate';
$url = 'https://api.cvit.ca';
$ch = curl_init($url);
curl_setopt($ch, CURLOPT_POST, 1);
curl_setopt($ch, CURLOPT_POSTFIELDS, $post_string);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
$response = curl_exec($ch);
curl_close($ch);
print($response);
?>
DELETE method - IS NOT YET IMPLEMENTED
|
__label__pos
| 0.969944 |
Acoustic Neuroma Association
600 Peachtree Parkway
Suite 108
Cumming, GA 30041
Keywords
ACOUSTIC
pertaining to hearing
ACOUSTIC NEUROMA (AN)
benign tumor of the eighth cranial nerve
ACOUSTIC NEUROMA ASSOCIATION OF CANADA (ANAC)
a registered non-profit organization in Canada with similar purposes to ANA
AUDIOGRAM
a chart of hearing acuity recorded during hearing tests
AUDIOLOGIST
a medical professional who assesses and manages hearing and balance-related disorders
AUDIOVESTIBULAR SPECIALIST
a medical professional specializing in the diagnosis and treatment of hearing, balance and communication problems, including tinnitus
AUDITORY BRAINSTEM IMPLANT (ABI)
a type of hearing device that bypasses the cochlea in the middle ear and the auditory nerve and is implanted in the brainstem
BENIGN
not malignant, non-cancerous: does not invade surrounding tissue or spread to other parts of the body
BILATERAL
pertaining to both sides of the body
BONE ANCHORED HEARING AID
a hearing device that works through bone conduction with a sound processor attached to a small titanium implant
The sound processor is placed on the deaf side, behind the ear and sound is transferred through the bone of the skull, stimulating the cochlea in the hearing ear. The brain is then able to distinguish between the sounds that it receives from the deaf side, via this system, from the sound that it receives directly from the hearing ear. This ultimately results in the sensation of hearing from the deaf side.
BRAINSTEM
connects the upper brain to the spinal cord; is less than three inches long
BRITISH ACOUSTIC NEUROMA ASSOCIATION (BANA)
a registered charity organization in the United Kingdom that is dedicated to promoting the exchange of mutual support and information among individuals affected by acoustic neuromas, with similiar purposes to ANA
CENTIMETER (cm)
.394 inch (2.54 cm equals one inch) - ten millimeters equal one centimeter
CEREBELLOPONTINE ANGLE
space bounded by the petrous bone, brainstem, and cerebellum, and through which cranial nerves 6-11 pass
CEREBELLUM
located behind the brainstem, extending from the brainstem out toward each mastoid bone; carries 11% of the brain's weight and controls muscular coordination
COCHLEAR IMPLANT (CI)
CI is a small, electronic device that is implanted within the inner ear to increase hearing capabilities. Cochlear implants may be helpful when the patient has significant hearing loss in both ears. The cochlear nerve and blood supply must be intact on the CI side - often not the case for post-treatment AN patients. CIs compensate for damaged or non-working parts of the inner ear, finds useful sounds and sends them to the brain.
CRANIAL NERVES
control the sensory and muscle functions around the eyes, face and throat- There are two sets each of twelve cranial nerves, one set for each side of the body.
CROS HEARING AID
Contralateral Routing of Sound - used with one-sided deafness. It receives sound on the deaf side, amplifies it, and carries it to the good ear.
CEREBROSPINAL FLUID (CSF)
a watery fluid, continuously being produced and absorbed, which flows in the ventricles (cavities) within the brain and around the surface of the brain and spinal cord
COMPUTERIZED TOMOGRAPHY (CT SCAN)
X-ray test which creates a cross-sectional picture of any part of the body - can distinguish among tissue, fluid, fat and bone
CYBERKNIFE (CK)
a robotic radiosurgery system that delivers multiple beams of radiation, used to treat benign tumors and cancers and other medical conditions located anywhere in the body in multiple sessions
EAR, NOSE AND THROAT (ENT) PHYSICIAN
also called an otolaryngologist, a physician specializing in the diagnosis and treatment of diseases of the head and neck, especially those involving the ears, nose and throat
ELECTRONYSTAGMOGRAM (ENG)
a recording of the eye movements, usually done to confirm the presence of involuntary eye movements; can also be done in cases of vertigo to determine if there is damage to the vestibular portion of the acoustic nerve or in cases of possible acoustic neuroma
FRACTIONATED STEREOTACTIC RADIATION (FSR)
any focused radiation treatment that requires more than one treatment delivery session
GADOLINIUM
a contrast material given at the time of MRI which concentrates in the tumor and makes it more visible
GAMMA KNIFE (GK)
a radiosurgical machine that contains 201 separate radioactive cobalt sources; gamma rays from each source are focused together at the tumor
INTENSITY MODULATED RADIATION THERAPY (IMRT)
an advanced mode of high-precision radiotherapy that utilizes computer-controlled linear accelerators to deliver precise radiation doses to a tumor or specific areas within the tumor
INTERNAL AUDITORY CANAL (IAC)
a short auditory canal in the petrous portion of the temporal bone, part of the base of the skull that extends from the ear towards the center of the head, through which pass the vestibulocochlear and facial neves
LINAC
a radiosurgical machine that produces x-rays electronically
MAGNETIC RESONANCE IMAGING (MRI)
a technique that uses a magnetic field and radio waves to create detailed images of the organs and tissues within the body
MIDDLE FOSSA
surgical approach to an acoustic neuroma primarily used for the purpose of hearing preservation
MILLIMETER (MM)
a metric unit of measure; 10mm=1cm
NEUROFIBROMATOSIS (NF)
a familial condition characterized by developmental changes in the nervous system, muscles, bones, and skin - the central form (Neurofibromatosis 2 - NF2) may produce bilateral acoustic neuromas
NEUROMA
benign growth originating on a nerve
NEUROTOLOGIST
a physician specializing in the neurological aspects of the auditory and vestibular apparatus
NEUROSURGEON
a physician with a surgical specialty concerned with the treatment of diseases and disorders of the brain, spinal cord and peripheral and sympathetic nervous system
OTOLARYNGOLOGIST (OTO)
a physician specializing in the diagnosis and treatment (including surgery) of head and neck disorders, especially those involving the ear, nose and throat (ENT)
OTOLOGIST
a physician specializing in the diagnosis and treatment of ear disorder only.
PONS
located at the base of the brain in front of the cerebellum, this section of the cranium is a mass of nerve tissue which coordinates the activities of the various lobes of the brain
POSTERIOR FOSSA
the cavity in the back part of the skull which contains the cerebellum, brainstem and cranial nerves 5-12
PROTON RADIATION
a therapy using protons, a positively charged particle, to treat AN
RADIOSURGERY (STEREOTACTIC RADIOSURGERY - SRS)
a treatment consisting of a single session of radiation treatment
RADIOTHERAPY (RT)
a treatment consisting of multiple sessions of radiation treatment
RETROSIGMOID/SUB-OCCIPITAL
a surgical approach for AN where an opening in the cranium behind the mastoid, close to the back of the head is used for access
SENSORINEURAL HEARING LOSS (SNHL)
deafness caused by failure of the acoustic nerve
SUDDEN SENSORINEURAL HEARING LOSS (SSHL)
a rapid loss of hearing that requires medical attention
SHUNT
a tube implanted in the cranium to balance the flow of cerebrospinal fluid and used in the treatment of hydrocephalus
TINNITUS
a common symptom of AN patients, a noise produced in the inner ear, such as ringing, buzzing, roaring, clicking, etc.
TRANSLABYRINTHINE
a surgical approach for AN where the mastoid bone and the bone in the inner ear (labyrinthine) are removed to access the tumor; this approach results in complete hearing loss on the tumor side
UNILATERAL
involving only one side
VERTIGO
a sensation of dizziness and loss of balance, associated particularly with looking down from a great height, or caused by disease affecting the inner ear or the vestibular nerve
VESTIBULAR
associated with the balance system
• Patient Event at the Johns Hopkins Hospital
• Seattle Support Group
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|
__label__pos
| 0.778868 |
Lung cancer screening: does pulmonary nodule detection affect a range of smoking behaviours?
Marcia E. Clark, Ben Young, Laura E. Bedford, Roshan das Nair, John F. R. Robertson, Kavita Vedhara, Francis Sullivan, Frances S. Mair, Stuart Schembri, Roberta C. Littleford, Denise Kendrick (Lead / Corresponding author)
Research output: Contribution to journalArticlepeer-review
2 Citations (Scopus)
Abstract
Background: Lung cancer screening can reduce lung cancer mortality by 20%. Screen-detected abnormalities may provide teachable moments for smoking cessation. This study assesses impact of pulmonary nodule detection on smoking behaviours within the first UK trial of a novel auto-antibody test, followed by chest x-ray and serial CT scanning for early detection of lung cancer (Early Cancer Detection Test-Lung Cancer Scotland Study).
Methods: Test-positive participants completed questionnaires on smoking behaviours at baseline, 1, 3 and 6 months. Logistic regression compared outcomes between nodule (n = 95) and normal CT groups (n = 174) at 3 and 6 months follow-up.
Results: No significant differences were found between the nodule and normal CT groups for any smoking behaviours and odds ratios comparing the nodule and normal CT groups did not vary significantly between 3 and 6 months. There was some evidence the nodule group were more likely to report significant others wanted them to stop smoking than the normal CT group (OR across 3- and 6-month time points: 3.04, 95% CI: 0.95, 9.73; P = 0.06).
Conclusion: Pulmonary nodule detection during lung cancer screening has little impact on smoking behaviours. Further work should explore whether lung cancer screening can impact on perceived social pressure and promote smoking cessation.
Original languageEnglish
Pages (from-to)600-608
Number of pages9
JournalJournal of Public Health
Volume41
Issue number3
Early online date29 Sep 2018
DOIs
Publication statusPublished - Sep 2019
Keywords
• lung cancer screening
• pulmonary nodules
• smoking behaviour
Fingerprint
Dive into the research topics of 'Lung cancer screening: does pulmonary nodule detection affect a range of smoking behaviours?'. Together they form a unique fingerprint.
Cite this
|
__label__pos
| 0.85657 |
12.6 C
London
HomeOtherHydrogen End-Use Applications
Hydrogen End-Use Applications
Integrating hydrogen end-use applications in industries such as automotive, marine, industrial, and aviation requires the development and deployment of hydrogen technologies specific to each sector. Here’s an overview of how hydrogen can be integrated into these industries.
• Hydrogen End-use Application in Aviation Sector:
Expand on the points related to hydrogen end-use applications in the aviation sector, focusing on hydrogen-powered aircraft and the necessary infrastructure development:
Hydrogen-Powered Aircraft:
Combustion Engines: Hydrogen can be used in combustion engines to propel aircraft. In this method, hydrogen combusts with oxygen to produce water vapor and heat, generating the necessary thrust for propulsion.
Fuel Cells: Another approach is using hydrogen fuel cells. Fuel cells electrochemically convert hydrogen into electricity, which then powers electric motors to drive the aircraft.
Environmental Benefits:
Reduced Carbon Emissions: Hydrogen-powered aircraft offer a promising solution for reducing carbon emissions in the aviation sector.
Infrastructure Development:
Hydrogen Storage Facilities: To facilitate the use of hydrogen in aviation, airports need to develop adequate storage facilities for hydrogen.
Hydrogen Refueling Systems: Specialized hydrogen refueling systems are required at airports to efficiently and safely refuel hydrogen-powered aircraft.
Aircraft Design Modifications: Existing aircraft designs may need modifications to accommodate the storage and distribution of hydrogen.
Technology Advancements:
Research and Development: Ongoing research and development efforts are crucial for advancing hydrogen propulsion technology in aviation.
Testing and Certification: Rigorous testing and certification processes are necessary to ensure the safety and reliability of hydrogen-powered aircraft.
Collaboration and Industry Support:
Public-Private Partnerships: Collaboration between governments, aviation industry stakeholders, and research institutions is essential to drive the development and adoption of hydrogen-powered aviation.
Incentives and Policy Support: Governments can incentivize the adoption of hydrogen in aviation through policies such as tax incentives, grants, and emissions reduction targets.
DOWNLOAD- https://www.marketsandmarkets.com/industry-practice/RequestForm.asp
· Hydrogen End-use Application in Industrial Sector:
Hydrogen for Industrial Processes:
Refineries: Hydrogen is a crucial element in the refining of crude oil. It is used in hydrocracking processes to remove impurities and produce high-quality fuels.
Petrochemicals: In petrochemical production, hydrogen is a feedstock for various processes, including hydrocracking and desulfurization.
Steel Production: Hydrogen is gaining attention as a cleaner alternative to coal in the production of steel. By replacing coke in blast furnaces with hydrogen, the steel industry can achieve a reduction in carbon emissions, moving towards a more sustainable and environmentally friendly steel manufacturing process.
Cement Manufacturing: Hydrogen can be used in cement production to replace traditional fuels in kilns. This can help decarbonize the cement industry, which is a significant source of carbon dioxide emissions.
Transition to Low-Carbon or Renewable Hydrogen:
Gray Hydrogen: Traditionally, hydrogen has been produced from fossil fuels, resulting in gray hydrogen. Transitioning from gray to low-carbon or renewable hydrogen is crucial for reducing the environmental impact of industrial processes.
Blue Hydrogen: In some cases, carbon capture and storage (CCS) can be applied to gray hydrogen production, resulting in blue hydrogen. This is a transitional step towards achieving a low-carbon hydrogen economy.
Green Hydrogen: Produced through the electrolysis of water using renewable energy sources, green hydrogen is considered the most environmentally friendly option. Its use in industrial processes aligns with broader sustainability goals.
On-site Hydrogen Production:
Electrolysis: Industries with high hydrogen demand can install on-site electrolysis facilities. Electrolysis involves splitting water into hydrogen and oxygen using an electric current.
Co-production: Some industries generate hydrogen as a byproduct of existing processes, such as chlor-alkali production or ammonia production.
Economic and Environmental Benefits:
Cost Savings: On-site hydrogen production can offer economic advantages by reducing transportation costs associated with the delivery of hydrogen.
Emissions Reduction: Shifting from fossil fuel-based hydrogen to low-carbon or renewable hydrogen helps industries meet emission reduction targets.
Investment and Policy Support:
Industry Collaboration: Collaboration between industrial stakeholders, governments, and research institutions is essential for advancing the adoption of hydrogen in industrial processes.
Government Incentives: Governments can provide financial incentives, grants, and supportive policies to encourage industries to invest in low-carbon and renewable hydrogen technologies.
Here are some examples of hydrogen integration in various industries:
Automotive Sector:
Examples:
Toyota Mirai: The Toyota Mirai is a hydrogen fuel cell electric vehicle (FCEV) that utilizes hydrogen to generate electricity, powering an electric motor for propulsion. It offers a range of over 500 kilometers and refueling times comparable to conventional vehicles.
Hyundai Nexo: The Hyundai Nexo is another hydrogen-powered FCEV that provides long-range capabilities and emits only water vapor. It has been deployed in several countries, including South Korea, the United States, and Europe.
Use Cases
Municipal Fleets: Municipalities can deploy hydrogen-powered vehicles in their fleets, such as buses and garbage trucks. These vehicles can operate on fixed routes and return to centralized refueling stations, making hydrogen a viable option for clean and efficient public transportation.
Long-Haul Trucks: Hydrogen fuel cell technology can be employed in long-haul trucks, offering zero-emission transportation for heavy-duty freight.
Marine Sector:
Examples:
Viking Energy: The Viking Energy is a hydrogen-powered offshore vessel being developed by Eidesvik Offshore, with hydrogen fuel cells providing propulsion.
MS Hydroville: The MS Hydroville is the first certified passenger vessel powered by hydrogen fuel cells in Belgium. It operates as a shuttle for commuters and tourists, demonstrating the feasibility and environmental benefits of hydrogen in the maritime sector.
Use Cases:
Passenger Ferries: Hydrogen can be utilized in passenger ferries operating in coastal areas and inland waterways. Hydrogen fuel cell systems enable zero-emission transportation for commuters and tourists, reducing the environmental impact of marine transport.
Offshore Support Vessels: Hydrogen-powered vessels can be employed in the offshore sector, supporting operations in the oil and gas industry, offshore wind farms, and other offshore installations.
READ MORE- https://www.marketsandmarkets.com/industry-practice/hydrogen/hydrogen-end-use-applications
explore more
|
__label__pos
| 0.987315 |
Stats
42 Contributors: 3 Tuesday, August 8, 2017
Licensed under: CC-BY-SA
Not affiliated with Stack Overflow
Rip Tutorial: [email protected]
Roadmap: roadmap
Download eBook
Design Patterns
Download android eBook
Introduction
Design patterns are formalized best practices that the programmer can use to solve common problems when designing an application or system.
Design patterns can speed up the development process by providing tested, proven development paradigms.
Reusing design patterns helps to prevent subtle issues that can cause major problems, and it also improves code readability for coders and architects who are familiar with the patterns.
Related Examples
|
__label__pos
| 0.956 |
GABAA receptor pharmacology evaluted in overexpressing HEK cells and primary astrocytes on QPatch - Sophion
GABAA receptor pharmacology evaluted in overexpressing HEK cells and primary astrocytes on QPatch
Author(s): Kim Boddum, Kadla Røskva Rosholm, Linda Blomster, Hervør Lykke Olsen, Naja Møller Sørensen, Göran Mattsson
The major inhibitory neurotransmitter of the central nervous system is γ-aminobutyric acid (GABA) and GABA is exerting its effect by binding to GABA receptors. The central role of GABA in the nervous system is underscored by the devastating consequences of pathophysiological changes in GABA signalling. Conversely, manipulation of GABA receptors can offer relief of a large group of neurological and psychiatric disorders. Pharmacological manipulation of GABAA has a large potential and ligands increasing the current will typically have anxiolytic, anticonvulsant, amnesic, sedative, hypnotic, euphoriant, and muscle relaxant effects.
GABAA receptors are ligand-gated ion channels, permeable to Cl- ions, consisting of 5 membrane-spanning subunits. 16 different subunits are identified in humans (α1-6, β1-3, γ1-3, δ, ε, θ, π) and the cellular GABA response is hence composed by a population of GABA receptors with significant different pharmacology. Here we demonstrate pharmacological GABA receptor evaluation in both a stably-transfected cell line containing only α5β3γ2 receptors and a primary cell culture of rat hippocampal astrocytes with a diverse GABA receptor population.
|
__label__pos
| 0.899718 |
Practical C++ Programming: Beginner Course | Zach Hughes | Skillshare
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Practical C++ Programming: Beginner Course
teacher avatar Zach Hughes
Watch this class and thousands more
Get unlimited access to every class
Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more
Watch this class and thousands more
Get unlimited access to every class
Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more
Lessons in This Class
27 Lessons (4h 24m)
• 1. Welcome
3:27
• 2. Installation of the Code - Blocks IDE
6:43
• 3. Anatomy of the Hello World Program
8:05
• 4. Data Types and Variables
13:46
• 5. Basic Output
12:06
• 6. Basic Input
11:34
• 7. Arithmetic
9:25
• 8. Concatenation
5:02
• 9. If Statements
13:56
• 10. Switch Statements
8:03
• 11. Practical Program #1
12:31
• 12. While and Do-While Loops
8:13
• 13. For Loops
6:50
• 14. Data Structures - Arrays
9:32
• 15. File Output
6:46
• 16. File Input
15:26
• 17. Advanced Input and Output Manipulation
12:05
• 18. Practical Program #2
16:59
• 19. Functions
7:01
• 20. Parameters
4:57
• 21. Pass by Reference
9:54
• 22. Function Overloading
8:22
• 23. String Functions
3:33
• 24. Random Number Generator
6:51
• 25. Project -Hangman (Part #1)
18:16
• 26. Project -Hangman (Part #2)
15:54
• 27. Project -Hangman (Part #3)
8:34
• --
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The level is determined by a majority opinion of students who have reviewed this class. The teacher's recommendation is shown until at least 5 student responses are collected.
2,088
Students
1
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About This Class
C++ is one of the most used programming languages. It is an object-oriented language, offering you the utmost control over interface, resource allocation and data usage.
This class covers the basics of programming in C++. Created for the beginner programmer, this class requires no prior knowledge of programming. The main aspects of the language are introduced in a logical, gradient manner with a step by step approach. This will provide you with a solid foundation for writing useful, correct, maintainable, and effective code.
By the end of this class you’ll have all the skills you need to start programming in C++. With this complete class, you’ll quickly learn the basics, and then move on to more advanced concepts.
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Zach Hughes
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Transcripts
1. Welcome: Hello. Welcome to Practical C Plus plus programming the beginner course. My name is Zak and I will be your instructor. Now, before we get started with this, syllabus might go ahead and tell you a little bit about myself and some of my credentials. I am currently a student at Carleton University, where I am on my way to earn a bachelor's degree in computer science and a miner's and associates degree in electrical engineering. My electrical engineering Associates degree is actually coming from a community calling in my area, and I do live in Texas if you couldn't tell by my accent. So I hope that doesn't bother you too much, but I'll try not to make it sound too Texan when I'm recording My background in programming involves a heavy use and heavy practice of C plus plus. I've taken many semesters and C plus plus at my university. I have actually taken three semesters in C plus plus total, and I've taken a semester in Matt Lab and engineering programming language and I, currently on the side programming Java for Android application development, and I've actually developed my own angel. Would applications for the Google play store. So that me and said, That's enough about me. Let's go ahead and look at what we're going to learning for this course. So if you look on screen, I've kind of listed everything that we will be for sure going over in this class. But just remember that the class is not limited to this syllabus. So there's gonna be things in between these these concepts right here that we will be going over, you know. So everything you see on screen is is not everything that you will learn. You will actually learn much more than just everything you see here. So, you know, if you're a complete beginner in programming, I would say that this course is definitely the perfect course for you because I'm not going to start with just the C plus plus principles. I'm going to actually introduce the basic programming principles in general to start the course off. So if you don't know anything about data tops and variables were actually going to cover that right at the beginning, of course. And then we're gonna move on to how to use these program and concepts in C plus plus and develop her own useful applications. And when I say useful, you know the course is called practical C plus plus programming. And that's because I think the C plus plus is a much fun or language to learn when you're using get in practical situations. So that's exactly what we're gonna do. We're going to develop a small business application, a simple calculator app and then at the very end of the course for a final project, we're going to develop a hangman game that you can show all your friends. And hopefully, if I get enough students for this course and enough people leave good reviews and tell me that they want to see an advanced course than that's then that's what we're gonna do. I'm actually going to make an advance C++ course. After this, we're going to object, wearing the design and everything like that. So stay tuned for this Siri's and I'm glad you're part of the course. Let's get started 2. Installation of the Code - Blocks IDE: Hello, everyone. My name is Zak and I'm here with practical C plus plus the beginner course. And in this tutorial, we will be going over the installation of the code blocks. I D e um I d e stand for interactive development environment. And ah, we will be using code blocks as our i d. For this entire course, our chose code box. Because, as I said in the introduction, it's actual what ice began not only learning c plus plus in but programming in general. So, uh, not only I mean that, but it's also free. So have fun is a really good choice to begin your programming. Um, you know your your programming going. So as you can see, I have a Web browser open. And if you go to Google on top in the search, more just code blocks. As so, um, the first link you will see is www dot code blocks dot work. And this is where you're gonna want Tokyo. You can either click on this first code blocks link and then click on downloads. Or you can do a common. Do you just click on the downloads link below when the page loads, You know, depending on my on how my Internet is ah, doing right now. But when the page does load, you're gonna be brought here and you will have several links. Like download the binary release, download the source code and that retrieve source code from SV end. You are going to want to click on download the binary release. They will bring you to this page. Now, depending on what operating system you're on, you're going to click on something different than I may be clicking on. You know, if you're on Lennix 32 bit or Lennox 64 bit, we're gonna be looking at these boxes right here from Mac OS X ray gun one scroll the way the bottom and they have a download link right here. Um, me, though I'm on windows seven. Salt will come up here. And if you look, it doesn't say Windows eight right here. But up here it says windows, you know, seven dash A. So these Ah, these binary builds should work on Windows eight neigh 80.1. In fact, I've actually downloaded on Windows eight and eight point warning, So I know for a fact that will work. If you look over here, there are two different links. There's burly OS and source forge dot net. I'm not that familiar with burly OS, Um, but I am familiar with source for it, and I use it for a lot of my downloads, so I would recommend using source foraged dot net. Now there are three different, um, types of binary releases that you can download now. When I first started, programming mind structure had us download this binary released right here. The 2nd 1 on the list, which is perfectly fine, works great. Um, but as I got in a more advanced C plus plus programming where I'll start doing concurrent threads and you know, different kind of ah, concurrent thread processing, multi threaded processing, I needed this GCC 4.8 point one for my co box toe work with threads. It's a specific compiler, so I would recommend if you plan on going Maurin debts with C plus plus and maybe taking a course after this to go ahead and download this one. Because if you start getting into threads and C plus plus, you will have to come back and download this compiler right here for toad blocks work. Um, otherwise, you know, this right here is a great auction is will. So either one is fine for this course, Um, you're gonna want to go ahead and click on source forge dot net to continue on either one of links and my click on the 2nd 1 It will take you to source foraged dot net and the countdown will begin for your download. And after the countdown, the XY foul should be downloaded. We'll give it a second, and right down here, you can see code blocks 13.1 point two Dottie XY, and, ah, it says they're still, you know, non 10 minutes left before it's done downloading on mind. So it might take a while for it to download. Um, I'm not going to sit through this tutorial on wait for it finished downloading simply because I already have it downloaded on my computer. But I will tell you when it does finish downloading, you're gonna want to launch the XY file, and a install wizard will pop up. And it's very simple. Install wizard. Basically, just click next on every single pop up window and it should install very easily with no problems. After it's done installing, you're gonna want to search for the program either by using your search like this, or it might have even put it in your task. Four like I have it right here. And it might even have a desktop shortcut. Whatever the case, you're gonna wanna launch code blocks, give it a little bit of time, especially on the first time launching it when the first time you launch it, it might take a little bit longer than you expect. Also, though, it seems like it's taken a while for it to download on my computer. My Internet connection is quite slow right now. I just moved, and ah, I have slow download speed as a right now because I haven't yet upgraded my Internet so that it's on your at your, you know, at your house. It might. It will probably go a lot faster than mine. I think I have, like, eight megabits download speed right now. This is what's going to pop up when code blocks launches and ah, in the next tutorial, we are going to create a new project and ah, we will discuss the hello world project that will be created and discuss the anatomy, uh, the our very first C plus plus program. So stay tuned, and I'll see you in the next tutorial. 3. Anatomy of the Hello World Program: Hello, everyone. Welcome to practical C plus plus beginner course. I'm Zak. And in this tutorial, we will be going over the hello world program. So if you open up code blocks, you're gonna want to click on, create a new project and then click on console application. Ah mahn, it's in the top right corner of the window on yours. It may be different, but you want to click on console application and then hit. Go then on the next window. You want to hit next until you get here, make sure you highlight C plus plus click next again and then give your project and name in mind. I'm just gonna call it tutorial wine and then specify folder to keep it in. Make sure it's a folder that you confined easily and then quit. Next, Leave all the default settings on this window right here. These air simply ah, direct directory and compiler settings Just hit finish and then your project is created. So right now you don't see anything. But if you go over here to the left and click on sources, you'll see the main dot CPP file, which stands for main dot C plus plus, and if you double click it, you'll see the code. Now, before we analyze this code, I want to go ahead and show you how to run it and what this code does. And to do that on Windows you can either hit F nine or if you're on a Mac or Lennox machine . You just go up here and hit, build and build and run. The code will compile, and then you'll see this console window. It prints the word hello world and then says process returned. Zero. Okay, so you can close out of this Now that we know what the code does, we're gonna look at how this code does what it does. Um, so beginning with kind of the main thing I want to show you in this tutorial other than you know how to run your first program and compile it is you know how to type out the skeleton of a C plus plus program. It's what I call the skeleton. And it's everything that you need Ah, for your code to rind, at least in this course anyways, for every program that we will be riding together. So if we go ahead I want to go ahead and take out this line because this line is not actually needed for this program to compile and Ryan. So if we take it out and we hit F nine and we build and run it again, we will get the process returned to zero. We just won't have hello world printed to the console, which means, I mean, that's fine. The program just ran and executed till it finished with no heirs, and it's a perfectly good program. So if we exit out of this, we're gonna now analyze everything that we need, which is everything that you see right here. It's a starting from the top you're going to see pound include Io stream. This line of code right here just simply tells the programme that it needs to include a C plus plus library known as I O Stream, which stands for input output stream. Now, on every program that we write, you will need this line of code, which is why I include it in our skeleton is because, you know, if you don't have this line of code, your program will lose its basic input and output functionality. So you do need this line of code and every program that we right moving on, you're gonna see using names based standard. Now, this line of code is not necessary for your program to rind. Okay, so if we took this code out right now, it should still run. Fine. We hit f non, everything goes fine and we still get the same result process returned to zero. However, I do want this lot of code in here for reasons I'll explain in the future For now. Just know that we do want to include it as part of our skeleton because it will make your life easier when we start riding more code and I will show you why in future tutorials. But for now, just know that you do need using name, space standard semi colon. Okay? And it doesn't with the cynical and I know right now you're saying, why does this line of code in with Semi Colon and this one doesn't? Well, we'll get to that in future tutorials again. It's all gonna become a habit for now. Just know this. This is the code that you will need in all of our C plus plus programs moving on to the next big chunk of code. This right here is known as your main function. And in every C plus plus program that we write, you will need a main function and type it up. You just simply right I into you extends for integer and then Maine open prints, sees close parentheses and then your brackets. What the return zero statement now and programming. There's two different conventions for writing these brackets. I'm gonna go ahead and show them to you now so you don't get confused later. If I do this one, convention is the way just saw it, which is like this where your brackets are open and closed down here. But the other way, you may see it is like this where your brackets opened at the top and closed down there, which is fine. There's no, um, difference in the code whatsoever. It will run just the same, so just know that it's ah, it's just a program and convention. There's no right or wrong way. Some people have their own opinions on why they do it a certain way, but just know, you know, it's all a matter of preference so before we in this tutorial, I want you go through and I want you to top this up with me so you can get in the habit of doing it. So what's the first thing we need to do? What we need to include the input output strings C plus plus library so that we can, you know, output stuff to the console window. To do that, we're going to hit pound, include i O Stream. Okay, No semi colon on this line again. We'll get in the habit of knowing where to put him when we're not to put him. But for now, you know, just know there's no cynical And at the end of this line now, though, we don't need it for this program You do want as part of your skeleton for this class. I do want you to get in the habit of riding, getting all of our programs. So let's go ahead and do it now. You want to use the standard name space and how do we do that? Remember, we topped using name Space Standard, and that one did have a cynical moving on them. Other really critical piece of code that we need for our program to run is the main function. And remember, that was proceeded with I and T. You extend for integer the name of the function main open princes, close parentheses and then our brackets, depending on what convention you decide to use will be different and then returned the value zero. This right here is working code. If you take out some of this code, such is that this code will return in air. It will not run. So for now, just know that everything in this code is needed. And in future tutorials, we will go into a discussion as to why they're needed and what exactly, they dio. But for now, let's move on to the next tutorial where we will be discussing data types and variables. Thank you. 4. Data Types and Variables: Hello, everyone. Welcome to practical C Plus plus the beginner course. I'm Zak. And in this tutorial, we will be discussing data types and variables before we get into the code blocks editor. I kind of wanna do this on a, um I'm no pad sheet real quick so that weaken we can show you. I want to kind discuss with you how these data types are declared and what they mean. So, a dead a type. What is the data type? What is a data type? Um, a data type is basically a description of what we are using. So, for instance, and the real world, um, if we were gonna use the this the letter B, for instance. Well, this is to us is known as a letter. This would be the data type in programming. Okay. Or what if we were talking about the number seven on our world? This is just called a number. This again is a data type. And, uh, you know, if we were talking about programming except in programming and in C plus plus, we don't call them letters and numbers. So how do what do we call them? Well, let's start with just a single letter. Let's let's to start with the character. Be okay again. This is just a letter, but in programming, this is called a character data type and the character data type. Sorry about that. The character data type is denoted or, um, kind of encapsulated with single quotation marks as so So That's how so and C plus place this letter B. It's called character data, and you have to declare it with single quotation marks. And when we get into example code you'll you'll understand what I mean by declare with single quotation marks. But let's move on to a number. So let's say seven and C plus Plus. This is called an integer, and an integer is denoted just as it seemed. No quotation marks, nothing special. What about multiple letters? So you know what? What about? You know, the name Bill? You know, that's that's four characters. But this thing is a whole What is it? Is that a word? You know, in our world, it's called a word, but, uh, what is it? What is it in C plus plus well in C plus plus, it is known as a string, which is Alfa numeric data. Okay, but for now, just know it's called a string data. And he did note string data with double quotation marks as so okay. And I kind of want to go over one last thing. Ah, and that's decimal numbers to, like, seven point 77 you know, Is that an integer? Well, no, it's not. It's not a whole number. So what is that called? Well, in this, it could be one of two things you know, actually could be multiple things, but for this course, we're gonna keep it simple. Know that it's either a float or a double. Okay? And it's denoted just as 7.77 for this class. We're going to use the word double on the reason why is because if you use the word float when you declare it as this 7.77 behind the scenes and code blocks code blocks automatically and converts it to a double anyways, so for now, we're just gonna call it a double data type. Okay, so let's move on to some actual code, okay? And we're gonna practice declaring these four major data types that I've shown you here So let's open up our code. And by the way, in the last tutorial, the code was a zoomed in. If you couldn't tell if it was hard for you to read in the last tutorial, Um, hopefully this will make it easier for you to read. Um, so right now, let's go ahead and practice what we learned in the last tutorial in this project. On a practice riding our skeleton, so to speak, everything we need for our code. Right. So we need to do basic input and output library. We need to include it. So lets include Io stream. Okay. We want to use the standard name space. Okay. And then we need to declare our main function. And we need a return value for this main function, which, as we said, we're gonna use zero. And this is our skeleton. This is everything we learned in the last tutorial again. If you If this isn't if you still haven't got this down yet, you know, she suggests his practice and get over and over until you get it down to where you can basically do right this code in your sleep to get your program to compile and rind, as with process returned zero. Let's get into our data types and variables. Okay, so we discussed. How do you know what the data types were? You know, it's kind of a description of what you're dealing with, but what's a variable? Well, a variable. It is kind of like a box. And your data type is a label on that box. That's how I want you picture this. So you have a box, and then you have a label on this box. Okay, so So let's say we put the letter B inside of box. Okay, So the letter B is character data, right? We discussed this earlier, so to declare the character data, we write C h a r, which stands for character. Okay, so that's our label character. C h a R. That we're putting on her box in our box is user to find meaning we can name it, whatever we want. So I'm going to name it, um, letter, because that's what's in this box is letter. It's a letter. OK, And then we need to declare what the letter is. We put a equal sign. I'm sorry about that. Guys don't know what that is popping up. They put me in equal sign. Okay? And then again, character data is denoted with single quotation marks. And then inside the single quotation marks, we put our letter be and then end the statement with a semi colon. And this right here is declaring a variable. What? The data type of character and that variable that character variable this letter B. Okay, so let's move on. Let's let's let's move on to the other. Now, if you don't know what this is used for yet, that's fine. In the next tutorial, we're gonna discuss how we use these variables and what exactly they're used for. But for now, we're just declaring them, and I'm kind of giving you a visual ization of what they are. Okay, so just for now, think of them is a A box with a label and then something inside the box. In this case, it's a box that holds characters and in the box is the character be okay? So let's move on to a number of box that holds numbers. So again, a number is gonna be an indicator if it's a whole number, so to declare a whole number integer value. We write the word I m t which he makes. You probably remember from here in Maine, which will get into why you need that again in future tutorials. But for now, let's focus on this, and we're gonna put a number in here. So let's call this variable number again. It's user to find. If I wanted to call it Jimmy, I could call it Jimmy. But you gotta want it. You know, it's convention to call it something that represents So we're gonna call it a number, and that's gonna equal with number seven and you end that with a semicolon. Remember, No quotation mark. Try here. Just the number. Okay, so that's That's another variable. Let's do the other two variables we did. Let's dio a, uh let's make a variable that holds the name Bill inside. So to do that, we write string because that's the data type. Remember, multiple letters is Alfa Alfa Numeric data is called string data. Okay, which you know I'll actually get into it later. String is technically a class, and I know you don't know what classes. So for now, just know it's it is a data type for now. Okay, string data we're gonna call it Name equals double quotation marks. Still semi colon. Okay, so there's your variable for a name or multi character data. OK, now and string don't don't get it confused with, um Onley being characters if I want to do Bill 99. Hyphen, hyphen, hyphen. Cynical on semi colon Inside these quotation marks, that's fine. Alphanumeric data will hold all these characters, and this will not cause an air. This is everything you can keep in this variable that is find. So I just know that, but for now, actually, let's call it Bill 99. Okay? So that you you don't forget that you can actually put numbers and string. If you wanted to do just 99. You could as long as you have the double quotation marks. It's still string data her case. But for now, it's Bill 99. And then for the last data type. Let's do a decimal number, Ricks. Remember? I said it could be flow or double, but for this class, we're going to use double, so it's a double. Let's call it decimal equals 7.7 seven. Cynical and all this is fine. If we run this program, it will not cause any heirs. Everything runs, process returns zero. Same outcome. All this stuff is happening behind the scenes. So of course you won't see anything in the console window when we run it. Anything different, you know, you'll still get the same result process returned. Zero Everything's fine. But the point is to just show you how to declare these variables and data types kind of discuss what they are, and, um, show that, you know, if you if you declare them right, you won't get a nadir, let me show you what will happen if you if you know if you call this character or let's say we call it, um, string okay, without the double quotation marks if you declare it wrong, which this is declared wrong because you're calling it a string and you don't have your double quotation marks. If you wanted to be right, you'd have to put in double quotation marks. But if you take those out, then this is declared wrong. When you try to build and run, we'll get an air, See this red box, And if you look down here in the log, you can scroll down and it says air conversion from double to non scaler. Top standard Colin Colin string requested. And that build failed one air. So they're right there will cause the program to crash. So when you change that back Teoh double. If we run it again building rind, everything will go fine. So that's it for this tutorial. Um, let's move on to the next tutorial and ah, we will learn about, um, input and output and ah, continue on after that some basic arithmetic and some more fun stuff. So thank you for watching. 5. Basic Output: Hello, everyone. Welcome to practical C plus plus for beginners. My name is Zak. And in this tutorial, we're going to be discussing basic input and output, and we will be using the stuff we learned in the previous tutorials to do. So, um, as you can see on screen, I've already got our, uh, basic code typed up. What we've been calling the skeleton. Do you need to add a return statement? Return zero, and ah, for this tutorial. You know, I kind of wanna I want to go into We're gonna start with output because I'll put it's going to be a little bit. We've kind of already seen it in the hello World program. So in the hello world program, we saw something along the lines of this. And when we ran it, the screen printed hello world before we saw her process return to zero. Okay, so for now, we're gonna go ahead and take out this end. L right here, because I just want to focus on something a little bit easier, and that's just basic output. And then we're going to basic input. So right now we're just going to see, um this right here hello world, and when we run it, you'll see a little bit of difference, but not much. The only difference is there's not much space here. It says Hello world, And then immediately process returned zero and later we'll get into Why that so when we took out that last code, But for now, let's just leave it as is. So So what is this? Well, this is an output stream. See out. That's where the out comes from, is output. So when you see out, you're referring to output to the console, and that's what the C stands for console output, and the console is the big black window that pops up when we were on our program. So when we say Consul Output and then these two operators right here and then we specify a string and this is a stream because there's double quotation marks, we specify. Hello, world. Um, the console will actually out plate the word hello world, and that's exactly what happens. Okay, so, you know, kind of just to show you we can play integer. We can put an integer here if we want to. We could say I'll put um nine and, ah, it'll output the number nine on the console. Okay, so that's kind of basic output. Um, but I kind of want to go into all come one throwing variables into this for a second. So the previous tutorial we discussed, you know, the variable letter. You know, we will call it the letter Z this time. Okay, So for one toe to declare a data type of one character and we want that character to be the letter z, we would do it as we specify C h a r character, and then we'll call it. You know, we're calling anything we want. We'll call it, You know, we can call it Ah, letter Z, um, equals And in single quotation marks, Z semi colon. So here's our variable right here. Well, if we wanted to output this variable, we would just say, see out operators, and then we re type the name of our variable letter Z. And if we run this, we'll get what you expect. We get the single characters e because what this is saying? This is saying, consul output, the variable letters E, which we call letters e. Okay, um, output, whatever this variable is holding and it's holding this letter and remember it just cause we called it letters. E Let's let's call it something else. Let's call it. Um, Let's just call it box. Okay? Let's say we called this box, okay? When we see out box, it's gonna go toe where we declared box at, which is right here the variable box. It's gonna look. What? What? It's holding what's holding a character Z. So when we consul output box, it outputs Z to the screen and the process returned to zero. Now let's back up for a second because I didn't really explain this. What are these right here? Well, these work hall ish Icahn stream operators there basically they're just the output stream and what it is is any time you say, see out, you're wanting to out. But you can't just say see, outbox, that we're returning air right there. That's an air. So what you do is use a output stream operator, which is just to less than in signs. So you have to say, see out to Leslie in signs Not this. That has to be to s and signs and then box. And if we do, I can. I just want to show you know more for practice. I really want you to do this on your own so you can see all the different possibilities. But if we do to these lines of code right after the other, you're gonna get just that you're gonna get to disease one after the other. Okay, so I got twosies because we wrote statement twice. So that means said, let's let's try something Here s O. So let's go into that right here in Del. What is that? Well, that stands for end line. And this is used a lot with basic output because it gives you spacing. So if we do output stream operator, which basically this is the same thing, it's saying, See outbox council output, end line. This will work the same way Z and then you'll see there's a space here because we added a blank line and if we do so, we could take this out. We can even add it right here. This is the same thing. We just did that. We're using it all in one line of code. Basically, you're gonna get the same exact results. I kind of wanted to show you that there's many ways you can do this. And I really want you to play with this on their own because you're gonna learn, you know, you you could say see outbox, um, en del box in Del. And then when you run it, you get just that you get a Z than on the next on Z and then a new lon. But if you know if you add another Z, you it gives you got there. So says Kind of you really just need play with this output because you're gonna get a lot of cool stuff. Let's move on below this. Let's leave that code there. Let's let's declare a new data type. Let's call it Let's do am a decimal about you. So remember, Double, we'll call it. Um, Box two. The variable name equals 89.47 semi colon groups semi colon. So that's our second variable. So let's had see out Box two in Del and let's run that. So what do you get? Will you get your twosies? Because he had box here, you ended the line On the next line, you put another box and you ended the lawn and then on the next line you put in Box two or what's in box to 89.47 That's why you 89.47 Then you ended the line and then you return zero process return zero So you can see if you play with this, you're gonna learn and you and I want you to do it. I want you to go through. I want you use different data types. Um, you know this this right now if we just did Ah, let's try another data type just before before we in this tutorial. Let's dio the string and we'll call a string Um, address equals 1400 College drive. That's an actress. And we had to use string because it's alphanumeric data, right? You know, if we wanted use, we couldn't use an integer because it obviously has Alfa characters in it, too. That's why we used string for the address. Variable. If we wanted Teoh, I'll put that you know, we could do you see out. We'll give it some space will do to new lines and then we'll say output address and then at another new on end line. Let's help put that. Let's see what that looks side. So we get our twosies, or box to variable 89.47 And if you look down here, we added two new lines. That's where the blank space comes in. And then we output in the address. Oops. So then we output address 1400 college drive, which we declared in this string variable right here. That's where that comes from, and then process return to zero. So there you go. I want you play with this when you try doing it with character data into your data string data and, um, double decimal point data. Alex, you practice out putting everything that we learned here, you know, maybe try ah, out putting your name and writing a sentence, you know, like, maybe try doing, you know, string name equals and then put your name. So my name is Zak and then try to output that So be, like, so in the line. And then output name and then output, um, is teaching a class okay. And then in the ill in the line. And then I'll put that and look at what you're gonna get you're going to get really, really cool output because you're getting a name that you declared in this variable string name. And you in that variable you're declaring it is the name Zack, which is output right here. So you're saying, um, you know, Consul Output, my name, which we declared here, Zack. And then immediately after that, don't in the line or anything. You know, output. This string that we're declaring as a raw string right here. This isn't an invariable. This is actually we're just putting this Straighten Your output stream is teaching a class and then in the line and look at everything. Look at all the cool stuff you can do with that. You'll surprise yourself. You know, I just really want your practice that originally we were going to do both input and output in this video. But we're gonna save input for the next video, I guess. You know, I really want you to practice this right now, declaring your variables, writing everything up and seeing how you can output different things to the stream on your own. But for now, that's all there is to this video. And I'll see you in the next tutorial. 6. Basic Input: Hello, everyone. Welcome, Teoh Practical C plus plus for beginners. My name is Zak, and in this tutorial, we will be going over basic input. So in the last tutorial, we discussed basic output. We discuss things such as, you know, if we declared a string with, um you know, we'll call it name equals Zach. And then we would declare, You know what? Let's say an integer value age equals 23. And then let's say we we wanted to output that, you know, we could say something like console output. See out the two less and signs, you know, don't forget that. And then we could say name to a saint signs and then is that's gonna be a raw string that were in putting into this ALPA stream right here. Then let's say I'm less than age. Let's say in less than years old and then we'll even throw in a new on end line out there when we printed that, you know, if you messed around with this enough. Oops. So you saw on air out there? I'm getting an air, and the reason lies because I wasn't paying attention, and I'm sure you caught it is You were watching me, but this right here is an integer, and I accidentally put in double quotation marks. So it's let's take Weathers. Double quotation marks turned this into an insecure. There we go. So now we shouldn't get an air. Whenever we build and run this someone build and run, it no airs, and we get the output. Zack, it's 23 years old, and, uh, if you practice this enough, this should seem pretty, uh, pretty easy for you to understand. Bullet. Let's let's move on to what we're talking about now. And that that is input console input. So, yes, if council output ISS see out. And what do you think Consul in played in is? Well, that's gonna be CNN, okay. And then see out deals with less and less than well, CNN deals with greater than greater than. Let's if you look at the difference between these two the stream operators and right now I know you're thinking, Wow, I'm going to get those mixed up a lot. Well, well, believe me, I got a mixed up all the time when I first started programming c++. But I promise you, after lots of practice, you will probably never, ever get a mixed up again because you'll get so used to using the right wine. And, uh, this is gonna have to be something that you practice a lot, though, because it's not something that you can get now and you know, immediately. So you have to remember seeing is greater than greater than see, how is less saying less than operator. So how do we you see? And, you know, console input? Well, if you're asking, let's take this out real quick, okay? Actually, no. Let let's leave this up here, okay? Let's just give us a little bit more space. Give us another new line and then let let's do at another name. Let's declare another variable. We'll call it String. Name two, and we won't. We won't. Uh, We won't give it an actual name. See how we have it appear we haven't declared as a Zack. We declared name. Is that right? Here were declining name, too. Is nothing named to isn't holding anything right now. And so what we can do is we can say something like we're gonna output enter name to. So what we're gonna do is basically we're gonna have Zach is 23 years old, output and then blow that. We're gonna get an output saying Inter name too. Well, it's asking for you to enter something. So to enter something and toe, let the user use the keyboard to type in a value u CN greater than greater than and then the variable name that you want to hold the employees in. So since we're telling them to intra name, we're gonna use this variable to hold the name that they enter. So let's say seeing greater than greater than name to and what that's going to new is when the when the user enters and a name, it's going to store whatever he enters into the variable name, too. Okay. And then we can say, see out, we're gonna end a few lines. We're gonna say you entered and then name, too, En Del, What this is going to do is whatever I enter and for this right here seeing name, too. It's gonna output you entered. And then whatever name too is holding at that point. So let's test it. Let real quick. Let's build and run this. So if you see we get. Zach is 23 years old. That is the result of this first declaration and output stream that we have going on in this first part of code that we did it beginning. But then it says Inter name, too. And you see a cursor blinking right here that shows that the consoles waiting for input. So right now, we're right here in the code seeing name, too. So it's saying Inter name, too, and the consul's waiting for us to input something. And whatever we input, it's gonna story into name, too. Okay, so if we enter, let's say Jimmy right here and I hit. Enter. They'll say you entered Jimmy because right here we consul output you entered in the name to and Name two is holding the value that we entered into. And that's where it says you entered Jimmy. So what happens if we want to do something like it age to you? Well, then, obviously, since we're declaring age two is an integer whenever we ask for input. Oops. I forgot to change this to age, to build and run it again. And then, of course, shot here. This needs to be H T. That's where these aiders are popping up, so fix that. Let's run it again. So it's asking Inter age too Well. Age, we declared, isn't integers. Obviously we need to enter an integer such as eight. I'll say you entered eight. What happens if we enter a character data Instead? I could be Were beat 39. That string data we'll send you entered zero because basically what's happening is is that's not a validated type. So it's giving us a garbage value right here. So any time you get a weird value, that's not what you're expecting. You not want to go look at your data types that you've declared and make sure that their matching the input of the user. So what I want you do is I want you to practice using the CNN Council input and maybe make you a little script or something, a little program that asks you what your name is and practice entering it in with different data types. You know, for instance, you could declare, you know, can't do first initial the character last initial and then say see, you know, Consul output, enter first initial, then you're going to a consul. Input first initial and then see out and then maybe give yourself some space with some in Dell's in lines and say, you know, enter second initial, and then you're gonna have to do another consul input, seeing greater than greater than last an issue initial and then say see out and then give yourself some space and say you know something along the lines of you know your initials are. And then first initial last initial. Then if you run that I will say in her first initial, the inner second initial are your first. Your initials rz are just kind of practice doing something like that, you know, entering different data and make sure you're declaring your data tops, right? And, uh, make sure you can do mix your input and output and get everything right. You know, practice using the stream operators because, you know, seeing is using greater than greater than and see out as usual left saying less. Man, I would say, Take a few hours, you know, practicing this and, uh, just doing different scenarios, you know, do your initials, a program that's for initials, and then do one that may be asked for your address and stuff like that and just practice entering in data and I'm action. Get with your data types. And in the next video, we're gonna We're gonna actually do something a little bit more practical. We're gonna use arithmetic. And then after that, Ah, you know, after we get the math down, we're gonna right, you're gonna make a calculator or something. So I look forward to that. I'll see you in the next tutorial. 7. Arithmetic: Hello, everyone. Welcome to practical C plus plus the beginner course. My name is Zak. And in this tutorial, we will be discussing arithmetic, which is all your basic math functionality in a C plus plus program. So to start off, I'm gonna go ahead and assume that all of you have been practicing declaring your variables and what not? Uh, so I'm not I'm not going to explain that stuff in debt time. I assume you already have this stuff down. So to begin, we're gonna start with simple addition and subtraction, and I just want you to follow along, and you should notice that it's fairly straightforward. So you could start by declaring your variable as so And then if to use these two variables and her arithmetic, you know, operation, you could do several things. You know, you could hold you could declare variable called result and hold the arithmetic value of the addition of numb one. Plus numb to know that will hold the value of this operation. So if we wanted to actually output that we could see what the value is after this operation , and you should see that it's 11 as so in the same thing if we wanted to do subtraction. You know, you just add a minus sign of hyphen, and when you put that, you should get negative. One negative went so you can see that in C plus plus addition of subtraction are fairly straightforward. And I do want to show you a few things you know, with respect to hard coating value. So if we want to do number one minus four, we can do that. We can hard code the value in there and C plus plus that's Bond will get one. And I also want to show you another outplayed trick. If we wanted to just output the result of numb one plus numb to we can do that, we can output. That result will get 11. So there's lots different things you can do with addition and subtraction and all your basic order of operations rules do apply here. So if we went to output, you know, number one plus numb to, you know, minus four, that's gonna It's gonna do the order of operations to do this so it's going to start in parentheses. Do this operation here that will result in 11 and then subtract four to give you seven. And we're just gonna output that all once and we get seven as so. So when you play went around with the addition and subtraction and you'll find it's fairly easy and that there's lots stuff you can do with it with respect to output in order of operations and hard coating values, etcetera. But let's move on to multiplication and division. Okay, So same thing with multiplication division, we're just gonna go ahead and put the whole operation in this output stream right here. So we're gonna see out number one and then for multiplication. It's not an X, as some of you may think, that section Asterix so number one times numb to which will give us 30 in this operation and we all put, we get 30. Multiplication is fairly easy. And you know I can set Order of operations again applies here. So if we went to put that there and then plus seven, we should get 37. And I do believe if you you know, if you if you remember this stuff from math class, you don't even have to have these parentheses here for this operation because multiplication will come before. Addition, Multiplication division first and then addition and subtraction come after that. So if we run that, we will still get 37 you know? So it's different. It's not gonna do no more implicit. You know, if we even put just to kind of prove to you if we put seven plus num wine, it's still going to do this operation first and then add seven. We'll get 37. So I just kind of wanna show the importance of order of operations and c++ because the rules still do apply. So let's do something like this. Let's let's change number 22 30 Okay? And let's let's cross some division and division you would just do numb to. And then the backslash is the division sign number one. And if this works rise could be 30 divided by five and it's gonna output six. That's fairly simple right there. And you could do the same thing again. Order of operations. You know, if I wanted to ad to to this will get eight, no matter where I put it. You know, I can't put it in between here because if I do something lets out do something like this. It's audio, you know, Numb one divided by five plus numb one that's gonna be It's gonna basically do this first. Actually, let's do this. You know, this is a cool order of operations, because here you have addition. But you have division first, so you might say, Well, division's gonna go first. Well, that's not true, because parentheses goes before multiplication and division. So here is going to do this operation 1st 5 plus Numb Boyne. So that's gonna turn into 10 and then it's going to the divisions. So 30 divided by 10 and it should output the number three. If we run that, that's exactly what we're gonna get. We're gonna get three. So this stuff, if you practice it enough, it's gonna become fairly straightforward. And you're gonna realize you could do a lot of cool stuff with this arithmetic operations. So one more main arithmetic operations on my show you is the module ist operator, which is the percent sign on the keyboard and what This days, this returns a remainder value from a division operation. So just to kind of show you, we're going to change number 2 to 11 and leave number one in five. And we're gonna hold. Result equals numb to module its operator NUM wine. Now, I want you to think about this. This right here. This operator is basically returning the value toe hold in result of the remainder of the division operation. And if we divide numb to buy numb wine, basically, we're gonna get 11. Divided by 55 will go into 11 2 times, with one being the remainder. So this operation will store the number one into results. And if we output result, you'll see. Oops. Hold on one second. If we output results, I think I hit the wrong key. Yeah, here we go. We get wine, which is the remainder. I kind of want to show you that. You know, if we do also, you know all the rules apply. You know you can do. I mean, I'm sure this becomes fairly clear to you, but you could do result plus four right here. See our result plus four. You know, you can you do That's going to do five. I just can't wait to show you that real quick. Okay. And, ah, going back to the module ist operator, you know, let's do another. How about let's do this? Let's let's do to modelo operators in a row. So let's do just so you can kind of see if you can guess what the value will be after this operation. Let's do the value. 14 here and let's output result module its operator Margallo. Whatever you wanna call it, Result module. Oh, let's do to you. Okay, so think about this. Results is holding the remainder of this operation, and then we're out putting the remainder of this operation. So think about that for a second. Now, I want you to try to guess without puts Gonna be. Now, if you guess 20 you're correct. Because what's happening is result is holding none to divide about number one and the remainder, which is going to be four. Okay, because five or go into 14 2 times with four left over, and then we're gonna output four divided by two in the remainder of that, Whether there isn't a remainder of four, divide by two, it zero because two goes into four evenly. So when we output this, we're going to get zero, and it's that easy. So that means said, that's all for arithmetic for this tutorial. And in the next tutorial, I'm not going to something called Concatenation, which is sort of like addition with strings. And ah, you will find that pretty interesting, too, I'm sure. So I'll see you in the next tutorial and thank you for watching. 8. Concatenation: Hello, everyone. Welcome to practical C plus plus the beginner course. My name is Zak, and in this tutorial we will be discussing concatenation now, before we actually get into Concoct Nation, which may sound like a difficult topic, which it's really not. I just kind of want to discuss this using name, space Standard one more time with you guys just to give you an idea of why it is in our code and the reason why I told you at the beginning of the Siri's not to worry about it. And that was just the reason why we're putting in our code is to make our lives easier. And I want to show you it's because when we do something as simple as cl hello world and we try to output that if we don't have this using name Space standard, all of a sudden our coat falls apart and we get an air right here in air and it says air Sea out was not declared in this scope Well, without getting into too much detail. This name space is including a function. Ah, see out the standard functions see out the output operators so we need using the name space standard just to do simple, you know, standard operations such as Output Hello World to the screen and then in the line. Now there is a way to get around this. Obviously, you could take this out and do something else to use this function, but I don't want to get into that yet because that's more of an advanced topic. I wouldn't consider that a good topic to discuss with absolute beginners and programming know that goes into using name space functions, which I consider, and advanced data structure. You know, it's similar to a class in a way which is going into object oriented design. And that's not something I want to get into in this series with you guys, because I just want to cover all the basics. And then when you get this down, maybe in a future, Siri's will go over advanced data structures and object oriented programming. But for now, we're gonna keep it simple, and we're just gonna keep using name space standard in our code. That being said, let's move on to concatenation, which is a simple and my opinion. It says it's a simple topic, even though it sounds difficult. And all concatenation is it's basically the addition of strings, and I want to show you that what I mean. So if we do string first name equals Tom and then string last name equals Jones, then we can actually out. Do you know something like string? Full name equals first name. Plus, let's add a space in there, plus last name, and we can output that we can output full name and it will output. Tom Space Jones. Let's run it. As you can see, Tom Jones appears in the Consul. So that being said, that's basically all there is to concoct nation. Now, there are a few rules. Um, you know, if you mess around with that, you're gonna find out you can't do stuff like output, Tom Place place Jones. When you output that you get an air, so you need to have a variable in between your raw strings. You need to be adding a raw string to a variable when you Dukan cat nation. Either that or two variables together. If you understand that, and if you don't, I would say cause practices concatenation topic and it will become ah, simple to understand? You know, when you can use concatenation when you can't and just messing around with it, you should, you know, get enough airs just playing around. You'll say, Oh, okay. I get what he's saying. You know, you have tohave, you know, if if I wanted to do output it Jimmy place, um, last name, I can do that. I can say Jimmy Jones, But if I want to say Jimmy Place Jones, I can't do that. That will throw in air. So that being said, that's basically all there is to strengthen Cat Nation. There are some, you know, built in library functions that you can use, but we'll gettinto built in functions later in this series. For now, I just want you to mess around with Can Cap Nation. And I wanted to show you while we have using the new space standard in our code. So thank you for watching and I'll see you in the next video 9. If Statements: Hello, everyone. Welcome to practical C Plus plus the beginner's course. My name is Zak, and in this tutorial, we will be discussing if statements now if statements are a very important part of programming and C plus plus programming. Um, because and if statement what you think of it as a way for a computer to make a decision based on certain conditions being mitt meaning if you think about like a small weather app , there might be an if statement. This says, you know, if it is raining, then show a cloud on screen. But if it is sunny, then show the sun on screen. And that's kind of what if statement is, it says if this is true, then do this. And I want to show you that I'm going to say if we're just gonna put true in here, which is a boolean variable, we're gonna go over that here in the second as well. We're going to say this code is ran. Okay, what's actually had an incline. And when we run this, I will say this code is ran because the condition inside these parentheses is true. And inside these parentheses is where you put your condition. So if we put false, okay, this code will not be ran. When we run it, you will not see this. It will just return zero and just kind of show you to kind of touch more on true and false . If you remember in the one the first tutorials over data tops, we talked about 1,000,000,000 debt data. And to do that, you talking bull because we're gonna declare a booing and data top. I don't I don't think we actually did an example of 1,000,000,000 data top, but I think we did discuss it may be and ah, I'd actually have to check and look. But boot Boolean data is a another day to top. That is either true or false. So we could say bullion, You know, um, var wine equals true and bull var too equals false. And then we could actually put the variable in here so we could say boulevard wine and which is holding the value true. And, ah, this coat will be ran if we run it. This code is ran, as you can see. So, uh, you can kind of see in this tutorial, you know, along with if statements were also kind of learning about Boolean variables, which are a very important part of C plus plus programming. It's will because all of these if statements are going, uh be focused on whether the condition inside these princes is true or false. Now, that being said, you don't have to have Boolean data in here. Um, necessarily. We could actually dio something like this. We could do it. Num wine equals five. And we can say if five is greater than three, then run this code and this is the greater the inside son. And if we do that, I will say this code is ran because actually, I hard coded five in here, but you could actually put number one is well, if number one is greater than three. And, um, I will say this code is rand the same token, though if you put if five is less than three, which it's not, then this condition is false because five is not less than three that returns false. So this code will not be ran. If we run it, you can see process return zero. This output statement was not ran, so that's pretty simple stuff. You know, you can kind of make an if statement Ryan based off of whether the condition inside here is true or false. And we're gonna go more in depth with that later in this section when we make our own at the practical add that you could use and I might even make some kind of number guessing game. I haven't decided yet, But either way, we're going to really show how we can use these if statements to make a really nice flowing code that makes decisions based on user input. So that being said, um, what if we wanted to add another part of this if statement basically saying, you know, if number one is less than three will say, um, well, actually say it will say Number one is less than three. But what if we want to say if numb one is greater than three will say Number one is greater than three. Well, to do that, we put else if number one is greater than three brackets, output number one is greater than three in line. To make it more interesting, we're going to say it numb one where I say enter a number and then put that number in them wine. And then depending on that, um, will be what code is ran. So you can say when we enter the number, it's gonna hold it in, um, wine. Okay. And if number one is less than three, this code is going to run. But it else if numb one is greater than three than this code is going to run. Watch. When we run it, what happens? Enter a number. We're gonna say 77 is greater than three. So this code down here should run. But this code right here should nine. When we hear enter, it says number one is greater than three and Onley this code ran. And so to run multiple if statements together based on one calculation, kind of you would use if and then else if and then if you wanted to basically do a default if all the above or false you add else and then brackets and then you don't add a condition to l statement. This basically says, if oil's fails, if all these return false, then do this no matter what. So just kind of thinking about it. What? What would be the default would could say, you know, if if number one is less than three, do this. If numb was greater than three, do this well. Otherwise, that would mean that number one is equal to three, right? So we could say else number point is equal to three. And to show you that we're on it and we'll enter three. I'll say Number one is equal to three because basically, we didn't even have to do a condition because its it knows that says, well, the way we did it, the way we coated it, We said, If this is false and this is false, then do this. You know, if all else fails, do this and that's what happened. But at that same token, you really don't even need this if this l statement, if you just wanted to do another else if you could say else. If numb one equals and just to This may seem confusing to you at first. But in an if statement when checking, if something is equal, you do need to equal signs. This may seem confusing at first, and it will probably take a little bit of practice, but that's just how C plus plus and even Java is coated. You know, you do need to equal signs to check a condition inside an if statement. So that's why I have to equal signs here rather than just wind, because that will actually return in air. So we need to equal signs here. So basically it's saying if number one is less than three, do this else. If number one is greater than three, do all this else if number If numb one is equal to three, then say Number one is equal to three. And that's the same thing is just saying else do this because those were really the only three possible outcomes. But you can see if you had a whole bunch of different statements, how you might just want an else statement at the end to return. A default value section is, um, you know, for instance, there's really not anything that would run this code. She would say you didn't enter a number, you know, because that's probably what would happen. Um, in fact, I think if we entered a string, it would actually return zero. It would just throw zero or a garbage value and no more in So So let's see. Let's just run it real quick if we enter three again, Um, this code will run. Number one is equal to three. But let's see if we don't enter if we enter something, the civilian try to get this code to run. Um, six years. If we Rhine just type in star or something, they will say Number one is less than three. And the reason why it's saying number one is less than three is because even though we entered Star is because Number one is a garbage value right now because we entered star and we're supposed to have an integer value inside our morning. So instead it through a garbage Valium in there, which is probably actually zero. It probably just defaulted to zero, and we can actually check at the end of everything we can actually output after all these if statements number one. So if we type in something like Zach says, number one is less than three. And the reason why, because the default value for no more and just happened to be zero. That was the value in the memory address for numb wind and That's why so So you could see how important it is that the user enters a number. Because if if he enters a string than the first branch of code is going to get, Rand, this branch right here, which may not be what you want toe happen to so it it may, you know, maybe. Ah, good idea. In a program like this, put emphasis on number. You know, um, there's obviously other ways you could handle it other than just putting emphasis on that. Um, for instance, you know, there's try and catch, um, pieces of code, but that's all advanced stuff again, so we won't be worrying about that. But when you get in, the more advanced programming you will be doing, trying catch causes and stuff like that and catching your exceptions that get thrown for whenever the user enters wrong data. So that means said, that's pretty much all there is to if statements, you know, I do think it's a good idea for you. Maybe go look up the operators that you can Do you know, for instance, um, if numb one is greater than you can also do greater than equal, which means greater than or equal to three. You can also dio less than or equal which. Basically, if number one is less than or equal to three, it will say Number one. It's less than three. If numb was greater than or equal to three, Number one is greater than three. And let's just run that and see if both codes get Rand. Because, basically, if we enter three, it will say Number one, it's less than three because Onley this code got rand but really could have ran if we just had, if instead of else, if because by putting else if it's basically adding it onto these if statements free under three again, I think all three of these election get rand. Um, let's check it real quick. I believe I d. But and I am I believe it's actually froze. But I do want you toe. I want you to play around with it right now and, um, just kind of look at all the different operators again. Um, you know, Ah, another good one to look at is not equal to exclamation point equal. That means not equal to so if it says if numb one is not equal to three run this code, And so before moving on to the next tutorial, really want your practices. If statements and really watch where you're code runs, you know when you can do stuff such as I want you to, you know, use different data tops to because you could say if, um, for instance, if you had a variable called name equals Jim, you know, string values. If you had a string in the name and if the name was GM, you could basically run this code. If name equals, Jim will say Welcome, Jim or something like that, So just mess around with it. Practice with ease if statements. And in the next tutorial, we're gonna look at an alternative to if statements called switch statements and you will kind of be ableto decide on your own, which ones you like using more in your code or when's the right situation to use? Which one is, and I think you'll find it pretty interesting. So stay tuned 10. Switch Statements: Hello, everyone. Welcome to practical C Plus plus programming the begin. Of course. My name is Zak. And in this tutorial, we will be discussing switch statement. Now, As I said in the previous tutorial, a switch statement is basically just an alternative to an if statement, but they are used in different scenarios. Now, I'm gonna go ahead and kind of give you an example of what a switch statement looks like and then discussed you how it works. So gonna go ahead and type out everything right here is where the switch statement starts. Okay, Groups, You put everything in brackets, go ahead and give you some room, and then you put your cases in. So here is our sweet statement. The basic functionality of a switch statement. Okay. And I attacked it out because I just kind of want you fall along. It's gonna be easier for me to explain it to you like this. So here we have a variable called Raid and our great isn't be okay. And then below that, we have our sweet statement defined in these brackets right here. Everything in these brackets. Okay, So, basically, to define a switch statement, you write the word switch and then in parentheses. Next to that, you put the variable that you are analyzing. And this Kate it In this case, it's grade. So we put grade here, and then in your brackets, you put your cases, so you put case and then what you're comparing the grade to. In this case, we're comparing it to different letter grades. So the 1st 1 is case A. And here you enter more brackets and you put the code that you want for case A. So you made a 90 or above. And then in Case V, you do the same thing. You can put your code here, you made a 80. Or, Abed, you can repeat that for each of these. And we're gonna go ahead and do it here so that you get full visual ization of how the switch statement works. In case F, you failed. So basically what's happening here is this code runs. We have a grade of a B, and then we look in the switch statement. We tell the switch statement to analyze grade, which is be so right here is grade. The variable that we're analyzing, we say, isn't in a well, no, it's not. Is it to be well, yes, it ISS So we're going to run this code. Is it a C? Well, no, it's not. Isn't NF well? No, it's not. So this is the only code that you get ran, but I want to show you something real quick so that you can You can see what happens when before I fix it. Let's go ahead and run this program and you can see it says you made an 80 or above you made a 70 or above You failed. Well, that's interesting because we made a B. It said everything except a But once he got to be, it basically did all the Cobell obi. And that's because, and a switch statement you need to add a break. And to do that after all your code, you say break and you do that at the end of each case to tell it to leave the sweet statement because switch statements have what I like to call a waterfall effect, meaning If you don't put your break and right here on this code to break from this switch statement, then once this is crew is gonna waterfall down into the rest of the code, the rest of the switch state and code. So if we took out this break statement right here, it should Ryan the B code and then run the C code before it breaks. Let's check it out. As you could see, it said you made an 80 year above you made a 70 year above because it didn't break from the sweet statement until it got right here. So what we need to change is, of course, if we just add a break here, you'll see that it says you made it a year. But because our greatest be and just to kind of show you how we can further this let's do grade and then we'll say something like Inter a grade, enter a letter grade, okay? And then let's you see in tow, hold our grade and then let's watch how this sweet state is He's injury letter. Great. We're gonna enter F. It says you failed because basically, it went through the sweet statement that put the grade we entered in here in the sweet statement, analyzed it. Look for the case is that in a no is it be? No. Is it a c? No. Is it in f? Yes, you failed. And that's how switch statement works. Okay, so that's right. One more time we're gonna enter in what's interim and eight and says he made a 90 year above. So that's the basic functionality of a switch statement, and you can see how it's very similar to an if statement by checking which condition is met . And like an else statement in an if statement, a switch statement also has something similar to else, which is called default. So if we wanted to take out this f, we can just say default. You made an Don't worry, you failed. Which is the same thing is saying is if none of these were true and obviously he failed, let's let's go to the default sweet statement, which is? He failed. If we run that and we enter in and f well, that's even entering and D. It'll say you failed because if we enter into D, obviously A, B and C aren't aren't a d. So it's going to go the default. Just say you failed, but you know, obviously find Rizzi. That same code is gonna run because it's either a, B or C. So if you want in tow to restrict the user to Onley entering the correct letter grades, what you would probably want to do is say something along lines of case. If and then output, you failed in line and then on the deep ball, you could say something like You entered an in valid letter grade. And now when you run the code, if it's not ABC, or if if you enter something like our will say you entered invalid letter grade because it's going to the default. So that's the basic functionality of switch statements, and I will see you in the next tutorial. 11. Practical Program #1: Hello. Welcome to Practical C Plus plus programming mining, Bizet. In this tutorial, we're going to be taking a look at her first practical program that we're gonna make together. And it's just gonna be a simple calculator at and to do it. The main focus of this is I want you kind of understand how we're gonna structure this program and use the concepts that we've already gone over to make it work. Like we wanted to. That being said, let's go ahead and begin and that the way I want to structure this is basically, we're gonna make a calculator that lets the user decide at the very beginning. If he wants to do addition, subtraction, multiplication or division. And to do that, we're gonna use a switch statement. Okay, so let's go ahead and structure. It s so we're gonna right switch. And then there we go. Sorry about that, guys. I am. The rest of my brackets got deleted. There we go. That's right. It just like that. Make sure you get your return statement, okay? And then in this switch statement is gonna be the variable that we're checking and the while I do it is, I basically wanna have the program opened up and have numbers. 134 ill say one addition to subtraction. Three multiplication and four division. And so to do that, we're gonna hold A variables were going to say it and we'll call it choice. Okay? And we're just gonna leave it like that. We're gonna say, see out, Enter a choice. Actually, the way we're going to do this, we want to let them know beforehand on what their options are. So we'll say Wine addition. Okay. In line to sub track in line three, Malta application in line and four division in line. So this is what they're going to see. They're going to see basically this on screen, and they're gonna have to make a choice on what they want to use. And then what we'll do is we'll say, at the very end, let's give a little bit more space, and we'll say, enter a choice, and then we're gonna hold that with a C and statement and choice or variable choice. So kinda if you need to push, pause and kind of breathe all this in exactly what we're doing. This is all stuff we've covered in the previous tutorials, and it should be fairly straightforward to you at this point. So at this point, we're holding the integer of the choice that the user selected into choice. So what we do is and switch, we need to put the variable that we're analyzing, which is our choice variable and then make cases. So obviously we'll have case wine case two case three case for let's go ahead and add a default as so okay. And obviously the deep all we can go ahead and add something like exiting you entered and in valid number eso because basically, we're going to say if they don't enter wine and they don't enter to or they don't under three or they don't enter four billion or something else and we're gonna say, exiting you entered something invalid and then it's just gonna go straight to this code return zero in the program will end. So that's how we're gonna handle that Now, in these, let's go ahead. And just so we don't forget, we're gonna add brackets to all of these as so that way, These are format and nice and easy, so that we can see. You know, Case three is gonna be right here. Case for is gonna be right here, and we're gonna go ahead and add or break statements so that we don't forget, because that's gonna be very important. These break statements were very important for this code, the way we're structuring it because you don't want multiplication and division to be run on the same at the same time. So let's go ahead and add or break statements, which is a good practice to do with sweet statements. They don't forget. I would recommend always adding your break statements first, if they're necessary. So there we go. So, as you can see, if you need push pause and kind of look at this, Matri had everything right. Go induce of now because this is how our program, our calculator, is gonna be structured. Is that so? With a switch statement? Okay, so that me and said Now that we have or choice entered, basically, everything is going to be the same at this point. So we'll say right here will say, Enter number one and just a the top. Let's go ahead and add our new variables. So we have choice here and if in you may see people to a different way. But the convention that I learned in school was always declare your variables at the top of your main function or at the top of any function that you're in, for that matter. So that's what we're gonna do. You just taken getting a good habit of doing it. Let's give us a little bit space. We have any choice, our sweet statement, and then we're gonna have We're gonna use doubles in case they decide they want to use, you know, floating point values for their calculations. So we're going to say double number wine. And then actually, before putting our seven corn here, I'm gonna Mills were going to say number two, which is a new way. This is another way you can declare your variables, and this basically says double number one and then double number two. You can declare him like this is the same. Both of these air doubled, and we're not initializing them to anything. We're just making two variables of the data Titan double, so you can try that out and again. It's a convention you could do. You could just write double number two down here if you wanted, but it's all a matter of preference. So we're gonna leave it like that for now. And then we're actually gonna add one more thing. We're gonna We'll just, uh, believe that just like that. Actually, we got, say, on each of these, we're gonna have him enter the number one in the number two. So the codes gonna be pretty repetitive, actually, on each one. So I'll say, see out into number one. Okay. And then CNN number one Okay. And then we're going to say we're gonna give a little bit of space, and we're gonna say Enter number to We're gonna get the input for a number two. Okay, Then we're gonna output. We're gonna give us plenty of rain. Okay? We're gonna tell him the result. We're gonna do it like this, cause so you can see the the numbers don't run off the board. We're going to say result equals Okay. And then we're gonna say since case one is addition, we're going to say number one plus number to and then break, and this code is gonna be pretty repetitive. So if you want You can just copy this because I could said it's gonna be the same for each one. Pretty much have pasted in here. Okay, True. Fix your formatting in the copy and paste mint stuff. Remember, Case to is subtraction. So the only one you're really gonna have to change is this change that to anonymous Case three was multiplication. So we'll just have to change the sedition to a multiply. A sign in case for was division will change that to a divisions on. So there you go. So now our program should run just like we want. We got our switch statements and everything. How it needs to be. Gwen. Save it will say build and run and let's see what happens. So there we go. As you can see on screen, we have our choices, Addition, subtraction, multiplication and division. Let's go ahead, Inter subtraction. I will say inter number one. So let's do five. Inter number 23 Well, say result equals to process ends. Have a subtraction. Let's go ahead and do ah, Division four else they enter number one. I kind of want to show you something with division. Um, because I don't know if we went over this in arithmetic. But let's say I dio that's how Do non Okay And then we do for number two, we do four, so they'll be non about about four. Obviously, there's a remainder there, but just straight division, it's not gonna give you the remainder. That's just gonna give you two, because it goes into it two times. Now you can see that the result was 2.25 And this is an interesting topic. If you if you practiced your arithmetic, the reason why we're actually getting a decimal. The actual answer is because we're doing double on double division. So we haven't since both of these or doubles the results gonna be in double. But let's change these two integers for a second. We're going to do that same That same problem there were going to four. We're going to non divided by four, and we should get to, As you can see, we can get to you even though the real answer is 2.25 And that's because reason whole number division and not allowing for a double, uh, result value. And that's really what we need to do if we you know, if you want into your just make it simple, you just change this to double And the other alternative would be If you want into, you could just leave these as integers and then hard code, a double result value. And basically, you could go down here and say result equals, um, you know, here you could say result equals number one plus number two and then say result equals result, and that will still give you in double value because you declare results in dough. But this was our first practical program. I just wanted to show you how we're gonna use everything that we learned throughout this course to actually actually applied to practical situations. This is the 1st 1 you know. We used our switch statement going. Thing we really didn't use to mention here was an if statement. But if you take the same token, if you want tried on herself, you can switch out the switch statement for an if statement. So to do you know, if choice equals Boyne, do this else if choice equals to do this and so on and so on. So I challenge you to do that are trying to try to this program with sweet statements and not telling you trying to it with, um if statements. Thank you for watching. And in the next tutorial, we're going to go on to more intermediate programming topics, So thank you for watching. 12. While and Do-While Loops: Hello. Welcome to Practical C Plus plus programming the beginner course. My name is Zak. And in this tutorial and this section in general, we're gonna be discussing a little bit more intermediate topics. And ah, starting off, we're gonna discuss looping such as while and do I leaps. And this should be a pretty short tutorial because we're not gonna go too much in depth about. We're just gonna kind of discuss how to use them. And once we get further into this section, you'll see the practical use of them and how often you will actually be using them real life situations. So let's go ahead and show how to define the loop. We're going to start with a Y, a leap such as this while you taught the word wild. And then you put your parentheses for your condition, and then your brackets and basically anything in your brackets right here will be run. As long as this condition is true. On this condition is true. This condition is checked. Better to say this condition is checked at the beginning of the loop, the code is ran and then it's checked again. And if it's still true, the code continues to write. So the best way I can say this is, um let's go ahead and do it like this. We're gonna say int um, Rhine equals 10. Okay? And then we're gonna basically say, See out, Ryen, see out, run, end line. Okay. And then we'll say brine equals run minus wine. Okay? And in this loop for the condemned for the condition, we're going to say, while run is greater than or equal to zero. Remember, we discussed this operator. That means greater than or equal to zero. Okay, so moving nine. Basically, this is going to say, while this is true, do this. And if you look at the end of our code, basically, we're saying we're setting run equal to run minus one. So the first time the code runs, um, run equals 2 10 and then the next time it equals two. Not until it gets to zero, and then it should quit running. So let's run it quit. And as you can see it that real quick, but it printed off because we're out putting get 10 9 all the way down to zero. And that's basically what we want. You to do? Um, one more. One thing I won't discuss that maybe we haven't discussed earlier in the arithmetic because there really is a lot of rithmetic things you can do and C plus. Plus, they're very interesting, and one of them I want to show you now over our while we can use it. Is this when we say run equals run minus wine Another way in C plus plus that weaken do that is, say, for it's actually easier less. Code writes, a run equals Monets wine. And what that does is that basically means run equals run minus one. And it would be the same thing if he said, run equals plus one. So say, Let's just run it real quick and you'll see that we get the same result. You're just gonna have to visualize. Okay, so it's wrong right there. I was wrong about that. So it may be it's minus equals. I believe it's minus equals wind. Yes, that's right. So I had it backwards. Sorry about that. So So this right here, this minus equals or if you did plus equals, is the same thing as saying Rhine equals run minus wine. So if we do that, obviously we're going to get the same thing is run equals run minus one. Right now, the one thing about that's dangerous about loops is that you could get caught in an infinite leap. Okay, so that means said, if we did, you know, Rhine plus equals one that sent and say run equals run plus point. Obviously, the variable will never get to zero, and this thing will be caught in an infinite loop. We're gonna go ahead and hit building, Ryan. So you can see what happens when this, uh when you get in this situation. And as you can see on screen, the number is just adding up very quickly. You can see how fast processor is clicking through these numbers going through this leap. I mean, it's almost instantaneous Will be 100,000. This Coby ran 100,000 right about now, you can see we've already ran this code over 100,000 times. And if you get caught in this, one thing that I would recommend doing is goes pushing control, see on windows, and that shuts it down. So if you get caught in an infinite loop on windows hit control C. I just come one to show you what that was. And, um so yeah, it infinitely hit control C and get out of it. And that's one thing you need to watch for. You know, when you're doing this is to make sure you get your code writing that you think about in your head first before you run it so that you don't get caught mothers infinite loops and have your computer accidentally crash. So there's a difference between minus equals and plus equals. You see how much different the code is, rather than getting in an infinite loop. It it quits because this statement is no longer true. Once run is less than one. So once it gets to negative oin, it doesn't run anymore and it goes ahead. It's return zero. So that that is a wild loop. Okay, so now I'm gonna introduce a do while loop and a dual leap basically says, Do this anything in here and then it checks the condition at the end, while Ryan is greater than or equal 20 Okay. And then you put a semi colon, so it's a little bit different syntax. You say do and then your brackets, and then while your condition and then it's semi colon. So let's see what the difference is between this having your having your condition checked at the end of the code rather than the beginning. But the difference is this. If we set and run equal to negative five, obviously it's not greater than or equal to zero. But the difference is, is this. This code will always Rhine at least annoyance, So I'll show you what I mean. Even though run is less than zero. When we run this, the code will still get Rand once and shows it's negative. Five. Okay, so that's the difference between a wall and and do well. But you can still have the same. You know, you say, Well, when would I use that? Well, when we get into more practical examples, you'll see these Duvall. And while loops are used interchangeably, depending on the situation, you know, if you want your code to run at least once, no matter what, then, obviously, and do all this more appropriate than a while loop. But just cachet that we can still get the same result out of this code is wildly. We're gonna go ahead and set, run equal to 10 leave our condition is the same and say run Mantis equals one. We should get the same result as before 10 all the way down through zero. So that's basically an introduction on loops and, uh, particularly just with a focus on while and do while loops in the coming tutorials, We're gonna go into four loops and more fun stuff like that, so stay tuned. 13. For Loops: Hello. Welcome to Practical C Plus plus programming the begin. Of course. My name is Zak. And in this tutorial, we will be discussing four leaps. Now, in the last tutorial, we discussed wild leaps, and I got to say four loops are quite different, and you'll see why here in a minute. So the way of four loops works. You're gonna go ahead and set it up the same. You're going to say four your condition and then the brackets, just as you would a while loop or an if statement. You just proceeded, you know, before the condition is where you write four. So the confusing part too many people is the condition inside the four leave and how it works, and I'm gonna go ahead and explain it to you all. But first time I set up a variable, call it value, and we'll say, um, equal. Actually, the way we're gonna do it is yeah, Well, say value equals zero. Okay, we're just set value equal to zero. And then here in the four Lee, we're going to declare an integer called index. We're gonna set it equal to zero, okay? And then put a semi Colon. Now just stick with me for a second, because it I know right now you're thinking What? That's that right there. You're declaring something in a condition. Well, this is not the whole condition. This is only 1/3 of the condition. So after you declare your variable int index equals 20 we're going to say index less Van 10 and then we'll say Index plus place. Now, I want you sit here and breathe this in for a second, because I know it looks complicated. Especially if this is your first time ever looking at afford a leap. So basically what we're doing in this four Lee, where declaring indexes variable in setting, get equal to zero. Okay. And then were saying this right here is basically our condition is what I would call the condition Assad from these other two. This middle part says, do this wild loop as long as index is less than 10. And then this third part is what I like to call the increment. The incremental part of the four loop. This is how much one increments each time. Baluch Rhines the variable that you're testing Index and I mentioned earlier in the last tutorial. I believe that minus equals was the same thing is saying index equals index minus one. Well in C plus place index plus plus is the same thing. That same index equals index plus wine. So easier way to write that is to just say index plus plus. And I know Ah, we did you know something like index minus equals one. Well, another way to do that, actually in the last tutorial could have just been index minus minus is well, so that's the same thing as saying index minus equals one. And that's also the same thing. And saying, index, um, monness index equals index minus one, Just as this is the same thing is saying index, this is the Sundays and index equals index plus one. So just keep that mind index plus plus were incriminating index by wine each time. The code inside this loop runs and what I want to do is I want to say, see out value en del, and what we'll do is we'll actually add, um will add five the value each time. So if we run this, you'll see we get 05 10 all the way up to 45 because this code is ran all the way until index equals 10. And each time this four loop is ran indexes getting incriminated by wine on one index equals 10. That means index is no longer less than 10. And the four loop jumps down here to return zero and you get process returned. Zero. So just to kind of take kind of show you a little bit more about it, let me let me actually take out value. We're gonna output index is what we'll do instead. And you can actually see what happens to index throughout the four loop. We're gonna hit Rhine. Let me, um let me get rid of this real quick. That was still open. We're gonna hit Ryen, and you can see what happens to indexes. It goes through the leap. Um, it goes from zero all the way up to nine and then loop ends. So I want you to practice with this four loops. See how you conduce different counting exercises and kind of cycle through numbers with this four loop and, you know, even even tried changing this operator Teoh index may be greater than 10 and see how that changes it. Um, you know, because obviously, if we ran this right now, it wouldn't even run. It would just return, because index start zero. So indexes never actually greater than 10. So this code never runs. And also, I want you to try, you know? You know, obviously we declared index right here. But what if we just said index equals 20 and we changed this value? We said index right here equals zero where we just said it index. Then we don't have to put right here. We can just right, index, and we can actually declare it like that. So there's different ways you can, um, kind of, I guess, declare this four loop. And I know right now, just counting through these loops like we have in and adding numbers doesn't seem very practical. But I promise you, by the end of this section, you will see very useful and practical examples of how we will use these four loops. So stay tuned for the next tutorial. Thank you. 14. Data Structures - Arrays: Hello. Welcome to Practical C Plus plus programming. My name is Zak. And in this tutorial, we will be discussing a raise, which is basically my introduction to data structures because I see an array as the most simplest, the most simple data structure that we can kind of delve into without getting too advanced . And I really went to introduce these arrays because you can use four loops and while leaves is a way of populating these rays and we'll get into that probably at the end of this tutorial. But to start off with what is an array? Well, an array is basically the best way I can explain. It is a list. So the way I was taught was think about when you go to the grocery store. Let's do this together. So we're going to declare an array, We're going to call it String. We're gonna make an array of strings, Okay, strength, and we're gonna call it grocery list. Okay, this is our grocery list. And this. I want you think of an array because it will give you a really good visualization. So we're going to the grocery store, and we need to buy several things we need by eggs when you buy milk, we need buy bread when you put all these things on the list. Well, to do that, we need to know how many items are gonna be on our list, First of all, with an array. And to do that, you put two brackets like this, not curly brackets, but straight brackets. And inside here we put a constant value. Variables are not allowed on the inside of the brackets during the declaration of an array , and that's very important to remember. So you have to know how many items you are going to populate your array with when you start . That means said, let's go ahead and assume we're going toe have only three items on our list. Okay, so we're gonna put the value three here. Now, the next part is to put these items in our list so he right equals and then curly brackets . And inside these curly brackets is where we write the items on our list. Now, obviously, we declared this grocery list is a string data type, which means we have to put string elements inside this array. So the first element is going to be Eggs were going to say we want eggs, Okay. And then you separate each element with a comma. We'll say we need milk. And then we'll say we need bread and then end three declaration with semi colon. So this right here is your first declaration of an array in C plus plus. And it's fairly simple. You just have to remember that you need to put the constant number of items right here in the brackets, and then you declare each item each element in the array, so to speak, and the curly brackets over here. Now, you may be thinking how when would I use an array? Well, you're going to use it all the time. We were gonna go into that later, but before we do that, I want to discuss this value right here. This three. Now that we wrote a three right here and we can't put variable. So, for instance, if we wrote into index equals three were not allowed to put index right here during our declaration. And I believe code blocks co blocks sometimes lets you get away with it, but actually it doesn't see, I just try to run it and I got on air. And that's why because we put a variable here, you're not allowed to put a variable there. However, I want to go ahead. And while we're on this topic discussed constants with you and a constant is different from a variable in that it never changes. Meaning what? All you have to do is add the constant keyword, which is CEO Seo in S T c o N s T. Const it index three. So now if we run, this index becomes a constant value in the program. The program runs now. Keep in mind when you add CONST. Right here. You are not allowed to change index later in the value. So if I try to say index equals to two all of a sudden or index plus plus, I will all of a sudden get an air because you are not allowed to change values that have the constant keyword in it because it's constant. It's not supposed to change throughout your program, so I just want you to keep that in mind. The other thing with Constance I want you to keep in mind is that it's often a convention to make constants in all capital letters and C plus plus so that when you look through the program, you automatically know what's a constant and what isn't. And so this is how you would probably see it most C plus plus programs. And though Index probably isn't a really good name, you would probably want to say something like Size Size is probably a a better word for the array constant. And that's usually how you'll see it in C plus plus programs when you're talking about a raise. Now I do want to go into we discuss four leaps last time in our last tutorial, and I want to go ahead and dive into how we're gonna use four loops and a raise together. And that's why I made a raise as our next tutorial after four leaps. So that would be fresh on your mind and you can see exactly how we're gonna use it. So let me go ahead and show you that Now we're gonna make a four leap, okay? And in this four loop were going to say integer index equals 20 and then we're going to say index less van size index plus plus. Now think about this for a minute. We're declaring a new variable called Index, and we're setting it equal to zero. Then we're saying, Index, this is our condition. We want to do this for loot. While index is less than size and size, we set to three, which is also the size of our stringer A. And there were incriminating index by one each time we run through this for lead. So this four loop If we ran it, we'll just see out index real quick so you can see it. It should only run three times. And if we run it, that's exactly what you'll see. 0123 times. And remember, index starts zero. Now when you look at a raise, this is a very important thing to know, because if you don't understand this concept, a race will get very confusing, and that is to access an array. So let's go ahead and access grocery list. We're going to say output grocery list, and then you put brackets okay and L and inside these brackets you put the number that you want to output. So that means said, let's let's go ahead and say, Um, we want to output eggs. No, let's say we want to output milk. Okay, So if we want to output milk, you would think you would enter to hear right. Well, that's wrong, because the thing with computers is in a race, especially, is a start counting it? Zero. So if you wanted to output the word milk, you would have to say grocery list and then put one in the brackets. Because this is index zero. This is index wine, and this is index, too, Which is why, oftentimes of four loops, you're going to see him start at zero. Because in a four Lee, oftentimes he used a raise or even vectors, and they all start counting it. Zero. Which is why you will always see most of the time in your program a career. These four loops start with a variable. There's initialized it. Zero. So that means said, we can actually put Index in this box to print out zero. The next time it'll count through will be one the next time it will be, too. So it printout eggs, milk and then bread. Now going back to a topic we discussed at the beginning, we said that you could only have constants in these brackets, and right here we have a variable. Well, the constant rule only applies when you're initializing the list. When you are actually accessing the array, you can use variables as we are right here. So let's go ahead and run this program and we'll see. We get eggs, milk and bread. It prints out the whole list for us. So, as you can see, this is a very practical example of using a for loop to iterated through a string, um, array, which we declared is a grocery list. And when we get further into this section, we're really gonna take this to the next level, and I think you're really going to enjoy it. So stay tuned and thank you for watching. 15. File Output: Hello. Welcome to Practical C Plus plus programming the beginner course. My name is Zak. And in this tutorial, we will be discussing file output. Now, In earlier sections, we talked about just basic console output, and I want to stress that you to not get too, um, concerned about file output because it it's actually a lot more simple than it's going to seem at first. It's gonna be a lot of new stuff, but if you just look at it and practice it, you will see how much more simple it actually is. Then it's gonna first appear that being said, the first thing we have to do when dealing with file output is include a new library. So we've been using this pound include Io String, which stands for input output stream. And we still need this library for for a programmer work. But we need to add a new one. And to do that, we're going to say pound include, and the new library that we're using is called F Stream, which stands for file stream. They have input output stream, and now we have filed stream. Okay, so the next thing we're going to do is declare an output stream, um, file, so to speak and output sharing file that we're going to use. And to do that, we say, oh, F Stream, which stands for output File stream, and then you give the output file, stream a name, and we're just gonna call it output file. Okay, now, output file. What you want to do is add parentheses and put a semi colon. And in these parentheses, I want you to declare the file name of the output file that you want to use now. That being said, if there's already an output file that is, that has been made, that is, in the current directory. Then, of course, you just want to enter that file name. Now, if it's in a different directory, you're gonna have to specify the full path. And to do that, you would say, you know, see, for the c drive. Um, Colt, you know, colon backslash, backslash users, backslash, backslash, and you do have to have double backslash when using the strings. And ah, without going to mention debt, you know, without cause I would really need to put that in a new tutorial. And ah, I will actually make a tutorial about that. But the first backslash window, almost strings, is considered an escape character. So just know that whenever you're specifying these files and you use a backslash, you need to put two of them in order for the 1st 1 to be read. So that's how I would do that. But for me, I'm just going to declare a new file, and it's gonna be in the current directory. I'm just going to call it names dot txt just like that. And if you run this, everything should run final, just return zero. And you know that you don't get any heirs now named Start txt is gonna be the name of our output file that just got created. So if we actually go to open right here, it should have created the name stuck txt file right here, as you can see for us since we just ran the program. So now that the file is created and everything is fine, we can actually start riding to that file now, before we do that, it's always good practice to have a branching statement in case something goes wrong with creating that file or finding it. And what I like to do is say, if no, I output file, which means output file, if not output file. It basically means if if the output file returns false meaning that it couldn't be created , then run this code. So that means if the help a foul name start txt could not be created nor found, then do this and we'll just output. The file could not be found, and then we'll say, return negative five and this can be any value, actually return negative seven. I could say Return on. I should put a negative value aknegative 10 or negative five. And that way, when we run, the program will run it right now if it says process returned. Negative. Five. I know that the file was not found, and you just use a a random number like that that you can easily associate with an air, and that would definitely been air. But since process returned zero, we know that names dot txt was created, so let's move on and let's actually write to this file. And to do that, let's create a string name equals Zach, and that's gonna be a fairly easy ah example. And whenever we make our practical program at the end of the section will see a more ah, in debt way of looking at file output. But for now, I'm just going to show you an easy that's in the simplest way I can, how it's performed. So we're gonna make string name equals AC. And now we're just gonna write that name to have fallen. To do that, we use our output stream name and the handle on that is called Output File. And then to write to it, you use the output operators as so let's say, unless saying, just like if we were doing see out the output to the screen, we used these less and less. Then instead, we're going to do output file, which is our output stream appear and we're using output file. That's the handle name you should say so output file less and less than name and then return zero. And if we run that everything should run, find with no heirs. And if we open, let's just open named start Txt year old Quit. We're gonna open it right here and you can see Zack was written to this file So that is a basic introduction on file output. And in the next tutorial, we're gonna discuss file input, which is actually a little bit more complicated. So stay tuned and thank you for watching. 16. File Input: Hello, everyone. Welcome to practical C Plus plus programming. My name is Zak, and in this tutorial, we will be discussing file input. Now, as we file output, the first thing we need to do is include the right library. That's gonna be the same library viol stream F stream, and then the We also need to declare a file handle that we're going to be in putting in using the if stream declaration. So in output stream, we use OEF stream and input stream. We're gonna use I f stream for input file string. Now the difference is in input. We need to already have a files. Specify that we're reading from. You know, you don't want the file to be empty. You wanna have a file that ah has data in its That's what we're gonna do. We're gonna use named start. Txt is from the last tutorial, and I'll go ahead and open it real quick so we can add some data we're gonna use names such as, Ah, will use Zack, Um, Troy, Sam, Jim, Mark and Kristen and Margaret Taylor, Um, Jake and, um, Sherry and Francis as the names for our We're just gonna basically make a name list. So these are all names. And let's say we're wanting to read these names in from this file and, um, store them in a variable. So to do that, we need to know the name of the file and declare a file. So we're gonna say will say input file. And we need to declare the name of the file that we're reading from, which was named dot txt. So we're reading from names dot txt. And as with output file, we just need to go ahead and say if night input file C L file not found in Taiwan and let's return a value like negative six is that we know it wouldn't be found. And let's go ahead and run it and we got process return zero. So named start txt was found, and it should be because it has been created. All these names in it. And remember, this is just a cautionary thing. So you your file for some reason comes up missing. It'll return negative six, and you'll know that it that it's gone. This is good again. Just a practical way to, ah code your program to look for heirs and problems with the with the code. So moving on, um, let's go ahead and create something to store these names in. And if you if you want to think about you know, the things that we've gone over, probably the perfect I think that we could use as a data structure sections an array, and we can just make one variable one array and then store all the names in that array. And so to do that, what we'll do is we'll say, um, you know one thing. One thing about an array is that we have. You could have an unknown number of, um, unknown number of names here, but and obviously, if that was the case, you would probably want to use a a different dead data structure. But since this is the beginning of beginner's course, we're gonna go ahead and use an array, or we're gonna assume that we know how many names are in the file. If I If I teach in advance course, but but you know, kind of depending on how how well this one does, if I teach in advance course, we're gonna definitely go into more advanced data structures and a better way to store this data whenever you know the amount of names is not known. But, like I said, for now, we're gonna go ahead and assume that we know how many names you're gonna be on this list. So it's countem ups. 123456789 10 11. So there's 11 names, So let's just let's go ahead and declare a constant value. Constant in size equals 11 and we use that for the size of our list. OK, and then let's go ahead and make an array. Like I said, I like to do. It's kind of a convention to declare your raised the beginning of the functions. That's what going to do. We're gonna declare the array, and it's it's gonna hold string values because these are all string values Alfa numeric, you know, uh, multiple character values. So, um, we're gonna use a string and will say, um, names list. We're gonna hold size for the value, and we're just going to declare just like that, as um, actually, I think the better way to do that would say equal and then brackets and then just do that just like that. And if we run that that should not get. Give us an air and it doesn't. And the reason why is because co blocks and most ID's will see this as instead of having to basically do empty strings 11 times to declare this array. If you just put one empty string in there, co box goes ahead and assumes to set all the default values to this an empty string, which is what we want. We want the strings to start out empty so that we can put new ones in their place later. So that's basically were initializing this array to a bunch of empty string values. That means said, Let's go ahead and get to where we can read in these thes file names. And to do that, the best way to do it is with a pre read and a post read within a wild leap, and you can play with this you want and kind of figure out the best way that you see fit. But when you do it with this wildly, you're going to realize that the pre read on Post Street is actually the best way to go about it because oftentimes, if you don't use a pre read and if you don't do a post read, you're gonna figure out that either you're going to read the last name twice or you won't read the first name at all. And this. That's why I like to use this strategy to read from a text files and I'll show you exactly what I mean in a second. So so beginning with the pre read This is gonna be the pre read and we'll go ahead and comments that I will say pre read. And if you didn't know, I know we haven't discussed this yet in any of the other tutorials to comment code. You just used double backslash that will comment code. So if I if I write double backslash, I can write whatever I want, and it won't affect the code at this point, so to kind of keep track of what you're doing, it might be a good idea to actually comment your code, especially when you get toe pretty, reads and Post reads, because it will make it easier to read when you go back and look at it, starting with the pre read you're gonna use the file handle, which is input file. And then they used the input stream operator, which is greater than greater than if you remember. I'm seeing operator. And then you're gonna store the the, um the the the name The string from named start. Txt. Sorry, I got my I got my words twisted there for a second. You're going to store the stream taken from this file into a variable declared here. Now, it's not advised to put this straight into her straight into an array. So what we're gonna do is we're gonna is gonna say strained. Um, tip name. And we're just gonna leave it like that. We're gonna put it into a variable called temp name. So the first time it reads is gonna read Zack, and it's going to store it here into temp name. Okay, Now, let's make a wild leap, and this will all make sense after we're done coding it and you'll see why. So then we make a while leave and we say, while not input file dot e o f parentheses and this dot e o f is a function and we'll go over more with functions and next tutorial, but it stains for end of file. So basically this condition says wow, input files, not at the end of its file. So basically went, reads through the cursor starts here. And as we read through it, the cursor is gonna move like this all the way down through this file as this wildly continues until the cursor gets to hear of the end of Francis. And that's considered an Indo file because there's no more text in this vile and ah, as long as it's not a day into the foul. This while loop is gonna keep looping. That's why I like to use for my loop. And while it's not at the end of file, go ahead and it interview times comment post read. We're going to say do the same thing is Thea pre greed just temp name and this right here. This pre read this post read in this while loop is your basic set up for file input, and I know right now you're saying that is really complicated. It doesn't make sense, but this is the best way to receive input from a file and you'll see why I want you to play with it and see if you can figure out a better way to do it. But I think I think once you play with it for a while, you're going to realize that this is definitely the cleanest way to receive foul from ah received text from an input file. So let's go ahead and continue on. And you always want the post re to be the last thing in your wild leap in your pre read to be the first thing. They're the last thing before your wild leap. So you never want anything in between your wildly like right here and you're pretty read. And he never won anything in between your post read and the end of your wild leap right here. And that's just a golden rule for foul input. So but anything anything that you want to do, um, you know, data processing wise congee Oh, in between here and that's exactly what we're gonna do. So we're going to say we're basically going to say input or we're going to say, How about this? The name was names list, so we'll say names list. Let's go ahead and declare an integer value so we'll say int index equals zero. We're going to say names, lists, indexes got started. The first Valiant Names list equals temp name. And then we'll say Index plus plus on. What that's going to do is it's gonna go through. Each name in this file is going to start with Zach, and it's going to start Index zero. Go to the first index and names list is gonna store Zach because Zach is me and held in temp name. There's gonna add one to index. It's going to the Post read. It's going to go back up to the top of this wild leap and is gonna put the next name on the list Troy into our array. And if you don't believe me, we're gonna hit run, you'll get no airs and nothing happened. So as of right now, there's no output. But I promise you, it just populated the whole array. Our whole names list array with the with the names in this violin to prove it to you. We're going to use a four loop on the outside of this wild leaps, so we're going to say four. It was called I equals 20 I less than size I plus place. And then we're just going to go through this Ah, this loop and prove to you that names list is populated with the names and are named start txt file. So we'll say names list I and a lot. And if we run this now it's gonna output are array. And right now it just says Francis. So let's take a look at what went wrong there. So something's gone wrong with our names dot txt. So we have. We have Francis here. Maybe. I believe it has something to do with the way we declared this array with this empty string . So let's see if we take this out. If that will fix it real quick, Let's hit run and it's still, says Francis. So we're having an issue with our with our, um, array declaration because I know for a fact we're getting the input file and we're storing it in Tempt name and them were using the index Teoh there. There's your problem right there. So So obviously, if indexes at the beginning of the while loop then and we're declaring it to zero that each time this while loop runs is gonna set index back to zero. So what we need to do is we need toe, take this index equals zero out and put it on the outside of our wildly. That way, it's only declared 20 once. Now, when we run our program, we will get all the names in our list as a minor is a minor mistake. Obviously it ah, completely changed output of the program. And so you really gotta look out for that stuff. And, um, if you didn't really get the mistake like I said, we had index equals zero at the top of our wild leaps. So every time this loop brand, it was setting index back to zero, which is why it has to be on the outside of our wildly. But I think you should definitely go through this program several times because when I first started C plus place, I found foul input is a pretty complicated topic. So go through this program in this tutorial several times, practice the pre read in the post read, and I promise you, when you get it down, it will make a whole lot of sense. And ah, it'll just be another thing to you will be very simple. So thank you for watching and stay teamed 17. Advanced Input and Output Manipulation: Hello. Welcome to Practical C Plus plus programming the beginner course planning Izet. In this tutorial, we will be discussing advanced input and output manipulation. And to do that, I've already got the code that we used from the last tutorial with foul input. If you remember, were just grabbing some names from this names dot txt file. And we're storing them in a variable when we read it in and are pre read. And then as long as we haven't reached the end of file, we are taking this temp variable and copying it to our array with the index of zero. And then we add one to it each time as we go through the output. It was just this names list that we have, but what I want discuss is these tips and tricks for advanced file input, output manipulation and these techniques are going to help you whenever you get in funny situations. The first technical when talking about is if you get a file with something like this. Now this is a header and many files have headers. But if we run this program right now and we change, for instance, we would have to change this to 12 before we did it. But if we ran it, it wouldn't crash. But we would get Well, let me make sure that I have this right. So Yeah, so So let me save it first, cause I haven't saved it. But if we save it now, now it's saved and we rain it again. Now we get names up here. So we read in this, But But what if we don't want that? We don't want to populate our ray with this header. We just want to ignore it. Well, that's what we're going to do. We're gonna use a function called ignore. So let's change this back to 11. We're gonna go down here and before our pre read, we're going to specify with a function that we want to ignore that header. And to do that we use we access the function with our input stream foul handle, which was input file diet. Ignore, which is a function in this function, has two parameters that we need to use. The first is the amount of characters that we want to ignore, which is 255 and the reason it's 255 is because in a console window in a C plus plus application and the console window, there is 255 characters on each line. So if we specify 255 then it will ignore this whole line in the console window, and the cursor will be moved to right here. Right before is that and the other, the parameter that we want specifies What's Nosa delimit? Er? And that's a character that says, If you reach this character, then go ahead and start reading. Quit ignoring. And this is a new line character and later in this tutorial, we're going to go over this and debts because this is also a formatting advanced formatting option that I want to discuss with you guys. But this is a new line delimit er, which means, basically, if you get to the end of this line, you're going to reach a new line character. I want you quit ignoring because when you get to a new on character, you're gonna end up right here right before. Is that so? That's what the delimit er does. So now if we run this program again, what they ignore function in we only pick up the names that we want and it ignores the header just like we wanted it to. So moving on. And once you go ahead and memorize this function because he will be using it a lot more than likely with foul input. But moving on, we're going to discuss this guy right here. These special formatting characters. We're gonna do that here at the bottom of our main function. We're going to say something along a lot. Let's give us some space. We're going to say this is a new blind character and then we're gonna put several of easy in those air Three new lot characters, backslash in bets, Leshy and back slash in. And essentially, what we're doing is by putting these backslash ends in with our screen, it's not going to pre peas. These are basically the same thing is saying this, but instead we can just use the formatting character with the backslash in and accomplish the same thing. And if we run it, we'll see exactly what I mean. We get three new lines right here below our output because we added these new on characters . Let's move on to another example of these special characters that we can use. We're gonna use the formatting tab character will say this is a tad character. Then we're gonna we're gonna enter in four tad characters which your back slash tes will say tan and then a couple of backslash ends to give us some space because remember, those are the same thing as New line characters. And these are gonna be our tab characters and you'll see what I mean. In a second, we run it, so we run it and you can see all this space in between. Tad and this is a tap character, and that's where these backslash teas coming to play. So that's a special formatting option you can use. The other one that I want to show you is quotation marks will say This is a quote and we'll be back slash quotation. I will say quote backslash, quotation and then a couple of new on characters in the quotation marks. What this backslash quotation does is it escapes the stream and it puts these in because if we take out this backslash and it messes up our stream, so we have to have that in there and this is just a special formatting option. And when we run it, you'll see be able to actually have quotations and or output and says This is a quote and then have quotation marks. Quote. So that would be useful to you in the future whenever you need to use these escape characters and there's plenty more of them that you can use, and I suggest that you probably go try to look up some of them and see what you can do with this output. Okay, moving, going. I want to go ahead and discuss one more thing with you are. Actually we're gonna do a couple more things, but but one more with foul input before we continue. And that is we're gonna go named start txt. And what if we had something like Troy Hodges here? Troy Hodge is. That's the last name. And if we change us to 12 I'll show you what happens when we actually run this so we'll run it. And instead of saying Troy Hodge's oops, we need to say that girl quick before we do that. Okay, so there we have it saved. Let's run it one more time. Instead of saying Troy Hodges, it says Troy and then Hodges on the next line. That's not what we want. We want the whole name on this same line. But what's happening is in a file as soon as the scanner and gets to a white space character. It seems that that's the end of this of what we're reading and puts that into our temporary variable. So we mean what we need is a function that will read this whole line and put that in a single variable, and that's what we're gonna use. So just like we used input file dot Ignore. We're going to use another function called get line. We're gonna put it in our pre read and post re and the gate line function. All you do is type get line. You specify the input stream that you're using, which were using input file that we specified. And then you specify the variable that you want to hold the string in, or the line in which we're calling Tim. We're gonna do that for both of these will say, get line input, vile Thomas Tim that changes or are pretty reading Poe Street, where it reads in the whole line instead of just a single string value on when we run it, you'll see the difference. Now we get Troy Hodges on one line, whereas before it was, it was separated into two separate variables. Now it's holding this whole value in one index of the array. So that's the get line. Functioned and then the other one. Last thing I want to show you is what's called the Iot Manipulation Library, which is the input output manipulation library. And if you include that include I o minute. You can do some really cool things without, but and I'll show you what I mean. So we'll come down here and we'll say, See, out left, which specifies a left alignment. I will say set precision to to and then fixed. And what this does is a specifies left alignment. Set the precision to to, and what that means is that decimal values will only hold two places. Were any value for that matter. So if you have the number 200 it's really just gonna look like 20 because it's not gonna hold that other zero Well, what fixed does fix says, Take this set precision and Onley apply it to after to the right of the decimal place. So now if you have 200 it will hold the whole number 200 plus 2000.0 Or if you have 200.134 it will only hold the value 200.13 So let me show you what I mean. We're gonna make a double value to 1.792 and we gotta give had a name, we'll call it double value equals and then we're just gonna output that will say, see out in a couple lines will say double value. And when we do that, you'll see that we only get to a 1.79 and not to a 1.792 And that's because we used this Iot manipulation technique to set the position to To after the decimal point. That's exactly what we did. And one last thing I want to show you is what's called the set wit. So we're gonna go ahead and make another value called insecure. Value equals 7 227 And what we'll do is we'll say, C l. We'll give it give us a little bit of space. Well, say cl set w specify 25 which is 25 characters. Then we'll say double value and will say set never you again. 25 And we'll say integer value and wine and what this does. This sets the width to 25 between each value of output and you'll see what I mean when we run it. Now we get to a 1.79 a width of 25 characters and then 227. And then, if you if you print it out something again, you would have another width of 225 in between it because we we specified to put another one right here, and that's exactly what that does. So I suggest you play around with these advanced output and input manipulation techniques, and you'll learn that you can do some really cool things. So thank you for watching and state aimed 18. Practical Program #2: Hello. Welcome to Practical C Plus plus programming the beginner course. And in this tutorial, we will be building our second practical program. And it's gonna be primarily a council application that you could maybe use in several business environments, and I'll show you what I mean. So I already have a employee stock txt file with two headers, name and salary and then the employee list on top and in several employees names with their salary on the right and you'll notice that has their first and last name. So that's gonna be a tricky part with this program that we're gonna have to pay attention to. But the primary focus of this program is to be able to read in this file and then display the contents of the file in the console window. And no matter what, if someone goes in and changes this file, maybe throws in a another name, Jake Long, and then adds another salary to it. So we added, you know, 82,000 to the salary. Whenever this final updates, we want our program to automatically know that it updates and be able to add that name to the console so that means said, that's something that we're gonna have to really focus on is we build this code and, ah, afterward I'm building, get, really advise you to try and see if you can go in and use file output and falling. Put together to create a program using everything that we've learned and, ah, modify this program toe where you can maybe, you know, push the number one in it and you can add a name to the list. Or, if you push number two, you can delete a name from the list. It will constantly be updating this employee list foul, and the idea is for it to seem like a human resource Ah, program that companies could use to kind of update all the employees that are on their payroll. So that being said, let's continue on and let's start this. Let's start coating this So the first thing we need to do is include our libraries, and we know that we're gonna be dealing with fouls a lot. So let's go ahead and include that library, the file stream library and then the other library we want to include, since since we're gonna be printing out this data in the console window, we're probably going to be doing a lot of output manipulation. So let's include the input output manipulation library, which is pound include io minutes, if you remember from the last tutorial. So now that we have all the libraries that we need, we're going to go ahead and continue on to the main function and ah, and set up our file input, so to speak. So So we need to include our foul input handle with the input file stream, Um, declaration and we're just gonna call it input or we'll call it. We'll give it a different name. We'll call it Employees file and will say employees dot txt for the constructor. And ah, we'll go over constructors, maybe in a future class if I do in advanced C plus plus tutorial on, you know, classes and advanced data structures. But basically, this is this is a function of if stream and this is the constructor. And all we're saying is we want to create a file handle called employee file and we want this file to automatically be associated with this text file. And I know we went over that in the file input tutorial, but I just kind of want to rehash your memory about that idea. So then let's set up or checking function to make sure that this file was found. Employee file. Okay, we got to say C l employees text file not found well in the line a few times just so we could throw those in there and then we'll return. Negative nine. Let's go ahead and run it. Make sure we don't get any heirs and that the file is found and it appears that the file was found cause the process returned zero rather than negative nine. So we're good to go and then let's go ahead and set up some variables to hold this data. So let's make a string employees name variable, and then let's make an ENT employee's salary variable. And these are the two variables that we're gonna use. And since we don't know, since we're gonna make this program toe where you can kind of update this program at any time and we don't want to have to go in and change the size of our arrays were not gonna use an array, but instead we're going to actually use our output in or wildly that we used to read in the input and I'll show you what I mean. So the first thing we need to do after declaring our variables with this program is get rid of these headers because we don't want to save these headers in any string. And to do that, we're gonna use our ignore function. So we're going to say employees file dot ignore And then the 255 characters in the new land a limiter that's for the first line. So that's gonna ignore this this first line and employees dot txt. Now we need to ignore the second line. So we're going to say employees vile diet. Ignore 255 new on the limiter Cole and Cole and then the semi colon. So now that we've got rid of these headers, we can actually start a pretty read in our post Read are wildly and start a real being in this data and printing it out. So to do that, let's go ahead and comment pre read. We're gonna put in our pre read right here, but remember, we want to hold the whole employee name in one variable. So we can't just read in one name at the time. We're gonna have to read in all this at a time without actually picking up this data so we can't use get line and actually read in the whole data. We're gonna actually have to make use of our delimit er and stream size with the get line function. So let's do that now. So for a pre read, we're gonna say get line. Our employee file is our our is our input string. And then for the stream size where we need to Assane that no name is pretty much gonna be 50 characters long because that's gonna be a really that's gonna be a really long name. So I 50 character, Long would probably name would probably come all the way out here all the way to the beginning of salary. Nearly so we're gonna use fifties are stream size limiter pretty much and ah, but first we need to hold it into a variable. So we're gonna say employees name employees name is our variable, and then we're gonna put in our stream size, which is 50. And if we run that we shouldn't get an air. OK, so we do get in there. But we need to, and it's probably cause we need to delimit er and so we're just going to throw in a new on delimited because we shouldn't reach that. Let's see if that runs and it still doesn't run. So let let's look at a functions. Let me ah, go back real quick and let's see what we can throw in here for a delimit er. So we start out with the input based extreme basic stream input and then character delimit er. So I'm pretty sure that the only thing we can do is that the limiter on the get line and to do that, let's go ahead and say, Let's see if this works. If we just put in a delimit er on get line and that does work just by putting in a delimit er so I don't I don't think, Let me Atacama. I think you can Maybe I was thinking maybe you could add in a size, but apparently that throws an air so you can't throw in a size, could only throw in a delimit er so the delimit er that we're gonna use. Let's go ahead and do that. The delimited er that will use is a comma. Okay, that will be the delimit er that we use. And we're gonna update our text, file this employee text file to separate the names, uh, and the salaries with the Kama. So, you know, we can just be like this. It doesn't even but only had to be lined up correctly or anything like that. But we will go ahead and do that. So So we have these this text file, right? And I kind of went, Let's make it a little bit prettier, so that so that it looks nice. And we have these commas separating the names and the salary because the get lan function as right now, just off the top of my head doesn't use a string sized a limiter like the ignore function does. So I could ignore function. We could ah, we could actually put in a string size and ignore up to a certain point. But on this one, we're going to use a delimit er, which is the comma. So basically, what's happening with get line is is that we're using the employee file stream were storing the very the data in the employee named String Variable and all the data that we're storing in there is all the day All the data up to this, um, comma right here, which is what we said is a delimit er So let's save that. And let's, uh, throw in our wild leave now. So we're going to say, While no employees filed on e o. F function, let's go ahead and set up a post read comment function so we know where it is. Our post read. It's gonna be the same as of right now, um, to store our data is going to Samos hard pre read. So we're gonna store employees. We're gonna store employees name and then the the limiter of a comma and then a semi colon . But remember, the pre and the Post Street isn't complete yet because we need the whole name. And then we also want to store the salary. And to do that, we need to add something in our pre read and post read, and that's exactly what we're gonna do. So we're gonna extend or pre read by a line, and we're going to say employee file, use the input stream operator and then say employees e salary and the same thing with our post read. So say sorry about that employee file input stream employee salary just like that. And that concludes our pre read and post read. And if we run that everything should run find with no heirs and we're actually reading in all the data into a variable. But the thing is, we're constantly overriding the same variable because we don't have an array set up yet. Well, we don't need an array because we're going to actually print out everything that we need in this, um, while leap. So let's go ahead and do that. We're gonna say we're going to set it up. We're going to say, see out and let's actually set up a We're gonna go ahead and set up or input where everything works, just like we want it. So at the top, we're going to set up a how we're going to manipulate our output with C out left and then we really don't need a set precision and fixed because we're not dealing with decimal values. So we'll just say, see out, left up there and then right here will say set ever you and will say 25 and then employees e name and then set W 25. Actually, we won't even need that one since we're using the left and left orientation so it gets a set every 25 employees name. And then I think if we just say employee salary, that should output just like we want in Milan and let's see how that works. And there you go. So we have the name and then the salary and then a new on the name and then a salary and then a new line, just like we wanted. But let's make this a little bit prettier. Let's go outside of the leap and let's add some. Ah, let's add some output So we'll say. Well, say something along the lines of right here before our ignore will say, See out, um, tab a few tabs and will say employees or will say Human resource is human resource is payroll list and we'll end a few lines and then we'll say C L set W 25 four name and then salary, and then we'll end a few lines again on. What this output will do is we'll make it look nice and set it up for us. So we have full name salary. Okay, so but salary needs to go over more. And I think the reason why is because we have this Jimmy Clark and binge Minutes actually picking up all the data to the comma. So just the way it looks right now, even though we have the same wit, it's not the same because because of the length of these variables, they're actually ah, lot longer than just Jimmy Clark because it's storing all the white space up to the comma that we used. It's a delimit er, so let's just push salary out a little bit further. It's going to kind of be a hit, miss top thing. For now, we'll just use, um, 35 see, see what that looks like on 35. It's nearly there, right over the top. We'll just add, you know, maybe 37 see what that looks like. 37. Perfect. So we have human resource is payroll list and then we have the header full name and then salary, and then we have a whole list of everyone on payroll in their salary. And if we update this, employees dot txt file. I just want to show you so we could say, um, Jimmie Johnson And we're just Atacama will give him, you know, 100. And we were just given 13,000 will save it. When we run this program again, Jimmie Johnson will be added because we put everything inside or a while loop. And that's kind of what I wanted Teoh do for this practical program was kind of show you how you can update instantly just by using these variables inside the wild leave and not having to use a data structure like an array. So I want you to play with this. I want you to figure out maybe a better way to do it on your own. If you can kind of dive into mawr output manipulation and see if maybe you can Ah, you know, if we obviously if we changed the delimit er if we change the comma like let's say we pulled the comma of way to right here, you know, in in our text file, that should change exactly how are ah, program looks so Let me just show you that real quick before I leave. Because I want to show you that just by changing your text file, it's gonna actually change the output of your program since we're using Akamas a delimit. Er let me just fix this real quick. And since we're using this commas a delimit er the variable like on Jake Long, for instance, is actually this long, this many characters, and when we save it, you should be able to see the difference in our program. So you see how much shorter since are variable names were actually shorter, that these numbers shift over, and that's kind of wives want to show you. And that's why we had to push salary over to kind of, ah, compensate for that delimit er being the comma and how far over it was in the text file. So that's it for this tutorial. Stay tuned for the next section when we start discussing functions and actually make our hangman game. Thank you 19. Functions: Hello. Welcome to Practical C Plus plus programming Beginner Course. My name is Zak. And in this tutorial we will begin discussing advanced topics and C plus plus, um, mainly all we're gonna be dealing with his functions and how to use functions. And though many people may say this isn't really an advanced topic, I do consider it as an introduction to advance topics, because in C Plus plus, you will be using functions all the time. And they can get pretty complicated when you start throwing in templates and using, you know, structures as a parameter and passing by pointers and returning a pointer of a pointer. So basically, I consider it this going to be an introduction to advance topics and ah, that's that's all we're gonna discuss. We're basically going to discuss everything that we need to know about functions to get started and toe build or Hank Mann gain. So that being said, let's get started. Um, from the very beginning, with we've always had this main function and this whole thing right here is the main function as well as we discussed. But I want to discuss the the anatomy of the function so to speak. So So what is this end right here? Will that end me? Is the return value of the main function and a function can either have a return value where it can not have a return value and just be void. The main function always returns an integer value. And you can see that here when says return zero and we can actually return whatever we want . So let me show you. For example, we could return eight right here. And when we run, this program will say Process returned eight. Well, that's because the main function is returning the value. Eight. So what? That same token? Let's go ahead and turn it back to zero. And let's create our own function that returns the value. So let's say, um, we make a function and we're gonna call it, um, we're just gonna make our first function, is not gonna return anything, and then we'll make a second function. The desert turned something, So our first function is going to be void, meaning it doesn't have a return type. And to do that, you type void. We'll call our function. You can call it anything you want. We'll call it print. Hello. Put the parent disease. That's where your parameters go. We'll discuss parameters in a later tutorial, and then put your brackets. And since this is avoid function, there is no need to type return and then a value that will actually throw an air. If we do that, because the return type is void, meaning we don't need a return type. And this function right now, this code will run just as his, you know. And ah, that's all it does. All it does is basically return zero, because this function never gets called. But let's let's give this function some code to run so we'll run. See out. Actually, it may go down a line real quick will say see out hello and then in the line. But when we run the code, we still won't get hello printed on screen. And the reason why is because we have to call that function. And to call this function, you simply just type the name of the function in your main and your main function. So you say print hello at your prints, sees and then add your semi colon. And so the code will always start running with your main function, it will say So come here. And the first thing this code days is look up the print hello function, go to it and run the code in this. And then it's gonna return Void so returned back to the main function and they'll say Process returned to zero because they know we're on this code. When we run it, that's what we get. We get the word hello and then process return. Zero. This is back in the main function, so let's make let's make one more function. Let's let's make one that returns a number So we're going to call it it. Or let's say double, um, get age and we'll say it returns the age Well, we can actually say we can do one thing we can say return age, which we could say is 23 or were what we're going to do. We're going to declare a variable. We're gonna say double age equals 23.0, what a semi colon and then we'll say return age. Okay, and then let's make one more function. This says string get name and will return a string value return. Zach just like that. So you're seeing various examples of functions being declared here, and in these functions they're returning something different, and they're doing it in different ways. So let's go to our main function and see if we can use these other functions to make a cool message. So first will call print Hello, and then we'll say, See out. I will say, Get name And then we'll say is get age years old implied. And this right here is basically saying print. Hello. So it's going to call the function print Hello, which will just say hello, it'll in the line. And then it's gonna output, get named the function, get name, and when you call, get name, it's gonna return the value, Zach. Which means basically, when you call, this is gonna return the string Zack. And it's basically just going toe output Zach right here since we're using it in an output stream. So say output. Zack is get age. This one returns age, which is 23. So say Zach is 23 years old and blind and let's run it. And that's exactly what we get we get. Hello, Zack is 23 years old. So that's all there is to functions right now. And the future tutorials. We're gonna go Maurin depth with these functions. But for now, once you have practice using the ideas we discussed in this tutorial and I'll see you in the next lecture Thank you. 20. Parameters: Hello. Welcome to practical C Plus plus programming. My name is Zak. And in this tutorial, we will be discussing function parameters. Now, function parameters are a pretty simple topic once you get your head wrapped around it. And in the previous tutorial, we discussed functions. And, you know, we did something like this would put, you know, string, print, name. And, um, basically, we would just say, you know, string name equals Zach, and then we would say Return name. And then down here, we did something like C l print name. And when we ran it, we got the name Zack to appear in the console. Well, with parameters, you can specify in the main function of value that you want to pass to the function that you're calling. So meaning basically, if we go up here if we want to return a name that, um, you know, or any names specified and we can add a parameter right here that says, you know, data type string because the name is probably gonna be a string and we'll call it name, Okay. And then this function will return the parameter name that is passed to it. So here If we say see out print name, we have to pass it a parameter a string value. We could call it Jim. And when we run this, it will print the name Jim because GM is being passed as name into this function and it's returning the name, which is Jim. And as I say, we can change this to Sam. And then Sam will be printed just like that. And just to kind of show, you know, you can you can pass in multiple parameters so we could pass in age. You know, if and if we just ran this as it is, we would get an air because we have to pass in another value. Units will pass in 17 and then it will run. Obviously, it's still just gonna print Sam because we aren't doing anything with age. But I just kind of wanted to show you on introduction on parameters and how to use them and functions. So, you know, maybe let's let me do one more example before we move on because I want to. I want to kind of give you a little bit more insight on how to use these. So we're going to say, um right here will say string, print name and it will pass in the name. And then let's make another one called, um, and age will pass in. We'll call it get age will pass in a double. We'll just call it X. We won't call it a judge just to show you, you know, that this is user defined. You can call it whatever you want and will return X Okay. And then here what we will do, we will say, um, you know, see out, enter a name, Let's give us Ah, let's say string name and then devil Age will say Enter a name and then we'll say CNN name . And then c l give it a couple of new lines with our new formatting characters that we learned will say Enter a age bitter and age the end age And then we'll say, See out, Give it a couple of lines will say your name is well, Say, get name. Is that what we called it? What we call it? We call print name. Okay, we'll say print name will pass in name that we got from the keyboard input, and then we'll say we'll go down here just to show you that you can do this and see in code blocks you can. This is all gonna be seen. It's one line of code, even though it's on two different lines. So I'll say print, name and then and you are get age will pass an age that we got from the user input and we'll say years old, just like that, when we run it, I will say, Enter a name that name will be You will just say Jim Intern age 23. I'll say your name is Jim and you're 23 years old and that was the result of passing those values to these functions as parameters. So that's it on parameters. Stay tuned and we'll get to talking about passing by reference and function. Overloading. Thank you. 21. Pass by Reference: Hello. Welcome to Practical C Plus plus programming the beginner course. And in this tutorial, we will be discussing passing by reference. So what is passing by reference? Well, let's go ahead and make a few Mawr functions. Let's just make one function and we'll call it, Um, I will say void, um, a git or will say void print, age, print, age and will pass in an integer value called X. Okay. And, um, go ahead and put it here and let me go ahead and show you before we do this because I was going to save this for a different tutorial. But, um, let's go ahead and show it to you here right now. If we run this program. Everything runs fine, because print ages declared above main. But if we move this function below main when we run it, it'll run fine until we call it when we call it. When we call print, age, past of value, we'll get an air, and that's because, um, the code the main function starts running. It says print age. It looks forward up here, and it's not there. So what we have to do is we have to prototype it. So I'm gonna go ahead and show you how to prototype that All you have to do is up here in the under using name space. You just type the name of the function. So you type void Print, age, ex semi colon. And now, whenever it gets to print age down here, it'll come up here and it will look at the prototype, look it up and then say OK, I know this function exists. I'm going to go find it, and that's exactly what what it does. So we're gonna start prototyping are functions for now on using this method instead of declaring our function above main. So if we run it right now, you can see everything will work Fine, because we've prototype function now. But let's me vine Teoh what we actually came here for and that's to learn what? Passing by references. So we have this function called print age. What we'll do is we'll just say C l X and ah, if we if we run this if we say print age seven, we will see that it prints out seven. But what happens if we say, for instance, let's make a value it age equal seven. We pass in age, Okay? And then here it's gonna print out the age it would pass in. It will print out seven. OK, but what happens if before we print out seven, we change. X equals it's five. So now we just past seven in for X, but we change it. We say X equals five. So let's run that seen out Prince L five instead of seven, even though we passed seven into the function. Okay, well, what happens if we if we're changing it here? The question is, if we're changing it in this function for change, because we're passing in seven is it getting changed in this function? Well, there's one way to find out. We can print out age after we run it, and I see what we get. And so we get five and then seven. So obviously age is not changing, except in this function. Well, what do we do if we want to change age in this function? But do it in this function, for instance, here we're changing X equal to five. What if we wanted to change the value that we're passing in 25 as well. Well, to do that, we have to pass by reference. And what that means is instead of passing a copy of age into this ah parameter, we're gonna pass the memory address of age into this parameter. And to do that, you simply just type in the ampersand symbol before your variable name. So now whenever we run it, we have toe type in our prototype up here is Wells. Don't forget to do that. So now when we run it both equal five. And the reason why is because pull this back up so you could see both at the same time. So here were saying into age equal seven. And as we declared in the prototype and in the function when we pass age and right here we're not passing in the value seven were passing in the memory address that this variable lies that in memory. So now in this function, X is equal to the value at the memory address of age, which is seven, and it changes it to five. Well, that means it's also changing wherever ages at the value at age 25 as well. And so it's changing both the copy and the actual variable 25 So So let me see if I can kind of show you what I mean. And one more quick example before we quit, cause that's gonna seem kind of confusing to you. Maybe at first. So we're going to do one more. That's maybe a little bit more clear. So we're going to say, believe this function we just won't call it will make another function called Ah, void change address. Okay, we'll pass in a string and we'll call it. Oops, we're passing a string and we'll call it address. Okay? And we need to prototype it appear remembers that will say string change, address, string. And you actually, in the prototype, you can take out this eggs. It just has to know that you're passing in a stringer and it in the prototype so you could actually make the variable name. You know why here and ex down here and it doesn't matter, but it's totally up to you if you want to leave it there. Just so you have a same look going out through the same look going throughout your code, that's fine. But often just leave the value the data type and then the ampersand symbol. If I'm passing by reference in the prototype parameter and here I'll do the same thing. I would just say String Ampersand, Which means we're passing in a string memory address here in this prototype. And then we'll add a cynical and finished prototype. Now, in the change address function, we're going to say Address equals four 1800 College drive. Okay, and that's all it's gonna do. It's gonna take the memory address that is provided. It's gonna look at the value at that memory address, and it's gonna change it to this. So let me show you again. So we'll say String. My address equals 24 18 Willow Road and then we'll say, See out, address before function, call my address in plan and then we'll say, will say change. Address will pass in my address, which is really the memory address, the memory location of this variable, and then we'll say, see out address after function, call my AG dress. So even though the variable my address is being declared, here's 24 18 Willow Road and it's not being changed anywhere in this function. We're passing it to this function as a memory address and changing it in here. Okay, so it's changing to 1400 college drive, and when we return, it's gonna be different. So watch. So we have void. Change address. Let's see what went wrong. Here we have string address equals 1400 College drive. It's saying, um, old declaration. So let's see. Let's make sure our pro top is fine And right here is What's wrong? So we have we need. We had a string return top here. Let's change it to avoid and let's run it again. Address before function Call is 24 18 Will a row. But the address after the function calls 1400 college derive so you can see how it's actually changing the my address Variable in this function, and just to prove it to you, we're gonna take this out that ampersand. We're gonna take out this ampersand, and we'll run it again without the ampersand and look at the difference. It says. Address before function. Call 24 18 will erode actress after function. Call 24 18 Willow Road, and that's basically the basics of passing by reference. And in the next tutorial, we're going to go over function overloading. So thank you for watching 22. Function Overloading: Hello. Welcome to Practical C Plus plus programming. My name is Zak. And in this tutorial, we will be going over function. Overloading, um functions. Overloading is an interesting topic, and I found it pretty easy to understand. Once you get your head wrapped around, it's not too difficult. And even though we probably won't be using it in our final Hang Man project, it's still something I think you should know as a beginner so that when you see it, you understand what is happening. So what we're gonna do is we're going to make a function that says, um, you know, void print salary, OK, and this function is going to take in an integer value and will make that function down here void print salary it x Well, basically, just say C l in Dillon Eggs in Dillon. And if we run that, we'll get just what we expect will say print salary will say 20,000 When we run it, it'll just print out 20,000 which is exactly what want. But what happens if we want to use the same function? But we want multiple ways to do it, for instance, will say something like c l enter your salary and we'll say, um, you know, CNN salary and salary could be, for instance, we don't know if they are going to enter a string, um, salary or let well safe right now. We'll say it's an end salary and will pass in salary out here. So whenever they enter it, it won't get put into the function. So if they enter, your salary will say 23 country male pronoun 2300. But what if you know, we're having, You know, in this function, it won't really make sense because you have to declare your data type meaning whatever you declare, they're going to have to enter it anyways. But what if you have, Whenever you start programming an object oriented design and, um, different tops of programming architecture, you're going to see that sometimes you don't know what kind of data is coming in. And even though that won't be the case here, we're gonna pretend like it is so that you can go ahead and get your head wrapped around the whole concept. And so let's assume that someone enters in a double salary all of a sudden, and when we run it and we enter the salary, you know, they enter in something like that. Well, it's only printing out this because it was supposed taking an injured your value. Well, what if we really wanted to print this value? But we can't with this print salary function because it's on Lee asking for an interview value. Or better yet, what if we went into, uh, say double salary and Pesce in a string on when it says into your salary? Now it doesn't even run because you can't pass in a string at all. But what if we did want to pass in a string and have him type out $2300 as a word? Well, to do that, we can solve this problem about something called function overloading. And to do that you simply make multiple prototypes of the same function, but with different parameters. For instance, will say void print salary and we'll say stream will make two will make double, Then we'll say, avoid print salary stream, and then we'll come down here. Copy this, and we're gonna make we're going to reprint them, and we're just gonna change these so this one will take a string value, and this one will take a double value. Well, now we have the same function, but we haven't overloaded with different parameters, so we can expect pretty much anything to be inputted. So now, before, when we got an air when we ran this, we can run it again and print salary We can actually enter in 20 three 100 dollars. I'll say 23. And I know it didn't say $100 because, technically, we didn't do get line. We're gonna go ahead and fix that real quick just so I can show you. Um, and I know you've seen this before in the previous tutorial, but probably not with CNN. So we'll say CNN with get line instead of before we did something like output file or input foul. And then we'll store it in salary. And now when we run this, if we can say, you know, 20 three 100 dollars, I'll say $2300. But that same token, you know, we can change this to a double, and it will use that same overloaded function and before you will use this function and before whenever we used. So let's see What's going on here? Says See out, double salary prince salary function Call to double And I believe let's just try to do this , I believe get lines only gonna work with string values anyways. So lets say, salary. And when we run it, we will say 2300.246 or whatever or 0246 We run it and it will actually get the get the whole value. And though it didn't do four sixes, because right now it's on default. Value of double precision point is 0.0.2 after the decibel. That's where the double comes from. If we actually changed double too flow, which is the same thing, is the devil's. Basically, they're both decimal point values. I just with they take them different bites and memory when we run it. Um, see here to make sure we have we have a prototype wrong appear change that float. I'm just kind of showing you how to overload different tops, and data went into your salary. Now 2300.2345 Again, we get the same thing, and I'm thinking that maybe it's because it's defaulting to a floating a precision point of 0.0.2 Let's just check that out real quick. Let's say you know, see out, Uh, set precision toe four fixed and let's see if that fixes it. We don't have Iona nip involved, so let's go ahead and include that. Let's run that. And there we go. So now we're getting the four position point. So we had to specify to set the precision point of four after the decimal because it's defaulting to no matter whether it's a float or a double data type. But, um, needless to say, you know, this tutorial was more about overloading functions, and that's basically what we did. We can enter in now when we run it. I know that you see when we're running it, we know that it's a float coming in because we had to declare it here. But what I want you to understand is that in future classes we get in the object oriented programming. You may not know what kind of data is coming in, and that's where function overloading is important because oftentimes you don't know if a string or a float is going to be passed into a function. And so you have to prepare for were all scenarios. So that's it for this tutorial in the coming tutorials. We're gonna go ahead and start building our hangman game and conclude this course. Thank you for watching. 23. String Functions: Hello. Welcome to Practical C Plus plus programming. My name is Zak, and in this tutorial, we will be discussing string functions. Now, I just want to kind of go over this because this is something that you will be using the lot throughout C plus plus. And I'm not gonna be able to show you every string function, obviously, because that would be a whole video Siri's. But I will show you the ones that you'll probably find yourself using quite a bit. And all the string function is Remember what I told you at the beginning of this quarter's that string wasn't really a data top, but was a class well, without diving too much in the classes. Um, what a class is basically is a It's an object that you can create an object of the class, and that object will have specific functions. Well, without, you know, you might not be able to wrap your head around that just yet. I want to kind of show you what I mean. Each time you create a variable of top string, for instance, name equals AC. This variable has several functions built into it because it is type string that we can use . For instance, we can say, you know, name diet saws and that will return the size of the variable name. So if we say C L name diet sighs when we run that it will print out the size of name, which is four characters. At that same token, there's another function called name dot length, which will do the exact sending thing I'll print out for. So, like I said, there's several string functions that you can use. And if you look through you, all you have to do is tapping name diet and all these functions pop up you that you can see . And if you just play around with them, you know, you can kinda see uh, what they dio. For instance, let's just use fine. We'll use dot find and will search for C. And I believe if we see out that if it runs, I will return to because that's the position of sea in the string. Because remember, 012 And if we type in H here, if we find H, it will return three. And that's what the find function does. You confined certain characters throughout the stream. But if we top in a character that isn't in the string, such as Jay, should return negative wine or a value sections. This some kind of garbage value because obviously that's not a position anywhere in the string. So you could, for instance, type in why, and you'll get another strange value. So we get another strange value. And that's kind of how you can decide for whether the whether the character was found. Nine. So that's basically all I kind of wanted to show you. Was it? Each of these name objects has for each of the string objects. Variables have their own built in functions that you can use, such as find size, length, replace in. You know, they're all down here. You can kind of stroll through him, Look at all of them, but that's all I want to show you for this tutorial. I know it wasn't very much, but it's something I want you to play with on Rhone, and, uh, I'll see you in the next tutorial 24. Random Number Generator: Hello. Welcome to Practical C Plus plus programming. My name is Zak. And then this tutorial, we will be discussing how to create our own random number generator. Now all the random number generator is is it's a function that returns a random number so that we can use it in our program. And the reason why I went to cover this is because, believe it or not, this is something that many people like to figure out how to do so that they can incorporate it into games or certain programs that require some level of randomness. Now, if if you if you're just trying to kind of figure out how to make a random number on her own, you would have to kind of create your own algorithm and would be quite a lengthy process. So what I recommend doing as a beginner c++ programmer is any time you're looking for some kind of kind of functionality, like, such as randomness, for example, like we're doing here, I recommend going to C plus plus dot com as you can see up here and just searching for what , what you're looking for in this case, I topped in random and I ended up with this function called Rand, and you can see it here is called Int rand void. And if you kind of just looked through these documents, you can see how you can use the this library in these libraries to create a random number generator. And it's really quite simple. And they spell it out for you out here. How easy it is to get your program to spit out a random number. Well, about his random is you can get anyways, you know, all computers. There's no way to really make them completely random, but you can at least make it appear and, um, to the user. So that's exactly what we're gonna do. You can kind of mark this reference down if you want to come back and read it later. But basically, in this program, all we're gonna be doing is everything that this reference page tells us to do to create our generator. So let's go back to our program, and we're first going to include the libraries that we need for a random number generator to work, and that is include standard library dot H file and then include time dot H file. You might be wondering what this time dot h libraries for? Well, our random number generator is going to be based off the internal clock of the machine, and it's gonna incorporate that into its algorithm to come up with a random number. And, uh, you'll you'll see what I mean here in a second. I mean, it won't be extremely clear, but that's basically how this algorithm works, is it? It gets the current time down to the millisecond and throws that into a function. And basically, that function is going to spit out a different number every single time. Because the time is changing constantly and depending on what the actual time is, the algorithm may spit our completely different number than the one it spit out one millisecond ago. So that being said, let's go ahead and create a function that's gonna generate our number will have it return an integer value because we want to return an integer. We'll call it, generate random number and won't take any arguments. And then down here, we're actually going Teoh right out or function so well, right the same thing. Generate random number and then in here is where we won't rather toad. And if you go to C plus plus dot com and look at it, it's actually quite simple. You just write, include your libraries and then write this small function right here, which initialize is the random seed Teoh the internal clock on the computer. And then you just simply spell out your your variable with this function, with this number always being the number between zero and then on this number to be all your random values that are possible and then plus wine. And this will return any number. For instance, right here. This I secret variable will spit out any number randomly between zero and 10 because they have 10 specified. Right here. Let me show you what I mean. So the first thing we have to do is type s rand parentheses. And in these parentheses for the constructor you type the time and then another constructor is no and a cynical one just like that. That may seem really un intuitive at first, but that's what the C plus plus dot com reference tells us to do. So that's exactly what we're gonna do for a random number generator and on certain functionality that you might need in your program. It's not that important to understand exactly how it works. You just need to know how to use it. And that's kind of what I'm showing you right here. So this is how you would used the Raynham random number generator library with the time dot H and standard library header files. So now let's go ahead and hold a um, we're gonna hold a value will just return. There's so there's two ways to do this. You can create a value called yet, and we'll just call it number. I will set it equal to Rand, But you're constructor Modelo operator. The number between zero and fifth and zero and then X basically that you want the highest number to be will put 50 down and then plus wine cynical. And this number when this code runs will be any number between zero and 50 on a basically picked randomly. So for you return number there and then here we just return. We call the function, generate random number. When we run the program, the main will call that function and you see processor turned 41 because it's calling in this return function of the main is calling generate random number. And when it returns, the value returned from generate random number. He returned 41 but if we run it again, it'll give us a different number. This time it returned 50. But we can keep running this over and over again. And every time it will be a different number between zero and 50 and all we have to do. If we want to change the spread, we could change this to 200. For example, there'll be any number between zero and 200. This time it was 96. So that's how you use a basic random number generator. I just kind of wanted to go over it with you so that you would know what we were doing when we make our final project, and also to kind of show you what C plus plus dot com is and how to use it to incorporate certain functionalities into your program. So thank you for watching, and I'll see you in the next tutorial. 25. Project -Hangman (Part #1): Hello. Welcome to Practical C Plus plus programming. My name is Zak. And in this tutorial, we're going to start our Hank man game. Now, in all the previous tutorials who learned about everything that we need to know to build this game, And I'm gonna actually spread this game out through a series of three different tutorials so that we can split it up nicely. And, um, you can really understand how we're going to unfold this process and, ah, build the application as a whole. So in this first tutorial, we're just going to start out by billing, building the main skeleton. So to speak of our entire program, we're gonna lay out all our functions and everything that we're going to need. That being said, let's go ahead and prototype all the functions that we we know we're gonna need. So one of the functions that we're gonna need is to get a word from a word bank and return it so meaning Basically, we need a function that opens a file, looks inside the file and grabs a word, and then uses. That is the word that we're gonna try to use and ah, use. That's a word that we're gonna use for our hang meaning. So to do that, we're just going to call a function with the return type is string. We'll just call it, get word, and we won't give. It ain't parameters, because that's that's just gonna do its own thing. Go inside a word bank and get us a word. And since that's working with files, let's go ahead and include the library that we're gonna need for that that function, which is include F string for file stream. Okay, now we have that. We're also gonna want a function that the prince, the board, the board that we're gonna need when I say board I mean the man. So we're gonna want a function that prints out kind of a you representation of how many lives that the user has lift. And to do that, we're just gonna call it avoid return top because it's not gonna return anything. It's just gonna print out on screen something, and, uh, we're just gonna call it print board, and this is going to take an integer value and that integer values basically just gonna be the amount of lobs that we have lived because depending on the amount of lives that the user has left, um is going to depend how much of our of the man is drawn. So that's what that parameter is all about. And speaking of printing or board, we also need a function that Prince blanks. Uh, you know, Prince, the amount of Blank's for the word that gets returned. And so to do that, we're just going to call a function that also, since all it does is printing something, it's just gonna be void. Return tight, We'll just call it Print Blank's. And we're gonna give that we're gonna give that function to arguments. And they're both gonna be of type string because the the first parameter is going to be a, um, the word that we get returned here and the second parameter is going to be the letters that the user has already guessed. And that's how it's gonna determine what blanks to print. And, uh, what letters to print. And we'll go over all this and the you know, as we go through these tutorials, go see exactly how it's gonna work, okay? And, ah, let's go ahead and make another function that generates random number because we're gonna use the random number that we generate to actually decide what word to grab out of our word bank. And we already did a random number generator tutorial. As you know, So this should be fairly familiar to you. We'll just call it, generate random number and cynical. And so these are prototypes that we're gonna use if I remember a function that we might need If we decide we want to create another function, we will. But for now, these are all the ones that I can think of off the top of my head that we're gonna need. So that being said, let's go ahead and ah, set up these functions so we'll say, you know, string, get word and set up or brackets. Then we'll set up a void print board, and we'll call it lives for the parameter. Avoid print blanks, and that's going to take two parameters. 1st 1 we'll just call it chosen word. And the 2nd 1 we will call letters guest. You'll see exactly why we're calling him that later, and the last one was our random number generator, and that one didn't take any parameters. So there we go. So this is the basic skeleton. Now, I also want to go ahead and add some stuff to our main function while we're here. And the way our main function is gonna work, we're gonna go ahead and initialize are our use. Your lives will call it user lives to seven. And then basically, we're going to say, Wow, use your lives is greater than zero. We want to do this, anything inside this leap. And basically this loop is just going to allow the It's just gonna allow the user to keep guessing letters as long as the lives are greater than zero. And then we'll set a break statement in there somewhere. If the if the word gets guessed correctly, then we'll do that. So But that's how we're gonna set up our main for now, and we'll add more stuff later. And the other thing that I want to go ahead and do is since we it's kind of fresh in our minds, go ahead and make our random number generator while we're in this tutorial. And to do that, all we do is say, you know, string or my bad. I was had the I had that get word on my mind. But the random number generator we just have to include to libraries. If you remember, one was standard library dot h, and the other one was time H. There were those of the two robberies we need. Now, let's go ahead and make a random number generator. We're just gonna say s friend time. No. And remember, this was the This is the function that we need according to C plus plus dot com and their reference that we used. And then we're going to say, um, we're pretty much just going to say return. And you know, there's two ways to do this. Um, you could say return rand percent. And then I don't know how many words we're gonna have in our World Bank. We'll just say we're gonna have 10 for now, 10 plus one. And ah, this right here will return an integer value random, insecure value right here if we do that. So that's what we'll do there. And we might have to come back later and change, In fact, just for tow, avoid the confusion. What we're gonna do is say, random number equals rand and you make this integer and random number equals rand. Ah, Markkula Operator 10 plus one. And we'll say return random the number. There we go. And if you see if we go up here and we, um let me just comment this out real quick. Actually, I can't do that. Sorry. This is another way to comment, by the way, is with the store backslash just like that. That's a new way of doing it. Just to kind of explain that. Let's go ahead and test a random number. Will say returned. Generate random number. And just make sure it's given us a random number. And it's not, Let's see what's reference. Generate random number J green number. It's gonna make sure thank you. Right. There we go. We called it random number. Generator. There were probably screaming at me whenever I called it. That s so we called it. Generate random number. There we go. And it says process returned. Three. Let's run it one more time. Process returned. Three If coincidence. There we go. Processor. Turn on. So we're getting a random number every time. And while we're in this tutorial, I want to go ahead and ah, I want to make us a word bank And, uh, back of them actually in this one. And we have employees, don't text he let me delete that real quick. There we go. And let's just make us a new a new document, Star Command. We'll just call it a word list. Txt. We're gonna open it. Well, word list. Txt. There we go open. I will just say no. Give it a header. We're list. We'll give it a couple of nights, you know, Words. So we'll say draft Rhino. Um Reavy, um you know, truck, um cricket grass Hopper Buzzer. Just thinking a random, you know, words off tough my head. There's not really a theme going on here. That's yummy. That's 12345678 Let's get to more will just say, Oh, Taito and one really good word. I will say it, Lennox, There we go. That's 10 words. Not really a overall theme there, but it's the 10 words will use Forward bank for now. And ah, let's go ahead. And in this tutorial, go ahead and make our get word function. Since ah, you know, following putting falling output. It's kind of a thing. We've been practicing for a while, so we'll go kind of quick. We need toe. Go ahead. Make are variable. Call it if stream format that it will be better if stream will call it input five, then wordless dot txt and ah, we'll know pretty much since this is returning a string. Well, no. Well, say, ah, you know, if not input fireable will print out on air, but we won't return because this is a stringer. Turn time. So we won't actually be able to return a an insecure value here anyways. But we'll say, you know, air Negative six. Um, you know, word lists not found, that will, uh, that will let us know, at least if the word list wasn't found, and then we'll say, Let's go ahead and make another variable here. We'll call it temp word, and then we'll just say so. Uh, probably an agent. Well, yeah. Temp. Where there we go. Ah, and actually, we're gonna need an array, so we'll say a string Garay. Um, we'll call it word lift or Ah, yeah, we're just called word list. And let's go ahead and make a constant value. Const. In word list, size equals 10. And you can put a comment here in the code to let you know, you know, change word list size here. Just let yourself No, later. You know, if you make a bigger word Maine. Now, if you want change it to 100 all you have to do is had 100 right there. We'll say we're list wordless size equals just initialize the whole thing, Teoh blank strings. And then, since we're gonna need a four Lee, we'll use this index variable set to zero because we're gonna use a four leap later in this function and that. So that's what we're gonna go ahead and do now we're going to a pre read and we do have a header. So don't forget about the header that we need get rid of. We're going to use our ignore function So we'll say input, file, die, ignore or 255 bites. And then our delimit er of nuan that will get rid of the header, remember? And then we'll do a ah pre read commented there just for habit. We'll say input. Final camp word and then are wildly, we'll say, Well, not input. Fouled on u F. But I didn't make a post read. Same thing, remember is the pre read input file temp word. And then we're going to store everything that we get from this viable into a word list or word list. And to do that, we basically just say we're going to use our index variable that we created up here. We'll just say a word list index has got started. Zero where list index zero equals temp word. And then we'll say index plus place and then down here. Once this loop is done, basically it's gonna populate our entire word list array with all the words in this word list. So what we can do is, since we need to return a string value, but it needs to be a random string. From our word list. We're gonna use a random number generator or generate random number function to return a random index of this array and ah, return a random word. And to do that, all we do say, return word list and then for index, since it needs to be ringing them will say generate random number. That's our function generate random number semicolon. And it will return a random index of this word list which is populated with these words. And just to kind of show you that if we go up to our main function here, we were returning a random number before we'll go ahead and put that back to return. Zero. We're just going to see out, get word and make sure it works. If we do that, we get the word blue. If we run it again one second that save it, we run it again. Oops. We get the word truck. Now we get the word linens so you can see we're getting We're getting new words every single time, Revie All from our word list, grasshopper. And that's how our hang me in game is gonna function. You know, it's gonna grab random words from this word list, and that's pretty much all I want to do for this tutorial. In the next tutorial, we're gonna go more in depth with printing our man and printing the blanks. But ah, I want you can go through this tutorial few times. And really look at this. Get word function and see how We're using this function in conjunction with the generate random number function to return a string of a index of this word list and, ah, you'll find it's actually may be a lot more simple than you were at first thinking. So thank you for watching and I'll see you in the next tutorial. 26. Project -Hangman (Part #2): into practical C plus plus programming. The beginning course. My name is Zak. And in this tutorial, we will be continuing or Hank Mann application. So and this tutorial, I've kind of already got the print, um, print board, um, code already program. And the reason why I went ahead and did it is because you really don't want to sit here for 25 minutes and watch me code out all this. You know, nitty gritty stuff that basically you can do on your own. All I'm doing is using my formatting tab operators. And I've kind of drawing out with these, you know, standard characters, this Hank Mann guy, and you can see as the lives the way I did it. The way I programmed it was in this function takes printer of lives, and I use a switch case. And as the lives go down to zero, the man is fully drawn, but is the lives go up to five. You know, the man isn't fully drawn. He's only halfway John. And when he has full lives, there's no man there at all. But basically, I mean, it's really easy code. You just you can draw it however you want. But for those of you who just want to use this, you know, I would say, Study it a little bit, but not too much because it's pretty simple stuff. It's just you can you can customize it yourself. You know, you could make it bigger if you want or whatever, but this is the way that I usually do it, and I will provide this code for you. And the resource is tad of this lecture so you can actually download this code and just copy and paste it into your program if you want, because, like I said, topping it, Alice kind of a hassle. And, uh, but if you want to do it yourself, that's perfectly fine. So it's up to you. But what I do want to do in this tutorials work on our print blanks function. And it's actually pretty simple function that's just gonna print out the blanks and in the letters of each word that we use. And, uh, we're gonna have to use quite a bit of our string functions that we discussed in a previous tutorial. Teoh, get what we want out of this print blanks function so to begin, All we're gonna do is create a four leap integer I equals zero. And then I was gonna be less than the parameter chosen word. And all chosen word is is gonna be a word from our word list that was chosen by our you know, our get word function with the generate random number generator and whatever word is chosen , we're gonna pass into this function has chosen word. Yes, we went I to be less than chosen word dot size. And remember, this was one of those string functions that we talked about in the previous tutorials. And then we'll just say I plus plus and open our four leap. And now inside this four live We want two things to happen. We went, If the if the chosen word If the letter is in debts, you know, zero of the chosen war, let's have the first letter of the chosen. Word is a and A is in the any of the indexes of letter guest that we want to print out a on screen. But if it's a is not in any of the indexes of letter guest, then we want a print of blame so to do that, we're gonna use more string functions. And what does say if we're gonna use letters? Guest dot find? Remember, this is a string function native to all the stream data types or string objects, and we're gonna find the letter of chosen word don't at I And what this is saying, this function, it's it was going to be really complicated at first. But these air all string functions that kind of told you to study and, ah, the earlier in this section and all its saying is we're gonna get this string, this word, this list of letters and we're gonna look in it. We're gonna find to see if this letter, you know, chosen word at I that's just gonna return a single letter. So if this is chosen, word dot at three is going to return the fourth letter of this chosen word. So if the chosen word was, you know, buzzard, it would return. Or let's say, let's say the chosen word was truck. Then an index I was three is going to return is going to return, see, because three is actually 0123 So it's the fourth letter which return, See? And all this is saying is find in letters guest John C. And if it's found that story turned something other than negative one. But if it's not found, it's gonna return negative wine. So to kind of determine whether it was found or not, We just say letters get stopped, find chosen word dot at I not equal to negative one. And that means it was found as long. As long as this operation doesn't return. Negative one that we know the letter was found somewhere in the function. And so this if statement is saying that the letter was found and so all we do is see out chose word dot at I Well, I had a space at the end of it just to give it some spacing. And that's just saying, you know, output. The letter, um, this word, it a certain index. So really, study that, And then the alternative will just say else, because the alternative is that it was negative one, which means it wasn't found it all. If that's the case, we want to print out a blame the space at the end, uh, to give it some space So you know, this is alternative. This means that the letter wasn't found in letters, guest. So we're gonna leave it blank. And that's all this is This is all there is to dysfunction, and if we can actually test it real quick, so we'll save this. Let's go up to our main function and let's test it. So we're gonna say, uh, you know, here we go. Main function is up here. No, we'll say, Well, go ahead and say string. Um, word equals Get word. Well, well, output where? Girl quicks at the top of the screen so that you know what the word is. But then I also want to run print, uh, print blanks, and we're gonna pass inward as the chosen word. And then let's just pass in some letters ourselves will pass in r S t l n e think those the most famous will fortune letters. So these air the letters guessed that we're saying our guest and if we if we run this program, we should get no errors. And ah, hold on. Stop working. If we run this program, here we go. The word was Lennix and since in was one of our letters. Guest. Uh, we get we get the letter in now, you might see that ILL is capital Ill. And we had el here. That's something I need to fix. Obviously, I think I have capital letters in my word list. So we're actually gonna change that? I'll change that in between tutorials, because obviously that's a bug in our program because we want we want this. We want all our words are wordless to be lover case because whenever we enter, you know, lower case ill. That's not going to show up because that's Capitol Hill, even though it should be there and s. So let's go ahead and run it one more time. Just show you so truck, You know, as you see the are was there because we haven't are the tea wasn't because this is a capital T in truck. We have a lower case t. So I will have to change that small bug and there's a way to get around it. You know, if you can check, basically say if you know if it's capital letter Lower case letter counted anyways, fill in the blank, but you just have to add more code. And if you want to do that, then I challenge you to go ahead and do that. But that's a basic functionality of print blanks. And while we're here, we're gonna go ahead and, uh, take out the comment of our while. Leap and let's go ahead and add some. Let's go ahead and add some basic stuff, too. So we'll say. Since we have all our functions planned out and everything that we need, we can basically go ahead and add the rest of what we need to this loop and all that is is basically say, you know, uh, print board will pass in, use your lives, which is seven. And then we'll say, Let's give it some. Let's give the user some instructions. I will say See out. You know, give it some new ons and we'll say, Uh, well, first of all, we we want to tell him what letters have been. Yes, I will say letters guest. Let's actually create a string for that. We'll say string letters. Guest equals Well, say, see out letters, guest letters, guest. There you go. Now they can see what letters have guessed, and then we'll say I'm see out, Um, enter a letter. Oops. Sorry, guys. It's the inner letter. When will you see? End? Well, say, um, straining. We're just calling. Guess. Well, so I see on guess There we go. And that right there were basically So we're just telling them what they've guess. We're printing out the board, which is gonna start out as a clean board. No man hanging from it. No letters. Yes. Will say in her letter will enter the guests. And then what? The first thing we need to do is say letters guest plus equals, Remember? That's just gonna add a string to it will add Guess to it now, Letters guest is gonna have a guess in it. And then ah, we need to. Well, first of all, we need to get our words. So that's another thing we need to add real quick. So let's say I will say string word equals get word there. I guess Now I have string where it equals Get word. And that's basically just gonna return. Remember, that's gonna return a word for more word bank. And we got a story inward, and then we need to do is print blanks and we actually want to do that before this. So we'll say Print board. And then we'll say, Um, now, let's give us a little bit of space and will say um, print Blank's I will say a word Letters guest that'll that'll print out all blanks the first time letters, gas plus equals guest. And then basically, if we want to check and see if But if it was in it. So let's go back now to do print blanks. Actually, I think we can actually do that in here. Yeah, there we go. And there we could add that end if we want. But actually, let's just go ahead and do that appear. We're gonna do that in the main function, but you could do it and neither one will say letters plus equals. Guests will say if, um I will say if you know word dot find. Yes, does not equal mega for Boyne continue, and basically this is just going to continue more into the loop and then we'll say else because this means that they got the got the guests right, and then it's automatically gonna update it will say else lives minus minus or I think we called it. Use your lives. User loves minus minus. There we go. Let's go ahead and run that and just make sure everything runs okay, So if you look at that, we got our four blank printed out are hanging man Elsa in her letter Hostess Interim are there We go and it goes ahead and draws it. If we enter in, let's say, B, we don't get calls. Another one to say l look, it adds it to our blank e There we go E ads for blank you b and C The world was blue, but, you know, obviously there's still some bugs because it's not letting us know when we win and keep going. I think, until we can enter multiple letters, no process return zero because our lives went out and, uh, we just fix a few things. Few minor things to make sure that who knows if it zero will print out of our full guy? But ah says that's pretty much we wanted to go over in this tutorial. I want to go ahead and add one more thing that you haven't seen before, and that is at the end of the at the end of each. Before we continue in each of these statements, I'm gonna add it line it says System CLS, which tells the console to clear. And it'll actually make our game look a lot better. So we will see what I mean. When we run it again, let's go ahead and run it and it'll say So what's Inter in the G and is you could see it's not running down the screen anymore like it was the last time. That's because, um, when we when we clear the screen, it's reprinting it all in the same exact spot. So it looks like it's it's not going anywhere. That's the kind of effect that we want That system CLS is something new, but something easy and something that you can use in order applications. I just want to go ahead and show you that. So in the next tutorial, we're gonna completely finish it up and then test job or application and conclude our class . So thank you for watching and I'll see you in the next tutorial 27. Project -Hangman (Part #3): Hello. Welcome to practical C Plus plus programming. My name is that and this is our final tutorial. So in this tutorial, I've kind of already topped up and fixed everything that we needed to fix. If you look are wordless dot txt you can see I changed everything toe lower case So we don't have any conflicts with our user input And, uh, the word that has chosen and ah to in order to what we really need to add was in order to decide whether the winner whether the user has won or lost during each gas And to do that, the first thing I had to do was declare a global variable called flag. Now, I'm not sure if we went over global and local variables, but all the global variable is is a variable that is declared outside of all the functions you can see I could not. In the main function, this variable has actually declared underneath all my prototypes and what this does it is. It allows this variable to be used in all of not functions across the board. Now, this normally isn't recommended. You definitely don't want to do this with all your variables for privacy reasons, but in this case, it's gonna work out perfectly for us. So I created a Boolean variable called Flag. You can name it, whatever you want. Values called it flag. And I declared it too false. For when the program first start starts up, it's gonna be declared false. And then if we scroll down a little bit in our wild leap, you can see that I have the flag set to true right at the beginning of the while loop and then I have a condition. This says, if flag equals true break. So I kind of want to show what this is doing. This is basically on the program when this wild loop starts setting flag to true and then after these functions run the flag is still true. It's gonna break from the while loop and I want to show you where this flag would get changed and that to my print blanks function. So let's go down to print lengths. I'll show you what happens. So in my print blanks function, all I did was I basically said that if this branch is executed, set flag to false. But if this branch never gets executed at all. Flag is going to stay true because every time this four lube runs is gonna be running this piece of code rather than this piece of code which basically means that throughout the life cycle of this four loop, if one blank gets printed, the flag is going to be set to false basically meaning that the puzzle has not yet been solved. Flag equals false meaning. The word has not been completed yet because they're still a blank. But let's say this four loop runs all the way through and not one blank gets printed. Then flag will never get set to false. And if we go back up to our main function, flag never gets set to fall. So flag is still true. After print blanks, it says a flag equals true break. What? That point if you break, you come out here outside the while loop and I added these two conditional statements. And basically it says, If use your lives equals zero, then obviously you broke out of this function because use your lives was zero. And if you recall, are wildly basically said, while use your lives is greater than zero keep doing this. But if usual lives equals zero, then break out of this function and come down here. And if you come down here after you break out a function with usual live zero, then you're going to run this code and it's gonna say you lose. The word was and they will tell you the word. But if you use your lives is greater than zero, then obviously you broke out this while loop in a different way, which was via the flag. So let's say use your lives is at three. Flag is set to true print length runs. No bread, no blanks were printed, basically meaning all the letters were guessed correctly. Flag is still true and you break out of this wild loop. Well, then you're breaking out. This wild loop while usual lives is three diffuser lives is three hence greater than zero than this code is gonna get ran instead of this code, and it's gonna say you win. And that's basically the functionality of this code. It's actually rather simple because basically, we just created one more variable, and we just we basically just you have to think about it a little bit. You know, we came down here and we just set the flag to false. If one blank got printed, basically a blanket all gets printed in the flag is gonna be false, and you will never be able to break out a while loop, Uh, be a the If statement checks that flag is true, you won't be able to break out. And the only other way breakout is if lives is equal to zero. And if laws equals zero, then you lose. And, uh, so that that's how this works. And if we run it, I want to go ahead and show you our finished product. When we run it, we'll go ahead and well, so this first word just cause we have 10 words and I kind of ran it a few times. I know what it is is linens. So, you know, let me get a couple of letters wrong. So it's starting to draw her main, and you can see, but if But if we get it all the letters, right, so we'll say I in you, x, and you can see it's telling us the letters that we've guessed, which is something I want to add in and they says You win. The word was Lennox Process return zero and you can see that that's how the game works. But let's run it again. I think we actually have the same word. No, we did. So this is a different word, not Lennox, because it picked a different word out the World Bank, and we're gonna try to get it wrong. So we'll just guess random letters. You could see it's drawing our man and more and says You lose The word was truck, and that's the basic functionality of this program. It's a basic hanged man game, but we used literally, you know, everything that we could and this course and so everything that we learned we got to utilize for this program. And that's kind of why I picked this project for end of the course final project. And so what I challenge you to do is is, you know, kind of convert this program into something more advance may be used file output or something else that we learned in this course, you know, to save high scorers, maybe use a word bank with 100 different words and then, you know, use the foul output to say, You know, each time you run it to save your score to this vile and then check with the high score is in that file and then print out with the high score is compared to your score and keep updating that file every time you run it. I would recommend trying that and just really getting good at thes beginner concepts and run through this run through these last few tutorials a few times so you can see how we use these functions. Because when you take an advance C plus plus scores, you're gonna have to be Ah, you're gonna have to really know all that stuff really well. So the last thing I ask is, if you really enjoyed this course and you learned a lot is to give me some good feedback and maybe leave a review if you can. And if you really, really locked it, you can. You can leave a review and just kind of tell me, you know, your doctor see an advanced course and I'll get enough people interested. I will definitely make an advanced course, and we'll do some cool. We'll do some more cold projects. But for now, thank you for watching and thank you for being a part of this course Goodbye.
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Brussels / 1 & 2 February 2020
schedule
IoT with CircuitPython
Look mam, no development environment.
Introduction to CircuitPython and how to make basic IoT without a development environment.
A brief history of CircuitPython CircuitPython vs MicroPython
Hello World demo: 1. Hello World in REPL 2. Hello World in a Python script 3. Blink (the electronic Hello World) 4. Cheerlights (the internet connectivity Hello World) 5. Hide and Seek (a BLE Hello World?)
Circuit Python supported hardware used for the IoT demo: * nRF52840 (Nordic Semiconductor) with build-in BLE * ATSAMD51 (Microchip) M4 with Airlift (ESP32 used as a Wifi Co-Processor)
Speakers
Photo of David Glaude David Glaude
Links
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| 0.732742 |
Fitness Tip: How To Hydrate and Replace Electrolytes When Working Out
Image result for Fitness Tip: How To Hydrate and Replace Electrolytes When Working Out
Water is important to live. some days while not it may end in death - it's that necessary. thus considering an association strategy, particularly once figuring out within the heat is important to overall health. we tend to lose water through respiration, sweating also as urinary and fecal output. Exercise hurries up the speed of water loss creating an intense exercise, particularly within the heat, a clear stage of resulting in cramping, lightheadedness and warmth exhaustion or heat stroke if adequate fluid intake is not met. Correct fluid intake is a very important priority for exercisers and non-exercisers within the heat. Water makes up hr of our bodies. thus it's improbably necessary to for several completely different roles within the body.
The Role of association within the Body:
Water has several necessary jobs. From a solvent to a mineral supply, water plays a district in many alternative functions. Here square measure a number of water's necessary jobs:
- Water acts as a solvent or a liquid which will dissolve alternative solids, liquids, and gases. It will carry and transport this stuff in a very variety of the way. 2 of water's most vital roles square measure the actual fact that water transports nutrients to cells and carries waste product far from cells.
- within the presence of water, chemical reactions will proceed once they could be not possible otherwise. thanks to this, water acts as a catalyst to hurry up catalyst interactions with alternative chemicals.
- drain the cup as a result of water acts as a lubricant! meaning that water helps lubricate joints and acts as a cushion for the eyes and medulla spinalis.
- Body association and fluid exchange facilitate regulate temperature. do not be afraid to sweat! It helps to regulate your temperature. after we begin to sweat, we all know that temperature has accrued. As sweat stays on the skin, it begins to evaporate that lowers the temperature.
- Did you recognize that water contains minerals? drinkable is very important as a supply of metallic element and metallic element. once drinkable is processed, pollutants square measure removed and lime or sedimentary rock is employed to re-mineralize the water adding the metallic element and metallic element into the water. as a result of re-mineralization varies reckoning on the situation of the quarry, the mineral content also can vary.
Which Factors confirm what proportion of Water we tend to Need:
What factors have an effect on what proportion water would like? All of the subsequent facilitate confirm what proportion of water we tend to need to require it.
Climate - hotter climates might increase water wants by a further five hundred cubic centimeter (2 cups) of water per day.
Physical activity demands - a lot of or a lot of intense exercises would require a lot of water - reckoning on what proportion exercise is performed, water wants may double.
How much we've sweated - the number of sweating might increase water wants.
Body size - Larger individuals can probably need a lot of water and smaller individuals would require less.
Thirst - additionally associate degree indicator of after we want water. Contrary to in style believe that after we square measure thirsty we'd like water, thirst is not sometimes perceived till 1-2% of body weight is lost. At that time, exercise performance decreases and mental focus and clarity might drop off.
We know why water is very important however will we set about hydrating correctly?
We get water not solely through the beverages we tend to consume however additionally through a number of the food we tend to eat. Fruits and vegetables in their raw kind have the very best proportion of water. lyonnaise or "wet" carbohydrates like rice, lentils, and legumes have a good quantity of water wherever fats like kookie, seeds and oils square measure terribly low in water content.
Fluid wants By Bodyweight:
One of the best thanks for confirming what proportion of water you would like is by weight. this may be the fundamental quantity you would like daily while not exercise. *Yes, you'll have to search out a metric device like this one to try and do the maths.
Water Needs: thirty - forty cubic centimeter of water per one kilogram of bodyweight
Example: if you weigh fifty kilograms (110 lb), you'd want one.5 L - two L of water per day.
Hydration Indicators:
You should be drinkable systematically (not all at one time) throughout the day. The body will solely absorb an explicit quantity of water at a time. Any rabid drinking could lead to health problems.
Thirst - As declared on top of, if you are thirsty, you are already dehydrated.
Urine - the color of your excretory product is additionally associate degree indicator of your association level.
colorless to slightly yellow - hydrous
soft yellow - hydrous
pale gold - hydrous
gold, dark gold or brown - doable lightweight to moderate dehydration
brown - dehydrated
Hydration + solution Strategy:
These straightforward steps can assist you to hydrate daily and before and once workouts.
1. confirm what proportion of water you would like to drink on routine mistreatment the weight formula on top of.
2. Pre-hydration - Drinking concerning two cups of water BEFORE intense exercise ensures adequate association to begin.
3. throughout Exercise - one cup (8 ounces) of water mixed with electrolytes (about 3/4 water to 1/4 electrolyte) each quarter-hour or so.
4. once Exercise - Fluid intake is needed to help in recovery. ill with a combination of water, supermolecule and carbs could be a nice plan additionally to electrolytes if required. Formula: or so 15g of supermolecule, 30g of carbs, electrolytes, and water.
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| 0.874409 |
opencv进行图像分割
时间: 2023-11-07 20:52:48 浏览: 28
OpenCV提供了一些图像分割算法，包括基于阈值的分割、边缘检测、区域增长、分水岭算法等。 下面简单介绍一下基于阈值的分割： 1. 将图像转换为灰度图像。 2. 选择一个合适的阈值进行分割，可以使用全局阈值或自适应阈值。 3. 将图像分成两个部分，一部分是高于阈值的像素，另一部分是低于阈值的像素。 4. 对于高于阈值的像素，将其赋予一个值，对于低于阈值的像素，将其赋予一个不同的值。 下面是一个基于阈值的图像分割的示例代码： ``` python import cv2 # 读取图像 img = cv2.imread('image.jpg') # 将图像转换为灰度图像 gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY) # 应用全局阈值分割 ret, thresh = cv2.threshold(gray, 127, 255, cv2.THRESH_BINARY) # 显示结果 cv2.imshow('image', img) cv2.imshow('gray', gray) cv2.imshow('thresh', thresh) cv2.waitKey(0) cv2.destroyAllWindows() ``` 在这个示例中，我们首先读取了一张图像，并将其转换为灰度图像。然后，我们使用cv2.threshold()函数对灰度图像进行全局阈值分割。最后，我们使用cv2.imshow()函数显示原始图像、灰度图像和分割图像。
相关问题
opencv 文字图像分割
OpenCV文字图像分割是一种用于将文本区域从图像中分离出来的计算机视觉技术。这个过程涉及到将图像中的文本区域与背景区域进行分离，从而为文本检测、识别或分析等后续步骤提供更好的基础。 在实现文本图像分割时，可以通过以下步骤进行： 1. 图像预处理：首先对图像进行预处理，包括灰度化、二值化、去噪等操作。这些操作有助于提取出文字和背景的区别。 2. 文字区域提取：使用形态学操作、连通组件分析等方法，可以提取出图像中的文字区域。这些方法能够找到文字的边界和连通区域。 3. 分割与去除：通过分析提取到的文字区域，可以进行进一步的分割和去除杂质操作。例如，可以利用图像的连通性和形状特征，去除非文字的干扰。 4. 文字区域重建：从分割后的文字区域中，可以进行重建和连接操作，以提高文本的连续性。例如，可以通过基于几何形状和布局的方法，对分割后的文字区域进行重建和连接。 在实际应用中，可以根据具体的场景和需求选择不同的方法和技术。例如，可以使用基于阈值分割的方法、基于深度学习的方法等来实现文字图像分割。而OpenCV提供了丰富的函数库和工具，可以方便地进行图像处理和分析。 总之，OpenCV文字图像分割是通过将图像中的文字区域与背景区域进行分离，提取出文字的技术。通过预处理、区域提取、分割与去除、文字区域重建等步骤，可以实现对文字图像的分割和处理。
opencv 图像分割算法
OpenCV（Open Source Computer Vision Library）是一个开源的计算机视觉库，它提供了许多图像处理和计算机视觉算法。图像分割是计算机视觉中的一个重要任务，它可以将图像分成多个区域或对象。OpenCV提供了许多图像分割算法，包括阈值分割、区域生长、聚类、水平集方法等。 以下是几个常见的OpenCV图像分割算法的介绍： 1. 阈值分割（Thresholding）：阈值分割是一种简单的图像分割方法，它通过将图像中的像素值与一个阈值进行比较，将像素分为不同的类别。OpenCV提供了多种阈值分割算法，如全局阈值和自适应阈值等。 2. 区域生长（Region Growing）：区域生长是一种基于像素的图像分割方法，它通过将具有相似属性的像素组合成一个区域，并将其他像素标记为背景。OpenCV提供了区域生长算法的实现，可以根据不同的应用场景选择不同的生长算法。 3. 聚类（Clustering）：聚类是一种无监督的图像分割方法，它通过将相似的像素组合成群集，并将其他像素标记为背景。OpenCV提供了多种聚类算法，如K-means、DBSCAN等。 4. 水平集方法（Level Set Method）：水平集方法是近年来发展起来的一种先进的图像分割方法，它通过将图像中的边界或轮廓进行跟踪和演化，将图像分割成不同的区域。OpenCV提供了水平集方法的实现，可以根据不同的应用场景选择不同的水平集算法。 在使用OpenCV进行图像分割时，通常需要先对图像进行预处理，如滤波、去噪、缩放等，然后再选择合适的算法进行分割。OpenCV还提供了许多工具和函数，用于处理图像数据和执行各种计算机视觉任务。使用OpenCV进行图像分割可以大大提高效率和准确性，适用于各种计算机视觉应用场景。
相关推荐
OpenCV中的图像分割算法是分水岭算法。该算法通过对图像进行预处理，使用cv2.watershed()函数实现分割。\[1\]在使用该函数之前，需要先对图像中的期望分割区域进行标注，将已确定的区域标注为正数，未确定的区域标注为0。分水岭算法将图像比喻为地形表面，通过标注的区域作为“种子”，实现图像分割。\[2\] 在OpenCV中，除了cv2.watershed()函数外，还可以借助形态学函数、距离变换函数cv2.distanceTransform()和cv2.connectedComponents()来完成图像分割的具体实现。\[3\]形态学函数用于对图像进行形态学操作，距离变换函数用于计算图像中每个像素点到最近边界的距离，而cv2.connectedComponents()函数用于将图像中的连通区域进行标记。 综上所述，OpenCV中的图像分割算法是分水岭算法，通过预处理和使用cv2.watershed()函数实现分割，同时还可以借助形态学函数、距离变换函数和cv2.connectedComponents()函数来完成图像分割的具体实现。 #### 引用[.reference_title] - *1* *2* *3* [OpenCV进行图像分割：分水岭算法（相关函数介绍以及项目实现）](https://blog.csdn.net/m0_62128864/article/details/124541624)[target="_blank" data-report-click={"spm":"1018.2226.3001.9630","extra":{"utm_source":"vip_chatgpt_common_search_pc_result","utm_medium":"distribute.pc_search_result.none-task-cask-2~all~insert_cask~default-1-null.142^v91^control_2,239^v3^insert_chatgpt"}} ] [.reference_item] [ .reference_list ]
通过使用OpenCV库和Python编程语言，可以实现图像分割的任务。下面是一种基于K-means聚类算法的图像分割方法的示例代码： python import cv2 import numpy as np # 读取图像 img = cv2.imread("path_to_image.jpg") # 将图像转换为灰度图 gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY) # 使用K-means聚类算法进行图像分割 Z = gray.reshape((-1, 1)) Z = np.float32(Z) criteria = (cv2.TERM_CRITERIA_EPS + cv2.TERM_CRITERIA_MAX_ITER, 10, 1.0) k = 2 # 聚类中心个数 ret, label, center = cv2.kmeans(Z, k, None, criteria, 10, cv2.KMEANS_RANDOM_CENTERS) center = np.uint8(center) res = center[label.flatten()] segmented_img = res.reshape((gray.shape)) # 显示分割结果 cv2.imshow("Segmented Image", segmented_img) cv2.waitKey(0) cv2.destroyAllWindows() 上述代码首先读取图像，并将其转换为灰度图像。然后使用K-means聚类算法对灰度图像进行分割，将像素值聚类为k个类别。最后，将分割结果可视化显示出来。 请注意，上述代码只是图像分割中的一种方法，其他图像分割方法也可以使用OpenCV中的不同函数来实现。具体选择哪种方法取决于实际需求和图像特征。123 #### 引用[.reference_title] - *1* *2* *3* [Python 计算机视觉（十二）—— OpenCV 进行图像分割](https://blog.csdn.net/qq_52309640/article/details/120941157)[target="_blank" data-report-click={"spm":"1018.2226.3001.9630","extra":{"utm_source":"vip_chatgpt_common_search_pc_result","utm_medium":"distribute.pc_search_result.none-task-cask-2~all~insert_cask~default-1-null.142^v93^chatsearchT3_2"}}] [.reference_item style="max-width: 100%"] [ .reference_list ]
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xcode precompiler header
In your xcode proj. You can see a other sources group. there is a xxx_Prefix.PCH. In this *.PCH, If you write the cpp code, you should add
#if defined __cplusplus
#include "MUtils.h"
#endif
Then the precompiler header function would be worked.
Reference:
http://www.facebook.com/note.php?note_id=220473491312585
http://forum.soft32.com/mac/Xcode-precompiled-header-files-problem-ftopict47632.html
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XmlStringSerializer Members
The XmlStringSerializer type exposes the following members.
Constructors
NameDescription
Public methodXmlStringSerializer()()()()
Creates the serializer based on XmlSerializer with default settings.
Public methodXmlStringSerializer(Func<(Of <<'(Type, XmlSerializer>)>>))
Creates the serializer that allows user to specify his own method instantiating XmlSerializer with desired settings.
Public methodXmlStringSerializer(XmlStringSerializer..::..XmlSerializerFactoryMethod)
Creates the serializer that allows user to specify his own method instantiating XmlSerializer with desired settings.
Methods
NameDescription
Public methodDeserialize<(Of <<'(_T>)>>)
Deserializes data into the specified type.
Public methodEquals
Determines whether the specified Object is equal to the current Object.
(Inherited from Object.)
Protected methodFinalize
Allows an Object to attempt to free resources and perform other cleanup operations before the Object is reclaimed by garbage collection.
(Inherited from Object.)
Public methodGetHashCode
Serves as a hash function for a particular type.
(Inherited from Object.)
Public methodGetType
Gets the Type of the current instance.
(Inherited from Object.)
Protected methodMemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.)
Public methodSerialize<(Of <<'(_T>)>>)
Serializes data to the xml string.
Public methodToString
Returns a String that represents the current Object.
(Inherited from Object.)
See Also
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__label__pos
| 0.928813 |
U.S. DEPARTMENT OF HEALTH AND HUMAN SERVICES
Public Health Service * National Institutes of Health
Summary
Occasional over indulgent Alcohol users and those that consume alcohol above the recommended average daily consumption (1 drink per day for women and up to 2 drinks per day for men) run the risk of diminished nutrient digestion and nutrient utilization through a number of complex mechanisms. Additionally, alcohol affects blood glucose levels leading to depravation of brain energy and function. Certain nutritional supplements (mainly; Vitamins B- 12, 6,5,3,2, C, D, E, folate, calcium, iron) may affect these implications and reverse the negative effects. An individual should consult with their doctor before subscribing to any nutritional supplement regimen. Abstinence from alcohol is preferred.
Alcohol and Nutrition
Nutrition is a process that serves two purposes: to provide energy and to maintain body structure and function. Food supplies energy and provides the building blocks needed to replace worn or damaged cells and the nutritional components needed for body function. Alcohol users often eat poorly, limiting their supply of essential nutrients and affecting both energy supply and structure maintenance. Furthermore, alcohol interferes with the nutritional process by affecting digestion, storage, utilization, and excretion of nutrients (1).
Impairment of Nutrient Digestion and Utilization
Once ingested, food must be digested (broken down into small components) so it is available for energy and maintenance of body structure and function. Digestion begins in the mouth and continues in the stomach and intestines, with help from the pancreas. The nutrients from digested food are absorbed from the intestines into the blood and carried to the liver. The liver prepares nutrients either for immediate use or for storage and future use.
Alcohol inhibits the breakdown of nutrients into usable molecules by decreasing secretion of digestive enzymes from the pancreas (2). Alcohol impairs nutrient absorption by damaging the cells lining the stomach and intestines and disabling transport of some nutrients into the blood (3). In addition, nutritional deficiencies themselves may lead to further absorption problems. For example, folate deficiency alters the cells lining the small intestine, which in turn impairs absorption of water and nutrients including glucose, sodium, and additional folate (3).
Even if nutrients are digested and absorbed, alcohol can prevent them from being fully utilized by altering their transport, storage, and excretion (4). Decreased liver stores of vitamins such as vitamin A (5), and increased excretion of nutrients such as fat, indicate impaired utilization of nutrients by alcohol users (3).
Alcohol and Energy Supply
The three basic nutritional components found in food--carbohydrates, proteins, and fats--are used as energy after being converted to simpler products. Some alcohol users ingest as much as 50 percent of their total daily calories from alcohol, often neglecting important foods (3,6).
Even when food intake is adequate, alcohol can impair the mechanisms by which the body controls blood glucose levels, resulting in either increased or decreased blood glucose (glucose is the body's principal sugar) (7). In nondiabetic moderate alcohol users, increased blood sugar, or hyperglycemia--caused by impaired insulin secretion--is usually temporary and without consequence. Decreased blood sugar, or hypoglycemia, can cause serious injury even if this condition is short lived. Hypoglycemia can occur when a fasting or malnourished person consumes alcohol. When there is no food to supply energy, stored sugar is depleted, and the products of alcohol metabolism inhibit the formation of glucose from other compounds such as amino acids (7). As a result, alcohol causes the brain and other body tissue to be deprived of glucose needed for energy and function.
Although alcohol is an energy source, how the body processes and uses the energy from alcohol is more complex than can be explained by a simple calorie conversion value (8). For example, alcohol provides an average of 20 percent of the calories in the diet of the upper third of drinking Americans, and we might expect many drinkers who consume such amounts to be obese. Instead, national data indicate that, despite higher caloric intake, drinkers are no more obese than nondrinkers (9,10). Also, when alcohol is substituted for carbohydrates, calorie for calorie, subjects tend to lose weight, indicating that they derive less energy from alcohol than from food (summarized in 8).
The mechanisms accounting for the apparent inefficiency in converting alcohol to energy are complex. (11), but several mechanisms have been proposed. For example, over drinking triggers an inefficient system of alcohol metabolism, the microsomal ethanol-oxidizing system (MEOS) (1). Much of the energy from MEOS-driven alcohol metabolism is lost as heat rather than used to supply the body with energy.
Research indicates that the majority of sometimes over indulgent drinkers may have detectable nutritional deficiencies. Because some alcohol users tend to eat poorly--often eating less than the amounts of food necessary to provide sufficient carbohydrates, protein, fat, vitamins A and C, the B vitamins, and minerals such as calcium and iron (6,9,26)--a major concern is that alcohol's effects on the digestion of food and utilization of nutrients may shift a well-nourished person towards a malnourished person and in some cases (daily alcohol use) severe malnutrition.
Summary
Occasional over indulgent Alcohol users and those that consume alcohol above the recommended average daily consumption (1 drink per day for women and up to 2 drinks per day for men) run the risk of diminished nutrient digestion and nutrient utilization through a number of complex mechanisms. Additionally, alcohol affects blood glucose levels leading to depravation of brain energy and function. Certain nutritional supplements (mainly; Vitamins B- 12, 6,5,3,2, C, D, E, folate, calcium, iron) may affect these implications and reverse the negative effects. An individual should consult with their doctor before subscribing to any nutritional supplement regimen. Abstinence from alcohol is preferred.
U.S. DEPARTMENT OF HEALTH AND HUMAN SERVICES
Public Health Service * National Institutes of Health
REFERENCES AVAILABLE UPON REQUEST: REPRINTS AVAILABLE
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__label__pos
| 0.796939 |
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Rails
【Rails】 9.Rubyのクラスとインスタンスを学ぼう
※ カリキュラムでは、Cloud9のターミナルやエディタを利用します。
まだ用意していない方は「AWS Cloud9を準備しよう」を参考に導入してください。
オブジェクト指向プログラミングの基礎知識
コンピュータに何かしてもらいたいときには、命令や指示を与えるプログラムが必要です。
プログラムとは、コンピュータに作業させる一連の手順を記述したものです。
プログラム
このプログラムを作ることを「プログラミング」といいます。プログラミングでは、コンピュータが理解できる機械語に翻訳しやすい言葉として、プログラミング言語を使います。その際、プログラマの意図した通りの順番でコンピュータが動作するように指示を与えます。
プログラムの表現方法にはいくつかありますが、その中でもRubyに大きく関わる「オブジェクト指向プログラミング」の基礎知識をおさえておきましょう。
オブジェクト指向プログラミングとは
ざっくり説明すると、「オブジェクト指向プログラミングとは、ものを組み立てるように表現するプログラムの記述手法」です。
オブジェクト指向プログラミングのイメージ
もう少し具体的に説明すると、まず「もの」は「オブジェクト」を指します。
それぞれのデータを「オブジェクト」という仮想の物体と捉え、「オブジェクト同士が互いに影響し合う」という関係性によってプログラムの処理を進めます。
オブジェクト指向プログラミングでは、オブジェクトの設計図である「クラス」と設計図に基づいて生成される「インスタンス」が必要になります。
クラスとインスタンスのイメージを図で掴もう
クラスは、オブジェクトの「設計図」に相当するものです。インスタンスは、設計図であるクラスに基づいて生成されるオブジェクトの実物体のことです。
クラスとインスタンス
設計図であるクラスには、オブジェクトの振る舞い（メソッド）や状態（オブジェクト毎のデータ）を定義します。
たとえとしてよく用いられるのは、「たい焼き器」と「たい焼き」です。
クラスとインスタンスの具体例
１つのたい焼き器（クラス）によって、見た目が同じ形のたい焼き（インスタンス）をどんどん作ることができます。
もちろん材料を入れることで、たい焼き毎に中身の味を変えられます。
インスタンス生成
たい焼きの見た目が同じ形でも、たい焼き（インスタンス）ごとに「味」や「値段」といった中身が異なる情報を持たせることができます。
この辺りは実際のソースコードをみた方が理解しやすいので、順を追って説明します。
用語を一旦整理してみよう
ここまで出てきた用語を整理してみましょう。
用語 説明
プログラミング プログラムを作ること
プログラム コンピュータに作業させる一連の手順を記述したもの
プログラミング言語 コンピュータが理解できる機械語に翻訳しやすい言葉
オブジェクト指向
プログラミング
ものを組み立てるように表現するプログラムの記述手法
オブジェクト Rubyで扱うことができる全てのデータ
振る舞い（動作） メソッドのこと
状態 オブジェクト毎のデータのこと
クラス オブジェクトの設計図に相当するもの
オブジェクトの振る舞いや状態を定義する
インスタンス クラスの定義に基づいて生成されるオブジェクトの実物体
ぴっかちゃん
たくさんあって覚えられるかなぁ。。。
このあと何回も出てくるから、いま覚える必要はないよ！
ぴかわかさん
クラスとインスタンスを作ってみよう
それでは、実際にコードをみながらクラスとインスタンスを作ってみましょう。
クラスとインスタンスの具体例
taiyaki.rbという名前のファイルを用意しておきましょう。
クラスを定義しよう
新しいクラスを定義するには、キーワードのclassを使います。
クラスの定義
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class クラス名
end
それでは、たい焼きの設計図である「Taiyakiクラス」を定義してみましょう。
以下のソースコードを「taiyaki.rb」に書いてみましょう。
taiyaki.rb | Taiyakiクラスを定義する
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class Taiyaki
end
クラスには、クラス名の先頭は大文字にするという決まりがあるので、taiyakiではなくTaiyakiと書きます。
ポイント
1. クラスは、オブジェクトの設計図に相当するもの
2. クラス名の先頭には、大文字を使う
インスタンスを生成しよう
インスタンスを生成するには、クラスに対して「newメソッド」を呼び出します。
クラスのインスタンスを生成する
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クラス名.new
それでは、たい焼きの設計図であるTaiyakiクラスから「たい焼き」を作ってみましょう。
以下のようにTaiyaki.newを「taiyaki.rb」に書いてみましょう。
taiyaki.rb | たい焼きの設計図からたい焼きを作成する
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class Taiyaki
end
Taiyaki.new
以下のようにTaiyaki.newによって「Taiyakiクラスのインスタンス」が生成されます。
Taiyakiクラスのインスタンス
また、たい焼き器からいくつでもたい焼きを作れるように、クラスからは１つのインスタンスだけではなく、newメソッドを繰り返し呼び出すことで、複数のインスタンスを生成することができます。
ポイント
クラスに対してnewメソッドを呼び出すと、そのクラスのインスタンスを生成することができる。
initializeメソッド
インスタンス生成したときに実行したい処理があれば、クラスの中に「initializeメソッド」という特殊なメソッドを定義しておきます。initializeメソッドとは、newメソッドでインスタンスを生成した際に自動で呼ばれるメソッドのことです。
initializeメソッドは、以下のようにクラスの中に定義します。
initializeメソッドの定義
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class クラス名
def initialize
#インスタンス生成時に実行したい処理を書く
end
end
それでは、Taiyakiクラスにinitializeメソッドを定義して、インスタンス生成時に自動で呼び出されることを確認してみましょう。
以下のソースコードをtaiyaki.rbに書き、rubyコマンドで実行してみましょう。
taiyaki.rb | initializeメソッドを定義する
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class Taiyaki
def initialize
puts "initializeメソッドが実行されました。"
end
end Taiyaki.new
動画のようにtaiyaki.rbを実行すると、ターミナルにはinitializeメソッドが実行されましたと出力されます。
initializeメソッドの動作を確認
前回の章では「メソッドを実行するには『メソッド呼び出し』が必要」と学びましたね。
しかし今回はTaiyaki.newが実行されると、initializeメソッドが自動で呼び出されるので、以下のような動作になります。
initializeメソッドの動作
初学者の方がよく混乱してしまうのが「newメソッド」と「initializeメソッド」だよ。２つのメソッドの違いをしっかりおさえておこう！
ぴかわかさん
ぴっかちゃん
newメソッドは「クラスのインスタンスを生成する」だよ。initializeメソッドは「newメソッドが実行されると自動で呼び出される」だよね！
initializeメソッドで引数を使う場合
initializeメソッドへの引数の指定は、以下のようにnewメソッドの引数で指定します。
引数を使う場合
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class クラス名
def initialize(仮引数)
#インスタンス生成時に実行したい処理を書く end end
クラス名.new(実引数)
それでは、initializeメソッドへの引数を指定して、インスタンス生成後にたい焼きの「味」と「値段」をターミナルへ出力できるようにしてみましょう。
以下のソースコードをtaiyaki.rbに書き、rubyコマンドで実行してみましょう。
taiyaki.rb | initializeメソッドに引数を使う
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class Taiyaki
def initialize(taste, price)
puts "#{taste}味のたい焼きは、#{price}円です。"
end end
Taiyaki.new("あんこ", 250)
動画のようにtaiyaki.rbを実行すると、あんこ味のたい焼きは、250円と出力されます。
initializeメソッドに引数を使う
newメソッドに指定した実引数（"あんこ", 250）は、以下のようにinitializeメソッドで指定する仮引数（taste, price）へそれぞれ渡されます。
initializeメソッドの中で仮引数を使うことで、渡された値を参照することができます。
initializeメソッドの引数を使う場合の動作
次のインスタンス変数で詳しく説明しますが、本来initializeメソッドはインスタンスの初期化の設定を行うことが望ましいです。
今回はinitializeメソッドの引数を説明するために、上記のような使い方をしています。
ポイント
1. クラスのインスタンスを生成する場合は、newメソッドを使う
2. newメソッドを呼び出すと、自動的にinitializeメソッドが呼び出される
3. newメソッドで指定した引数は、initializeメソッドに渡される
空のクラスとインスタンス
newメソッドは、指定したクラスのインスタンスを生成して返します。
Taiyakiクラスのインスタンスは、#<Taiyaki:0x00000000014580e8>のような形式で表現されます。数値の部分はインスタンスごとに異なります。
サンプルコード | newメソッドの戻り値を確認する
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class Taiyaki
end
Taiyaki.new
#=> #<Taiyaki:0x00000000014580e8>
上記のTaiyakiクラスのように、設計図に何も書いていない状態でTaiyaki.newしても、見た目はたい焼きの形だけど皮しかないたい焼き、つまり何も情報をもたないインスタンスが作られるだけです。
何も情報をもたないたい焼き
このままでは、たい焼きの「味」や「値段」が分かりません。
たい焼き（インスタンス）に情報をもたせるには、「インスタンス変数」を使います。
ポイント
1. newメソッドは、指定したクラスのインスタンスを生成して返す
2. クラスの中身が空の状態だと、何も情報をもたないインスタンスが生成される
インスタンス変数について学ぼう
インスタンス変数とは、名前の先頭に@がついた変数のことです。
サンプルコード | インスタンス変数に値を代入する
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@taste = "あんこ"
initializeメソッドの中でインスタンス変数を初期化することで、インスタンス生成時に情報をもたせることができます。
サンプルコード | initializeメソッドの中でインスタンス変数を初期化する
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class クラス名
def initialize
@taste = "あんこ"
end end
たい焼きを作るときに「味」の情報をもたせたい場合は、以下のようにinitializeメソッドの中で@tasteというインスタンス変数を引数を使って初期化します。
引数を使うことで、固定ではなく味を変えてたい焼きを作ることができます。
サンプルコード | 引数を使ってインスタンス変数を初期化する
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class Taiyaki
def initialize(taste)
@taste = taste #@tasteには、引数に渡された値が代入される
end end #newメソッドの引数に指定した値は、initializeメソッドの引数に渡される
Taiyaki.new(@tasteに代入する値を指定する)
ポイント
1. インスタンス変数とは、名前の先頭に@がついた変数のこと
2. initializeメソッドでは、インスタンス変数の初期化を行う
あんこ味のたい焼きを作ることを考えてみよう
例として、あんこ味のたい焼きを作ることを考えてみましょう。
インスタンス変数の@tasteに代入したい値は"あんこ"になるので、以下のようにnewメソッドの引数には"あんこ"を指定します。
サンプルコード | あんこ味のたい焼きを作る
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class Taiyaki
def initialize(taste)
@taste = taste
end
end
Taiyaki.new("あんこ")
# => #<Taiyaki:0x0000000002dbf518 @taste="あんこ"> #生成されたインスタンス
Taiyaki.new("あんこ")によって生成されたインスタンスを確認すると、数値のあとには@taste="あんこ"が設定されていますね。
このインスタンスには、たい焼きの「味はあんこ」という情報があります。
あんこ味のたい焼き
このようにinitializeメソッドでインスタンス変数を初期化すると、インスタンス生成時に何かしらの情報を持たせることができます。
実際に手を動かしながら、インスタンスが情報を持てるようにしよう！
ぴかわかさん
インスタンス変数を定義しよう
それでは、たい焼きを作るときに「味」や「値段」の情報をもてるように、Taiyakiクラスのinitializeメソッドの中にインスタンス変数を定義してみましょう。
以下のソースコードをtaiyaki.rbに書き、rubyコマンドで実行してみましょう。
taiyaki.rb | たい焼き作成時に味と値段の情報をもてるようにする
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end end
p Taiyaki.new("あんこ", 250)
動画のようにtaiyaki.rbを実行すると、生成されたインスタンスが出力されます。
※数値の部分はインスタンス毎に異なるので、一致している必要はありません
インスタンス変数を使う
処理の流れを１つ１つ確認してみましょう。
newメソッドに指定した実引数（"あんこ", 250）は、initializeメソッドで指定する仮引数（taste, price）へ渡され、インスタンス変数@taste@priceにそれぞれ代入されます。
インスタンス変数に値を代入する
続いて、以下のようにTaiyakiクラスのインスタンスが戻り値として呼び出し元に返ります。
newメソッドの戻り値
処理の流れを整理
1. newメソッドの実引数の値は、initializeメソッドの仮引数に渡される
2. 仮引数の値は、それぞれのインスタンス変数に代入される
3. 生成されたインスタンスには、インスタンス変数によって値が保持される
空のクラスと比べてみよう
中身が空のクラスに対してインスタンス生成した場合と比べてみます。
クラスに何も書かれていない状態では、以下のようにTaiyaki.newしても何も情報をもたないインスタンスが生成されるだけでしたね。
空のクラスからインスタンス生成する
しかし、クラスのinitializeメソッドの中でインスタンス変数を初期化すれば、生成するインスタンスに情報を持たせることできます。
今回は@taste@priceを使って、味と値段の情報をインスタンス毎に持たせます。
味と値段の情報をインスタンス毎に持たせる
このようにインスタンスに何か情報を持たせたい場合は、インスタンス変数を使います。
インスタンス変数の特徴として、他にも「クラス内の異なるメソッド間でも値を受け渡せる」「インスタンス毎に固有の値を保持できる」などがあります。
この特徴は、次のインスタンスメソッドを学ぶことによって理解することができます。
ポイント
1. 中身が空のクラスの場合は、情報をもたないインスタンスが生成される
2. インスタンス変数を使うと、インスタンスに値を保持できるようになる
3. initializeメソッドの中でインスタンス変数を初期化すれば、インスタンス生成時に値を設定することができる
次に進む前に必要のない処理を削除しよう
taiyaki.rbのソースコードを編集しましょう。
pメソッドは削除して、変数のanko_taiyakiTaiyaki.new("あんこ", 250)を代入しておきましょう。
taiyaki.rb | 変数に代入する
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
end
anko_taiyaki = Taiyaki.new("あんこ", 250)
newメソッドによって生成されたTaiyakiクラスのインスタンスは、繰り返し利用できるように変数のanko_taiyakiに代入しておきます。
インスタンスメソッドを定義しよう
インスタンスメソッドとは、クラスのインスタンスに対して呼び出すことができるメソッドのことです。クラスの中でキーワードのdefを使って普通に定義したメソッドは、インスタンスメソッドとして扱われます。
インスタンスメソッドを定義する
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class クラス名
def メソッド名
# 処理
end
end
インスタンスメソッドの呼び出しは、以下のように対象のインスタンスにドット（.）をつなげてメソッド名を指定します。
インスタンスメソッドの呼び出し
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対象のインスタンス.メソッド名
それでは、Taiyakiクラスにインスタンスメソッドを定義していきましょう。
インスタンスメソッドと共にインスタンス変数の特徴も学んでいくよ！
ぴかわかさん
ぴっかちゃん
たしかインスタンス変数の特徴は「クラス内の異なるメソッド間でも値を受け渡せる」「インスタンス毎に異なる値を扱える」だったね
インスタンス変数を使って処理を書いてみよう
インスタンス変数は、クラス内の異なるメソッド間でも値を受け渡せる」ので、インスタンスメソッドの中でもインスタンス変数を利用することができます。
以下のソースコードをtaiyaki.rbに書き、rubyコマンドで実行してみましょう。
taiyaki.rb | インスタンスメソッドのshow_infoを定義する
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
def show_info
puts "#{@taste}味のたい焼きは#{@price}円です。"
end
end anko_taiyaki = Taiyaki.new("あんこ", 250)
anko_taiyaki.show_info
show_infoはインスタンス変数の@taste@priceを利用して、インスタンスの保持するたい焼きの味や値段を表示するインスタンスメソッドです。
動画のようにtaiyaki.rbを実行すると、あんこ味のたい焼きは250円と出力されます。
インスタンスメソッドを呼び出す
anko_taiyakiには、以下のように@taste@price"あんこ"250が設定されたTaiyakiクラスのインスタンスが代入されています。
anko_taiyakiに代入されるインスタンス
このインスタンスが持つ情報は、インスタンス変数を使うことでクラス内の異なるメソッド間でも扱うことができます。
以下のようにanko_taiyakiに対してshow_infoメソッドを呼び出すと、show_infoメソッド内のインスタンス変数は、anko_taiyakiが持つ値をそれぞれ参照します。
インスタンスメソッドを呼び出す流れ
このようにインスタンスメソッドの中でインスタンス変数を使うと、対象のインスタンスが持つ情報を利用することができます。
インスタンスが保持している情報は、クラス内の異なるメソッドでも共有できます。
ポイント
1. インスタンス変数は、クラス内の異なるメソッド間でも値を受け渡せる
2. インスタンスメソッドの中でインスタンス変数を使うと、インスタンスが持つ情報を利用できる
インスタンス毎に保持される情報を確かめてみよう
インスタンス変数は、インスタンス毎に異なる値を扱うことができます。
先ほどのanko_taiyakiには、@taste = "あんこ" @price = 250の情報をもつTaiyakiクラスのインスタンスが代入されていますね。
anko_taiyakiに代入されるインスタンス
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anko_taiyaki = Taiyaki.new("あんこ", 250)
# => #<Taiyaki:0x00000000026b4840 @taste="あんこ", @price=250>
anko_taiyakiに代入されるインスタンス
これはインスタンス生成時に、newメソッドの引数の値でインスタンス変数を初期化しているからでしたね。
以下のようにnewメソッドの引数に"あんこ"250を指定しています。
サンプルコード | インスタンス変数の初期化
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
#...以下省略 end #インスタンス変数に代入する値は、newメソッドの引数に指定
anko_taiyaki = Taiyaki.new("あんこ", 250)
newメソッドの引数に指定する値を変更すると、インスタンス変数に"あんこ"250とは異なる情報を設定することができます。
例として、「カスタード味のたい焼き 300円」で考えてみましょう。
newメソッドの引数に"カスタード"300を指定することで、@taste = "カスタード" @price = 300の情報をもつTaiyakiクラスのインスタンスが生成されます。
custard_taiyakiに代入されるインスタンス
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custard_taiyaki = Taiyaki.new("カスタード", 300)
# => #<Taiyaki:0x00000000009ef980 @taste="カスタード", @price=300
custard_taiyakiには、この情報を持つインスタンスが代入されます。
カスタード味のたい焼き
このようにインスタンス変数は、インスタンス毎に異なる値を保持することができます。
インスタンス毎にインスタンスメソッドを呼び出してみよう
anko_taiyakiに対してshow_infoメソッドを呼び出した場合は、「あんこ味のたい焼きは250円です。」と出力されましたね。
anko_taiyakiに対して、show_infoメソッドを呼び出す
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anko_taiyaki.show_info
# 出力結果 あんこ味のたい焼きは250円です。
それではcustard_taiyakiに対してshow_infoメソッドを呼び出すと、どのような結果になるでしょうか。
インスタンス毎にインスタンスメソッドを呼び出す
実際にカスタード味のたい焼きを作成して確かめてみましょう。
以下のソースコードをtaiyaki.rbに書き、rubyコマンドで実行してみましょう。
taiyaki.rb | カスタード味のたい焼きを作成する
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
def show_info
puts "#{@taste}味のたい焼きは#{@price}円です。"
end
end
anko_taiyaki = Taiyaki.new("あんこ", 250)
anko_taiyaki.show_info
custard_taiyaki = Taiyaki.new("カスタード", 300)
custard_taiyaki.show_info
custard_taiyakiに対してshow_infoメソッドを呼び出した場合は、動画のように「カスタード味のたい焼きは300円です。」と出力されます。
custard_taiyakiに対してインスタンスメソッド呼び出し
anko_taiyaki.show_infoの出力結果とは違いますよね。
出力結果
インスタンスメソッドは、インスタンス毎に異なる処理を行うことができます。それはインスタンス変数によって、インスタンス毎に異なる情報を持っているからです。
以下のようにanko_taiyakiに対してshow_infoを呼び出せば、インスタンスメソッドであるshow_infoのインスタンス変数は"あんこ"250をそれぞれ参照します。
ソースコード1
custard_taiyakiに対してshow_infoを呼び出した場合は、以下のようにインスタンス変数はそれぞれ"カスタード"300を参照します。
ソースコード2
このようにインスタンス変数は、生成されたインスタンス毎に共有される変数です。
インスタンス変数
ポイント
1. インスタンスメソッドは、クラスのインスタンスに対して呼び出せるメソッド
2. インスタンス変数は、インスタンス毎に異なる値を保持することができる
次に進む前に必要のない処理をコメントアウトしよう
taiyaki.rbのソースコードを編集しましょう。
次のアクセサメソッドではshow_infoは使わないので、以下のように３行をコメントアウトしておきましょう。
taiyaki.rb
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
def show_info
puts "#{@taste}味のたい焼きは#{@price}円です。"
end
end
anko_taiyaki = Taiyaki.new("あんこ", 250)
#anko_taiyaki.show_info
#custard_taiyaki = Taiyaki.new("カスタード", 300)
#custard_taiyaki.show_info
アクセサメソッドについて学ぼう
Rubyでは、クラス外部からインスタンス変数に対して直接操作することができません。
以下のようにクラス外部から@tasteを参照したり、変更を行おうとすると、エラーが発生してしまいます。
インスタンス変数をクラス外部から参照した場合
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
end
anko_taiyaki = Taiyaki.new("あんこ", 250)
anko_taiyaki.@taste
#taiyaki.rb:9: syntax error, unexpected instance variable anko_taiyaki.@taste
インスタンス変数をクラス外部から変更した場合
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anko_taiyaki.@taste = "栗あん"
# taiyaki.rb:11: syntax error, unexpected instance variable anko_taiyaki.@taste = "栗あん"
クラスの外部からインスタンス変数を参照・変更するには、それを目的としたインスタンスメソッドを定義する必要があります。
ゲッターメソッド
クラスの外部からインスタンス変数の値を参照するインスタンスメソッドのことを「ゲッターメソッド」と呼びます。
ゲッターメソッド名は、それぞれ@なしのインスタンス変数名を付けます。
クラスの外部から@tasteの値を参照するには、以下のようにゲッターメソッドのtasteを定義し、呼び出します。
ゲッターメソッドを経由して@tasteの値を参照する
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
#ゲッターメソッド（@tasteを外部から参照するためのメソッド）
def taste
@taste #"あんこ"を呼び出し元へ返す
end
end anko_taiyaki = Taiyaki.new("あんこ", 250)
anko_taiyaki.taste #=> "あんこ"
インスタンス変数は「クラス内の異なるメソッド間でも値を受け渡せる」と学びましたね。
上記ではanko_taiyakiに対してtasteを呼び出しているので、tasteメソッド内の@tasteでは"あんこ"を参照することができます。
ぴっかちゃん
ゲッターメソッドを経由してインスタンス変数の値を参照できるんだね！
セッターメソッド
インスタンス変数の値をクラスの外部から変更するためのインスタンスメソッドのことを「セッターメソッド」と呼びます。
セッターメソッド名は、@なしのインスタンス変数名で末尾に=をつけます。
クラスの外部から@tasteの値を変更するには、セッターメソッドのtaste=を以下のように定義し、呼び出します。
セッターメソッドを経由して@tasteの値を変更する
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
#セッターメソッド（@tasteを外部から変更するためのメソッド）
def taste=(taste)
@taste = taste
end
end anko_taiyaki = Taiyaki.new("あんこ", 250)
anko_taiyaki.taste = "栗あん"
#変更されたか確認する anko_taiyaki.taste #=> "栗あん"
上記のanko_taiyaki.taste = "栗あん"では、anko_taiyaki.taste"栗あん"を代入しているようにみえますよね。
実際には、以下のように引数に"栗あん"を指定してtaste=を呼び出しています。
サンプルコード
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#代入しているようにみえるが...
anko_taiyaki.taste = "栗あん"
#実際にはtaste=メソッドを呼び出している
anko_taiyaki.taste=("栗あん")
セッターメソッドのtaste=のように、メソッド名の末尾に=(イコール)をつけると、代入式のようにメソッドを呼び出すことができます。
ぴっかちゃん
セッターメソッドを経由してインスタンス変数の値が変更できるんだね！
実際に手を動かして、ゲッター / セッターメソッドを定義してみよう
ぴかわかさん
ゲッターメソッド / セッターメソッドを定義してみよう
まずは、クラスの外部から@taste@priceの値を参照できるようにゲッターメソッドを定義しましょう。
以下のソースコードをtaiyaki.rbに書き、rubyコマンドで実行してみましょう。
taiyaki.rb | ゲッターメソッドを定義して@tasteと@priceの値を参照する
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
def taste
@taste
end
def price
@price
end
def show_info puts "#{@taste}味のたい焼きは#{@price}円です。" end end anko_taiyaki = Taiyaki.new("あんこ", 250)
p anko_taiyaki.taste
p anko_taiyaki.price
#anko_taiyaki.show_info #custard_taiyaki = Taiyaki.new("カスタード", 300) #custard_taiyaki.show_info
taiyaki.rbを実行すると、動画のように"あんこ"250がターミナルへ出力されます。
ゲッターメソッド
ゲッターメソッドのtastepriceによって、各インスタンス変数の値を参照できるようになりましたね。
ぴかわかさん
続いて、クラスの外部から@taste@priceの値を変更できるようにセッターメソッドを定義しましょう。
anko_taiyakiがもつ各インスタンス変数の値を"栗あん"350に変更できるように、以下のソースコードをtaiyaki.rbに書き、rubyコマンドで実行してみましょう。
taiyaki.rb | セッターメソッドを定義して@tasteと@priceの値を変更する
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
def taste
@taste
end
def price
@price
end
def taste=(taste)
@taste = taste
end
def price=(price)
@price = price
end
def show_info puts "#{@taste}味のたい焼きは#{@price}円です。" end end anko_taiyaki = Taiyaki.new("あんこ", 250) p anko_taiyaki.taste p anko_taiyaki.price
anko_taiyaki.taste = "栗あん"
anko_taiyaki.price = 350
p anko_taiyaki.taste
p anko_taiyaki.price
#anko_taiyaki.show_info #custard_taiyaki = Taiyaki.new("カスタード", 300) #custard_taiyaki.show_info
taiyaki.rbを実行すると、動画のように"あんこ"250の後に変更された "栗あん"350がターミナルに出力されます。
セッターメソッド
セッターメソッドであるtaste=price=によって、各インスタンス変数の値を変更できるようになりましたね。
ポイント
1. Rubyでは、クラス外部からインスタンス変数に対して直接操作することができないので、ゲッター / セッターメソッドを経由する必要がある
2. ゲッターメソッドは、インスタンス変数の読み取り用のインスタンスメソッド
3. セッターメソッド、インスタンス変数の書き込み用のインスタンスメソッド
attr_accessorメソッド
ゲッターメソッドとセッターメソッドは、参照・変更したいインスタンス変数が増える度に定義する必要があるので、コード量が増えてしまいます。そこで「attr_accessorメソッド」を使うことで、まとめてアクセサメソッドを定義することができます。
attr_accessorメソッドとは、ゲッター / セッターメソッドを自動で定義してくれるメソッドのことです。
以下のように参照と変更したいインスタンス変数名をシンボル、もしくは文字列で指定します。カリキュラムでは、シンボルの指定方法で進めます。
attr_accessorメソッドの書き方
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attr_accessor :インスタンス変数名
クラスの外部から@ageを参照・変更できるようにしたい場合は、以下のようにクラスの中に定義します。
サンプルコード
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def クラス名
attr_accessor :age #インスタンス変数名は@なしで指定する
end
上記によって自動で定義されるゲッター / セッターメソッドは、以下の通りです。
attr_accessor :ageによって自動で定義されるメソッド
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# ゲッターメソッド
def age
@age
end
# セッターメソッド
def age=(age)
@age = age
end
複数行にかけて定義していたメソッドが１行で済むので、コードもスッキリとしますね。
これまでTaiyakiクラスでは、@taste@priceのゲッター / セッターメソッドをそれぞれ定義しているので、以下のように冗長なソースコードになっています。
taiyaki.rb | attr_accessorメソッド使用前
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class Taiyaki
def initialize(taste, price)
@taste = taste
@price = price
end
def taste
@taste
end
def price
@price
end
def taste=(taste)
@taste = taste
end
def price=(price)
@price = price
end
def show_info
puts "#{@taste}味のたい焼きは#{@price}円です。"
end
end
anko_taiyaki = Taiyaki.new("あんこ", 250)
p anko_taiyaki.taste
p anko_taiyaki.price
anko_taiyaki.taste = "栗あん"
anko_taiyaki.price = 350
p anko_taiyaki.taste
p anko_taiyaki.price
#anko_taiyaki.show_info
#custard_taiyaki = Taiyaki.new("カスタード", 300)
#custard_taiyaki.show_info
attr_accessorメソッドで、Taiyakiクラスの中をスッキリさせよう！
ぴかわかさん
taiyaki.rbを以下のソースコードのように編集しましょう。
クラスの外部から参照・変更したいインスタンス変数は、@taste@priceの２つあるので、attr_accessorメソッドに指定する際にカンマ（,）で区切ります。
taiyaki.rb | attr_accessorメソッドを使う
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class Taiyaki
attr_accessor :taste, :price
def initialize(taste, price) @taste = taste @price = price end def show_info puts "#{@taste}味のたい焼きは#{@price}円です。" end end anko_taiyaki = Taiyaki.new("あんこ", 250) p anko_taiyaki.taste p anko_taiyaki.price anko_taiyaki.taste = "栗あん" anko_taiyaki.price = 350 p anko_taiyaki.taste p anko_taiyaki.price #anko_taiyaki.show_info #custard_taiyaki = Taiyaki.new("カスタード", 300) #custard_taiyaki.show_info
attr_accessorメソッドを使うことで、Taiyakiクラスのコード量がずいぶんと減りましたね。
最後に問題なく動くかどうかを確かめるために、taiyaki.rbを実行してみましょう。
attr_accessorメソッドを使う
動画のようにターミナルに出力されたら問題ありません。
ポイント
1. attr_accessorメソッドとは、ゲッター / セッターメソッドを自動で定義してくれるメソッドのこと
2. 参照と変更したいインスタンス変数名をシンボル、もしくは文字列で指定する
クラス変数を定義しよう
クラス変数とは、名前の先頭に@が２つ付いた変数のことです。
サンプルコード | クラス変数に値を代入する
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@@name = "ぴっかちゃん"
クラス変数には、全てのインスタンスで共有される共通の情報をもたせることができます。
インスタンス変数は「生成されたインスタンス毎に共有される変数」だったのに対して、クラス変数は「全てのインスタンスで共有される変数」です。
全てのインスタンスで共有される変数
たい焼きが全部で何個作られたかわかるように、クラス変数を定義してみましょう。
以下のソースコードのようにtaiyaki.rbにクラス変数の@@total_taiyaki_countを定義して、たい焼きを作る度にカウントアップさせましょう。
taiyaki.rb | クラスの定義部分
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class Taiyaki
attr_accessor :taste, :price
@@total_taiyaki_count = 0
def initialize(taste, price) @taste = taste @price = price
@@total_taiyaki_count += 1
end #...以下省略 end
上記では、Taiyaki クラスの中で@@total_taiyaki_count0で初期化し、initializeメソッドで@@total_taiyaki_countの値をひとつ加算しています。
initializeメソッドは、newメソッドを呼び出すと自動で呼び出されるメソッドでしたね。
つまり、newメソッドでTaiyakiクラスのインスタンスを生成する（たい焼きを作成する）度に@@total_taiyaki_countの値がひとつ加算されます。
次のクラスメソッドで、たい焼きが全部で何個作られたかを確認するよ
ぴかわかさん
ポイント
1. クラス変数とは、名前の先頭に@が２つ付いた変数のこと
2. 全てのインスタンスで共有される共通の情報をもたせることができる
3. インスタンス変数はインスタンス毎、クラス変数は全てのインスタンスで共有される違いがある
クラスメソッドを定義しよう
同じクラスから生成されるインスタンスが共通で使用する処理は、クラスの中に「クラスメソッド」というメソッドを定義します。クラスメソッドとは、クラスに対して呼び出すことができるメソッドのことです。
クラスメソッドは、２つの定義方法があります。
クラスメソッドの定義方法１
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class
def self.クラスメソッド名
# 処理
end
end
クラスメソッドの定義方法２
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class クラス名
class << self
def クラスメソッド名
# 処理
end
end
end
クラスメソッドの定義は、どちらの方法でも構いませんが、クラスメソッドをたくさん定義する場合は、定義方法２の方がクラスメソッド名に毎回self.と付けなくてよいです。
クラスメソッドの呼び出しは、対象のクラスにドット（.）をつなげてクラスメソッド名を指定します。
クラスメソッドの呼び出し
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クラス名.クラスメソッド名
例としてHello!と挨拶するHelloクラスのクラスメソッドであるgreetingを定義し、呼び出すには以下のように書きます。
サンプルコード | 定義方法1でクラスメソッドを定義する
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class Hello
# クラスメソッドを定義
def self. greeting
puts "Hello!"
end
end
# クラスメソッドを呼び出す
Hello.greeting #=> Hello!
クラス変数を使って処理を書いてみよう
それでは、たい焼きが全部で何個作られたかを案内してくれるクラスメソッドのshow_all_countを定義してみましょう。
たい焼きが作成された個数は、先ほど学んだクラス変数の@@total_taiyaki_countが保持しています。以下のようにtaiyaki.rbにクラスメソッドを追加しましょう。
taiyaki.rb | クラスの定義部分
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class Taiyaki
attr_accessor :taste, :price
@@total_taiyaki_count = 0
def initialize(taste, price)
@taste = taste
@price = price
@@total_taiyaki_count += 1
end
def show_info
puts "#{@taste}味のたい焼きは#{@price}円です。"
end
def self.show_all_count
puts "たい焼きは全部で#{@@total_taiyaki_count}個作成されました。"
end
end
続いて、Taiyakiクラス外でnewメソッドを呼び出してインスタンスを生成し、クラスメソッドを呼び出してみましょう。
Taiyakiクラス外の処理は一旦全てコメントアウトして、以下の内容を追加しましょう。
taiyaki.rb | Taiyakiクラス外に処理を書く
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Taiyaki.new("あんこ", 250)
Taiyaki.new("カスタード", 300)
Taiyaki.new("抹茶", 350)
Taiyaki.show_all_count
動画のようにrubyコマンドでtaiyaki.rbを実行してみましょう。
たい焼きが全部で何個作成したかを確かめる
newメソッドによってTaiyakiクラスのインスタンスを全部で３つ生成したので、ターミナルにはたい焼きは全部で3個作成されました。と出力されます。
このようにクラスメソッドにクラス変数を使うことで、同じクラスのインスタンスで共有する情報の処理を行えます。
ポイント
1. クラスメソッドとは、クラスに対して呼び出すことができるメソッドのこと
2. 同じクラスから生成されるインスタンスが共通で使用する処理に使う
3. インスタンスメソッドは、インスタンス毎に保持されるデータ（インスタンス変数）の処理に使う
完成したtaiyaki.rbのソースコード
お疲れ様でした！これで「クラスとインスタンス」は終了です。完成のtaiyaki.rbのソースコードをコメントアウトの解説付きで載せておきますので、比較して確認してみてください。
taiyaki.rb
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#たい焼きの設計図を作成
class Taiyaki
#@tasteと@priceのゲッター・セッターメソッドを定義（クラス外から参照・変更可能になる）
attr_accessor :taste, :price
#クラス変数を0で初期化
@@total_taiyaki_count = 0
#newメソッドでインスタンスを生成した際に自動で呼ばれるメソッド
def initialize(taste, price)
#インスタンス変数を引数の値で初期化
@taste = taste
@price = price
#クラス変数の値をひとつ加算させる
@@total_taiyaki_count += 1
end
#インスタンスメソッド
def show_info
puts "#{@taste}味のたい焼きは#{@price}円です。"
end
#クラスメソッド
def self.show_all_count
puts "たい焼きは全部で#{@@total_taiyaki_count}個作成されました。"
end
end
#Taiyakiクラスの外
#Taiyakiクラスのインスタンスを生成
anko_taiyaki = Taiyaki.new("あんこ", 250)
#クラスの外からanko_taiyakiが保持する@tasteと@priceの値を参照
p anko_taiyaki.taste
p anko_taiyaki.price
#クラスの外からanko_taiyakiが保持する@tasteと@priceの値を変更
anko_taiyaki.taste = "栗あん"
anko_taiyaki.price = 350
#クラスの外からanko_taiyakiが保持する@tasteと@priceの値を参照（変更できたか確認）
p anko_taiyaki.taste
p anko_taiyaki.price
#anko_taiyakiに対してインスタンスメソッド呼び出し
anko_taiyaki.show_info
#Taiyakiクラスのインスタンスを生成
custard_taiyaki = Taiyaki.new("カスタード", 300)
#custard_taiyakiに対してインスタンスメソッド呼び出し
custard_taiyaki.show_info
#クラスメソッド呼び出し
Taiyaki.show_all_count
※「クラス変数を使って処理を書いてみよう」で追加したコードは、以下のようにハイライト箇所を不要なため削除してあります。
「クラス変数を使って処理を書いてみよう」で追加したコード
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Taiyaki.new("あんこ", 250)
Taiyaki.new("カスタード", 300)
Taiyaki.new("抹茶", 350)
Taiyaki.show_all_count
最後にこれまで出てきた用語を整理していこう
ぴかわかさん
オブジェクト指向プログラミングの用語整理
ここまでたくさんの用語が出てきましたが、１つ１つ整理してみましょう。
用語 意味/役割
クラス オブジェクト指向における設計図
インスタンス クラスから生成された実物体
インスタンス変数
@変数名
インスタンス毎に値を保持できる
クラス変数
@@変数名
クラス内の全てのインスタンスで共有される値
インスタンスメソッド インスタンスに対する操作を行うためのメソッド
クラスメソッド クラスに対する操作を行うためのメソッド
initializeメソッド 主にインスタンスの初期化の設定を行うためのメソッド
ゲッターメソッド クラスの外部からインスタンス変数の値を参照するインスタンスメソッド
セッターメソッド インスタンス変数の値をクラスの外部から変更するためのインスタンスメソッド
attr_accessorメソッド ゲッター / セッターメソッドを自動で定義してくれるメソッド
アクセサメソッド インスタンス変数の参照と変更ができるメソッドの総称
この記事のまとめ
• クラスには、設計図としてオブジェクトの振る舞いや状態を設定できる
• インスタンスは、クラスに基づいて生成されたオブジェクトのこと
• クラスに対してnewメソッドを呼び出すと、インスタンス生成できる
0
わかった！
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Erectile Dysfunction and Its Link to Sleep Disorders
Erectile
and Health
Erectile dysfunction (ED) is a common male sexual health problem and a major concern for many men. ED is a condition where a man cannot achieve or maintain an erection during sexual activity. It affects up to 30 million men in the United States and is estimated to affect up to 52% of all men over 40 years of age. Recent studies have revealed a strong link between ED and sleep disorders, such assleep apnea, insomnia, and narcolepsy.
Sleep Apnea
Sleep apnea is a disorder in which a person’s breathing pauses multiple times during sleep and can last up to 10 seconds at a time. It can cause restless sleep and can lead to serious health issues such as heart problems, stroke, and high blood pressure. It can also lead to ED. Several studies have found a significant correlation between sleep apnea and erectile dysfunction. In one study, men with sleep apnea were found to have double the rate of erectile dysfunction compared to men without sleep apnea.
See also The Impact of Testosterone Replacement Therapy on Sexual Health and Libido
Insomnia
Insomnia is defined as difficulty falling and staying asleep. It can cause fatigue, mood swings, and mental health issues. Studies have found that men with chronic insomnia have a three-fold increased risk of developing ED. This is because insomnia disrupts testosterone production, which plays an important role in male sexual health and sexual performance.
See also low testosterone in men
Narcolepsy
Narcolepsy is a sleep disorder which causes excessive fatigue and daytime sleepiness. It can also lead to difficulty with sexual arousal and erection. In one study, men with narcolepsy were found to have a six-fold increased risk of ED compared to the general population.
How to Reduce the Risk of ED
Given the link between sleep disorders and ED, it is important for men to take steps to reduce the risk of erectile dysfunction. The most important step is to get a good night’s sleep each night. This means sticking to a consistent sleep schedule and avoiding things that can disrupt sleep, such as caffeine, alcohol, and electronics. For men who are struggling with a sleep disorder, it is important to seek help from a medical professional. Additionally, eating a healthy diet, exercising regularly, and avoiding smoking can also help reduce the risk of ED.
See also Low Testosterone and Its Impact on Muscle Mass and Strength
Conclusion
Erectile Dysfunction is a common problem that affects many men. Recent studies have revealed a strong link between ED and sleep disorders, such as sleep apnea, insomnia, and narcolepsy. It is important for men to take steps to reduce their risk of ED and seek help from a medical professional if they are suffering from a sleep disorder.
Keywords: Erectile Dysfunction, ED, Sleep Disorders, Sleep Apnea, Insomnia, Narcolepsy, Testosterone, Sexual Health, Sexual Performance.
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Web Attacks and Countermeasures
Web Attacks and Defense
1. Introduction
What is a web application? Why web applications are the first target for hackers? What are the attacks Web applications usually face, how to prevent from these attacks. Lets start from the various web application attacks. This article is divided into three areas including types of attacks, countermeasures and risk factor.
2. ATTACKS
Following are the most common web application attacks.
a. Remote code execution
b. SQL injection
c. Format string vulnerabilities
d. Cross Site Scripting (XSS)
e. Username enumeration
Remote Code Execution
As the name suggests, this vulnerability allows an attacker to run arbitrary, system level code on the vulnerable web application server and retrieve any desired information contained therein. Improper coding errors lead to this vulnerability. At times, it is difficult to discover this vulnerability during penetration testing assignments but such problems are often revealed while doing a source code review. However, when testing Web applications is important to remember that exploitation of this vulnerability can lead to total system compromise with the same rights as the Web server itself is running with.
SQL Injection
SQL injection is a very old approach but it’s still popular among attackers. This technique allows an attacker to retrieve crucial information from a Web server’s database. Depending on the application’s security measures, the impact of this attack can vary from basic information disclosure to remote code execution and total system compromise.
Format String Vulnerabilities
This vulnerability results from the use of unfiltered user input as the format string parameter in certain Perl or C functions that perform formatting, such as C’s printf().
A malicious user may use the %s and %x format tokens, among others, to print data from the stack or possibly other locations in memory. One may also write arbitrary data to arbitrary locations using the %n format token, which commands printf() and similar functions to write back the number of bytes formatted. This is assuming that the corresponding argument exists and is of type int *.
Format string vulnerability attacks fall into three general categories: denial of service, reading and writing.
Cross Site Scripting
The success of this attack requires the victim to execute a malicious URL which may be crafted in such a manner to appear to be legitimate at first look. When visiting such a crafted URL, an attacker can effectively execute something malicious in the victim’s browser. Some malicious JavaScript, for example, will be run in the context of the web site which possesses the XSS bug.
Username enumeration
Username enumeration is a type of attack where the backend validation script tells the attacker if the supplied username is correct or not. Exploiting this vulnerability helps the attacker to experiment with different usernames and determine valid ones with the help of these different error messages.
3. Countermeasures
Username enumerations:
Display consistent error messages to prevent disclosure of valid usernames. Make sure if trivial accounts have been created for testing purposes that their passwords are either not trivial or these accounts are absolutely removed after testing is over – and before the application is put online.
Cross site scripting:
Input validation, secure programming and usage of good language for dynamic web applications.
SQL Injection:
Avoid connecting to the database as a super user or as the database owner. Always use customized database users with the bare minimum required privileges required to perform the assigned task. Perform input validation and do not give error response on client side.
Format String:
Edit the source code so that the input is properly verified.
Remote code execution:
It is an absolute must to sanitize all user input before processing it. As far as possible, avoid using shell commands. However, if they are required, ensure that only filtered data is used to construct the string to be executed and make sure to escape the output
4. Risk Factors
SQL Injection:
Rating: Moderate to Highly Critical
Remote Code Execution:
Rating: Highly Critical
Cross Site Scripting:
Rating: Less Critical
User Name Enumeration
Rating: Less
5. Summary
This is the short article to develop awareness on web attacks and countermeasures, these are common web application attacks.
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LAnnotation::SetRestrictToContainer - Raster imaging C++ Class Library Help
LAnnotation::SetRestrictToContainer
#include "ltwrappr.h"
virtual L_INT LAnnotation::SetRestrictToContainer(bRestrict, uFlags)
L_BOOL bRestrict;
flag that specifies whether to restrict the object to the container
L_UINT uFlags;
flags that determine which objects to process
Restricts one or more annotation objects to a container.
Parameter Description
bRestrict Boolean value that specifies whether to restrict the object to the container. Possible values are:
Value Meaning
TRUE Restrict the object to the container.
FALSE Do not restrict the object to the container.
uFlags Flags that determine which objects to process. Most of the flags apply only to container objects. You can combine values when appropriate by using a bitwise OR ( | ). The following are valid values:
Value Meaning
0 Process only the specified object.
ANNFLAG_SELECTED [0x0001] Process only objects that have the selected property set to TRUE. For getting and setting the selected property, use the LAnnotation::IsSelected and LAnnotation::SetSelected functions.
ANNFLAG_NOTTHIS [0x0004] Process only one level of objects within the specified container, not the container itself. If there are containers within the container, they are modified, but the objects within them are not.
ANNFLAG_RECURSE [0x0008] Process objects within a container, and within any subcontainers, down to any level.
ANNFLAG_NOTCONTAINER [0x0002] (Used with ANNFLAG_RECURSE) Process objects within containers, not the containers themselves.
ANNFLAG_NOINVALIDATE [0x0010] Do not invalidate the affected rectangle in the window. Use this to avoid generating unwanted paint messages.
ANNFLAG_CHECKMENU [0x0020] Process objects only if the ANNAUTOTEXT_MENU_UNGROUP menu item has been selected.
Returns
SUCCESS
The function was successful.
< 1
An error occurred. Refer to Return Codes.
Comments
If an object is restricted to the container, then no part of that object can be moved outside of the container using automation.
All annotation objects support restriction.
Required DLLs and Libraries
LTANN
For a listing of the exact DLLs and Libraries needed, based on the toolkit version, refer to Files To Be Included With Your Application.
Platforms
Win32, x64.
See Also
Functions:
Class Members, LAnnContainer::Convert, LAnnContainer::HitTest, LAnnContainer::RestrictCursor, LAnnotation::GetRestrictToContainer, LAnnotation::Define2, LAnnotation::Define, LAnnotation::SetAutoSnapCursor, LAnnotation::GetAutoSnapCursor
Topics:
Annotation Functions: Object Properties
Implementing Annotations
Automated User Interface for Annotations
Annotation Functions: Creating and Deleting Annotations
Types of Annotations
Annotation Functions: Implementing Custom Annotations
Annotation Functions: Creating Custom Annotations
Displaying and Manipulating Annotation Objects
Example
// This example toggles the "restrict to container" property of the annotation to the container // If an object is restricted to the container, no part of it can be moved outside the container.
L_INT LAnnotation_SetRestrictToContainerExample(LAnnotation * pLObject)
{
L_INT nRet;
L_BOOL bRestrict;
L_TCHAR *pszMsg;
nRet = pLObject->GetRestrictToContainer(&bRestrict);
if (nRet == SUCCESS)
{
bRestrict = !bRestrict;
nRet = pLObject->SetRestrictToContainer(bRestrict, 0);
if(nRet != SUCCESS)
return nRet;
pszMsg = bRestrict ? TEXT("Restricted to container") : TEXT("Not restricted to container");
MessageBox(NULL, pszMsg, TEXT(""), MB_OK);
}
else
return nRet;
return SUCCESS;
}
Help Version 19.0.2017.10.27
Products | Support | Contact Us | Copyright Notices
© 1991-2017 LEAD Technologies, Inc. All Rights Reserved.
LEADTOOLS Raster Imaging C++ Class Library Help
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ASP.NET Razor Code Expressions
Razor syntax is widely used with C# programming language. To write C# code into a view use @ (at) sign to start Razor syntax. We can use it to write single line expression or multiline code block. Let's see how we can use C# code in the view page.
The following example demonstrate code expression.
// Index.cshtml
Produce the following output.
Output:
ASP Razor code expression 1
Implicit Razor Expressions
Implicit Razor expression starts with @ (at) character followed by C# code. The following example demonstrates about implicit expressions.
// Index.cshtml
It produces the following output.
Output:
ASP Razor code expression 2
Explicit Razor Expressions
Explicit Razor expression consists of @ (at) character with balanced parenthesis. In the following example, expression is enclosed with parenthesis to execute safely. It will throw an error if it is not enclosed with parenthesis.
We can use explicit expression to concatenate text with an expression.
// Index.cshtml
It produces the following output.
Output:
ASP Razor code expression 3
Razor Expression Encoding
Razor provides expression encoding to avoid malicious code and security risks. In case, if user enters a malicious script as input, razor engine encode the script and render as HTML output.
Here, we are not using razor syntax in view page.
// Index.cshtml
It produces the following output.
Output:
ASP Razor code expression 4
In the following example, we are encoding JavaScript script.
// Index.cshtml
Now, it produces the following output.
Output:
ASP Razor code expression 5
This time razor engine encodes the script and return as a simple HTML string.
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Flotation equipment is binq made excellent mining crushing machinery, we offer you the best of the equipment and services . . .
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Gravity Flotation and Dissolved Air Flotation
get prices
Gravity Flotation
Gravity flotation is used, sometimes in combination with sedimentation and sometimes alone, to remove oils, greases, and other flotables such as solids that have a low specific weight. Various types of “skimmers” have been developed to harvest floated materials, and the collection device to which the skimmers transport these materials must be properly designed. Figures 8-98(a)–(f) are photographs of different types of gravity flotation and harvesting equipment.
Dissolved Air Flotation
Dissolved air flotation (DAF) is a solids separation process, similar to plain sedimentation. The force that drives DAF is gravity, and the force that retards the process is hydrodynamic drag. Dissolved air flotation involves the use of pressure to dissolve more air into wastewater than can be dissolved under normal atmospheric pressure, then releasing the pressure. The “dissolved” air, now in a supersaturated state, comes out of solution, or “precipitates,” in the form of tiny bubbles. As these tiny bubbles form, they become attached to solid particles within the wastewater, driven by their hydrophobic nature. When sufficient air bubbles attach to a particle to make the conglomerate (particle plus air bubbles) lighter than water (specific gravity less than one), the particle is carried to the water surface.
A familiar example of this phenomenon is a straw in a freshly opened bottle of a carbonated beverage. Before the bottle is opened, its contents are under pressure, having been pressurized with carbon dioxide gas at the time of bottling. When the cap is taken off, the pressure is released, and carbon dioxide precipitates from solution in the form of small bubbles. The bubbles attach to any solid surface, including a straw, if one has been placed in the bottle. Soon, the straw rises up in the bottle.
In a manner similar to the straw, solids having a specific gravity greater than one can be caused to rise to the surface of a volume of wastewater. Solids having a specific gravity less than one can also be caused to rise to the surface at a faster rate by using DAF than without it. Often, chemical coagulation of the solids can significantly enhance the process, and in some cases, dissolved solids can be precipitated, chemically, then separated from the bulk solution by DAF.
“Dissolution” of Air in Water Examination of the molecular structures of both oxygen and nitrogen reveals that neither would be expected to be polar, therefore, neither would be expected to be soluble in water.
Dalton’s law of partial pressures states further that, in a mixture of gases, each gas exerts pressure independently of the others, and the pressure exerted by each individual gas, referred to as its “partial pressure,” is the same as it would be if it were the only gas in the entire volume. The pressure exerted by the mixture, therefore, is the sum of all the partial pressures. Conversely, the partial pressure of any individual gas in a mixture, such as air, is equal to the pressure of the mixture multiplied by the fraction, by volume, of that gas in the mixture.
The consequence of this equation is that, by way of the process of diffusion, molecules of any gas, in contact with a given volume of water, will diffuse into that volume to an extent that is described by Henry’s Law, as long as the quantity of dissolved gas is relatively small. For higher concentrations, Henry’s constant changes somewhat. This principle holds for any substance in the gaseous state, including volatilized organics. The molecules that are forced into the water by this diffusion process exhibit properties that are essentially identical to those that are truly dissolved. In conformance with the second law of thermodynamics, they distribute themselves uniformly throughout the liquid volume (maximum disorganization), and they will react with substances that are dissolved.
An example is the reaction of molecular oxygen with ferrous ions. Unlike dissolved substances, however, they will be replenished from the gas phase with which they are in contact, up to the extent described by Henry’s law, if they are depleted by way of reaction with other substances, or by biological metabolism. The difference between a substance existing in water solution as the result of diffusion and one that is truly dissolved can be illustrated by the following example.
Consider a beaker of water in a closed space—a small, airtight room, for instance. An amount of sodium chloride is dissolved in the water, and the water is saturated with oxygen; that is, it is in equilibrium with the air in the closed space. Now, a container of sodium chloride is opened, and at the same time, a pressurized cylinder of oxygen is released.
The concentration of sodium chloride will not change, but, because the quantity of oxygen in the air within the closed space increases (partial pressure of oxygen increases), the concentration of dissolved oxygen in the water increases. The oxygen molecules are not truly dissolved; that is, they are not held in solution by the forces of solvation, or hydrogen bonding by the water molecules. Rather, they are forced into the volume of water by diffusion, which is to say, by the second law of thermodynamics. The molecules of gas are constantly passing through the water-air interface in both directions. Those that are in the water are constantly breaking through the surface to return to the gas phase, and they are continually being replaced by diffusion from the air into the water. An equilibrium concentration becomes established, described by Henry’s law. All species of gas that happen to exist in the “air” participate in this process: nitrogen, oxygen, water vapor, volatilized organics, or whatever other gases are included in the given volume of air.
The concentration, in terms of mass of any particular gas that will be forced into the water phase until equilibrium becomes established, depends on the temperature and the concentration of dissolved substances such as salts and the “partial pressure” of the gas in the gas phase.
As the temperature of the water increases, the random vibration activity, “Brownian motion,” of the water increases. This results in less room between water molecules for the molecules of gas to “fit into.” The result is that the equilibrium concentration of the gas decreases. This is opposite to the effect of temperature on dissolution of truly soluble substances in water, or other liquids, where increasing temperature results in increasing solubility.
Some gasses are truly soluble in water because their molecules are polar, and these gases exhibit behavior of both solubility and diffusivity. Carbon dioxide and hydrogen sulfide are examples. As the temperature of water increases, solubility increases, but diffusivity decreases. Also, because each of these two gases exists in equilibrium with hydrogen ion when in water solution, the pH of the water medium has a dominant effect on their solubility, or rather, their equilibrium concentration, in water.
In the previous example, where a beaker of water is in a closed space, if a flame burning in the closed space depletes the oxygen in the air, oxygen will come out of the water solution. If all of the oxygen is removed from the air, the concentration of “dissolved oxygen” in the beaker of water will eventually go to zero (or close to it), and the time of this occurrence will coincide with the flame extinguishing because of lack of oxygen in the air.
Dissolved Air Flotation Equipment The dissolved air flotation (DAF) process takes advantage of the principles described earlier. Figure 8-99 presents a diagram of a DAF system, complete with chemical coagulation and sludge handling equipment. As shown in Figure 8-99, raw (or pretreated) wastewater receives a dose of a chemical coagulant (metal salt, for instance), then proceeds to a coagulation-flocculation tank. After coagulation of the target substances, the mixture is conveyed to the flotation tank, where it is released in the presence of recycled effluent that has just been saturated with air under several atmospheres of pressure in the pressurization system shown. An anionic polymer (coagulant aid) is injected into the coagulated wastewater just as it enters the flotation tank.
The recycled effluent is saturated with air under pressure as follows: A suitable centrifugal pump forces a portion of the treated effluent into a pressure-holding tank. A valve at the outlet from the pressure-holding tank regulates the pressure in the tank, the flow rate through the tank, and the retention time in the tank, simultaneously. An air compressor maintains an appropriate flow of air into the pressure-holding tank. Under the pressure in the tank, air from the compressor is diffused into the water to a concentration higher than its saturation value under normal atmospheric pressure. In other words, about 23 ppm of “air” (nitrogen plus oxygen) can be “dissolved” in water under normal atmospheric pressure (14.7 psig). At a pressure of six atmospheres, for instance, (6 × 14.7 = about 90 psig), Henry’s law would predict that about 6 × 23, or about 130 ppm, of air can be diffused into the water.
In practice, dissolution of air into the water in the pressurized holding tank is less than 100% efficient, and a correction factor, f, which varies between 0.5 and 0.8, is used to calculate the actual concentration.
After being held in the pressure-holding tank in the presence of pressurized air, the recycled effluent is released at the bottom of the flotation tank, in close proximity to where the coagulated wastewater is being released. The pressure to which the recycled effluent is subjected has now been reduced to one atmosphere, plus the pressure caused by the depth of water in the flotation tank. Here, the “solubility” of the air is less, by a factor of slightly less than the number of atmospheres of pressure in the pressurization system, but the quantity of water available for the air to diffuse into has increased by a factor equal to the inverse of the recycle ratio.
Practically, however, the wastewater will already be saturated with respect to nitrogen but may have no oxygen because of biological activity. Therefore, the “solubility” of air at the bottom of the flotation tank is about 25 ppm, and the excess air from the pressurized, recycled effluent precipitates from “solution.” As this air precipitates in the form of tiny, almost microscopic, bubbles, the bubbles attach to the coagulated solids. The presence of the anionic polymer (coagulant aid), plus the continued action of the coagulant, causes the building of larger solid conglomerates, entrapping many of the adsorbed air bubbles. The net effect is that the solids are floated to the surface of the flotation tank, where they can be collected by some means, thus removed from the wastewater.
Some DAF systems do not have a pressurized recycle system, but rather, the entire forward flow on its way to the flotation tank is pressurized. This type of DAF is referred to as “direct pressurization” and is not widely used for treating industrial wastewaters because of undesirable shearing of chemical flocs by the pump and valve.
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| 0.943307 |
I hope zope-dev is the right list to post this -
I'm runining zope 2.2.2 on linux with python 1.5.2
Here's the set-up. I create a python class to act as a base class for
a ZClass.
I create a folder in Products called bugtest. Create a python class called
bugtest. create and __init__.py file to register the base class.
Restart Zope, see I have a new product called bugtest. Enter the bugtest
product and create a ZClass the has bugtest as a base class. Everything
works fine. Edit the python file bugtest.py - type in 'import x' (import
something that isn't int the PYTHONPATH).
Restart Zope.
The bugtest product isn't shown as broken. Open the folder and everything
in the folder has the 'product' icon. Try and create a new ZClass anywhere,
and it fails.
This is a sneaky bug because nowhere does zope tell you where or what the
problem is, and it has the side-effect that you can't create a new ZClass.
Could someone confirm this?
_______________________________________________
Zope-Dev maillist - [EMAIL PROTECTED]
http://lists.zope.org/mailman/listinfo/zope-dev
** No cross posts or HTML encoding! **
(Related lists -
http://lists.zope.org/mailman/listinfo/zope-announce
http://lists.zope.org/mailman/listinfo/zope )
Reply via email to
|
__label__pos
| 0.846935 |
Boost C++ Efficiency Top Performance Tips Unveiled
Boost C++ Efficiency: Top Performance Tips Unveiled
In the fast-paced realm of programming, the need for optimized code is paramount. As a C++ enthusiast, you’re likely familiar with the importance of performance. Let’s delve into the secrets that can turbocharge your C++ code and elevate your programming prowess.
Mastering the Basics: Lay a Solid Foundation
First and foremost, ensure your grasp on the fundamentals is rock-solid. Efficient C++ programming starts with a clear understanding of language basics, syntax, and data structures. Familiarize yourself with the intricacies of memory management, pointers, and smart pointers. A strong foundation sets the stage for high-performance code.
Smart Usage of Containers: Choose Wisely
C++ offers a plethora of container classes, each with its strengths and weaknesses. When optimizing for performance, selecting the right container becomes crucial. Vector, list, map, and unordered_map have distinct use cases. Understand the characteristics of each and employ them judiciously based on your specific requirements.
Const-Correctness: Embrace the Immutable
Embracing const-correctness not only enhances code readability but also contributes to performance improvements. Declare variables and functions as const wherever applicable. This not only communicates intent but also enables the compiler to make certain optimizations, leading to more efficient code execution.
Inline Functions: Unleash the Speed
Inlining functions can be a game-changer for C++ performance. By eliminating the overhead of function calls, you can significantly reduce execution time. However, exercise caution – indiscriminate inlining can lead to code bloat. Identify critical functions and strategically inline them to strike the right balance between speed and code size.
Optimized Memory Usage: Mind Your Footprint
Efficient memory usage is a cornerstone of C++ performance. Be mindful of your data structures and their memory requirements. Choose the smallest data type that meets your needs, and avoid unnecessary dynamic memory allocations. This not only speeds up your program but also minimizes the risk of memory-related issues.
Algorithmic Efficiency: Choose the Right Tool for the Job
C++ boasts a rich set of algorithms in its Standard Template Library (STL). When optimizing for performance, understanding the time complexity of these algorithms is essential. Select algorithms that align with the specific requirements of your task. Choosing the right tool for the job can lead to substantial improvements in execution speed.
Parallelism and Concurrency: Harness the Power
Modern processors often come equipped with multiple cores, and C++ provides mechanisms to harness this parallel processing power. Explore threading and concurrency to break down tasks into parallel units. Be cautious with synchronization to avoid pitfalls, but when employed correctly, parallelism can unlock substantial performance gains.
Compiler Optimization Flags: Unveil the Compiler Magic
Compilers are not just translators; they are sophisticated tools that can optimize your code during compilation. Familiarize yourself with compiler flags tailored for performance. Experiment with optimization levels, explore inlining options, and delve into architecture-specific optimizations. Unleashing the power of compiler flags can lead to significant speed enhancements.
Profile and Benchmark: Measure, Analyze, Optimize
To truly understand and improve the performance of your C++ code, profiling and benchmarking are indispensable. Identify
Maximize C Language Efficiency with Proven Coding Tips
Maximizing C Language Efficiency with Proven Coding Tips
Programming in C can be a challenging yet rewarding endeavor. To truly master this language and unlock its full potential, developers must embrace proven coding tips that enhance efficiency and streamline their projects. In this article, we’ll explore essential strategies and techniques to elevate your C programming skills.
Essential Foundations: Mastering C Language Basics
Before diving into advanced tips, it’s crucial to reinforce the fundamentals. Ensure a solid understanding of C syntax, data types, and basic programming constructs. Building a strong foundation sets the stage for implementing more sophisticated coding techniques.
Efficiency Unleashed: Tips for Optimal Code Performance
One key aspect of C programming lies in optimizing code for peak performance. Utilize efficient algorithms, minimize resource consumption, and leverage the full power of C to create applications that run seamlessly. These tips ensure your code operates at its best, even in resource-intensive scenarios.
Invaluable Insights: Navigating Challenges with Practical Tips
Every programmer encounters challenges, and C is no exception. Gain invaluable insights into problem-solving and debugging techniques specific to C programming. Learn to navigate common pitfalls and emerge with a deeper understanding of your codebase.
Code Like a Pro: Mastering Advanced C Coding Techniques
Elevate your coding prowess by delving into advanced techniques. Explore pointers, memory management, and complex data structures. Mastering these advanced concepts empowers you to write more sophisticated and efficient code, giving you a competitive edge in the world of C programming.
Uncovering Secrets: Pro Tips for C Coding Success
Unlock the secrets of C coding success with expert tips. From best practices to lesser-known tricks, these insights provide a deeper understanding of the language. Discover how to write cleaner, more maintainable code that stands the test of time.
Diving Deep: In-Depth Tips for Coding Excellence
To truly excel in C programming, go beyond the surface. Delve into in-depth tips that cover nuances, optimizations, and lesser-known features of the language. This exploration allows you to push the boundaries of what’s possible and develop a deeper connection with the intricacies of C.
Proven Proficiency: Essential C Language Mastery Tips
Achieve proven proficiency with essential mastery tips. Enhance your coding style, adhere to industry standards, and adopt a mindset of continuous improvement. These tips are the building blocks for becoming a proficient and respected C programmer.
Revolutionizing Your Approach: Game-Changing C Tips
Revolutionize your coding approach by embracing game-changing tips. From adopting new paradigms to exploring innovative libraries, stay open to transformative ideas that challenge the status quo. Revolutionizing your approach keeps your coding style dynamic and adaptable.
Empowering Your Skills: Must-Know C Coding Insights
Empower your C coding skills with must-know insights. Stay updated on the latest language features, tools, and community trends. This constant learning process ensures you remain at the forefront of C programming, ready to tackle new challenges as they arise.
Unlocking New Dimensions: Transformative C Tips
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Boost Your C Skills Essential Tips for Efficient Coding
Boost Your C Skills: Essential Tips for Efficient Coding
Mastering the Basics: Building a Solid Foundation
Before we dive into the advanced tips and tricks, let’s start with the basics. Understanding the core elements of C, such as syntax, data types, and fundamental constructs, lays the groundwork for efficient coding. Even experienced developers benefit from revisiting these essentials, ensuring a strong foundation to build upon.
Elevating Your Code: Proven Tips for Optimal Performance
Efficiency is the name of the game in C programming. To boost your skills, focus on optimizing your code for peak performance. Implementing efficient algorithms, minimizing resource usage, and harnessing the full power of C will make your applications run seamlessly, even in resource-intensive environments.
Insider Insights: Navigating Challenges in C Programming
Every coder faces challenges, and C is no exception. Gain invaluable insights into problem-solving and debugging techniques specific to C programming. Learning to navigate common pitfalls equips you with the skills needed to overcome hurdles and build robust, error-free code.
Code Smarter: Proven Tips for Efficient C Development
Efficient coding is not just about speed but also about writing clean and maintainable code. Discover proven tips for smarter coding that enhances not only the speed of development but also the clarity of your code. Adopting best practices ensures your codebase remains manageable and scalable.
Advanced C Techniques: Coding Mastery Unleashed
Now it’s time to level up your coding prowess. Delve into advanced C techniques, exploring pointers, memory management, and complex data structures. Mastery of these concepts empowers you to write more sophisticated and efficient code, setting you apart as a skilled C programmer.
Unlocking C Secrets: Expert Tips and Tricks Revealed
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Dive Deep into C: Essential Tips for Coding Excellence
To truly excel in C programming, you need to go beyond the surface. Dive deep into essential tips that cover nuances, optimizations, and lesser-known features of the language. This exploration allows you to push the boundaries of what’s possible and develop a deeper connection with the intricacies of C.
Code Optimization: Mastering C Language Efficiency
Optimizing your code is not just about making it run faster; it’s about making it run better. Learn the art of code optimization in C by fine-tuning your algorithms, minimizing memory usage, and enhancing overall efficiency. Mastering these skills ensures your code performs at its best.
Transformative Tips: Enhancing Your C Coding Prowess
Ready to transform your coding approach? Embrace tips that challenge the status quo. From adopting new paradigms to exploring innovative libraries, stay open to transformative ideas. Revolutionizing your approach keeps your coding style dynamic and adaptable in the ever-evolving landscape of C programming.
Must-Know Tips: Navigating Challenges in C Coding
Every coder encounters challenges, but not everyone knows how to navigate them effectively. Arm yourself with
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| 0.998756 |
Introduction to R
Basic Computation
Welcome!
Welcome to your first tutorial for coding in R! In this tutorial set, we'll discuss how to set up calculations, create and use basic data structures, and run several basic descriptor commands.
Keep in mind...
Throughout this tutorial, you will see code chunks like this:
2+2
Often, these code chunks will be completed and ready to go for demonstration. You should run them to see what happens.
You should also feel free to play around with them too and run them with other entries! Don't worry, you won't break the tutorial by changing the contents. :)
Some code chunks will be challenges for you to fill. In these, use the provided hints...the last hint will be the suggested solution. You can also use submit to check that your output is correct!
Arithmetic
First lets practice basic computations with R. Addition, subtraction, multiplication, division, and exponents use symbols that are likely already familiar to you.
look at the examples provided for simple computations and then produce some of your own:
56+1
66-60
45*2
81/9
5^2
sqrt(144)
Adding Parentheses
We can also use parentheses to complete multiple calculations at once.
When implmenting computations into R, keep in mind order of operations (PEMDAS), thus adding () into a certain portion of your math problem in R is essential if calculating multiple operations at once.
(25-5)/4
((6*3)-12)^2
Your Turn!
Output the code 8 plus 6, all divided by 2. The solution is available for reference:
(8 + 6)/2
#Did you use parentheses around 8+6?
(8+6)
Vectors in R
Introducing Vectors
A vector is a collection of items (for example, a list of numbers) that are tied together into one structure. To create a vector, we will use our first R function, c (which is short for "concatenate")
functions in R are usually a letter or name, followed by parentheses that include inputs for that function.
The following vector could represent the heights (in inches) of 13 adults. The entries are placed inside the function like an input, and then when I run this function, it outputs the same list of numbers, but tied together as a vector.
c(65,71,63,68,67,72,64,61,67,71,72,68,64)
Characters
An example of a character vector might be storing responses to a question that produces categorical responses. Notices that character entries should be in quotation marks (whereas numbers should typically be listed without quotation marks).
c("yes","yes","no","yes","no","yes","yes","yes","no","yes")
Sequences
In some cases (like plots), we might wish to create a sequence of equally placed numbers. There is a special function named "seq" that allows us to make a sequence from a starting value to a final value, by intervals of our choice.
Notice that this function now has multiple arguments to fill. We will define the three listed here.
seq(from = 2, to = 20, by = 2)
Leaving out the argument names
Keep in mind that in R, we don't have to fill in all of the argument names. If we list our inputs in this order, R will assume the order is...from, to, by...in that order.
seq(2, 20, 2)
Default entries
Something else to keep in mind--we don't have to fill in every possible argument to a function. Only the necessary ones. For example, if we leave the "by" argument empty, R will assume a default value of 1. Try running this to see!
seq(2, 20)
In case you're curious, you can always check out the documentation for a function by running ? in front of the name. This will give you info about what argument options are available, and what default entries are used if left undefined. It's a bit technical and confusing at first, but as you become more coding experienced, it can be very helpful to reference for new (to you) functions.
?seq
Creating Variables
We can also save vectors to a variable name--this is helpful when we might want to summarize or use this vector in a later command.
heights = c(65,71,63,68,67,72,64,61,67,71,72,68,64)
heights
breaks = seq(0,100,5)
breaks
Operations on a variable
We can complete arithmetic operations on vectors, as well as calculate various summary statistics if working with data.
Take a look at the following example, where we take our height vector and multiply it by 2.54 to convert these values from inches to centimeters.
Try changing 2.54 to a different number to observe what happens!
height = c(65,71,63,68,67,72,64,61,67,71,72,68,64)
height_cm = height*2.54
height_cm
Practice!
Give it a try! Create a sequence from 3 to 24 by 3's. Name this as Vector, and then divide Vector by 3. It should produce a vector from 1 to 8 by 1's after this division.
______ = ___(from = __, to = __, by = __)
Vector/__
Vector = seq(from = 3, to = __, by = __)
Vector/__
Vector = seq(from = 3, to = 24, by = 3)
Vector/3
More Practice!
Now, try creating a vector with the following data representing inches of precipitation for 12 months in Champaign.
Save this data as a vector named Temp_2019
3.85, 1.90, 5.09, 4.89, 6.08, 2.82, 3.38, 2.19, 3.36, 5.00, 1.91, 1.82
FYI: Weather data for the Champaign_Urbana area can be found here: https://stateclimatologist.web.illinois.edu/data/champaign-urbana/
3.85, 1.90, 5.09, 4.89, 6.08, 2.82, 3.38, 2.19, 3.36, 5.00, 1.91, 1.82
Temp_2019 = c(...)
Temp_2019 = c(3.85, 1.90, 5.09, 4.89, 6.08, 2.82, 3.38, 2.19, 3.36, 5.00, 1.91, 1.82)
Temp_2019
Data Frames (and Tibbles) in R
Introducing Data Frames
A data frame in R is a collection of vectors, where each vector represents one variable of data. Typically, each column of a data frame is a variable, and each row represents one observation (set of measurements from one individual at one point in time).
In an upcoming software video, we'll see how to use RStudio to import data into a session (since most of the time, we're working with data in a spreadsheet or some other file), but for now, we'll focus on data we create directly in R, or some named datasets that exist online in the R universe already for learning purposes.
Upload the Prostate Data frame with Package
In the following code, we will upload a data frame named "prostate." This data is saved in a package named "faraway." Packages are ways that R users can create code structures or data frames and share them with others! We'll use packages many times throughout the course.
Note that if using a package on your personal computer, you'll need to install it before librarying it. So if you want to replicate this next bit on your own computer, be sure to run the following: install.packages("faraway")
Once installed, you can activate any package for use in your current session of R by running library(package_name). In this case, the package name is faraway, so we will run that here!
library(faraway)
prostate
Note that library(faraway) calls on the location of this data, and then prostate is one (of many!) data frames in this package that we can access. By running just the name, we get a snapshot of this data frame in our output.
A Little Exploration
We can use different functions on a data frame to learn more about it. Here are a couple basic ones.
"Number of rows (observations)"
nrow(prostate)
"Number of coloumns (variables)"
ncol(prostate)
Create a Data Frame Manually
We can also create a data frame manually by entering named vectors that we want to tie together. We will use the command "data.frame", which concatenates vectors that we list separated by commas.
Class = data.frame(
heights = c(65,71,63,68,67,72,64,61,67,71,72,68,64),
responses = c("yes","yes","no","yes","no","yes","yes","yes","no","yes","no","no","yes")
)
Class
New Lines to Improve Readability
Notice in the code chunk above, we hit "enter" after each comma to list each variable in a new line. With most functions in R, you can insert line breaks to improve readability without changing the operation! We could list all of that in one long line, and it would run exactly the same, but it is now very difficult to read!
As you are learning to code, please please please make line breaks where appropriate! It will make it much easier for you and for those of us who might be helping you. :)
Data Frames with Multiple Variables
Now, can you try creating a data frame with two variables? Let's report the test scores of 5 fictional students, as well as their Names.
Scores: 90, 81, 87, 98, 78
Names: "Jose", "Maddie", "Peter", "Amy", and "Kara"
Let's call this data frame "Results."
Then be sure to call up this data frame at the end.
Don't forget to put a comma at the end of the Scores line!
______ = data.frame(
Scores = ...
Names = ...
)
Results
Results = data.frame(
Scores = c(90, 81, ...),
Names = c("Jose", ...)
)
Results
Results = data.frame(
Scores = c(90, 81, 87, 98, 78),
Names = c("Jose", "Maddie", "Peter", "Amy", "Kara")
)
Results
And Tibbles Too
You should also be aware that "tibbles" are another data structure that you may encounter. Tibbles behave exactly like data frames in basically every way--the only real difference is how they display data when called on.
In this R tutorial, you won't see a difference. In fact, this tutorial purposely displays data frames like a tibble! But if using R on your personal computer, you'll notice that data frames display clunkier. They might display as many as 1,000 rows of data, while tibbles display a truncated version, plus some additional variable info. Tibbles just give you an efficient run down!
The more data you work with in R, the more you'll notice the difference, and probably realize why tibbles are easier to work with than data frames.
We can actually take the same data from earlier and save it as a tibble.
Class = tibble(
heights = c(65,71,63,68,67,72,64,61,67,71,72,68,64),
responses = c("yes","yes","no","yes","no","yes","yes","yes","no","yes","no","no","yes")
)
Class
Summarizing Data
Summarizing Data
When analyzing data, we are often interested in summarizing certain variables in our data.
The summary command is a quick way to produce several helpful summary statistics for all of our variables at once. Summary produces the 5-number summary and the mean for all variables.
library(faraway)
summary(prostate)
We can also produce specific summaries for specific variables using commands like mean, sd, and median. Just make sure you call on specific variables by using the $ operator. This allows you to access a specific element of the data frame.
sd(prostate$lweight)
mean(prostate$lweight)
median(prostate$age)
Exploring the diabetes data frame
Now Lets take a look at a new dataset
library the faraway package again, and then call up the data frame named diabetes to display.
library(_______)
________
library(faraway)
diabetes
Calculate the numbers of observations from the dataset:
nrow(___)
nrow(diabetes)
Summary
Now, run a summary of the diabetes data frame.
summary(____)
summary(diabetes)
More Statistics
And lastly, calculate the standard deviation of the age variable (within diabetes).
sd(diabetes$____)
sd(diabetes$age)
This tutorial was created by Brandon Pazmino (UIUC '21) with editing and maintenance by Kelly Findley. We hope this experience was helpful for you!
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Manduca Sexta Lab Report
967 Words 4 Pages
Metabolism is a complex process in organisms that convert the food into energy for organism to function properly. Metabolic rate is typically defined as the amount of energy the organism is currently using to maintain the activity (Perkin, 2015). Since metabolism is a key in sustaining the organisms’ life, we decided to investigate the experiment about metabolic rate caterpillar, Manduca sexta. There are many techniques used to measuring the metabolic rate, but there are only best method that can be done in laboratory: “quantification of heat production and quantification of gas consumption” (Perkin, 2015).Our investigation is about measuring the oxygen consumption of the caterpillar in the trial duration when it was placed in sealed chamber along with chemical absorbing CO2 produced. Metabolism are mostly varies in …show more content…
Since we need to find that whether there is a statistically correlation between the body mass and metabolic rate of Manduca sexta, Regression test was used to analyze our data. The independent variable is the metabolic rate and the dependent variable is body mass.
3. Result The body mass of caterpillar (Manduca sexta) varied from 0.25g to 1.55g. The measured temperature inside of the chamber varied in different caterpillar and ranged from 22.0°C to 23.1°C. The volume of O2 consumption in 15 min ranged from 0.0mL to 0.60mL. The metabolic rate value (mL/min) was 0 to 0.04. From the Regression test, the P value is 0.079, coefficient value is 0.019 and 95% confidence interval value is from -0.0033 to 0.0405. It can be seen the correlation between body mass of Manduca sexta and its metabolic rate based on the graph (Fig.1) Figure 1: The correlation between body mass of Manduca sexta and its metabolic rate 4.
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are identical twins completely identical
Whereas for identical twins since one egg is splitting into two, the two cells have the same exact DNA make up and chromosomes. While we Do identical twins have identical characteristics? btk5093 September 17, 2015 at 4:47 pm. Monozygotic twins, or identical twins, are formed by a single zygote that splits itself into two blastocysts. Some of the differences can be caused by the environment. In the Hensels case, the egg separation process started but it did not finish, leaving a partially divided egg. Twins have long been the darlings of genetic research. That actually can happen two twins born, 1 on New Years Eve 11:59pm and another New Years Day 12:00am. These fraternal twins are no more alike than any other siblings in Although identical twins share the same genetic makeup, one womb, physical characteristics, and many more, they are not necessarily indistinguishable. The twins were born in South Korea in 1985 and adopted by different Jewish American families. In other words, the photons formed twins even though they were born completely independently of one another. Identical twins share the same genes. After: 182 pounds. The team analysed 3,046 DNA samples from identical twins in the UK, Australia, the Netherlands and Finland, and compared this to a control group of 3,396 non-identical twins. The mixed race twins with DIFFERENT colour skin and eyes: Amelia and Jasmine become UK's first sisters to be genetically identical but don't look the same. The researchers identified a link this is completely normal. Di di twins can be identical or non-identical, but are actually more likely to be non-identical, because of the way this type of twin occurs in your womb. Identical twins start out as a singleton pregnancy, and are also known as monozygotic twins. Your response should be able to distinguish Identical twins have the same genes or DNA. They are nurtured in equal prenatal conditions. If homosexuality is caused by genetics or prenatal conditions and one twin is gay, the co-twin should also be gay. Because they have identical DNA, it ought to be 100%, Dr. Whitehead notes. But the studies reveal something else. Before: 5.9 mmol/liter. Identical twins are genetically the same, but the laws of genetics dont completely determine your physical appearance. Conjoined Twins. Pt 1: 1.What exactly are twins, and how do they arise? But this does not mean the twins will be identical in every way. How does this So even if identical twins are genetically similar, the pressure faced by the fetus in the womb can affect their fingerprints. 75% of conjoined twins are female.. As a slight curve ball, there have also been a handful of cases of semi-identical twins, which are thought to occur due to simultaneous fertilization of the egg It occurs very rarely when two sperm fertilizes a single egg which then splits. These twins share the same genetic material. Not All Twins Are Identical (Even Identical Ones) - Medium Twins are two offspring produced by the same pregnancy. genetics reproduction mitosis twins.
NOTE: This section is reserved for twins who do not look alike at all; brother-sister twins who look alike go to Half-Identical Twins.. Anime. Even the difference in the length of umbilical cord can make changes to the fingerprints. While twins may confuse us humans, canines can sniff out their differences. This story is just completely insane I mean both were adopted as babies and both were named James. Non-identical twins are uniquely separate individuals who just happen to be gestating at the same time and place as each other. Identical twins Genetic materials called chromosomes in both babies are completely identical. Nevertheless, these isogenic individuals are not completely identical, but show phenotypic discordance for many traits from birth weight to a range of complex diseases . They start with identical genes, because each is formed from a single fertilised egg that splits into two embryos. Non-identical twins are created when a woman produces two eggs at the same time and both are fertilised, each by a Mentor Program Coordinator. Non-identical twins are also known as fraternal twins or dizygotic twins Fraternal twins can be different genders because they are two completely different eggs getting fertilized; but even two same gender fraternal twins do not look completely alike. Associate Editor. If we assume that identical twins are exactly identical, then if we make a clone of a twin will all three be exactly identical? On average, identical twins are more similar in personality traits and (especially) IQ than non-identical twins or other siblings. This is one of the main pieces of evidence for their being some genetic influence on IQ and personality. However, identical twins can be quite dissimilar in these characteristics. Identical twins look alike and share the same DNA, but they aren't completely identical. The most common type of twins are non-identical twins, which can be the same or different sexes. Because the egg was fertilized by one sperm, identical twins have an almost identical genetic code, and their gender is always the same. The fact that they were separated at birth and still They moved to Los Angeles, California when they were 16 years old. Korean identical twins met for the first time in Florida on their 36th birthday after being separated at birth. Nov. 22, 2021 Identical twins share the same DNA, but one twin can suffer from type 2 diabetes while the other twin does not develop the disease. Share. Because monozygotic twins were thought to be genetically identical, they were perfect for sorting out which traits Learn vocabulary, terms, and more with flashcards, games, and other study tools. ; In Corsair, fraternal twins Aura and Leti are clearly related yet distinct, with Leti taking much more after their father's side and Aura taking more after their mother's side. 12 MAY 2020. Identical twins or monozygotic twins are developed from a single zygote that splits into two embryos. Even mammals form natural clones: identical twins are a common example in many species. Factors that increase your chances of having twins include: 4. Start studying identical twins. While most identical twins do share almost completely identical DNA, some do not. Non-identical twins form from two separate eggs which are fertilized by two completely separate sperm. This can change the physical appearance of the twins and cause a size discrepancy between them. This is the rarest, making up less than .1% of all pregnancies, according to Columbia University. Identical twins form from the same egg and get the same genetic material from their parents but that doesn't mean they're genetically identical by the time they're born. It's common especially in drawn or animated media, where the creator has complete control over the appearance of the characters to use brother-sister twins as being each other's Distaff Counterpart and Spear Counterpart. When identical twins are conceived, the fertilized egg splits into two, causing two separate embryos to grow. The most common type of twins are non-identical twins, which can be the same or different sexes. Twins are defined as two offspring produced from the same pregnancy, they can be either identical or fraternal. But for the most part, basic biology says identical twins share the same DNA. Identical twins form from the same egg and get the same genetic material from their parents but that doesn't mean they're genetically identical by the time they're born. If Identical Twins Married Identical Twins, How Genetically It completely slipped my mind. The zygote divides into two or more embryos early in development. MZ twins arise from the same single cell and therefore share almost all of their genetic variants (Figure 1). Again, because the embryos develop independently after the zygotes split, identical twins
Through their production company, True Image Productions, Inc., the Merrell Twins produced and released a completely self-funded, original scripted series called "Prom Knight".
Out of 381 pairs of identical twins involved in the new study, 39 had more than 100 differences in their DNA. Identical twins are never completely identical. However, "such genomic differences between identical twins are still very rare, on the order of a few differences in 6 billion base pairs," with base pairs being the building blocks of After: 4.9 mmol/liter. The Identical Twin ID Tag trope as used in popular culture. Identical twins will share the same genetic information so the same genetic markers can be identified. 2. Thus identical twins, though they start with the same genes, likely develop different personalities in the same environment partially based on how they interact with their This causes the babies to be born conjoined. These separate zygotes go on to form embryos. In a 2011 study published in the journal PLOS One, German shepherd police dogs were presented with the scents of identical twins.Then, they were then able to find the exact matches among jars that contained scents from other people that were meant to distract them. Identical twins have identical DNA fingerprints because their DNA codes are basically cloned copies of each other. Identical, or monozygotic, twins come from the same fertilized egg. There are always small physical differences that you can use to tell one twin from another. Trusted Source. That means a different genetic code and the possibility that fraternal twins will not look that much alike.
Answer (1 of 2): Any two siblings of the same family may resemble each other but they cannot be completely identical.Maternal and paternal genes undergo shuffling during the process of Identical, or monozygotic, twins occur when a single egg, fertilised by a single sperm, divides and makes two babies. Humans have always been fascinated by identical twins. Your twins being in separate sacs means that your pregnancy is Fraternal twins occur when two egg cells are each fertilized by a different sperm cell in the same menstrual cycle. So, at some point during cell division (before 14 days post-conception), identical twin embryos share It is interesting to note that although Ross gained greater muscle mass with a traditional diet, Hugo reported no symptoms of weakness and had a feeling of similar strength while eating a vegan diet. These can be used to tell twins apart. 7 thoughts on Are identical twins really identical? It has been reported that most parents of identical twins actually believe their children are fraternal twins because they are not identical in every way. Do identical twins always look alike? If we break this word down, mono means one, and zygote means So ya, identical twins could fool everybody with their looks, but they aint fooling the fingerprint test! Thus, the twins share the same DNA from their mother but each gets a slightly different version of their father's DNA. Answer (1 of 10): Identical twins have exactly the same genetic sequence (well, there could be a small number of somatic mutations that distinguish them, but in general we can assume that Dizygotic twins, or fraternal twins, are formed by two different zygotes fertilized by two sperm. Why do identical twins come out at the same time? Why cant the twins be born years apart and still look completely identical?!? Identical twins predominantly have the same sex. But, there have been extremely rare instances where the monozygotic twins are of different sexes. This scenario is so rare that there have been only a few reported cases so far and it is unlikely that you will come across such twins in your lifetime. Parents of identical twins Even though identical twins are from the same sperm and egg and therefore have exactly the same set of chromosomes and therefore genes, Another big factor why identical twins aren't necessarily completely identical is the environment which each of them were raised in. When two different eggs are fertilized by two different sperm, the twins resulting from this are fraternal. Just how their individuality emerges has remained a bit of a mystery. Since identical twins develop from one zygote, When a mother gives birth to twins, the offspring are not always identical or even the same gender. Mostly, newborn twins are identical but once they get out into the world and start forming an identity of their own, the physical and mental changes that they undergo are clearly visible. To narrow it down, there are two major factors that are responsible for identical twins not looking identical; Environmental differences and DNA differences . ENVIRONMENTAL DIFFERENCES There are various environmental influences that can affect the genes of identical twins. 20 surprising facts about identical twins. Conclusion. Twins can be either monozygotic ('identical'), meaning that they develop from one zygote, which splits and forms two embryos, or dizygotic ('non-identical' or 'fraternal'), meaning that each twin develops from a separate egg and each egg is fertilized by its own sperm cell. They should be The differences between identical and fraternal twins are due to how they are conceived. Being identical twins is awesome! This is only possible with identical twins. When their fascinating case came to light, scientists saw how very valuable they could be to the study of reunited twins. (Zappys Technology Solutions) The original fertilized egg
It is possible to have triplets where two of the babies are identical twins (and may share one placenta, and even one sac) and the third baby is non-identical (with completely separate placenta and sac). Epigenetic patterns can separate twins over time. Although they share similar DNA, it is not completely the same. The DNA replication If we have a child that is from You have an identical best friend and worst enemy. Molly Sinert and Emily Bushnell embraced each other for the first time at Hyatt Centric Las Olas Fort Lauderdale on March 29, according to Good Morning America. Improve this question. Research published on Non-identical twins form from two completely separate eggs which are fertilised by two completely separate sperm. Di Di identical twins: the early years. Shutterstock. When identical twins are conceived, the fertilized egg splits into two, causing two Of 381 pairs of identical twins studied and two sets of identical triplets, scientists found that 15% of them had a substantial number of mutations specific to one twin but not the other, the researchers write. But a new study says This is because both babies come from the same egg and sperm. Fraternal Twins. A new type of twinning was identified in 2007. More types of twins exist than previously thought. Monochorionic-Monoamniotic (Mono-Mono): Both twins share the same amniotic sac and the same placenta. The Jim twins are so interesting. Even though identical twins come from the same fertilized egg, in the end each twin has slightly different DNA. BONUS FACT: Identical twins are always the same gender, and only fraternal twins can be different genders. This is why identical twins can have differing fingerprints. Twin pregnancies have unique risks and outlooks. Answer (1 of 10): Identical twins have exactly the same genetic sequence (well, there could be a small number of somatic mutations that distinguish them, but in general we can assume that the sequences are identical). Non-identical twins are no more alike than any other brothers or sisters. Fuck, Im too high for this. Conjoined twins are formed when a woman produces one egg which doesnt fully separate after fertilization. On February 9, 1979, the Jim Twins were finally reunited. Tyler Howard Winklevoss (born August 21, 1981) is an American investor, founder of Winklevoss Capital Management and Gemini cryptocurrency exchange, and Olympic rower.Winklevoss co-founded HarvardConnection (later renamed ConnectU) along with his brother Cameron Winklevoss and a Harvard classmate of theirs, Divya Narendra.In 2004, the Winklevoss Identical twins are formed from the splitting of a zygote formed from one egg and one sperm. These are known as conjoined twins.There are two possibilities for the formation of conjoined twins- either the single fertilized egg does not split completely during the formation of identical twins, or two fertilized eggs fuse together earlier during the development. IDENTICAL Twins? Identical twins will have the same blood type and even though they are extremely similar, they may not be exactly identical due to environmental factors. Identical Twins Not So Identical Environmental influences separate twins over time 5 Jul 2005 By Cathy Tran Not so similar. Non-identical twins form from two completely separate eggs which are fertilised by two completely separate sperm. But from that moment onwards, their DNA begins diverging. Veronica and Vanessa are identical twins born on August 6, 1996 in Kansas City, Missouri. As it turns out, The change continues as the twins grow up into adults. Having identical twins is genetic. She is Mom to 17-year-old identical twins girls and a 12-year-old son. Despite the name, identical twins are rarely completely identical. So there technically have 2 birthdays and born different years. Identical twins are formed after zygote, a fertilised egg splits into two embryos and shares the same genes. The idea of having a duplicate lies at the origin of many myths and beliefs.
If a starfish is chopped in half, both pieces can regenerate, forming two complete, genetically identical individuals. Look for things like birthmarks, freckles, moles, and other distinguishing features. Female identical twins can have differences in which X chromosomes one from each parent are active. Fraternal twins or dizygotic twins, on the other hand, are developed from separate eggs that are fertilised by different sperm cells. Identical light particles (photons) are important for many technologies based on quantum physics. Beyond identical and fraternal, there's a rare third type. Kirio and Kirika from Kamichama Karin. View identical twins Megan Iversen from BZ 350 at Colorado State University, Fort Collins. These twins are always identical and can be conjoined. Also read: Determination of Sex Sometimes the identical twins are physically connected. When fraternal twins are conceived, two eggs are fertilized at the same time. A set of twins who look and act for all the world like they're identical, except for the miiiinor detail that one's male and the other's female.. How many identical triplets are there in the US? Weight: Before: 185 pounds.
The differences between identical and fraternal twins are due to how they are conceived. Recent studies have shown that identical twins have very "similar" not "identical" DNA, but for the most part, according to basic biology it is identical. These fraternal twins are no more alike than any other siblings in a family with the same biological mother and father. Identical twins, meanwhile, result when one egg is fertilized Namely, identical twins are formed when a fertilized egg separates into two embryos during the first few weeks of pregnancy. Despite having the same genetic makeup, identical twins have their own distinctive personalities. In the animal world, the eggs of female aphids grow into identical genetic copies of their motherwithout being fertilized by a male. Image by Lorilee Alanna via Pixabay. Another If two identical twins grow up to be completely different from one another, we can assume that their environments were more influential in their behaviour than genetics. Known as fraternal twins, they represent a longstanding The reason the zygote splits is thought to be inherited, which may be why some families have a few sets of identical twins. Muscle Mass: Before: 153 pounds. These twins do not share the same genetic material. 3. Their skin tone, weight, height or personality, to name a few characteristics, may be different. Identical twins come from the same egg, making their genetic makeup the same, while fraternal twins share half of their genes since they form from different eggs. One twin may have a particular medical condition, while the other does not. , German researchers examining 40 genetically identical twin mice found they could develop very distinct personalities. Because identical twins come from a single After: 152 pounds. The reason twins to not end up as identical clones of each other lies in Very rarely, the zygote splits around day 13-15, making it impossible for the twins to separate fully. These twins will be the same sex and share the same The stereotype of identical twins is that they are exactly the same: they look alike, they dress in matching outfits, they share the same likes and dislikes. The
When mom has identical twins, it means one fertilized egg splits in More research is being done to investigate this.
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CIVIL ENGINEERING 365 ALL ABOUT CIVIL ENGINEERING
Optogenetic stimulation was used on mESC-derived MEBs to implement training regimens during two important stages of neural development: neurogenesis (while still in suspension) and synaptogenesis (seeded on functionalized glass or MEAs) (Fig. 1a). Training regimens consisted of periodic stimulation with 5 ms pulses at 20 Hz in 1 s intervals for an hour (Supplementary Fig. 1a). This regimen has been shown to enhance axonal growth30, and thus would suggest that it could lead to a shift in structural potentiation in a neural network. The regimen was repeated every 24 h as differentiation occurred within the EBs, with an expectation that consistent repetition would enhance the potentiation and cause long-term changes in the firing patterns of the network. Following established differentiation protocols of mESC towards mature motor neurons31,32,33, the described training regimen was started at D2 of differentiation, at which point stem cells have been induced towards neuronal lineages, and specialization and maturation of motor neurons has been shown to take place in the subsequent 7 days (Fig. 1b). Since one of the transcription factors that drove differentiation, retinoic acid, is light sensitive, media was changed every single day immediately after stimulation to ensure that stimulation effects on MEBs were not artifacts (i.e. false positives) caused by photodegradation of factors (Supplementary Fig. 1b)34. Furthermore, since the differentiation was monitored with the expression of the motor neuronal marker Hb9 through a GFP reporter, we used the plateau of GFP expression between D8 and D9, as an indicator that D9 was an appropriate time point for seeding the MEBs on glass (Supplementary Fig. 1c). Thus, after these 7 days (D2-D9) of differentiation, stimulated (S) and non-stimulated (NS) cultures were seeded on MEA chips (Fig. 1c). Careful seeding practices were applied to ensure that ~ 20 MEBs were seeded within the sensing area of the MEAs for a ~ 50% coverage by the MEBs (Supplementary Fig. 2). Seeding in this manner ensured empty space between clusters for the extension of processes, even though some nearby clusters would start fusing into larger clusters. The resulting two groups of samples seeded on MEAs were further subdivided into two more experimental groups, referring to whether or not a training regimen was continued during network formation on chip for the consequent 15 days (D10-D25). For ease of discussion, S or NS prior to a colon (e.g. S:X or NS:X) will refer to the presence or lack thereof of stimulation, during neurogenesis, while S or NS written after a colon (e.g. X:S or X:NS), indicates the presence or absence of stimulation during synaptogenesis (Fig. 1a).
Figure 1
Approach to training mESC-derived motor neuronal embryoid body networks during neurogenesis and synaptogenesis. a Representative diagram of experimental setup combining differentiating ChR2 mESC’s and MEAs. b Representative diagram of ChR2 mESC differentiation toward motor neuronal embryoid bodies monitored by the expression of GFP guided by the motor neuronal specific Hb9 promoter (scale bar: 200 µm). c Representative image of fabricated MEA chip. d Representative spontaneous spike trains from MEA recordings of cultured embryoid body networks.
Figure 2
figure2
Intact MEBs indicate formation of internal networks and form active networks between them a (i) Scanning electron micrograph of two embryoid bodies. (scale bar: 200 µm) and (ii) confocal image showing dense clusters of synaptophysin between cultured embryoid bodies (scale bar: 50 µm). b (i) MEB cryosections showing usual internal structure. (Scale bar: 50 µm) with (ii) zoom in of internal structure of a sectioned embryoid body (scale bar: 15 µm). c Representative confocal image of MEB cryosection stained for GAD65/67 and vGlut. Triangles show GAD65/67 clusters d. Representative confocal image of entire field of view for neural culture grown on the MEA sensing area (scale bar: 200 µm) with scanning electron micrograph zoom in of embryoid bodies extending processes atop of sensing electrodes. e. Bar graph for average firing rate of 15 active electrodes for cultured embryoid body networks exposed to known neuronal signaling molecules at sequential addition of tonic baths of 10, 100 and 250 µM. Glut Glutamate, ACh Acetylcholine, cAMP cyclic AMP, cGMP cyclic GMP, NE norepinephrine, GABA gamma-aminobutyric acid) across 5 min of recording/exposure (n = 15; error bar represents SEM, * p < 0.05; ANOVA with Tukey post-hoc test).
The electrical activity of the resulting neuronal cultures was measured with the MEA system and the raw data was filtered to remove low frequencies (< 200 Hz), to remove undesired voltage artifacts (e.g. stimulation artifacts), and extract action potentials recorded as spiking events (Fig. 1d). A two-step procedure was used to remove false positives from the analyzed data: (1) the detection threshold was set at a value at which no positives would be detected from the ground electrode, then (2) the recorded spikes at each electrode were inspected to ensure that the detected spikes had the appropriate voltage phases relating to action potentials: depolarization, repolarization and refractory period.
MEB cultures form active neural networks with excitatory and inhibitory populations
In this work, neural networks were cultured from intact MEBs, in contrast to growing them as a monolayer after dissociation. The long-term goal of our study is the modulation of electrical activity of the MEBs towards downstream implantation in in-vivo or in-vitro experimental systems and modulating the functionality of such systems through the resulting interaction. When cultured in their intact form, MEBs tend to keep their spheroid shape, while extending processes which contain neurites that form networks as they undergo synaptogenesis (Fig. 2a). Furthermore, dense web-like neurite structures form within the spheroid itself (Fig. 2b) and both excitatory (vGlut) and inhibitory (GAD65/67) receptors stain positively (Fig. 2c).
Network formation was validated by exposing MEB cultures grown on MEAs (Fig. 2d) to varying concentrations of commonly used exciting and inhibiting signaling molecules for 5 min: L-glutamate, acetylcholine, cyclic AMP, cyclic GMP, norepinephrine and GABA. (Fig. 2e). As expected, L-glutamate evoked a statistically significant (repeated measures ANOVA with a Greenhouse–Geisser correction, n = 15; F(1.28,17.89) = 18.78, p = 1.88E-4) response in the network. A post hoc Tukey test showed a statistically significant positive difference at p < 0.05 between 0 µM to 10 µM, while higher concentrations, 100 µM and 250 µM, showed a decrease in firing rate with the latter showing a statistically significant negative difference to the spontaneous firing rate, most likely related to excitotoxicity35. Other excitatory signaling molecules, acetylcholine and cyclic AMP, evoked a continuously excitatory response (repeated measures ANOVA; ACh (with Greenhouse–Geisser correction), n = 15: F(2.13,29.78) = 16.14, p = 1.31E-5 and cAMP: F(3,42) = 125.49,p = 4.20E-15) continued a gradual increase in firing rate with increasing concentrations. Cyclic GMP, another cyclic nucleotide similar in function as cAMP, failed to evoke any statistically significant effect on firing rate (repeated measures ANOVA with a Greenhouse–Geisser correction, n = 15; F(2.08,29.18) = 2.86, p = 0.07). On the other hand, the inhibitory neurotransmitters evoked statistically significant effects on the MEB-derived networks, with norepinephrine (repeated measures ANOVA, n = 15; F(3,42) = 81.43, p = 1.53E-17), showing a statistically significant decrease at p < 0.05 in a post hoc Tukey test from 0 µM to 10 µM, and 100 µM to 250 µM, while GABA (repeated measures ANOVA, n = 15; F(3,42) = 191.55, p = 1.60E-24) showed a statistically significant decrease in firing rate at p < 0.05 in post hoc Tukey test at each concentration. The responses corroborated the development of endogenously active neural networks expressing different kinds of receptors. The observations that MEBs extend processes within the body itself while responding to both excitatory and inhibitory signaling molecules would lead to the hypothesis that these MEBs could be forming intrabody circuits which could be “trained” during differentiation and have these changes last after network formation.
Stimulation during neurogenesis results in morphological changes in MEB cultures
The effects of stimulation during differentiation were initially observed in neurite extension and presynaptic protein clustering. While it has been reported that neurite outgrowth could be enhanced if neural populations simultaneously underwent optogenetic stimulation30, it was not clear if effects of the stimulation on MEBs done in suspension would still result in an increase of neurite extension when later seeded on chips, as this would indicate some stable long-term changes in the neuronal system. To quantify this, S:NS and NS:NS MEBs were seeded at low confluence on gridded coverslips and imaged 6 times every two hours on D10 (1 DIV) to quantify the number of extending neurites (Fig. 3a). Observations showed a consistently statistically significant positive difference (ANOVA, n = 20; 14hrs: F(1,38) = 215.44, p = 0.0; 16hrs: F(1,38) = 148.40, p = 1.08E-2; 18hrs: F(1,38) = 257.32, p = 0.0; 20hrs: F(1,38) = 199.14,p = 1.11E-2; 22hrs: F(1,38) = 221.35, p = 0.0; 24hrs: F(1,38) = 76.11,p = 1.31E-2) of number of neurites extended for S:NS samples, compared to NS:NS, for each of the six hours the two groups were measured and compared. This indicates an increased rate of neurite extension as a result of the stimulation during neurogenesis (Fig. 3b). Next, we wanted to observe the effect of stimulation during differentiation on the propensity of the network to form synapses. To quantify this, the clustering of presynaptic synaptophysin stained with anti-SY38, was counted along individual neurites as well as per unit area between the groups NS:NS and S:S (Fig. 3c). By D11 (2 DIV) S:S samples showed a statistically significant ~ twofold increase (ANOVA, n = 10; F(1,18) = 24.58, p = 1.02E-4) of synaptophysin clusters per neurite than NS:NS samples (Fig. 3d). This increase of pre-synaptic clusters per neurite combined with the increase in neurite extension resulted in S:S samples presenting a statistically significant higher synaptophysin clusters per unit area than NS:NS counterparts at D11 (ANOVA, n = 10; F(1,18) = 40.18, p = 5.68), D13 (ANOVA, n = 10; F(1,18) = 131.58, p = 1.04E-9) and D15 (ANOVA, n = 10; F(1,18) = 74.87, p = 7.88E-8) (Fig. 3e). When monitoring the difference of pre-synaptic clusters per unit area at D13 and D15, the statistically significant difference indicated that optogenetic stimulation during neurogenesis evoked physiological responses on two important aspects of neural network development: neurite extension and presynaptic clustering (Fig. 3e).
Figure 3
figure3
Stimulation during neurogenesis affects key morphological parameters of network formation. a. Representative phase contrast images of neurite extension along the periphery of embryoid bodies between non-stimulated (NS) and stimulated during neurogenesis (S) samples (scale bar: 50 µm). b. Bar graphs representing the average number of neurites protruding from the periphery of embryoid body normalized by the perimeter of the embryoid body at a given time after seeding. Each point signifies the number of extending neurites normalized by the perimeter of an individual embryoid body (n = 20; error bar represents SEM, *p < 0.05, ANOVA with Tukey post-hoc test). c. Representative fluorescence images of synaptic puncta stained against SY38 at D11 along a neurite. Arrow denote presynaptic puncta. (scale bar: 5 µm). d. Bar graphs representing the average number of presynaptic puncta along the length of neurites for D11. Each point corresponds to the average number of synaptic puncta along a neurite normalized the length of the neurite per field of view (n = 10; error bar represents SEM, *p < 0.05, ANOVA with Tukey post-hoc test). e. Bar graphs representing the average number of presynaptic puncta per unit area for D11-D15. Each point corresponds to the average number of synaptic puncta per unit area in an individual field of view (n = 10; error bar represents SEM, *p < 0.05, ANOVA with Tukey post-hoc test).
MEB network synchronicity is amplified by stimulation during neurogenesis and synaptogenesis
Network synchrony is a common parameter used to characterize a developing neural network, as it gives information on the network’s plasticity and connectivity. Various studies have successfully shown that the presence of chronic stimulation results in improved network synchrony36,37,38. In our study, we wanted to observe the long-term effects of stimulation regimens on the network synchrony and determine if these effects were amplified or shifted when the training regimen during neurogenesis was extended during synaptogenesis. From the raster plots of the spontaneous activity recorded at D21, the increased level of synchronous activity was notable between NS:S and S:S samples versus S:NS and NS:NS (Fig. 4a). This can be appreciated by the peaks above the raster plots, which correspond to a summation of the activity across all electrodes, where synchronous networks would result in discrete peaks whereas in samples that lacked coordinated firing, the resulting line plot seemed to lack any peaks.
Figure 4
figure4
MEB network synchronicity is amplified by stimulation during neurogenesis and synaptogenesis. a. Representative raster plots of MEB cultures at D25 showing network synchrony by line plots of the sum of active electrodes for each time point. b. The average correlation value (χ) was calculated for active electrodes across time for an average value for each electrode, then mapped to their respective spatial position on the MEA array. c. Bar graphs representing the mean correlation value across the culture for the MEA cultures at the different days of recording. The correlation value for the culture was calculated using active electrodes during spontaneous time of each culture for each day of recording. Each point corresponds to the correlation value across electrodes for each MEA culture. (n = 3; error bar represents SEM, *p < 0.05, ANOVA with Tukey post-hoc test).
Similarity between electrode recordings was quantified with cross-correlation in order to quantify synchronous behavior. Values for the similarity across the network were obtained by calculating cross-correlation for all electrode combinations (Supplementary Fig. 3). For this analysis, only spontaneous recordings of active electrodes (electrodes detecting at least 10 spikes/min) were used to quantify the long-term effects of the training regimen on steady state synchrony. When average correlation values per electrodes were mapped to their position on the chip, NS:S and S:S samples showed high synchrony level ((stackrel{-}{chi }) > 0.5) across the entire network for spontaneous recordings at D21 (Fig. 4b). This showed that synchronous behavior extended across the entire network and was markedly higher for networks that were stimulated during synaptogenesis.
Interestingly, when the network wide mean synchronicity was calculated for each recording day, a trend of higher synchrony was observed for samples that had been exposed to some form of training regimen (NS:S, S:NS or S:S) but no statistical significance was observed at D11 (ANOVA, n = 3; F(3,8) = 3.42, p = 0.073) and D13 (ANOVA, n = 3; F(3,8) = 1.77, p = 0.23). At D15, a statistically significant difference (ANOVA, n = 3; F(3,8) = 7.47, p = 0.010) was observed, with a post hoc Tukey test performed at p < 0.05 showing statistical significance between NS:S and S:NS (stackrel{-}{chi }) values. Subsequently, while no statistical significance was observed for D17 (ANOVA, n = 3; F(3,8) = 3.88, p = 0.055), D19 (ANOVA, n = 3; F(3,8) = 3.58, p = 0.066) and D21 (ANOVA, n = 3; F(3,8) = 3.61, p = 0.065), a gradual trend was observed for the synchronicity of networks undergoing training during synaptogenesis (NS:S and S:S) being larger than their counterparts (NS:NS and S:NS). At D23, there was a statistically significant difference among the experimental groups (ANOVA, n = 3; F(3,8) = 8.73, p = 6.6E-3). Post hoc comparisons using Tukey test at p < 0.05 indicated that the (stackrel{-}{chi }) value for NS:S and S:S were higher than both NS:NS and S:NS groups. This statistically significance was sustained for D25 (ANOVA, n = 3; F(3,8) = 6.46, p = 0.016), with the post hoc Tukey test showing significant difference between (stackrel{-}{chi }) for S:S and (stackrel{-}{chi }) for NS:NS as well as S:NS. (Fig. 4c).
Spectral density elucidates changes in steady state firing
Conventionally, electrophysiological behavior is characterized by firing rate during set epochs and burst parameters (Supplementary Fig. 4). However, when analyzing these parameters during spontaneous firing, there was no discernable trend in the change of long-term firing rate or burst parameters between experimental groups. However, when observing the spike data during steady state of a more mature neural network (D25), there were deviations on how the spike firing clustered into bursts, despite the fact that no clear change in the number of spikes was observed (Fig. 5a). We accredited this seeming conflict between the quantitative and qualitative data to the selection method of the burst detection parameters (See Quantification and statistical analysis). In order to avoid arbitrariness in the selection of these parameters, we decided to characterize the data in the frequency domain. For this reason, we focused on characterizing spontaneous firing recorded on MEAs by comparing changes in the power spectrums of recorded signals calculated through Fourier transforms (Fig. 5b). To obtain spectral profiles, binned spike counts were divided into 10-s-long contiguous windows and transformed to the frequency domain, thus representing the power spectrum as a function of time (Fig. 5b). When initially calculating the power spectral density (PSD) and observing between the DC frequency and the Nyquist frequency, we noticed that most of the components appeared below 7 Hz for all samples. For this reason, we compared samples between 0.1 Hz (to remove DC component) and 5 Hz. Focusing between 0.1–5 Hz, all samples except S:S, showed frequency profiles of their respective firing patterns with components across the entire bandwidth of interest. This spontaneous heterogeneous firing patterns can be expected from these cultures formed from MEBs, as they are a super-network composed of individual networks from within each MEB. On the other hand, S:S samples show a clear change in their frequency profile, where most of the spectral power fell within 0.1-1 Hz.
Figure 5
figure5
Stimulating training regimens modulates firing patterns in the frequency domain. a. Fifteen second representation of spontaneous voltage recording from NS:NS, NS:S, S:NS and S:S samples for D25. b. Smoothened (3 point moving average) and normalized (AUC) power spectra was calculated for contiguous 10 s windows across the 4 min of spontaneous recording NS:NS, NS:S, S:NS and S:S. Resulting matrices were averaged across samples. c. Bar graph for the sum of power spectral density magnitude from (b) across the spontaneous recording time between 0.1 Hz and 1 Hz (n = 3; error bar represents SEM, *p < 0.05, ANOVA with Tukey post-hoc test).
Moreover, if the signal power is summed between the frequency range of 0.1-1 Hz, the training regimen pattern had a statistically significant effect at p < 0.05 on the power magnitude within this frequency interval (ANOVA, n = 3; F(3,8) = 20.15, p = 4.37E-4). Post hoc comparisons using Tukey test at p < 0.05 showed a statistically significant difference between power magnitude withing 0.1-1 Hz of samples non stimulated during synaptogenesis (NS:NS, S:NS) and samples stimulated throughout development (S:S) (Fig. 5c). Moreover, the post hoc Tukey test indicated a statistically significant difference between power spectra values between NS:S and S:S, implying that combined stimulation of both neurogenesis and synaptogenesis had an amplified effect on modulating the power spectra of the networks than just stimulation during synaptogenesis. This statistical significance was not observed in the mature networks (D25: ANOVA, n = 3; F(3,8) = 0.063, p = 0.98) if the power was summed for the whole frequency interval of interest (0.1-5 Hz) (Supplementary Fig. 5).
Neurogenetic stimulation changes the opto-response of MEB networks
Another aspect of consideration on the effect of training MEBs during neurogenesis was whether the early stage perturbation had some effects on how the later-stage network would respond to the same perturbation. To study this, we recorded responses to optogenetic stimulation from sets of samples that had not undergone the training regimen during neurogenesis (Fig. 6a) and compared them to those set that had undergone such regimen (Fig. 6b). Initial observation showed a difference between how the networks responded when stimulated early in the network development (D11) versus more mature networks (D25). For example, when early networks, which had a low spontaneous firing rate (D11) were stimulated, there would be a very notable evoked response during stimulation followed by a quiescent state, where the network would barely fire before returning to the baseline spontaneous firing rate. In contrast, more mature networks (D25), would still show an evoked response during stimulation but would automatically return to baseline firing rate right after stimulation ceased. What was interesting was that the quiescent time after stimulation for early S:S networks were notably shorter than those from the NS:S samples (Fig. 6a-b). Moreover, at D25, while NS:S samples would return to the same baseline firing rate right after stimulation stopped, S:S samples showed a transient change in firing rate for several seconds after the stimulation stopped (Fig. 6a-b).
Figure 6
figure6
Stimulation during neurogenesis alters response to stimulation during network formation. Summed spike counts per each 100 ms for all active electrodes across the 20 min of recording were graphed for D11 and D25 for one representative sample from NS:S (a) and S:S (b). c. Zoom-in of a for 1 min, centered around the 20 s of stimulation at D25 for sample NS:S, the arrows represent the firing rate interval prior to stimulation (FRpre), the firing rate during stimulation (FRstim) and the firing rate after stimulation (FRpost). d. Bar graphs showing the mean firing rate increase between Frstim/Frpre for D11-D25. (n = 9; error bar represents SEM, *p < 0.05, ANOVA with Tukey post-hoc test)). e. Bar graphs showing the firing rate increase between Frpost/Frpre for D11-D25. (n = 9: error bar represents SEM, *p < 0.05, ANOVA with Tukey post-hoc test)). f. Raster plot of average correlation value for each electrode during 10 s bins across the entire recording time. g. Ratio of average correlation value prior to stimulation during recording and correlation value post stimulation (χpost/ χpre). (n = 3; error bar represents SEM, *p < 0.05, ANOVA with Tukey post-hoc test).
To quantify this behavior, the evoked firing rate during stimulation (FRstim) and the post-response firing rate (FRpost) were compared to the firing rate prior to stimulation (FRpre) for the three instances of stimulation within recording for each of the three MEA networks for both experimental groups (Fig. 6c). While the fold-change increase of firing rate FRpre to FRstim decreased with time for both NS:S (repeated measures ANOVA with Greenhouse–Geisser correction, n = 3; F(1.48, 11.83) = 14.79, p = 1.12E-3) and S:S (repeated measures ANOVA with Greenhouse–Geisser correction, n = 3; F(1.88, 15.02) = 11.02, p = 1.31E-3 (because more mature networks would have a higher baseline firing rate), when comparing the amount of evoked action potentials during stimulation (FRstim/FRpre), S:S samples seemed to respond more strongly to stimulation than NS:S samples (Fig. 6d). One-way ANOVA determined a statistically significant difference between NS:S and S:S FRstim/FRpre values for D13 (n = 9; F(1, 16) = 5.55, p = 0.031), D15 (n = 9; F(1,16) = 5.90, p = 0.027), D17 (n = 9; F(1,16) = 11.30, p = 4E-3), D19 (n = 9; F(1,16) = 8.78, p = 9.2E-3), D23 (n = 9; F(1,16) = 10.81, p = 4.6E-3) and D25 (n = 9; F(1,16) = 9.94, p = 6.2E-3), while only showing a trend (not statistically significant) of higher S:S FRstim/FRpre values for D11 (n = 9; F(1,16) = 4.48, p = 0.05) and D21 (n = 9; F(1,16) = 1.1, p = 0.31).
Additionally, the quiescent state response post-stimulation observed in early days (D11, D13 and D15), reflected itself in FRpost being less than FRpre, resulting in FRpost/FRpre < 1 for NS:S and S:S samples. We observed that this transient decrease in firing rate was statistically significantly shorter for the S:S samples than the NS:S for D11 (ANOVA, n = 9; F(1,16) = 19.95, p = 3.9E-4) and D13 (ANOVA, n = 9; F(1,16) = 9.49, p = 7.2E-3) (Fig. 6e). Repeated measured ANOVA indicated that FRpost/FRpre ratios increased for both NS:S (Greenhouse–Geisser corrected, n = 9; F(3.06, 24.48) = 36.92, p = 2.69E-9) and S:S (n = 9; F(7,56) = 5.66, p = 5.63E-5). Furthermore, at later days of network development, it was notable that FRpost/FRpre was ~ 1 for NS:S, meaning that the steady state firing rate was indistinguishable from that immediately following the termination of stimulation. On the other hand, S:S samples showed FRpost/FRpre values above 1 from D17 forward, indicating that the network would transiently increase in firing rate right after stimulation. One-way ANOVA showed that this increase between FRpost/FRpre values for S:S and NS:S was statistically significant for D17 (n = 9; F(1,16) = 12.19, p = 3E-3), D21 (n = 9; F(1,16) = 6.94, p = 0.018) and D23 (n = 9; F(1,16) = 9.91, p = 6.23E-3), while only showing a non-statistically significant trend for D19 (n = 9; F(1,16) = 2.16, p = 0.16) and D25 (n = 9; F(1,16) = 3.76, p = 0.071). It is relevant to mention that these effects were observed while there was no perceivable change in efficiency of the blue light to activate the ChR2 ion channels and evoke a response in the networks (Supplementary Fig. 6). These observations were corroborated by repeated measures ANOVA performed at p < 0.05, which showed no statistically significance change in efficiency (repeated measures ANOVA, n = 12; F(2,22) = 1.25, p = 0.31).
To further study how the training regimens affected network response, we also quantified the evoked response reflected in the network’s synchronicity for the initial stimulation done on the initial spontaneous interval of recording. For this purpose, raster-plots of the average values of cross-correlation (as calculated for the analysis in Fig. 4) were calculated using 10 s bins across the entire 20 min of recording (Fig. 6f). When quantifying the short term effect of stimulation during recording had on network synchronicity, by comparing (stackrel{-}{chi }) post to (stackrel{-}{chi }) pre, a trend was observed where the presence of a training regimen during neurogenesis seemed to cause the correlation fold-change ((stackrel{-}{chi }) post/(stackrel{-}{chi }) pre) for S:S samples to be higher than NS:S samples. One-way ANOVA detected a statistically significant difference between (stackrel{-}{chi }) post/(stackrel{-}{chi }) pre for S:S and NS:S for days D19 (n = 3; F(1,4) = 16.49, p = 0.015) and D23 (n = 3; F(1,4) = 11.12, p = 0.029) (Fig. 6g).
Changes evoked by stimulation during neurogenesis result in genetic changes
Given the effects on neurite extension, presynaptic clustering, frequency profiles and network response to stimulation that were observed as a result of the presence of training regimens on MEBs during neurogenesis, we proceeded to determine genetic changes that could provide possible mechanistic explanations. Total messenger RNA sequencing was performed and analyzed for stimulated (S) and non-stimulated (NS) MEBs at D9, as well as EBs at D2. The differentially expressed genes in MEBs that underwent training regimens during neurogenesis were compared to those that did not, both with respect to the genetic expression of EBs sampled prior to differentiation (at D2). A total of 749 differentially expressed genes between S and NS with p < 0.05 were detected and clustered and color coded with respect to the differential expression of D2 (Fig. 7a). There were 200 genes that were upregulated during control differentiation, but this upregulation was lessened for samples that underwent training regimen (black bar), while the upregulation of 172 genes was amplified for those same samples (red bar). On the other hand, there were 202 genes whose downregulation was stagnated for samples with training regimen (yellow bar). For 173 genes, the control downregulation was further amplified after stimulation (blue bar). Something important to note was that this observed differential expression did not include changes in phenotype populations, matching the immunostaining observations (Supplementary Fig. 7). This indicated that training regimen during differentiation did not seem to noticeably disrupt the rate of phenotype specification or generation of the neural populations that generally result from the differentiation protocol (Table 1). This suggests that training regimens affected other functional pathways rather than altering the differentiation of populations. For further analysis, a more stringent threshold (p < 0.0005) was set to detect the most promising genes as key factors for the behavioral changes seen in stimulated MEB cultures. This threshold resulted in 97 differentially expressed genes for the black cluster (Fig. 7b), 63 differentially expressed genes for the red cluster (Fig. 7c), 77 differentially expressed genes for the yellow cluster (Fig. 7d) and 71 differentially expressed genes for the blue cluster (Fig. 7e). From this pool, a thorough literature study was used to identify gene targets that had been reported to be related to known neural development and function (Table 2, Supplementary Fig. 9).
Figure 7
figure7
RNA Sequencing shows differential expression as a result of optical stimulation during neurogenesis. a. Heat map of standard deviation of differential expression for genes with p < 0.05 (n = 749). Genes were primarily clustered for: (1) genes that would overexpress during differentiation and underexpressed due to stimulation, (2) genes that would overexpress during control differentiation and overexpressed further due to stimulation, (3) genes that would underexpress during control differentiation and stimulation minimized that underexpression and (4) genes that would underexpress during control differentiation and stimulation amplified that underexpression. (first color column in order: black, red, yellow, blue). Significantly differentially regulated genes, with p < 0.0005 (n = 307) were extracted as column plots for: b. black, c. red, d. yellow and e. blue clusters.
Table 1 Expression comparisons for phenotypic gene targets.
Table 2 Significantly (p < 0.0005) differentially expressed genes reported in literature as regulators of neural development.
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__label__pos
| 0.882319 |
Mitochondrial DNA polymorphism in the Vietnamese population
Authors
R. Ivanova, Hôpital Saint Louis, Centre Hayem, 1, Ave Cl Vellefaux, 75475, Paris, Cedex 10, France.
Abstract
The mitochondrial DNA variation was screened in a sample of 50 unrelated individuals of the Vietnamese population originating from Hanoi. A combination of long and standard PCR and restriction endonuclease digests with the enzymes HpaI, BamHI, HaeII, MspI, AvaII and HincII were used to reveal mtDNA variation. Twenty enzyme morphs were detected, three of which (HaeII-13Viet, MspI-19Viet and MspI-20Viet) are new and are produced by a single mutational event in already known enzyme morphs. Ten already known and four new mitotypes [93Viet (1-1-2-4-1), 94Viet (2-1-13Viet-1-1), 95Viet (2-1-13Viet-19Viet-1) and 96Viet (1-1-2–20Viet-12)] were found in the Vietnamese population. The 9-bp deletion occurring in the COII/tRNALys region of the mitochondrial genome was also analysed and 10 samples were found to have this deletion. The comparison of the Vietnamese with other East Asian populations showed a close genetic relationship of the population under investigation with other Orientals. However, the Vietnamese population can be differentiated by the significantly higher frequency of the enzyme morph HincII-5 and by seven new markers. These results strongly support the hypothesis of a dual ethnic origin of the Vietnamese population from the Chinese and Thai–Indonesian populations based on HLA markers and linguistic evidence.
Ancillary
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__label__pos
| 0.800392 |
ComputerSome computer scientists create packages to manage robots. Computer, the flagship publication of the IEEE Computer Society, publishes peer-reviewed articles written for and by computer researchers and practitioners representing the full spectrum of computing and data know-how, from hardware to software and from emerging research to new applications.
Most jobs for computer and knowledge research scientists require a master’s degree in computer science or a associated discipline. Computers (ISSN 2073-431X) is a global scientific peer-reviewed open entry journal of computer science, together with computer and network architecture and computer-human interaction as its essential foci, revealed quarterly online by MDPI.
An inventory of instructions is called a program and is stored on the computer’s arduous disk Computers work by means of the program by using a central processing unit , and they use fast memory known as RAM as an area to store the instructions and information while they are doing this.
Computer scientists build algorithms into software packages that make the information easier for analysts to use. If you happen to fail to heed this caution, the program could stop working with your browser, operating system or device at any time and you could not be capable to recover your account or your tax info.
Computer packages that be taught and adapt are a part of the rising field of synthetic intelligence and machine learning Artificial intelligence based mostly products usually fall into two major classes: rule based techniques and pattern recognition programs.
Categories: Computer
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__label__pos
| 0.717889 |
Spatiotemporal variability in fatty acid profiles of the copepod Calanus marshallae off the west coast of Vancouver Island
Date
2015-04-21
Authors
Bevan, Daniel
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Factors affecting energy transfer to higher trophic levels can determine the survival and production of commercially important species and thus the success of fisheries management regimes. Juvenile salmon experience particularly high mortality during their early marine residence, but the root causes of this mortality remain uncertain. One potential contributing factor is the food quality encountered at this critical time. The nutritionally vital essential fatty acids (EFA) docosahexaenoic acid (DHA, 22:6n-3) and eicosapentaenoic acid (EPA, 20:5n-3) are essential to all marine heterotrophs, and their availability has the potential to affect energy transfer through a limitation-driven food quality effect. Assessing variability in DHA and EPA in an ecologically important prey species of juvenile salmon could give insight into the prevalence and severity of food quality effects. On the west coast of Vancouver Island (WCVI), one such species is the calanoid copepod Calanus marshallae. This omnivorous species possesses a high grazing capacity and the ability to store large amounts of lipids. As it is also an important prey item for a diverse array of predators, including juvenile Pacific salmon, C. marshallae plays a key role in energy transfer from phytoplankton to high-trophic iv consumers. This study quantified spatiotemporal variability in the quality of C. marshallae as prey for higher trophic levels using three polyunsaturated fatty acid indicators: DHA:EPA, %EFA and PUFA:SFA (polyunsaturated fatty acids to saturated fatty acids). Samples were collected on the WCVI in May and September of 2010 and May 2011. The environmental parameters included in the analysis were the phase of the Pacific Decadal Oscillation (PDO), sea surface temperature (SST), latitude, station depth, and season (spring versus late summer). Despite a phase shift in the PDO from positive to negative, overall means of the fatty acid indicators did not vary between May 2010 and May 2011. Same-station %EFA values rarely fluctuated more than 5%. DHA:EPA ratios were more variable but without a discernable pattern, while PUFA:SFA ratios decreased in shelf stations and increased offshore. Contrary to expectations, fatty acid indicators showed a weak positive correlation or no relationship with SST, nor was there a relationship with latitude. The narrow temperature range observed across all stations suggests that temperature may not play a significant role in PUFA availability off the WCVI. There were, however, significant relationships between the fatty acid indicators and bottom depth and season. Shelf and slope stations showed significantly higher %EFA and PUFA:SFA than did offshore stations (depth >800 m), with this gradient appearing stronger in May than September. While the food quality represented by C. marshallae was consistent across all shelf stations, the lower food quality observed offshore could potentially affect juvenile salmon growth along the WCVI where the shelf narrows to less than 5 km.
Description
Keywords
Polyunsaturated fatty acids, DHA, EPA, Copepod, Juvenile salmon, Food quality, Calanus marshallae
Citation
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__label__pos
| 0.97969 |
File Include 01
This exercise is one of our challenges on File Include vulnerabilities
PRO
Tier
Medium
< 1 Hr.
10228
Many web applications need to include files for loading classes or sharing templates across multiple pages. "File Include" vulnerabilities occur when user-controlled parameters are used in file inclusion functions like `require`, `require_once`, `include`, or `include_once` without proper filtering. This can allow an attacker to manipulate the function to load and execute arbitrary files.
In this lab, you will explore both Local File Include (LFI) and Remote File Include (RFI) vulnerabilities. By injecting special characters or using directory traversal techniques, you can read and execute files, potentially gaining control over the server. The lab also demonstrates how PHP's configuration option `allow_url_include` can enable remote file inclusion, leading to severe security risks.
Want to learn more? Get started with PentesterLab Pro! GOPRO
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__label__pos
| 0.908307 |
Python String to int
By | September 28, 2021
Python String to int
In Python, integer numbers are represented as the whole numbers with no fraction value. In Python using the int and str data types we can store integer values however the arithmetic operators do not work on str datatype. In this tutorial, we have provided the python code snippets which can convert Python str datatype to an int datatype.
Represent an Integer value in Python
An integer value can be stored as str and int data type, but it always suggested to save it as int.
Integer value as an str data type
>>> a = "200"
Integer value as an int data type
>>> a = 200
Convert str to int
In Python, we have a built-in function int() which can convert str datatype integer value to int datatype integer value. But while using the int() function we need to make sure that the string contains only integer or whole number, nothing else.
Vamware
Example
s = "400"
i = int(s)
print(s)
Output
Vamware
400
<Note>:The string value which we want to convert into an integer value must contain only whole or integer number else the int() function return an error.
Example
s = "200k"
i = int(s)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
ValueError: invalid literal for int() with base 10: '200k'
int() function parameters
The int() function can accepts two parameters:
• value.
• base.
Syntax
int(value , base)
Here the base signifies the base value of the passed value, and by default, the base value is 10(decimal).
There are 4 major number systems we can use to represent integer and whole numbers.
• Binary (base 2)
• octal (base 8)
• decimal (base 10)
• hexadecimal (base 16)
Binary to int
When we want to convert a binary number to an integer value, then we need to pass its base value 2.
Example
>>>i = int(30.23)
>>> b = "1000"
>>> n = int(b, base=2)
>>> n
8
Octal to int
To convert an octal number to an integer, we need to mention the base 8. Octal numbers can only be represented using 0, 1, 2, 3, 4, 5, 6, and 7 numbers.
10 in octal means 8 in decimal or integer.
Example
>>> o = "10"
>>> i = int(o, base=8 )
>>> i
8
Decimal numbers
By default the int() function change all the valid values to decimal values.
Example
>>> d = "129"
>>> n = int(d, base= 10)
>>> n
129
Hexadecimal to int()
To change a hexadecimal number to an integer, we need to pass the base value as 16. Hexadecimal can represent a number using 0 to 9 and A-F, which make its base count to 16.
Example
>>> hd = "1F"
>>> n = int(hd, base= 16)
>>> n
31
Python convert int to str
Similar to the int() function we have str() function in Python which can convert the passed value to string datatype.
Example
>>> f = 30.23
>>> s = str(f)
>>> s
'30.23'
>>> type(s)
<class 'str'>
<Note> The type() function returns the data type of the variable or object.
Summary
• The inbuilt int() function can convert the passed string value to its corresponding integer value.
• The string value we passed in the int() function must be an integer value or whole number, the function returns an error.
• The base parameter of int() function represents the number system base of the value.
• The str() function can convert any value to a string data value.
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__label__pos
| 0.999865 |
Infinite sets of points in the Euclidean plane, even discrete sets, do not always have Euclidean minimum spanning trees. For instance, consider the points with coordinates
\[\left(i, \pm\left(1+\frac1i\right)\right),\]
for positive integers \(i\). You can connect the positive-\(y\) points and the negative-\(y\) points into two chains with edges of length less than two, but then you have to pick one edge of length greater than two to span from one chain to the other. Whichever edge you choose, the next edge along would always be a better choice. So a tree that minimizes the multiset of its edge weights (as finite minimum spanning trees do) does not exist for this example. And as the same example shows, the sum of edge weights may be infinite, so how can we use minimization of this sum to define a tree?
Discrete infinite set of points with no Euclidean minimum spanning tree
Despite that, here’s a construction that works for any compact set, even one with infinitely many components, and that generalizes easily to higher-dimensional Euclidean spaces. I think it deserves to be called the Euclidean minimum spanning tree. Given a compact set \(C\), consider every partition \(C=A\cup (C\setminus A)\) of \(C\) into two disjoint nonempty compact subsets. For each such partition, find a line segment \(s_A\) of minimum length with endpoints in \(A\) and \(C\setminus A\), breaking ties lexicographically by coordinates. By the assumed compactness of \(A\) and \(C\setminus A\), such a line segment exists. Let \(T_C\) be the union of \(C\) itself and of all line segments obtained in this way. For example, the union of a triangle, square, and circle shown below has three partitions into two nonempty compact subsets, separating one of these three shapes from the other two. Two of these partitions choose the diagonal pink segment as their shortest connection, and the third partition chooses the horizontal pink segment. So in this case, \(T_C\) consists of the three blue given shapes and two pink segments.
Minimum spanning tree of a circle, square, and triangle
When \(C\) is a finite point set, \(T_C\) is just a Euclidean minimum spanning tree. When \(C\) has finitely many connected components, like the example above, \(T_C\) is again a minimum spanning tree, for the component-component distances. In the general case, \(T_C\) still has many of the familiar properties of Euclidean minimum spanning trees:
• It consists of the input and a collection of line segments connecting pairs of input points, by construction.
• It is a connected set. Topologically, this means that it cannot be covered by two disjoint open sets that both have a nonempty intersection with it. (This is different from being path-connected, a stronger property.) Any nontrivial open disjoint cover of \(C\) would be spanned by a line segment from one set to the other, and no new disjoint covers can separate these line segments from their endpoints.
• For any added segment \(s_A\), the intersection of two disks with that segment as radius (a “lune”) has no point of \(C\) in its interior. Any interior point would form one end of a shorter connecting segment between \(A\) and \(C\setminus A\), with the other end at an endpoint of \(s_A\). No two added segments can cross without violating the empty lune property.
The empty lune of an edge
• For any added segment \(s_A\), the open rhombus with angles \(60^\circ\) and \(120^\circ\) having \(s_A\) as its long diagonal is disjoint from the rhombi formed in the same way from the other segments. Any two overlapping rhombi would allow the longer of the two segments they come from to be replaced by a shorter segment crossing the same compact partition, on a three-segment path connecting its endpoints via the other segment endpoints. Because these non-overlapping rhombi cover a region of bounded area, the squared segment lengths have a bounded sum, and only finitely many segments can be longer than any given length threshold.
An infinite minimum spanning tree and its empty rhombi
• The union of \(C\) with any subset of added segments is compact. If \(p\) is a limit point of a sequence \(\sigma_i\) of points in this union, it must either lie in the empty rhombus of a segment (in which case it can only be a point of the same segment), or it is a limit point of a sequence of points in \(C\), obtained by replacing each point in \(\sigma_i\) that is interior to a segment by the nearest segment endpoint. This replacement only increases the distance from the replaced point to \(p\) by a constant factor, which does not affect convergence. By compactness the replaced sequence converges to a point in \(C\).
• For any \(i\), the set \(T_i\) of the largest \(i\) added segments (with the same tie-breaking order) are edges of a minimum spanning tree for a family of \(i-1\) sets. To construct these sets, find the components of the union of \(C\) with all shorter segments, and intersect each component with \(C\). None of these components can cross between \(A\) and \(C\setminus A\) for any edge \(s_A\in T_i\). Because adding \(T_i\) connects all these components, there can be at most \(i-1\) components. Each edge in \(T_i\) is shortest (with a consistent tie-breaking rule) across some partition of the components, one of the ways of determining the edges in a finite minimum spanning tree. In particular, \(T_C\) is minimally connected: removing any edge \(s_A\in S_i\) separates some of the components from each other.
• \(T_C\) has the minimum sum of squared edge lengths of all collections of line segments between points of \(C\) that connect \(C\). To see this, consider any other connecting set \(X\) of line segments with a finite sum of squared edge lengths. Truncate the sorted sequence of edges of \(T_C\) to a finite initial sequence \(T_i\) such that the rest of the sequence has negligible sum of squares. Because \(T_i\) is a minimum spanning tree of its components, and \(X\) connects those same components (perhaps redundantly), the sequence of edge lengths in \(T_i\) is, step for step, less than or equal to the sorted sequence of lengths in \(X\).
There may exist other sets of line segments that connect \(C\) with the same sum of squared edge lengths but they all are minimally connected, with the same sequence of edge lengths, the same empty lune and empty rhombus properties, and the same property that their initial sequences form finite minimum spanning trees of their components.
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Transparency and patterned drawing
256-color transparency
In paletted video modes, translucency and lighting are implemented with a 64k lookup table, which contains the result of combining any two colors c1 and c2. You must set up this table before you use any of the translucency or lighting routines. Depending on how you construct the table, a range of different effects are possible. For example, translucency can be implemented by using a color halfway between c1 and c2 as the result of the combination. Lighting is achieved by treating one of the colors as a light level (0-255) rather than a color, and setting up the table appropriately. A range of specialised effects are possible, for instance replacing any color with any other color and making individual source or destination colors completely solid or invisible. Color mapping tables can be precalculated with the colormap utility, or generated at runtime. Read chapter "Structures and types defined by Allegro" for an internal description of the COLOR_MAP structure.
Truecolor transparency
In truecolor video modes, translucency and lighting are implemented by a blender function of the form:
unsigned long (*BLENDER_FUNC)(unsigned long x, y, n);
For each pixel to be drawn, this routine is passed two color parameters x and y, decomposes them into their red, green and blue components, combines them according to some mathematical transformation involving the interpolation factor n, and then merges the result back into a single return color value, which will be used to draw the pixel onto the destination bitmap.
The parameter x represents the blending modifier color and the parameter y represents the base color to be modified. The interpolation factor n is in the range [0-255] and controls the solidity of the blending.
When a translucent drawing function is used, x is the color of the source, y is the color of the bitmap being drawn onto and n is the alpha level that was passed to the function that sets the blending mode (the RGB triplet that was passed to this function is not taken into account).
When a lit sprite drawing function is used, x is the color represented by the RGB triplet that was passed to the function that sets the blending mode (the alpha level that was passed to this function is not taken into account), y is the color of the sprite and n is the alpha level that was passed to the drawing function itself.
Since these routines may be used from various different color depths, there are three such callbacks, one for use with 15-bit 5.5.5 pixels, one for 16 bit 5.6.5 pixels, and one for 24-bit 8.8.8 pixels (this can be shared between the 24 and 32-bit code since the bit packing is the same).
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Water (molecule)
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Water (H2O, HOH) is the most abundant molecule on Earth's surface, composing of about 70% of the Earth's surface as liquid and solid state in addition to being found in the atmosphere as a vapor. It is in dynamic equilibrium between the liquid and vapor states at standard temperature and pressure. At room temperature, it is a nearly colorless, tasteless, and odorless liquid, with a hint of blue. Many substances dissolve in water and it is commonly referred to as the universal solvent. Because of this, water in nature and in use is rarely clean, and may have some properties different from those in the laboratory. However, there are many compounds that are essentially, if not completely, insoluble in water. Water is the only common, pure substance found naturally in all three common states of matter—for other substances, see Chemical properties.
Forms of water
See the Water#Overview of types of water
Water can take many forms. The solid state of water is commonly known as ice (while many other forms exist; see amorphous solid water); the gaseous state is known as water vapor (or steam, though this is actually incorrect, since steam is just condensing liquid water droplets), and the common liquid phase is generally taken as simply water. Above a certain critical temperature and pressure (647 K and 22.064 MPa), water molecules assume a supercritical condition, in which liquid-like clusters float within a vapor-like phase.
Heavy water is water in which the hydrogen is replaced by its heavier isotope, deuterium. It is chemically almost identical to normal water. Heavy water is used in the nuclear industry to slow down neutrons.
Physics and chemistry of water
Water is the chemical substance with chemical formula H2O: one molecule of water has two hydrogen atoms covalently bonded to a single oxygen atom. Water is a tasteless, odorless liquid at ambient temperature and pressure, and appears colorless in small quantities, although it has its own intrinsic very light blue hue. Ice also appears colorless, and water vapor is essentially invisible as a gas.[1] Water is primarily a liquid under standard conditions, which is not predicted from its relationship to other analogous hydrides of the oxygen family in the periodic table, which are gases such as hydrogen sulfide. Also the elements surrounding oxygen in the periodic table, nitrogen, fluorine, phosphorus, sulfur and chlorine, all combine with hydrogen to produce gases under standard conditions. The reason that oxygen hydride (water) forms a liquid is that it is more electronegative than all of these elements (other than fluorine). Oxygen attracts electrons much more strongly than hydrogen, resulting in a net positive charge on the hydrogen atoms, and a net negative charge on the oxygen atom. The presence of a charge on each of these atoms gives each water molecule a net dipole moment. Electrical attraction between water molecules due to this dipole pulls individual molecules closer together, making it more difficult to separate the molecules and therefore raising the boiling point. This attraction is known as hydrogen bonding. Water can be described as a polar liquid that dissociates disproportionately into the hydronium ion (H3O+(aq)) and an associated hydroxide ion (OH(aq)). Water is in dynamic equilibrium between the liquid, gas and solid states at standard temperature and pressure, and is the only pure substance found naturally on Earth to be so.
Water, ice and vapor
Heat capacity and heat of vaporization
Water has the second highest specific heat capacity of any known chemical compound, after ammonia, as well as a high heat of vaporization (40.65 kJ mol−1), both of which are a result of the extensive hydrogen bonding between its molecules. These two unusual properties allow water to moderate Earth's climate by buffering large fluctuations in temperature.
Density of water and ice
The solid form of most substances is more dense than the liquid phase; thus, a block of pure solid substance will sink in a tub of pure liquid substance. But, by contrast, a block of common ice will float in a tub of water because solid water is less dense than liquid water. This is an extremely important characteristic property of water. At room temperature, liquid water becomes denser with lowering temperature, just like other substances. But at 4 °C, just above freezing, water reaches its maximum density, and as water cools further toward its freezing point, the liquid water, under standard conditions, expands to become less dense. The physical reason for this is related to the crystal structure of ordinary ice, known as hexagonal ice Ih. Water, lead, uranium, neon and silicon are some of the few materials which expand when they freeze; most other materials contract. It should be noted however, that not all forms of ice are less dense than liquid water. For example HDA and VHDA are both more dense than liquid phase pure water. Thus, the reason that the common form of ice is less dense than water is a bit non-intuitive and relies heavily on the unusual properties inherent to the hydrogen bond.
Generally, water expands when it freezes because of its molecular structure, in tandem with the unusual elasticity of the hydrogen bond and the particular lowest energy hexagonal crystal conformation that it adopts under standard conditions. That is, when water cools, it tries to stack in a crystalline lattice configuration that stretches the rotational and vibrational components of the bond, so that the effect is that each molecule of water is pushed further from each of its neighboring molecules. This effectively reduces the density ρ of water when ice is formed under standard conditions.
The importance of this property cannot be overemphasized for its role on the ecosystem of Earth. For example, if water were more dense when frozen, lakes and oceans in a polar environment would eventually freeze solid (from top to bottom). This would happen because frozen ice would settle on the lake and riverbeds, and the necessary warming phenomenon (see below) could not occur in summer, as the warm surface layer would be less dense than the solid frozen layer below. It is a significant feature of nature that this does not occur naturally in the environment.
Nevertheless, the unusual expansion of freezing water (in ordinary natural settings in relevant biological systems), due to the hydrogen bond, from 4 °C above freezing to the freezing point offers an important advantage for freshwater life in winter. Water chilled at the surface increases in density and sinks, forming convection currents that cool the whole water body, but when the temperature of the lake water reaches 4 °C, water on the surface decreases in density as it chills further and remains as a surface layer which eventually freezes and forms ice. Since downward convection of colder water is blocked by the density change, any large body of fresh water frozen in winter will have the coldest water near the surface, away from the riverbed or lakebed. This accounts for various little known phenomena of ice characteristics as they relate to ice in lakes and "ice falling out of lakes" as described by early 20th century scientist Horatio D. Craft.
The following table gives the density of water in grams per cubic centimeter at various temperatures in degrees Celsius:[2]
Temp (°C) Density (g/cm³)
30 0.9956502
25 0.9970479
22 0.9977735
20 0.9982071
15 0.9991026
10 0.9997026
4 0.9999720
0 0.9998395
−10 0.998117
−20 0.993547
−30 0.983854
The values below 0 °C refer to supercooled water.
Freezing point
A simple but environmentally important and unusual property of water is that its usual solid form, ice, floats on its liquid form. This solid state is not as dense as liquid water because of the geometry of the hydrogen bonds which are formed only at lower temperatures. For almost all other substances the solid form has a greater density than the liquid form. Fresh water at standard atmospheric pressure is most dense at 3.98 °C, and will sink by convection as it cools to that temperature, and if it becomes colder it will rise instead. This reversal will cause deep water to remain warmer than shallower freezing water, so that ice in a body of water will form first at the surface and progress downward, while the majority of the water underneath will hold a constant 4 °C. This effectively insulates a lake floor from the cold. The water will freeze at 0 °C (32 °F, 273 K), however, it can be supercooled in a fluid state down to its crystal homogeneous nucleation at almost 231 K (−42 °C)[3]. Ice also has a number of more exotic phases not commonly seen (go to the full article on Ice).
Density of saltwater and ice
The density of water is dependent on the dissolved salt content as well as the temperature of the water. Ice still floats in the oceans, otherwise they would freeze from the bottom up. However, the salt content of oceans lowers the freezing point by about 2 °C and lowers the temperature of the density maximum of water to the freezing point. That is why, in ocean water, the downward convection of colder water is not blocked by an expansion of water as it becomes colder near the freezing point. The oceans' cold water near the freezing point continues to sink. For this reason, any creature attempting to survive at the bottom of such cold water as the Arctic Ocean generally lives in water that is 4 °C colder than the temperature at the bottom of frozen-over fresh water lakes and rivers in the winter.
As the surface of salt water begins to freeze (at −1.9 °C for normal salinity seawater, 3.5%) the ice that forms is essentially salt free with a density approximately equal to that of freshwater ice. This ice floats on the surface and the salt that is "frozen out" adds to the salinity and density of the seawater just below it, in a process known as brine rejection. This more dense saltwater sinks by convection and the replacing seawater is subject to the same process. This provides essentially freshwater ice at −1.9 °C on the surface. The increased density of the seawater beneath the forming ice causes it to sink towards the bottom.
Miscibility and condensation
Water is miscible with many liquids, for example ethanol in all proportions, forming a single homogeneous liquid. On the other hand water and most oils are immiscible usually forming layers according to increasing density from the top.
Red line shows saturation
As a gas, water vapor is completely miscible with air. On the other hand the maximum water vapor pressure that is thermodynamically stable with the liquid (or solid) at a given temperature is relatively low compared with total atmospheric pressure. For example, if the vapor partial pressure[4] is 2% of atmospheric pressure and the air is cooled from 25 °C, starting at about 22 °C water will start to condense, defining the dew point, and creating fog or dew. The reverse process accounts for the fog burning off in the morning. If one raises the humidity at room temperature, say by running a hot shower or a bath, and the temperature stays about the same, the vapor soon reaches the pressure for phase change, and condenses out as steam. A gas in this context is referred to as saturated or 100% relative humidity, when the vapor pressure of water in the air is at the equilibrium with vapor pressure due to (liquid) water; water (or ice, if cool enough) will fail to lose mass through evaporation when exposed to saturated air. Because the amount of water vapor in air is small, relative humidity, the ratio of the partial pressure due to the water vapor to the saturated partial vapor pressure, is much more useful. Water vapor pressure above 100% relative humidity is called super-saturated and can occur if air is rapidly cooled, say by rising suddenly in an updraft.[5]
Vapor Pressures of Water
Temperature (°C) Pressure (torr)
0 4.58
5 6.54
10 9.21
12 10.52
14 11.99
16 13.63
17 14.53
18 15.48
19 16.48
20 17.54
21 18.65
22 19.83
23 21.07
24 22.38
25 23.76
[6]
Compressibility
The compressibility of water is a function of pressure and temperature. At 0 °C in the limit of zero pressure the compressibility is 5.1×10-5 bar−1.[7] In the zero pressure limit the compressibility reaches a minimum of 4.4×10-5 bar−1 around 45 °C before increasing again with increasing temperature. As the pressure is increased the compressibility decreases, being 3.9×10-5 bar−1 at 0 °C and 1000 bar. The bulk modulus of water is 2.2×109 Pa.[8] The low compressibility of non-gases, and of water in particular, leads to them often being assumed as incompressible. The low compressibility of water means that even in the deep oceans at 4000 m depth, where pressures are 4×107 Pa, there is only a 1.8% decrease in volume.[8]
Triple point
The various triple points of water[9]
Phases in stable equilibrium Pressure Temperature
liquid water, ice I, and water vapour 611.73 Pa 273.16 K
liquid water, ice Ih, and ice III 209.9 MPa 251 K (-22 °C)
liquid water, ice Ih, and gaseous water 612 Pa 0.01 °C
liquid water, ice III, and ice V 350.1 MPa -17.0 °C
liquid water, ice V, and ice VI 632.4 MPa 0.16 °C
ice Ih, Ice II, and ice III 213 MPa -35 °C
ice II, ice III, and ice V 344 MPa -24 °C
ice II, ice V, and ice VI 626 MPa -70 °C
The temperature and pressure at which solid, liquid, and gaseous water coexist in equilibrium is called the triple point of water. This point is used to define the units of temperature (the kelvin and, indirectly, the degree Celsius and even the degree Fahrenheit). The triple point is at a temperature of 273.16 K (0.01 °C) by convention, and at a pressure of 611.73 Pa. This pressure is quite low, about 1/166 of the normal sea level barometric pressure of 101,325 Pa. The atmospheric surface pressure on planet Mars is remarkably close to the triple point pressure, and the zero-elevation or "sea level" of Mars is defined by the height at which the atmospheric pressure corresponds to the triple point of water.
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water phase diagram
Y-axis = Pressure in Pascal (10n),
X-axis = Temperature in Kelvin.
S = Solid
L = Liquid
V = Vapour
CP = Critical Point
TP = Triple point of water
The triple point of water (the single combination of pressure and temperature at which pure liquid water, ice, and water vapor can coexist in a stable equilibrium) is used to define the kelvin, the SI unit of thermodynamic temperature. As a consequence, water's triple point temperature is a prescribed value rather than a measured quantity: 273.16 kelvins (0.01 °C) and a pressure of 611.73 pascals (approximately 0.0060373 atm). This is approximately the combination that exists with 100% relative humidity at sea level and the freezing point of water.
Although it is commonly named as "the triple point of water", the stable combination of liquid water, ice I, and water vapour is but one of several triple points on the phase diagram of water. Gustav Heinrich Johann Apollon Tammann in Göttingen produced data on several other triple points in the early 20th century. Kamb and others documented further triple points in the 1960s.[10][9][11]
Mpemba effect
The Mpemba effect is the surprising phenomenon whereby hot water can, under certain conditions, freeze sooner than cold water, even though it must pass the lower temperature on the way to freezing. However, this can be explained with evaporation, convection, supercooling, and the insulating effect of frost.
Hot ice
Hot ice is the name given to another surprising phenomenon in which water at room temperature can be turned into ice that remains at room temperature by supplying an electric field on the order of 106 volts per meter.[12]
The effect of such electric fields has been suggested as an explanation of cloud formation. The first time cloud ice forms around a clay particle, it requires a temperature of −10 °C, but subsequent freezing around the same clay particle requires a temperature of just −5 °C, suggesting some kind of structural change.[13]
Surface tension
Water drops are stable, due to the high surface tension of water, 72.8 mN/m, the highest of the non-metallic liquids. This can be seen when small quantities of water are put on a surface such as glass: the water stays together as drops. This property is important for life. For example, when water is carried through xylem up stems in plants the strong intermolecular attractions hold the water column together. Strong cohesive properties hold the water column together, and strong adhesive properties stick the water to the xylem, and prevent tension rupture caused by transpiration pull. Other liquids with lower surface tension would have a higher tendency to "rip", forming vacuum or air pockets and rendering the xylem water transport inoperative.
Electrical properties
Pure water containing no ions is an excellent insulator, however, not even "deionized" water, is completely free of ions. Water undergoes auto-ionisation at any temperature above absolute zero. Further, because water is such a good solvent, it almost always has some solute dissolved in it, most frequently a salt. If water has even a tiny amount of such an impurity, then it can conduct electricity readily, as impurities such as salt separate into free ions in aqueous solution by which an electric current can flow.
Water can be split into its constituent elements, hydrogen and oxygen, by passing a current through it. This process is called electrolysis. Water molecules naturally dissociate into H+ and OH ions, which are pulled toward the cathode and anode, respectively. At the cathode, two H+ ions pick up electrons and form H2 gas. At the anode, four OH ions combine and release O2 gas, molecular water, and four electrons. The gases produced bubble to the surface, where they can be collected. It is known that the theoretical maximum electrical resistivity for water is approximately 182 ·m²/m (or 18.2 MΩ·cm²/cm) at 25 °C. This figure agrees well with what is typically seen on reverse osmosis, ultrafiltered and deionized ultrapure water systems used for instance, in semiconductor manufacturing plants. A salt or acid contaminant level exceeding that of even 100 parts per trillion (ppt) in ultrapure water will begin to noticeably lower its resistivity level by up to several kilohm-square meters per meter (a change of several hundred nanosiemens per meter of conductance).
Electrical conductivity
Pure water has a low electrical conductivity, but this increases significantly upon solvation of a small amount of ionic material water such as hydrogen chloride. Thus the risks of electrocution are much greater in water with the usual impurities not found in pure water. Any electrical properties observable in water are from the ions of mineral salts and carbon dioxide dissolved in it. Water does self-ionize where two water molecules become one hydroxide anion and one hydronium cation, but not enough to carry enough electric current to do any work or harm for most operations. In pure water, sensitive equipment can detect a very slight electrical conductivity of 0.055 µS/cm at 25 °C. Water can also be electrolyzed into oxygen and hydrogen gases but in the absence of dissolved ions this is a very slow process, as very little current is conducted. While electrons are the primary charge carriers in water (and metals), in ice (and some other electrolytes), protons are the primary carriers (see proton conductor).
Dipolar nature of water
model of hydrogen bonds between molecules of water
An important feature of water is its polar nature. The water molecule forms an angle, with hydrogen atoms at the tips and oxygen at the vertex. Since oxygen has a higher electronegativity than hydrogen, the side of the molecule with the oxygen atom has a partial negative charge. A molecule with such a charge difference is called a dipole. The charge differences cause water molecules to be attracted to each other (the relatively positive areas being attracted to the relatively negative areas) and to other polar molecules. This attraction is known as hydrogen bonding, and explains many of the properties of water. Certain molecules, such as carbon dioxide, also have a difference in electronegativity between the atoms but the difference is that the shape of carbon dioxide is symmetrically aligned and so the opposing charges cancel one another out. This phenomenon of water can be seen if you hold an electrical source near a thin stream of water falling vertically, causing the stream to bend towards the electrical source.
Although hydrogen bonding is a relatively weak attraction compared to the covalent bonds within the water molecule itself, it is responsible for a number of water's physical properties. One such property is its relatively high melting and boiling point temperatures; more heat energy is required to break the hydrogen bonds between molecules. The similar compound hydrogen sulfide (H2S), which has much weaker hydrogen bonding, is a gas at room temperature even though it has twice the molecular mass of water. The extra bonding between water molecules also gives liquid water a large specific heat capacity. This high heat capacity makes water a good heat storage medium.
Hydrogen bonding also gives water its unusual behavior when freezing. When cooled to near freezing point, the presence of hydrogen bonds means that the molecules, as they rearrange to minimize their energy, form the hexagonal crystal structure of ice that is actually of lower density: hence the solid form, ice, will float in water. In other words, water expands as it freezes, whereas almost all other materials shrink on solidification.
An interesting consequence of the solid having a lower density than the liquid is that ice will melt if sufficient pressure is applied. With increasing pressure the melting point temperature drops and when the melting point temperature is lower than the ambient temperature the ice begins to melt. A significant increase of pressure is required to lower the melting point temperature —the pressure exerted by an ice skater on the ice would only reduce the melting point by approximately 0.09 °C (0.16 °F).
Electronegative Polarity
Water has a partial negative charge (σ-) near the oxygen atom due to the unshared pairs of electrons, and partial positive charges (σ+) near the hydrogen atoms. In water, this happens because the oxygen atom is more electronegative than the hydrogen atoms — that is, it has a stronger "pulling power" on the molecule's electrons, drawing them closer (along with their negative charge) and making the area around the oxygen atom more negative than the area around both of the hydrogen atoms.
Adhesion
Dew drops adhering to a spider web
Water sticks to itself (cohesion) because it is polar. Water also has high adhesion properties because of its polar nature. On extremely clean/smooth glass the water may form a thin film because the molecular forces between glass and water molecules (adhesive forces) are stronger than the cohesive forces. In biological cells and organelles, water is in contact with membrane and protein surfaces that are hydrophilic; that is, surfaces that have a strong attraction to water. Irving Langmuir observed a strong repulsive force between hydrophilic surfaces. To dehydrate hydrophilic surfaces—to remove the strongly held layers of water of hydration—requires doing substantial work against these forces, called hydration forces. These forces are very large but decrease rapidly over a nanometer or less. Their importance in biology has been extensively studied by V. Adrian Parsegian of the National Institute of Health.[14] They are particularly important when cells are dehydrated by exposure to dry atmospheres or to extracellular freezing.
Surface tension
This daisy is under the water level, which has risen gently and smoothly. Surface tension prevents the water from submerging the flower.
Water has a high surface tension caused by the strong cohesion between water molecules. This can be seen when small quantities of water are put onto a non-soluble surface such as polythene; the water stays together as drops. Just as significantly, air trapped in surface disturbances forms bubbles, which sometimes last long enough to transfer gas molecules to the water. Another surface tension effect is capillary waves which are the surface ripples that form from around the impact of drops on water surfaces, and some times occur with strong subsurface currents flow to the water surface. The apparent elasticity caused by surface tension drives the waves.
Capillary action
Capillary action refers to the process of water moving up a narrow tube against the force of gravity. It occurs because water adheres to the sides of the tube, and then surface tension tends to straighten the surface making the surface rise, and more water is pulled up through cohesion. The process is repeated as the water flows up the tube until there is enough water that gravity can counteract the adhesive force.
Water as a solvent
Water is also a good solvent due to its polarity. When an ionic or polar compound enters water, it is surrounded by water molecules (Hydration). The relatively small size of water molecules typically allows many water molecules to surround one molecule of solute. The partially negative dipole ends of the water are attracted to positively charged components of the solute, and vice versa for the positive dipole ends.
In general, ionic and polar substances such as acids, alcohols, and salts are relatively soluble in water, and nonpolar substances such as fats and oils are not. Nonpolar molecules stay together in water because it is energetically more favorable for the water molecules to hydrogen bond to each other than to engage in van der Waals interactions with nonpolar molecules.
An example of an ionic solute is table salt; the sodium chloride, NaCl, separates into Na+ cations and Cl- anions, each being surrounded by water molecules. The ions are then easily transported away from their crystalline lattice into solution. An example of a nonionic solute is table sugar. The water dipoles make hydrogen bonds with the polar regions of the sugar molecule (OH groups) and allow it to be carried away into solution.
Solvation
High concentrations of dissolved lime make the water of Havasu Falls appear turquoise.
Water is a very strong solvent, referred to as the universal solvent, dissolving many types of substances. Substances that will mix well and dissolve in water (e.g. salts) are known as "hydrophilic" (water-loving) substances, while those that do not mix well with water (e.g. fats and oils), are known as "hydrophobic" (water-fearing) substances. The ability of a substance to dissolve in water is determined by whether or not the substance can match or better the strong attractive forces that water molecules generate between other water molecules. If a substance has properties that do not allow it to overcome these strong intermolecular forces, the molecules are "pushed out" from the water, and do not dissolve. Contrary to the common misconception, water and hydrophobic substances does not "repel", and the hydration of a hydrophobic surface is energetically, but not entropically, favorable.
Amphoteric nature of water
Chemically, water is amphoteric — i.e., it is able to act as either an acid or a base. Occasionally the term hydroxic acid is used when water acts as an acid in a chemical reaction. At a pH of 7 (neutral), the concentration of hydroxide ions (OH) is equal to that of the hydronium (H3O+) or hydrogen (H+) ions. If the equilibrium is disturbed, the solution becomes acidic (higher concentration of hydronium ions) or basic (higher concentration of hydroxide ions).
Water can act as either an acid or a base in reactions. According to the Brønsted-Lowry system, an acid is defined as a species which donates a proton (an H+ ion) in a reaction, and a base as one which receives a proton. When reacting with a stronger acid, water acts as a base; when reacting with a stronger base, it acts as an acid. For instance, it receives an H+ ion from HCl in the equilibrium:
HCl + H2O Template:Unicode H3O+ + Cl
Here water is acting as a base, by receiving an H+ ion.
In the reaction with ammonia, NH3, water donates an H+ ion, and is thus acting as an acid:
NH3 + H2O Template:Unicode NH4+ + OH
Acidity in nature
In theory, pure water has a pH of 7 at 298 K. In practice, pure water is very difficult to produce. Water left exposed to air for any length of time will rapidly dissolve carbon dioxide, forming a dilute solution of carbonic acid, with a limiting pH of about 5.7. As cloud droplets form in the atmosphere and as raindrops fall through the air minor amounts of CO2 are absorbed and thus most rain is slightly acidic. If high amounts of nitrogen and sulfur oxides are present in the air, they too will dissolve into the cloud and rain drops producing more serious acid rain problems.
Hydrogen bonding in water
A water molecule can form a maximum of four hydrogen bonds because it can accept two and donate two hydrogens. Other molecules like hydrogen fluoride, ammonia, methanol form hydrogen bonds but they do not show anomalous behaviour of thermodynamic, kinetic or structural properties like those observed in water. The answer to the apparent difference between water and other hydrogen bonding liquids lies in the fact that apart from water none of the hydrogen bonding molecules can form four hydrogen bonds either due to an inability to donate/accept hydrogens or due to steric effects in bulky residues. In water local tetrahedral order due to the four hydrogen bonds gives rise to an open structure and a 3-dimensional bonding network, which exists in contrast to the closely packed structures of simple liquids. There is a great similarity between water and silica in their anomalous behaviour, even though one (water) is a liquid which has a hydrogen bonding network while the other (silica) has a covalent network with a very high melting point. One reason that water is well suited, and chosen, by life-forms, is that it exhibits its unique properties over a temperature regime that suits diverse biological processes, including hydration.
It is believed that hydrogen bond in water is largely due to electrostatic forces and some amount of covalency. The partial covalent nature of hydrogen bond predicted by Linus Pauling in the 1930s is yet to be proven unambiguously by experiments and theoretical calculations.
Quantum properties of molecular water
Although the molecular formula of water is generally considered to be a stable result in molecular thermodynamics, recent work started in 1995 has shown that at certain scales, water may act more like H3/2O than H2O at the quantum level.[15] This result could have significant ramifications at the level of, for example, the hydrogen bond in biological, chemical and physical systems. The experiment shows that when neutrons and electrons collide with water, they scatter in a way that indicates that they only are affected by a ratio of 1.5:1 of hydrogen to oxygen respectively. However, the time-scale of this response is only seen at the level of attoseconds (10-18 seconds), and so is only relevant in highly resolved kinetic and dynamical systems.[16][17]
Heavy Water and isotopologues of water
Hydrogen has three isotopes. The most common, making up more than 95% of water, has 1 proton and 0 neutrons. A second isotope, deuterium (short form "D"), has 1 proton and 1 neutron. Deuterium, D
2
O
, is also known as heavy water and is used in nuclear reactors as a neutron moderator. The third isotope, tritium, has 1 proton and 2 neutrons, and is radioactive, with a half-life of 12.32 years. T
2
O
exists in nature only in tiny quantities, being produced primarily via cosmic ray-driven nuclear reactions in the atmosphere. D
2
O
is stable, but differs from H
2
O
in in that it is more dense - hence, "heavy water" - and in that several other physical properties are slightly different from those of common, Hydrogen-1 containing "light water". D
2
O
occurs naturally in ordinary water in very low concentrations. Consumption of pure isolated D
2
O
may affect biochemical processes - ingestion of large amounts impairs kidney and central nervous system function. However, very large amounts of heavy water must be consumed for any toxicity to be apparent, and smaller quantities can be consumed with no ill effects at all.
Transparency
Water's transparency is also an important property of the liquid. If water were not transparent, sunlight, essential to aquatic plants, would not reach into seas and oceans.
History
The properties of water have historically been used to define various temperature scales. Notably, the Kelvin, Celsius and Fahrenheit scales were, or currently are, defined by the freezing and boiling points of water. The less common scales of Delisle, Newton, Réaumur and Rømer were defined similarly. The triple point of water is a more commonly used standard point today.[18]
The first scientific decomposition of water into hydrogen and oxygen, by electrolysis, was done in 1800 by William Nicholson, an English chemist. In 1805, Joseph Louis Gay-Lussac and Alexander von Humboldt showed that water is composed of two parts hydrogen and one part oxygen (by volume).
Gilbert Newton Lewis isolated the first sample of pure heavy water in 1933.
Polywater was a hypothetical polymerized form of water that was the subject of much scientific controversy during the late 1960s. The consensus now is that it does not exist.
Pseudoscience concept is water memory.
Systematic naming
The accepted IUPAC name of water is simply "water", although there are two other systematic names which can be used to describe the molecule.
The simplest and best systematic name of water is hydrogen oxide. This is analogous to related compounds such as hydrogen peroxide, hydrogen sulfide, and deuterium oxide (heavy water). Another systematic name, oxidane, is accepted by IUPAC as a parent name for the systematic naming of oxygen-based substituent groups,[19] although even these commonly have other recommended names. For example, the name hydroxyl is recommended over oxidanyl for the –OH group. The name oxane is explicitly mentioned by the IUPAC as being unsuitable for this purpose, since it is already the name of a cyclic ether also known as tetrahydropyran in the Hantzsch-Widman system; similar compounds include dioxane and trioxane.
Systematic nomenclature and humor
Dihydrogen monoxide or DHMO is an overly pedantic systematic covalent name of water. This term has been used in parodies of chemical research that call for this "lethal chemical" to be banned. In reality, a more realistic systematic name would be hydrogen oxide, since the "di-" and "mon-" prefixes are superfluous. Hydrogen sulfide, H2S, is never referred to as "dihydrogen monosulfide", and hydrogen peroxide, H2O2, is never called "dihydrogen dioxide".
Some overzealous material safety data sheets for water list the following: Caution: May cause drowning![citation needed]
Other systematic names for water include hydroxic acid or hydroxylic acid. Likewise, the systematic alkali name of water is hydrogen hydroxide—both acid and alkali names exist for water because it is able to react both as an acid or an alkali, depending on the strength of the acid or alkali it is reacted with (amphoteric). None of these names are used widely outside of DHMO sites.
See also
References
1. Braun, Charles L. (1993). "Why is water blue?" (HTML). J. Chem. Educ. 70 (8): 612. Unknown parameter |coauthors= ignored (help)
2. Lide, D. R. (Ed.) (1990). CRC Handbook of Chemistry and Physics (70th Edn.). Boca Raton (FL):CRC Press.
3. P. G. Debenedetti, P. G., and Stanley, H. E.; "Supercooled and Glassy Water", Physics Today 56 (6), p. 40–46 (2003).
4. The pressure due to water vapor in the air is called the partial pressure(Dalton's law) and it is directly proportional to concentration of water molecules in air (Boyle's law).
5. Adiabatic cooling resulting from the ideal gas law.
6. Brown, Theodore L., H. Eugene LeMay, Jr., and Bruce E. Burston. Chemistry: The Central Science. 10th ed. Upper Saddle River, NJ: Pearson Education, Inc., 2006.
7. Fine, R.A. and Millero, F.J. (1973). "Compressibility of water as a function of temperature and pressure". Journal of Chemical Physics. 59 (10): 5529. doi:10.1063/1.1679903.
8. 8.0 8.1 R. Nave. "Bulk Elastic Properties". HyperPhysics. Georgia State University. Retrieved 2007-10-26.
9. 9.0 9.1 Template:Cite paper
10. Template:Cite paper
11. William Cudmore McCullagh Lewis and James Rice (1922). A System of Physical Chemistry. Longmans, Green and co.
12. Choi, Eun-Mi; Yoon, Young-Hwan; Lee, Sangyoub; Kang, Heon. "Freezing Transition of Interfacial Water at Room Temperature under Electric Fields". Physical Review Letters. 95 (8): 085701. doi:10.1103/PhysRevLett.95.085701.
13. Connolly PJ, Saunders CPR, Gallagher MW, Bower KN, Flynn MJ, Choularton TW, Whiteway J, Lawson RP (2005). "Aircraft observations of the influence of electric fields on the aggregation of ice crystals". Quarterly Journal of the Royal Meteorological Society, Part B. 131 (608): 1695–1712. Unknown parameter |month= ignored (help)
14. Physical Forces Organizing Biomolecules (PDF)
15. Phil Schewe, James Riordon, and Ben Stein (31 Jul 03). "A Water Molecule's Chemical Formula is Really Not H2O". Physics News Update. Check date values in: |date= (help)
16. C. A. Chatzidimitriou-Dreismann, T. Abdul Redah, R. M. F. Streffer and J. Mayers (1997). "Anomalous Deep Inelastic Neutron Scattering from Liquid H2O-D2O: Evidence of Nuclear Quantum Entanglement". Physical Review Letters. 79 (15): 2839. doi:10.1103/PhysRevLett.79.2839.
17. C. A. Chatzidimitriou-Dreismann, M. Vos, C. Kleiner and T. Abdul-Redah (2003). "Comparison of Electron and Neutron Compton Scattering from Entangled Protons in a Solid Polymer". Physical Review Letters. 91 (5): 057403–4. doi:10.1103/PhysRevLett.91.057403.
18. http://home.comcast.net/~igpl/Temperature.html
19. Leigh, G. J. et al. 1998. Principles of chemical nomenclature: a guide to IUPAC recommendations, p. 99. Blackwell Science Ltd, UK. ISBN 0-86542-685-6
External links
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To Terminate or Attenuate?
To Terminate or Attenuate?
Terminations and attenuators can handle high power levels at microwave frequencies, and advanced materials are enabling them to do so in smaller packages.
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Attenuators and terminations are commonly-used components in high-frequency systems, used to adjust or absorb power, respectively. In many ways, the two types of components are similar, since they are both designed to stop RF/microwave power. Attenuators decrease some portion of the power, in fixed or variable amounts, while terminations stop power applied to them altogether.
Both types of components are available in various forms, from miniature chips to higher-power coaxial components and the highest-power waveguide assemblies. Both types of components play important roles in high-frequency circuits and systems, especially when high-power signals must be managed.
Fig. 1
Aettnuators and terminations must both handle high power levels, with some key differences.
Attenuators and, in particular, high-power terminations are usually specified with size, weight, power-handling capability, and frequency range as essential parameters for comparison. Power-handling capability is generally a function of size, with the highest-power components occupying the greatest amount of volume in a design.
A termination is a one-port component meant to absorb all the power applied to it, while an attenuator is a two-port component that reduces the level of the power passing through it by a fixed or variable amount. Attenuators can reduce signal power by a fixed amount or can provide an adjustable range of attenuation. For the most part, adjustments are continuously variable or switched in discrete steps.
Terminations are typically connected at an unused port in a system, such as an unused port of a power divider that is splitting off signal power to other parts of the system. In addition, terminations are used when a passive component (such as a filter or a coupler) is being matched to 50 Ω for measurement purposes, as when testing for return loss or power-handling capability. Terminations used for establishing reference impedances at high power levels are usually referred to as dummy loads.
Matching an attenuator or termination to an application is a matter of understanding the main operating parameters and making the best choice of component for a particular set of requirements. Attenuators are available with fixed attenuation values for a particular frequency range or with a range of attenuation settings that can be set in steps or under continuously variable control.
Whether fixed or variable, attenuators can be compared in terms of bandwidth, attenuation flatness across the frequency range, insertion loss, return loss or VSWR, power-handling capability, operating temperature range, size, and weight. For more on the fundamental operating parameters of RF/microwave attenuators, see “Know When To Add Attenuation.”
Terminating Power
As with attenuators, terminations are available in many form factors. These include miniature chips, coaxial packages, and high-power waveguide components, generally with power ratings to match their sizes. Terminations are characterized by fewer parameters than attenuators, since they do not exhibit amplitude responses as a function of frequency. Rather, the frequency range of a termination is the span of frequencies over which it can maintain an impedance match with a system’s characteristic impedance—usually 50 Ω, but sometimes 75 Ω for broadcast applications or other impedances for specialized uses.
An important function of an RF/microwave termination, especially for high-power models, is its capability to dissipate heat. Any type of power-absorbing component, such as a termination, can dissipate heat by means of conduction, convection, or radiation. Conduction takes place by means of physical contact of different materials, such as a flange-mounted termination to a heat sink.
Conduction occurs when heat is dissipated as it moves from areas of higher energy to areas of lower energy. Convection is a dissipation of heat from a source by means of a flowing liquid (such as water) or a flowing gas (including air, as in fan-cooled terminations). Thermal radiation occurs when a source emits EM waves that carry the heat energy—e.g., infrared (IR) radiation, as used in space heaters.
Any resistive element, including attenuators and terminations, will generate heat that must be dissipated to minimize temperature-related stress and ensure the long-term reliability of a component, circuit, or system. For that reason, terminations are usually fabricated from or packaged in a material with high value of emissivity or heat radiation efficiency.
An ideal thermal radiator would have an emissivity value of 1. While no materials exhibit that thermal radiating efficiency, aluminum comes close, with an emissivity of 0.9. For that reason, aluminum is often used to construct extremely high-power terminations, dummy loads, and attenuators.
Terminations are somewhat simpler to specify than RF/microwave attenuators, since the primary goals of any termination are to establish a good match with the system characteristic impedance and to absorb and dissipate a certain amount of power. As for attenuators, the number of suppliers for high-frequency terminations is large, with package styles ranging from tiny chip terminations to much larger waveguide terminations. As noted, heat must be dissipated, so the power-handling capabilities of these different terminations are related to physical size and connections to surrounding circuitry.
For example, American Technical Ceramics, which supplies both attenuators and terminations, supplies circuit-board-mountable components but in different packages and with different power ratings. The firm’s leaded and surface-mount-technology terminations are well suited for densely packed PCBs. However, these tiny components cannot match the power-handling and thermal-management capabilities of slightly larger flange-mount terminations and their larger cross-sectional mounting connections for effective thermal dissipation.
Res-Net Microwave builds its chip terminations and resistors on thermally dissipative beryllium oxide (BeO) substrate material, allowing for relatively large power-handling capabilities in small component sizes. The firm supplies terminations in most major package styles (see figure). These include conduction- and convection-cooled coaxial terminations with SMA connectors for use at power levels to 250 W from DC to 4 GHz, and the same power rating through 3 GHz with Type-N and TNC coaxial connectors.
The power-handling capabilities drop with increasing frequency, to about 50 W for SMA terminations operating to 18 GHz. The firm offers chip terminations based on its BeO substrates rated to 15 W at microwave frequencies.
Another material building block for high-power terminations is aluminum oxide, A2O3, also known as alumina, long a favorite substrate for high-power passive RF/microwave components. As an example, the chip resistors fabricated by US Microwaves on alumina substrates can also be used as chip terminations at power levels beyond 100 W through microwave frequencies.
The material supports a wide operating temperature range, from -65 to +200°C. Similarly, aluminum nitride material is effective for thermal dissipation, and is often used in packaging for high-power attenuators and terminations.
In spite of the thermal advantages of composite materials, higher power levels will require larger terminations to safely dissipate heat from a high-frequency design. Material advances have made possible some impressive power ratings for chip and SMT resistors, terminations, and attenuators. Nevertheless, higher-power applications, such as communications transmitters and radar systems, will still require the largest terminations and attenuators, usually with waveguide flanges for consistent dissipation of power levels that often exceed 1 kW CW.
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Connecting to a SQL Server CE Database
SQL Server 2000
Before you can manipulate information in a database, you must open a connection to a valid data source. The Connection object is used to represent a connection to a data source. To open a connection to a data source, create a variable that represents the connection, and then create a Microsoft® ActiveX® Data Object for Windows® CE (ADOCE) Connection object by using the Set statement and CreateObject function. The following example shows how to do this:
Dim cn As ADOCE.Connection
Set cn = CreateObject("ADOCE.Connection.3.1")
Note When you use the CreateObject function to create a reference to the ADOCE 3.1 control, you must include the version number. If the version number is omitted from the string, an earlier version of the control is used. If no earlier version of the control exists on the device, an error is returned. Microsoft SQL Server™ 2000 Windows CE Edition (SQL Server CE) can be accessed only through ADOCE 3.1 or later.
After a Connection object is created, you can use the properties and methods of the Connection object to open, close, and manipulate a connection. The following example shows how to open a connection to a database on the device by using the Open method:
cn.ConnectionString = "Provider=Microsoft.SQLSERVER.OLEDB.CE.2.0; data source=\Northwind.sdf"
cn.Open
Caution You must specify the SQL Server CE provider string when you open a SQL Server CE database. If you do not specify a provider string in the Open method, Open defaults to using the proprietary Windows CE data source and creates a new Windows CE data source file named Test.sdf. This is the equivalent of specifying CEDB for the Provider property in the connection string.
In the previous sample, the connection string property is set before the Open method is executed. The Open method is used without any parameters. A connection string can also be used as a parameter of the Open method. When connecting to a SQL Server CE database, you must specify both the provider and data source properties in the connection string. The data source property must be set with the full path and database name.
Disconnecting from a Database
After you make modifications and save them to the database, close the connection to the data source. The following example shows how to use the Close method to close a connection:
cn.Close
Set cn = Nothing
Note You can have only one open connection to a SQL Server CE database at a time, and this connection must be closed before starting replication or remote data access (RDA).
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Lymphatic Circulation
Lymph travels through a network of small and large channels that are in some ways similar to the blood vessels. However, the system is not a complete circuit. It is a oneway system that begins in the tissues and ends when the lymph joins the blood (see Fig. 12-1).
Lymphatic Capillaries
The walls of the lymphatic capillaries resemble those of the blood capillaries in that they are made of one layer of flattened (squamous) epithelial cells. This thin layer, also called endothelium, allows for easy passage of soluble materials and water (Fig. 12-3). The gaps between the endothelial cells in the lymphatic capillaries are larger than those of the blood capillaries. The lymphatic capillaries are thus more permeable, allowing for easier entrance of relatively large protein particles. The proteins do not move back out of the vessels because the endothelial cells overlap slightly, forming one-way valves to block their return. Unlike the blood capillaries, the lymphatic capillaries arise blindly; that is, they are closed at one end and do not form a bridge between two larger vessels. Instead, one end simply lies within a lake of tissue fluid, and the other communicates with a larger lymphatic vessel that transports the lymph toward the heart (see Figs. 12-1 and 12-2).
Some specialized lymphatic capillaries located in the lining of the small intestine absorb digested fats. Fats taken into these lacteals are transported in the lymphatic vessels until the lymph is added to the blood.
The lymphatic system in relation to the cardiovascular system
Pathway of lymphatic drainage in the tissues
Figure 12-2 Pathway of lymphatic drainage in the tissues. Lymphatic capillaries are more permeable than blood capillaries and can pick up fluid and proteins left in the tissues as blood leaves the capillary bed to travel back toward the heart.
Structure of a lymphatic capillary
Figure 12-3 Structure of a lymphatic capillary. Fluid and proteins can enter the capillary with ease through gaps between the endothelial cells. Overlapping cells act as valves to prevent the material from leaving.
Lymphatic Vessels
The lymphatic vessels are thin walled and delicate and have a beaded appearance because of indentations where valves are located (see Fig. 12-1). These valves prevent back flow in the same way as do those found in some veins. Lymphatic vessels (Fig. 12-4) include superficial and deep sets. The surface lymphatics are immediately below the skin, often lying near the superficial veins. The deep vessels are usually larger and accompany the deep veins. Lymphatic vessels are named according to location. For example, those in the breast are called mammary lymphatic vessels, those in the thigh are called femoral lymphatic vessels, and those in the leg are called tibial lymphatic vessels. At certain points, the vessels drain through lymph nodes, small masses of lymphatic tissue that filter the lymph. The nodes are in groups that serve a particular region. For example, nearly all the lymph from the upper extremity and the breast passes through the axillary lymph nodes, whereas lymph from the lower extremity passes through the inguinal nodes. Lymphatic vessels carrying lymph away from the regional nodes eventually drain into one of two terminal vessels, the right lymphatic duct or the thoracic duct, both of which empty into the bloodstream.
Figure 12-1 The lymphatic system in relation to the cardiovascular system. Lymphatic vessels pick up fluid in the tissues and return it to the blood in vessels near the heart.
Figure 12-4 Vessels and nodes of the lymphatic system. (A) Lymph nodes and vessels of the head. (B) Drainage of right lymphatic duct and thoracic duct into subclavian veins.
The Right Lymphatic Duct The right lymphatic duct is a short vessel, approximately 1.25 cm (1/2 inch) long, that receives only the lymph that comes from the superior right quadrant of the body: the right side of the head, neck, and thorax, as well as the right upper extremity. It empties into the right subclavian vein near the heart (see Fig. 12-4 B). Its opening into this vein is guarded by two pocket-like semilunar valves to prevent blood from entering the duct. The rest of the body is drained by the thoracic duct.
The Thoracic Duct The thoracic duct, or left lymphatic
duct, is the larger of the two terminal vessels, measuring approximately 40 cm (16 inches) in length. As shown in
Figure 12-4, the thoracic duct receives lymph from all parts of the body except those superior to the diaphragm on the right side. This duct begins in the posterior part of the abdominal cavity, inferior to the attachment of the diaphragm. The first part of the duct is enlarged to form a cistern, or temporary storage pouch, called the cisterna chyli. Chyle is the milky fluid that drains from the intestinal lacteals, and is formed by the combination of fat globules and lymph. Chyle passes through the intestinal lymphatic vessels and the lymph nodes of the mesentery (membrane around the intestines), finally entering the cisterna chyli. In addition to chyle, all the lymph from below the diaphragm empties into the cisterna chyli, passing through the various clusters of lymph nodes. The thoracic duct then carries this lymph into the bloodstream. The thoracic duct extends upward through the diaphragm and along the posterior wall of the thorax into the base of the neck on the left side.
Here, it receives the left jugular lymphatic vessels from the head and neck, the left subclavian vessels from the left upper extremity, and other lymphatic vessels from the thorax and its parts. In addition to the valves along the duct, there are two valves at its opening into the left subclavian vein to prevent the passage of blood into the duct.
Movement of Lymph
The segments of lymphatic vessels located between the valves contract rhythmically, propelling the lymph along. The contraction rate is related to the volume of fluid in the vessel the more fluid, the more rapid the contractions. Lymph is also moved by the same mechanisms that promote venous return of blood to the heart. As skeletal muscles contract during movement, they compress the lymphatic vessels and drive lymph forward. Changes in pressures within the abdominal and thoracic cavities caused by breathing aid the movement of lymph during passage through these body cavities.
Contacts: [email protected] www.encyclopedia.lubopitko-bg.com Corporation. All rights reserved.
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How to plot a graph in Python
Python provides one of a most popular plotting library called Matplotlib. It is open-source, cross-platform for making 2D plots for from data in array. It is generally used for data visualization and represent through the various graphs.
Matplotlib is originally conceived by the John D. Hunter in 2003. The recent version of matplotlib is 2.2.0 released in January 2018.
Before start working with the matplotlib library, we need to install in our Python environment.
Installation of Matplotlib
Type the following command in your terminal and press enter.
The above command will install matplotlib library and its dependency package on Window operating system.
Basic Concept of Matplotlib
A graph contains the following parts. Let's understand these parts.
How to plot a graph in Python
Figure: It is a whole figure which may hold one or more axes (plots). We can think of a Figure as a canvas that holds plots.
Axes: A Figure can contain several Axes. It consists of two or three (in the case of 3D) Axis objects. Each Axes is comprised of a title, an x-label, and a y-label.
Axis: Axises are the number of line like objects and responsible for generating the graph limits.
Artist: An artist is the all which we see on the graph like Text objects, Line2D objects, and collection objects. Most Artists are tied to Axes.
Introduction to pyplot
The matplotlib provides the pyplot package which is used to plot the graph of given data. The matplotlib.pyplot is a set of command style functions that make matplotlib work like MATLAB. The pyplot package contains many functions which used to create a figure, create a plotting area in a figure, decorates the plot with labels, plot some lines in a plotting area, etc.
We can plot a graph with pyplot quickly. Let's have a look at the following example.
Basic Example of plotting Graph
Here is the basic example of generating a simple graph; the program is following:
Output:
How to plot a graph in Python
Ploting Different Type of Graphs
We can plot the various graph using the pyplot module. Let's understand the following examples.
1. Line Graph
The line chart is used to display the information as a series of the line. It is easy to plot. Consider the following example.
Example -
Output:
The line can be modified using the various functions. It makes the graph more attractive. Below is the example.
Example -
2. Bar Graph
Bar graph is one of the most common graphs and it is used to represent the data associated with the categorical variables. The bar() function accepts three arguments - categorical variables, values, and color.
Example -
3. Pie Chart
A chart is a circular graph which is divided into the sub-part or segment. It is used to represent the percentage or proportional data where each slice of pie represents a particular category. Let's understand the below example.
Example -
Output:
How to plot a graph in Python
4. Histogram
The histogram and bar graph is quite similar but there is a minor difference them. A histogram is used to represent the distribution, and bar chart is used to compare the different entities. A histogram is generally used to plot the frequency of a number of values compared to a set of values ranges.
In the following example, we have taken the data of the different score percentages of the student and plot the histogram with respect to number of student. Let's understand the following example.
Example -
Output:
How to plot a graph in Python
Let's understand another example.
Example - 2:
Output:
How to plot a graph in Python
5. Scatter Plot
The scatter plot is used to compare the variable with respect to the other variables. It is defined as how one variable affected the other variable. The data is represented as a collection of points. Let's understand the following example.
Example -
Output:
How to plot a graph in Python
Example - 2:
Output:
How to plot a graph in Python
In this tutorial, we have discussed all basic types of graph which used in data visualization. To learn more about graph, visit our matplotlib tutorial.
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BIOGRAPHY
Professor and Director, Division of Otolaryngology, University of Mississippi Medical Center, Jackson, 1963–79. Coauthor of Voice, Speech, Language; Clinical Communicology.
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Persons with profound hearing impairment rely on cues from sight, sound, and touch for communication.
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Metallomacrocycles as ligands: Synthesis and characterisation of aluminium-bridged bisglyoximato complexes of palladium and iron
Paul Kelley, Madalyn R. Radlauer, Abraham J. Yanez, Michael W. Day, Theodor Agapie
Research output: Contribution to journalArticlepeer-review
5 Scopus citations
Abstract
Dialuminiummacrocycles based on bisglyoximato moieties were prepared and their coordination chemistry with Fe II and Pd II was investigated. The bridging aluminium centers were supported by several types of tetradentate diphenoxide diamine ligands. The nature of the ancillary ligands bound to aluminium was found to affect the overall geometry and symmetry of the metallomacrocycles. Enantiopure, chiral diphenoxide ligands based on the (R,R)-trans-1,2-diaminocyclohexane backbone afforded cleanly one metallomacrocycle isomer. The size and electronic properties of remote substituents on aluminium-bound ligands affected the binding mode and electronic properties of the central iron. A structurally characterized iron complex shows trigonal prismatic coordination mode, with phenoxide bridges between iron and aluminium. Increasing the size of the phenoxide substituents led to square bipyramidal coordination at iron. Employing p-NO 2- instead of p-tBu-substituted phenoxide as supporting ligands for aluminium caused a 0.27 V positive shift of the Fe III/Fe II reduction potential. These results indicate that the present synthetic approach can be applied to a variety of metallomacrocycles based on bisglyoximato motifs to affect the chemistry at the central metal.
Original languageEnglish (US)
Pages (from-to)8086-8092
Number of pages7
JournalDalton Transactions
Volume41
Issue number26
DOIs
StatePublished - Jul 14 2012
Externally publishedYes
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Transform Data
Transform data between time and frequency domains
Functions
fftTransform iddata object to frequency domain data
ifftTransform iddata objects from frequency to time domain
etfeEstimate empirical transfer functions and periodograms
spaEstimate frequency response with fixed frequency resolution using spectral analysis
spafdrEstimate frequency response and spectrum using spectral analysis with frequency-dependent resolution
Examples and How To
Transform Time-Domain Data in the App
Transform time-domain data to frequency-domain or frequency-response data.
Transform Frequency-Domain Data in the App
Transform frequency-domain input-output data to time-domain or frequency-response data.
Transform Frequency-Response Data in the App
Transform frequency-response data to frequency-domain input-output data or to frequency-response data with a different frequency resolution.
Concepts
Supported Data Transformations
Transform between time-domain and frequency-domain data at the command line.
Transforming Between Time and Frequency-Domain Data
Transform between time-domain and frequency-domain iddata objects at the command line.
Transforming Between Frequency-Domain and Frequency-Response Data
Transform between iddata and idfrd objects at the command line.
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video播放器全屏兼容方案
Github上有两个video的插件维护的比较积极，在Github里搜索video，排序选择最高star的，关于video播放器的分别是video.jsmediaelement，虽然video.js的数目很多，但我想只是因为它这个项目的名称起得好，所以大家搜索video的内容时，Github总是第一位推荐，而mediaelement却没法出现在Github的那个推荐搜索里。
我们的网站使用的是mediaelement集成的，里面有很多很实用的插件，其中关于video全屏的方案兼容性做得很好，比起video.js那个插件，不支持ios Safari全屏播放，考虑PC方面的比较多，而mediaelement的那个Fullscreen.js就写得比较全了。我提取了里面的一些代码，总结如下：
1.首先检查是否支持浏览器自带的全屏方法。
github上有一个fullscreen.js的api很全，mediaelement就是使用这个类似的方法，当然video.js这个也是使用上面这个。
用法很简单，引入js后，screenfull 就是一个全局变量。我们可以通过定义一个按钮点击后出发这个全屏api.代码如下：
if (screenfull.enabled) {
screenfull.toggle(target);
screenfull.on('change', () = >{
if (screenfull.isFullscreen) {
$('body').addClass('fullscreen');
} else {
$('body').removeClass('fullscreen');
}
});
}
还有更多用法可以参考官网：https://github.com/sindresorhus/screenfull.js
2.当不支持上面的用法时，我们还可以继续检测是否支持苹果Safari自带的video全屏api。
var element = $('#video video')[0]; //video DOM
if (element.webkitEnterFullscreen || element.enterFullScreen) {
element.webkitEnterFullscreen && element.webkitEnterFullscreen();
element.enterFullScreen && element.enterFullScreen();
}
为什么这个代码有用而且必须加呢？因为iphone上，微信和Safari都不支持第一种的浏览器api，但支持这个全屏api，所以我们使用这个api实现了iphone下面video标签的全屏。
3.当然是以上两种方法都不支持的情况下，我们就只能模拟video全屏了，模拟的意思就是只能实现样式上看起来全屏，但实际浏览器自带的头部和尾部都没法隐藏，不像上面这两种api，当全屏状态下，浏览器的上下导航是会隐藏的。
function mockFullscreen(curEl) {
var wrapperEl = $('#video .video_wrap');
var playerEl = $('#video video');
if (curEl.hasClass('normal')) {
playerObj.fullscreen = false;
$('body').removeClass('fullscreen');
curEl.removeClass('normal');
} else {
playerObj.fullscreen = true;
$('body').addClass('fullscreen');
curEl.addClass('normal');
}
}
所以，模拟video控件的全屏，就是上面这三种写法了，结合起来就能实现各个平台的全屏效果了。
完整代码：
<div id="video">
<button class="fscreen J_play">播放视频</button>
<button class="fscreen J_fscreen">video全屏</button>
<div class="video_wrap"><video id="video_js_palyer" preload="auto" autoplay="autoplay" playsinline="true" webkit-playsinline="true" x-webkit-airplay="true" x5-video-player-type="h5" x5-video-player-fullscreen="true" x5-video-orientation="portraint" x5-video-ignore-metadata="true" style="width: 100%; object-fit: contain;" src="//auto.pcvideo.com.cn/pcauto/vpcauto/2018/07/05/1530785606876-vpcauto-78188-1_3.mp4"></video></div>
</div>
<script type="text/javascript" src="https://js.3conline.com/pcvideo/2017/wap/live/v2/fullscreen.js" charset="gbk"></script>
<script src="https://js.3conline.com/min/temp/v1/lib-jquery1.10.2.js"></script>
<script type="text/javascript">
var Features = {};
var target = $('#video')[0]; // Get DOM element from jQuery collection
var element = $('#video video')[0];
var NAV = window.navigator;
var UA = NAV.userAgent.toLowerCase();
Features.IS_IOS = /ipad|iphone|ipod/i.test(UA) && !window.MSStream;
// iOS
Features.hasiOSFullScreen = (element.webkitEnterFullscreen !== undefined);
// W3C
Features.hasNativeFullscreen = (element.requestFullscreen !== undefined);
// OS X 10.5 can't do this even if it says it can :(
if (Features.hasiOSFullScreen && /mac os x 10_5/i.test(UA)) {
Features.hasNativeFullscreen = false;
Features.hasiOSFullScreen = false;
}
var IS_CHROME = /chrome/i.test(UA);
if (IS_CHROME) {
Features.hasiOSFullScreen = false;
}
var playerObj = {};
$('.J_play').on('click',function(){
var self = $(this);
if(!self.hasClass('playing')){
element.play();
self.addClass('playing');
}else{
element.pause();
self.removeClass('playing');
}
});
element.addEventListener('pause',function(){
if(!$('.J_play').hasClass('playing')){
$('.J_play').addClass('playing');
}
});
element.addEventListener('pause',function(){
if($('.J_play').hasClass('playing')){
$('.J_play').removeClass('playing');
}
});
element.addEventListener('ended',function(){
if($('.J_play').hasClass('playing')){
$('.J_play').removeClass('playing');
}
})
$(document).on('click','.J_fscreen',function(){
var curEl = $(this);
curEl.html('触发全屏');
if(!$('.J_play').hasClass('playing')){
$('.J_play').trigger('click');
}
enterFullScreen();
function enterFullScreen(){
if(Features.IS_IOS && Features.hasiOSFullScreen && typeof element.webkitEnterFullscreen === 'function' && element.canPlayType('video/mp4')){
// alert(2);
Features.isiOSFullScreen = true;
console.log('ios全屏');
setTimeout(function(){
element.webkitEnterFullscreen();
},0);
return;
}
fakeFullScreen();
}
function fakeFullScreen(){
if(Features.isiOSFullScreen) return;
if (screenfull.enabled) {
console.log('浏览器全屏');
screenfull.toggle(target);
screenfull.on('change', () => {
if(screenfull.isFullscreen){
playerObj.isFullScreen = true;
$('body').addClass('body_fullscreen');
}else{
playerObj.isFullScreen = false;
$('body').removeClass('body_fullscreen');
}
});
}else{
console.log('伪全屏');
_mockFullscreen();
}
}
function exitFullscreen(){
fakeFullScreen();
}
function _mockFullscreen() {
var wrapperEl = $('#video .video_wrap');
var playerEl = $('#video video');
if (curEl.hasClass('fullscreen_on')) {
playerObj.isFullScreen = false;
$('body').removeClass('body_fullscreen');
curEl.removeClass('fullscreen_on');
} else {
playerObj.isFullScreen = true;
$('body').addClass('body_fullscreen');
curEl.addClass('fullscreen_on');
}
}
})
</script>
演示：http://caibaojian.com/demo/2018/8/video-fullscreen.html
原创文章：video播放器全屏兼容方案 ，未经许可，禁止转载，©版权所有
原文出处：前端开发博客 (http://caibaojian.com/video-screenfull.html)
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Skip to content
IFileInfo
This lets you iterate every available tag in a handle without knowing their name in advance.
Example
var handle = fb.GetFocusItem();
var obj = {};
var f = handle.GetFileInfo();
for (var i = 0; i < f.MetaCount; i++) {
var name = f.MetaName(i).toUpperCase();
obj[name] = [];
var num = f.MetaValueCount(i);
for (var j = 0; j < num; j++) {
obj[name].push(f.MetaValue(i, j));
}
}
console.log(JSON.stringify(obj, null, 4));
Example output
{
"ALBUM": [
"Chairlift at 6:15"
],
"ARTIST": [
"Chairlift"
],
"DATE": [
"2012-10-28"
],
"MUSICBRAINZ_ALBUMID": [
"bc96af2e-11e9-4abe-a75b-2b91a5eff027"
],
"MUSICBRAINZ_ARTISTID": [
"a3cd61ef-7fd4-44af-a27f-99641a82b22b"
],
"MUSICBRAINZ_RELEASEGROUPID": [
"cb2114a7-87fb-44ea-8931-766b75840683"
],
"MUSICBRAINZ_RELEASETRACKID": [
"008578a0-3188-31e5-887a-27194cecb069"
],
"MUSICBRAINZ_TRACKID": [
"57632bd4-185d-40be-9c95-c3d690c697af"
],
"RELEASETYPE": [
"EP"
],
"TITLE": [
"I Belong in Your Arms (Japanese version)"
],
"TOTALTRACKS": [
"6"
],
"TRACKNUMBER": [
"6"
]
}
Properties
MetaCount number read
InfoCount number read
Example
console.log(f.MetaCount);
console.log(f.InfoCount);
Methods
Dispose()#
No return value.
InfoFind(name)#
Arguments
name string
Returns a number to indicate the info index or -1 on failure.
InfoName(idx)#
Arguments
idx number
Returns a string.
InfoValue(idx)#
Arguments
idx number
Returns a string.
MetaFind(name)#
Arguments
name string
Returns a number to indicate the metadata index or -1 on failure.
MetaName(idx)#
Arguments
idx number
Returns a string.
Note
The case of the tag name returned can be different depending on tag type so using toLowerCase() or toUpperCase() on the result is recommended.
MetaValue(idx, vidx)#
Arguments
idx number
vidx number
Returns a string.
MetaValueCount(idx)#
Arguments
idx number
Returns a number.
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This page in other versions: Latest (6.9) | 6.8 | 6.7 | 6.6 | 6.5 | Development
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Navigation
Code Snippets
This document contains code for some of the important classes, listed as below:
PgAdminModule
PgAdminModule is inherited from Flask.Blueprint module. This module defines a set of methods, properties and attributes, that every module should implement.
class PgAdminModule(Blueprint):
"""
Base class for every PgAdmin Module.
This class defines a set of method and attributes that
every module should implement.
"""
def __init__(self, name, import_name, **kwargs):
kwargs.setdefault('url_prefix', '/' + name)
kwargs.setdefault('template_folder', 'templates')
kwargs.setdefault('static_folder', 'static')
self.submodules = []
self.parentmodules = []
super(PgAdminModule, self).__init__(name, import_name, **kwargs)
def create_module_preference():
# Create preference for each module by default
if hasattr(self, 'LABEL'):
self.preference = Preferences(self.name, self.LABEL)
else:
self.preference = Preferences(self.name, None)
self.register_preferences()
# Create and register the module preference object and preferences for
# it just before the first request
self.before_app_first_request(create_module_preference)
def register_preferences(self):
# To be implemented by child classes
pass
def register(self, app, options):
"""
Override the default register function to automagically register
sub-modules at once.
"""
self.submodules = list(app.find_submodules(self.import_name))
super(PgAdminModule, self).register(app, options)
for module in self.submodules:
module.parentmodules.append(self)
if app.blueprints.get(module.name) is None:
app.register_blueprint(module)
app.register_logout_hook(module)
def get_own_stylesheets(self):
"""
Returns:
list: the stylesheets used by this module, not including any
stylesheet needed by the submodules.
"""
return []
def get_own_messages(self):
"""
Returns:
dict: the i18n messages used by this module, not including any
messages needed by the submodules.
"""
return dict()
def get_own_javascripts(self):
"""
Returns:
list: the javascripts used by this module, not including
any script needed by the submodules.
"""
return []
def get_own_menuitems(self):
"""
Returns:
dict: the menuitems for this module, not including
any needed from the submodules.
"""
return defaultdict(list)
def get_panels(self):
"""
Returns:
list: a list of panel objects to add
"""
return []
def get_exposed_url_endpoints(self):
"""
Returns:
list: a list of url endpoints exposed to the client.
"""
return []
@property
def stylesheets(self):
stylesheets = self.get_own_stylesheets()
for module in self.submodules:
stylesheets.extend(module.stylesheets)
return stylesheets
@property
def messages(self):
res = self.get_own_messages()
for module in self.submodules:
res.update(module.messages)
return res
@property
def javascripts(self):
javascripts = self.get_own_javascripts()
for module in self.submodules:
javascripts.extend(module.javascripts)
return javascripts
@property
def menu_items(self):
menu_items = self.get_own_menuitems()
for module in self.submodules:
for key, value in module.menu_items.items():
menu_items[key].extend(value)
menu_items = dict((key, sorted(value, key=attrgetter('priority')))
for key, value in menu_items.items())
return menu_items
@property
def exposed_endpoints(self):
res = self.get_exposed_url_endpoints()
for module in self.submodules:
res += module.exposed_endpoints
return res
NodeView
The NodeView class exposes basic REST APIs for different operations used by the pgAdmin Browser. The basic idea has been taken from Flask’s MethodView class. Because we need a lot more operations (not, just CRUD), we can not use it directly.
class NodeView(View, metaclass=MethodViewType):
"""
A PostgreSQL Object has so many operaions/functions apart from CRUD
(Create, Read, Update, Delete):
i.e.
- Reversed Engineered SQL
- Modified Query for parameter while editing object attributes
i.e. ALTER TABLE ...
- Statistics of the objects
- List of dependents
- List of dependencies
- Listing of the children object types for the certain node
It will used by the browser tree to get the children nodes
This class can be inherited to achieve the diffrent routes for each of the
object types/collections.
OPERATION | URL | HTTP Method | Method
---------------+-----------------------------+-------------+--------------
List | /obj/[Parent URL]/ | GET | list
Properties | /obj/[Parent URL]/id | GET | properties
Create | /obj/[Parent URL]/ | POST | create
Delete | /obj/[Parent URL]/id | DELETE | delete
Update | /obj/[Parent URL]/id | PUT | update
SQL (Reversed | /sql/[Parent URL]/id | GET | sql
Engineering) |
SQL (Modified | /msql/[Parent URL]/id | GET | modified_sql
Properties) |
Statistics | /stats/[Parent URL]/id | GET | statistics
Dependencies | /dependency/[Parent URL]/id | GET | dependencies
Dependents | /dependent/[Parent URL]/id | GET | dependents
Nodes | /nodes/[Parent URL]/ | GET | nodes
Current Node | /nodes/[Parent URL]/id | GET | node
Children | /children/[Parent URL]/id | GET | children
NOTE:
Parent URL can be seen as the path to identify the particular node.
i.e.
In order to identify the TABLE object, we need server -> database -> schema
information.
"""
operations = dict({
'obj': [
{'get': 'properties', 'delete': 'delete', 'put': 'update'},
{'get': 'list', 'post': 'create'}
],
'nodes': [{'get': 'node'}, {'get': 'nodes'}],
'sql': [{'get': 'sql'}],
'msql': [{'get': 'modified_sql'}],
'stats': [{'get': 'statistics'}],
'dependency': [{'get': 'dependencies'}],
'dependent': [{'get': 'dependents'}],
'children': [{'get': 'children'}]
})
@classmethod
def generate_ops(cls):
cmds = []
for op in cls.operations:
idx = 0
for ops in cls.operations[op]:
meths = []
for meth in ops:
meths.append(meth.upper())
if len(meths) > 0:
cmds.append({
'cmd': op, 'req': (idx == 0),
'with_id': (idx != 2), 'methods': meths
})
idx += 1
return cmds
# Inherited class needs to modify these parameters
node_type = None
# Inherited class needs to modify these parameters
node_label = None
# This must be an array object with attributes (type and id)
parent_ids = []
# This must be an array object with attributes (type and id)
ids = []
@classmethod
def get_node_urls(cls):
assert cls.node_type is not None, \
"Please set the node_type for this class ({0})".format(
str(cls.__class__.__name__))
common_url = '/'
for p in cls.parent_ids:
common_url += '<{0}:{1}>/'.format(str(p['type']), str(p['id']))
id_url = None
for p in cls.ids:
id_url = '{0}<{1}:{2}>'.format(
common_url if not id_url else id_url,
p['type'], p['id'])
return id_url, common_url
def __init__(self, **kwargs):
self.cmd = kwargs['cmd']
# Check the existance of all the required arguments from parent_ids
# and return combination of has parent arguments, and has id arguments
def check_args(self, **kwargs):
has_id = has_args = True
for p in self.parent_ids:
if p['id'] not in kwargs:
has_args = False
break
for p in self.ids:
if p['id'] not in kwargs:
has_id = False
break
return has_args, has_id and has_args
def dispatch_request(self, *args, **kwargs):
http_method = flask.request.method.lower()
if http_method == 'head':
http_method = 'get'
assert self.cmd in self.operations, \
'Unimplemented command ({0}) for {1}'.format(
self.cmd,
str(self.__class__.__name__)
)
has_args, has_id = self.check_args(**kwargs)
assert (
self.cmd in self.operations and
(has_id and len(self.operations[self.cmd]) > 0 and
http_method in self.operations[self.cmd][0]) or
(not has_id and len(self.operations[self.cmd]) > 1 and
http_method in self.operations[self.cmd][1]) or
(len(self.operations[self.cmd]) > 2 and
http_method in self.operations[self.cmd][2])
), \
'Unimplemented method ({0}) for command ({1}), which {2} ' \
'an id'.format(http_method,
self.cmd,
'requires' if has_id else 'does not require')
meth = None
if has_id:
meth = self.operations[self.cmd][0][http_method]
elif has_args and http_method in self.operations[self.cmd][1]:
meth = self.operations[self.cmd][1][http_method]
else:
meth = self.operations[self.cmd][2][http_method]
method = getattr(self, meth, None)
if method is None:
return make_json_response(
status=406,
success=0,
errormsg=gettext(
'Unimplemented method ({0}) for this url ({1})').format(
meth, flask.request.path
)
)
return method(*args, **kwargs)
@classmethod
def register_node_view(cls, blueprint):
cls.blueprint = blueprint
id_url, url = cls.get_node_urls()
commands = cls.generate_ops()
for c in commands:
cmd = c['cmd'].replace('.', '-')
if c['with_id']:
blueprint.add_url_rule(
'/{0}{1}'.format(
c['cmd'], id_url if c['req'] else url
),
view_func=cls.as_view(
'{0}{1}'.format(
cmd, '_id' if c['req'] else ''
),
cmd=c['cmd']
),
methods=c['methods']
)
else:
blueprint.add_url_rule(
'/{0}'.format(c['cmd']),
view_func=cls.as_view(
cmd, cmd=c['cmd']
),
methods=c['methods']
)
def children(self, *args, **kwargs):
"""Build a list of treeview nodes from the child nodes."""
children = self.get_children_nodes(*args, **kwargs)
# Return sorted nodes based on label
return make_json_response(
data=sorted(
children, key=lambda c: c['label']
)
)
def get_children_nodes(self, *args, **kwargs):
"""
Returns the list of children nodes for the current nodes. Override this
function for special cases only.
:param args:
:param kwargs: Parameters to generate the correct set of tree node.
:return: List of the children nodes
"""
children = []
for module in self.blueprint.submodules:
children.extend(module.get_nodes(*args, **kwargs))
return children
BaseDriver
class BaseDriver(object):
"""
class BaseDriver(object):
This is a base class for different server types.
Inherit this class to implement different type of database driver
implementation.
(For PostgreSQL/EDB Postgres Advanced Server, we will be using psycopg2)
Abstract Properties:
-------- ----------
* Version (string):
Current version string for the database server
* libpq_version (string):
Current version string for the used libpq library
Abstract Methods:
-------- -------
* get_connection(*args, **kwargs)
- It should return a Connection class object, which may/may not be
connected to the database server.
* release_connection(*args, **kwargs)
- Implement the connection release logic
* gc()
- Implement this function to release the connections assigned in the
session, which has not been pinged from more than the idle timeout
configuration.
"""
@abstractproperty
def version(cls):
pass
@abstractproperty
def libpq_version(cls):
pass
@abstractmethod
def get_connection(self, *args, **kwargs):
pass
@abstractmethod
def release_connection(self, *args, **kwargs):
pass
@abstractmethod
def gc_timeout(self):
pass
BaseConnection
class BaseConnection(object):
"""
class BaseConnection(object)
It is a base class for database connection. A different connection
drive must implement this to expose abstract methods for this server.
General idea is to create a wrapper around the actual driver
implementation. It will be instantiated by the driver factory
basically. And, they should not be instantiated directly.
Abstract Methods:
-------- -------
* connect(**kwargs)
- Define this method to connect the server using that particular driver
implementation.
* execute_scalar(query, params, formatted_exception_msg)
- Implement this method to execute the given query and returns single
datum result.
* execute_async(query, params, formatted_exception_msg)
- Implement this method to execute the given query asynchronously and
returns result.
* execute_void(query, params, formatted_exception_msg)
- Implement this method to execute the given query with no result.
* execute_2darray(query, params, formatted_exception_msg)
- Implement this method to execute the given query and returns the result
as a 2 dimensional array.
* execute_dict(query, params, formatted_exception_msg)
- Implement this method to execute the given query and returns the result
as an array of dict (column name -> value) format.
* def async_fetchmany_2darray(records=-1, formatted_exception_msg=False):
- Implement this method to retrieve result of asynchronous connection and
polling with no_result flag set to True.
This returns the result as a 2 dimensional array.
If records is -1 then fetchmany will behave as fetchall.
* connected()
- Implement this method to get the status of the connection. It should
return True for connected, otherwise False
* reset()
- Implement this method to reconnect the database server (if possible)
* transaction_status()
- Implement this method to get the transaction status for this
connection. Range of return values different for each driver type.
* ping()
- Implement this method to ping the server. There are times, a connection
has been lost, but - the connection driver does not know about it. This
can be helpful to figure out the actual reason for query failure.
* _release()
- Implement this method to release the connection object. This should not
be directly called using the connection object itself.
NOTE: Please use BaseDriver.release_connection(...) for releasing the
connection object for better memory management, and connection pool
management.
* _wait(conn)
- Implement this method to wait for asynchronous connection to finish the
execution, hence - it must be a blocking call.
* _wait_timeout(conn, time)
- Implement this method to wait for asynchronous connection with timeout.
This must be a non blocking call.
* poll(formatted_exception_msg, no_result)
- Implement this method to poll the data of query running on asynchronous
connection.
* cancel_transaction(conn_id, did=None)
- Implement this method to cancel the running transaction.
* messages()
- Implement this method to return the list of the messages/notices from
the database server.
* rows_affected()
- Implement this method to get the rows affected by the last command
executed on the server.
"""
ASYNC_OK = 1
ASYNC_READ_TIMEOUT = 2
ASYNC_WRITE_TIMEOUT = 3
ASYNC_NOT_CONNECTED = 4
ASYNC_EXECUTION_ABORTED = 5
ASYNC_TIMEOUT = 0.2
ASYNC_WAIT_TIMEOUT = 2
ASYNC_NOTICE_MAXLENGTH = 100000
@abstractmethod
def connect(self, **kwargs):
pass
@abstractmethod
def execute_scalar(self, query, params=None,
formatted_exception_msg=False):
pass
@abstractmethod
def execute_async(self, query, params=None,
formatted_exception_msg=True):
pass
@abstractmethod
def execute_void(self, query, params=None,
formatted_exception_msg=False):
pass
@abstractmethod
def execute_2darray(self, query, params=None,
formatted_exception_msg=False):
pass
@abstractmethod
def execute_dict(self, query, params=None,
formatted_exception_msg=False):
pass
@abstractmethod
def async_fetchmany_2darray(self, records=-1,
formatted_exception_msg=False):
pass
@abstractmethod
def connected(self):
pass
@abstractmethod
def reset(self):
pass
@abstractmethod
def transaction_status(self):
pass
@abstractmethod
def ping(self):
pass
@abstractmethod
def _release(self):
pass
@abstractmethod
def _wait(self, conn):
pass
@abstractmethod
def _wait_timeout(self, conn, time):
pass
@abstractmethod
def poll(self, formatted_exception_msg=True, no_result=False):
pass
@abstractmethod
def status_message(self):
pass
@abstractmethod
def rows_affected(self):
pass
@abstractmethod
def cancel_transaction(self, conn_id, did=None):
pass
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Home iPhone How to turn on the LED notification light on your iPhone?
How to turn on the LED notification light on your iPhone?
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By activating the LED Flash for Alerts bulunan feature on the iPhone, you can make the flash blink when the phone rings or when a notification is received. Although the flash light for notifications is actually for the visually impaired, it is very useful for many people. For iOS 13 and later, we will explain in detail how to use the flash light as a notification light.
After the iOS 13 update, the Accessibility menu was removed from the General tab and became a stand-alone menu. The LED Flash feature for alerts is also carried under a different menu. In this article, we will illustrate how to turn on the flash on the iPhone and iPad models using iOS 13 and above, when the phone rings or when the notification arrives.
How to turn on the LED notification light on your iPhone?
1. Open the Settings section.
2. Select Accessibility.
3. Select Audio / Visual.
4. Activate the LED Flash option for alerts. This will cause the backlight to blink when the phone rings or when you receive a notification.
5. If you select Blink When Quiet in the same menu, the flash will also flash when your phone is silent.
Using the flash light as a notification light will often not miss calls and notifications. It also adds visual elegance.
If you are having problems with the flash when the iPhone is playing, you can ask us in the comments section.
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The OSI Model: An Essential Foundation to Networking
Updated: Mar 26
No technology that is connected to internet is un-hackable. It's only a matter of time.
Introduction
What is the OSI Model? The OSI, short for (the Open Systems Interconnection model) is a conceptual framework for understanding how network communication works. It was the first standard model adopted by all major computer and telecommunication companies in the early 1980s.
The OSI Model (aka ISO-OSI, i.e., International Organization of Standardization – Open System Interconnection) divides the communication process between two devices into seven layers. It provides a standard reference model that allows different networking technologies and protocols to interoperate and communicate.
Scenario
Imagine you have two servers that need to share information. The message doesn't just magically teleport from an application on the first machine to the application on the other. Instead, it transits down the layers and eventually reaches the transmission line. Once it jumps across the gap to the other device, it has to repeat the process in reverse by ascending layers until it reaches the receiving application.
Core Definition
For any starting number N representing a layer that transmits a message, the OSI model can be used to explain the transmission few key concepts:
• Protocol Data Units (PDUs) are abstracted messages that include payloads, headers, and footers.
• Service Data Units (SDUs) are equivalent to the payloads.
At each subsequent transition from some layer N to some layer N-1, a layer-N PDU becomes a new N-1 SDU. This payload gets wrapped up in a layer N-1 PDU with the relevant headers and footers. On the opposite end, the data passes up the chain, unwrapping at each relevant stage until it's just a payload that the corresponding layer-N device can consume.
The 7 Layers of OSI
7 Layers of the OSI Model
7 Layers of the OSI Model
We'll describe OSI layers "top-down" from the application layer that directly serves the end user to the physical layer.
7. The Application Layer
• The application layer is the highest layer of the OSI Model and is responsible for providing the interface between the network and the end user's application.
• Standard network services such as file transfer, email, and web browsing are provided at the application layer. Protocols such as HTTPS (Hypertext Transfer Protocol Secure) and, FTP (File Transfer Protocol), SMTP (Simple Mail Transfer Protocol) operate at this layer, allowing users to access and transfer files and other resources over the network.
Functions of the Application Layer
Functions of the Application Layer
• The application layer also provides the interface for user authentication and authorization. Protocols such as LDAP (Lightweight Directory Access Protocol) and Kerberos are used to verify the identity of users and grant them access to specific resources or services on the network.
6. The Presentation Layer
• The presentation layer is responsible for formatting and encoding data in a standardized way independent of the application or system being used. It includes protocols like SSL (Secure Sockets Layer) that provide secure communication.
• It deals with issues such as data compression and encryption.
• An example of a presentation service would be converting an extended binary-coded decimal interchange code text computer file to an ASCII-coded file. The presentation layer could translate between multiple data formats using a standard format if necessary.
Functions of the Presentation Layer
Functions of the Presentation Layer
5. The Session Layer
• The session layer establishes, maintains, and terminates connections between devices. Some standard protocols that operate at the session layer include Remote Procedure Call (RPC), NetBIOS (Network Basic Input Output System), and Windows Internet Name Service (WINS).
Functions of the Session Layer
Functions of the Session Layer
Some standard functions of the session layer include :
• Setting up and tearing down communication sessions between devices.
• Synchronizing the flow of data between devices.
• Resuming communication after a temporary interruption or fault.
• Negotiating the options and parameters for a communication session.
• Managing access to shared resources during a communication session.
4. The Transport Layer
• The transport layer provides end-to-end communication services and error recovery for the application layer. It includes protocols like TCP (Transmission Control Protocol) and (UDP) User Datagram Protocol that provides error correction, flow control, and data segmentation and reassembly.
• Every protocol uses a unique decimal number to ensure that the data is sent and received on the intended application as it passes through the network or Internet.
Functions of the Transport Layer
Functions of the Transport Layer
• TCP is a connection-oriented protocol that guarantees the delivery of the message, while UDP is a connectionless protocol that sends the data without error correction. Under the TCP and UDP are port numbers used to distinguish the specific type of application.
3. The Network Layer
• The network layer is responsible for routing data between different networks. It includes protocols like (IP) Internet Protocol, (IPX) Internetwork Packet Exchange, and AppleTalk. These protocols provide the necessary functions for routing data across a network and ensuring it reaches its destination.
• It is responsible for determining the best path for data as it travels from its source to its destination. The network layer also assigns logical addresses to devices on the network, which are used to identify the devices and route data to them.
Functions of the Network Layer
Functions of the Network Layer
• The network layer is often considered the "heart" of the OSI model because it plays a central role in the operation of a network. It is a critical component of modern computer networks and is essential for allowing devices to communicate with each other and exchange information.
2. The Data Link Layer
• The data link layer links two devices on the same physical network, such as a local area network (LAN). It ensures that data is transmitted correctly and without errors.
• It includes protocols like (SDLC) Synchronous Data Link Protocol, (HDLC) High-Level Data Link Protocol, (SLIP)Serial Line Interface Protocol, (PPP)Point - to - Point Protocol, (LCP) Link Control Protocol, and (NCP) Network Control Protocol.
• This layer comprises two parts—Logical Link Control (LLC), which identifies network protocols, performs error checking, and synchronizes frames. Media Access Control (MAC) uses MAC addresses to connect devices and define permissions to transmit and receive data.
Functions of the Data Link Layer
Functions of the Data Link Layer
• Overall, the data link layer is crucial in ensuring data's reliable and efficient transmission over a network.
1. The Physical Layer
• The physical layer is responsible for transmitting raw data over a communication channel, including the hardware, cables, and other components that make up the network.
• It defines the physical characteristics of the communication channel, including the signaling used, the frequency range, and the data rate.
Functions of the Physical Layer
Functions of the Physical Layer
• The physical layer ensures that data is transmitted accurately and reliably from one device to another.
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