Bending of electron beams due to electrostatic interactions with matter
Electron diffraction is a generic term for phenomena associated with changes in the direction of electron beams due to elastic interactions with atoms. It occurs due to elastic scattering, when there is no change in the energy of the electrons.Chpt 4Chpt 5 The negatively charged electrons are scattered due to Coulomb forces when they interact with both the positively charged atomic core and the negatively charged electrons around the atoms. The resulting map of the directions of the electrons far from the sample is called a diffraction pattern, see for instance Figure 1. Beyond patterns showing the directions of electrons, electron diffraction also plays a major role in the contrast of images in electron microscopes.
This article provides an overview of electron diffraction and electron diffraction patterns, collective referred to by the generic name electron diffraction. This includes aspects of how in a general way electrons can act as waves, and diffract and interact with matter. It also involves the extensive history behind modern electron diffraction, how the combination of developments in the 19th century in understanding and controlling electrons in vacuum and the early 20th century developments with electron waves were combined with early instruments, giving birth to electron microscopy and diffraction in 1920–1935. While this was the birth, there have been a large number of further developments since then.
There are many types and techniques of electron diffraction. The most common approach is where the electrons transmit through a thin sample, from 1 nm to 100 nm (10 to 1000 atoms thick), where the results depending upon how the atoms are arranged in the material, for instance a single crystal, many crystals or different types of solids. Other cases such as larger repeats, no periodicity or disorder have their own characteristic patterns. There are many different ways of collecting diffraction information, from parallel illumination to a converging beam of electrons or where the beam is rotated or scanned across the sample which produce information that is often easier to interpret. There are also many other types of instruments. For instance, in a scanning electron microscope (SEM), electron backscatter diffraction can be used to determine crystal orientation across the sample. Electron diffraction patterns can also be used to characterize molecules using gas electron diffraction, liquids, surfaces using lower energy electrons, a technique called LEED, and by reflecting electrons off surfaces, a technique called RHEED.
There are also many levels of analysis of electron diffraction, including:
Electron diffraction is similar to x-ray and neutron diffraction. However, unlike x-ray and neutron diffraction where the simplest approximations are quite accurate, with electron diffraction this is not the case.Sec 3Chpt 5 Simple models give the geometry of the intensities in a diffraction pattern, but dynamical diffraction approaches are needed for accurate intensities and the positions of diffraction spots.
A primer on electron diffraction.
All matter can be thought of as matter waves,Chpt 1-3 from small particles such as electrons up to macroscopic objects – although it is impossible to measure any of the "wave-like" behavior of macroscopic objects. Waves can move around objects and create interference patterns,Chpt 7-8 and a classic example is the Young's two-slit experiment shown in Figure 2, where a wave impinges upon two slits in the first of the two images (blue waves). After going through the slits there are directions where the wave is stronger, ones where it is weaker – the wave has been diffracted.Chpt 1,7,8 If instead of two slits there are a number of small points then similar phenomena can occur as shown in the second image where the wave (red and blue) is coming in from the bottom right corner. This is comparable to diffraction of an electron wave where the small dots would be atoms in a small crystal, see also note. Note the strong dependence on the relative orientation of the crystal and the incoming wave.
Close to an aperture or atoms, often called the "sample", the electron wave would be described in terms of near field or Fresnel diffraction.Chpt 7-8 This has relevance for imaging within electron microscopes,Chpt 3Chpt 3-4 whereas electron diffraction patterns are measured far from the sample, which is described as far-field or Fraunhofer diffraction.Chpt 7-8 A map of the directions of the electron waves leaving the sample will show high intensity (white) for favored directions, such as the three prominent ones in the Young's two-slit experiment of Figure 2, while the other directions will be low intensity (dark). Often there will be an array of spots (preferred directions) as in Figure 1 and the other figures shown later.
History.
The historical background is divided into several subsections. The first is the general background to electrons in vacuum and the technological developments that led to cathode-ray tubes as well as vacuum tubes that dominated early television and electronics; the second is how these led to the development of electron microscopes; the last is work on the nature of electron beams and the fundamentals of how electrons behave, a key component of quantum mechanics and the explanation of electron diffraction.
Electrons in vacuum.
Experiments involving electron beams occurred long before the discovery of the electron; ēlektron (ἤλεκτρον) is the Greek word for amber, which is connected to the recording of electrostatic charging by Thales of Miletus around 585 BCE, and possibly others even earlier.
In 1650, Otto von Guericke invented the vacuum pump allowing for the study of the effects of high voltage electricity passing through rarefied air. In 1838, Michael Faraday applied a high voltage between two metal electrodes at either end of a glass tube that had been partially evacuated of air, and noticed a strange light arc with its beginning at the cathode (negative electrode) and its end at the anode (positive electrode). Building on this, in the 1850s, Heinrich Geissler was able to achieve a pressure of around 10−3 atmospheres, inventing what became known as Geissler tubes. Using these tubes, while studying electrical conductivity in rarefied gases in 1859, Julius Plücker observed that the radiation emitted from the negatively charged cathode caused phosphorescent light to appear on the tube wall near it, and the region of the phosphorescent light could be moved by application of a magnetic field.
In 1869, Plücker's student Johann Wilhelm Hittorf found that a solid body placed between the cathode and the phosphorescence would cast a shadow on the tube wall, e.g. Figure 3. Hittorf inferred that there are straight rays emitted from the cathode and that the phosphorescence was caused by the rays striking the tube walls. In 1876 Eugen Goldstein showed that the rays were emitted perpendicular to the cathode surface, which differentiated them from the incandescent light. Eugen Goldstein dubbed them cathode rays. By the 1870s William Crookes and others were able to evacuate glass tubes below 10−6 atmospheres, and observed that the glow in the tube disappeared when the pressure was reduced but the glass behind the anode began to glow. Crookes was also able to show that the particles in the cathode rays were negatively charged and could be deflected by an electromagnetic field.
In 1897, Joseph Thomson measured the mass of these cathode rays, proving they were made of particles. These particles, however, were 1800 times lighter than the lightest particle known at that time – a hydrogen atom. These were originally called "corpuscles" and later named electrons by George Johnstone Stoney.
The control of electron beams that this work led to resulted in significant technology advances in electronic amplifiers and television displays.
Waves, diffraction and quantum mechanics.
Independent of the developments for electrons in vacuum, at about the same time the components of quantum mechanics were being assembled. In 1924 Louis de Broglie in his PhD thesis "Recherches sur la théorie des quanta" introduced his theory of electron waves. He suggested that an electron around a nucleus could be thought of as standing waves, and that electrons and all matter could be considered as waves. He merged the idea of thinking about them as particles (or corpuscles), and of thinking of them as waves. He proposed that particles are bundles of waves (wave packets) that move with a group velocityChpt 1-2 and have an effective mass, see for instance Figure 4. Both of these depend upon the energy, which in turn connects to the wavevector and the relativistic formulation of Albert Einstein a few years before.
This rapidly became part of what was called by Erwin Schrödinger "undulatory mechanics", now called the Schrödinger equation or wave mechanics. As stated by Louis de Broglie on September 8, 1927, in the preface to the German translation of his theses (in turn translated into English):v"M. Einstein from the beginning has supported my thesis, but it was M. E. Schrödinger who developed the propagation equations of a new theory and who in searching for its solutions has established what has become known as “Wave Mechanics”."
The Schrödinger equation combines the kinetic energy of waves and the potential energy due to, for electrons, the Coulomb potential. He was able to explain earlier work such as the quantization of the energy of electrons around atoms in the Bohr model, as well as many other phenomena. Electron waves as hypothesizedChpt 1-2 by de Broglie were automatically part of the solutions to his equation, see also introduction to quantum mechanics and matter waves.
Both the wave nature and the undulatory mechanics approach were experimentally confirmed for electron beams by experiments from two groups performed independently, the first the Davisson–Germer experiment, the other by George Paget Thomson and Alexander Reid; see note for more discussion. Alexander Reid, who was Thomson's graduate student, performed the first experiments, but he died soon after in a motorcycle accident and is rarely mentioned. These experiments were rapidly followed by the first non-relativistic diffraction model for electrons by Hans Bethe based upon the Schrödinger equation, which is very close to how electron diffraction is now described. Significantly, Clinton Davisson and Lester Germer noticed that their results could not be interpreted using a Bragg's law approach as the positions were systematically different; the approach of Hans Bethe which includes the refraction due to the average potential yielded more accurate results. These advances in understanding of electron wave mechanics were important for many developments of electron-based analytical techniques such as Seishi Kikuchi's observations of lines due to combined elastic and inelastic scattering, gas electron diffraction developed by Herman Mark and Raymond Weil, diffraction in liquids by Louis Maxwell, and the first electron microscopes developed by Max Knoll and Ernst Ruska.
Electron microscopes and early electron diffraction.
In order to have a practical microscope or diffractometer, just having an electron beam was not enough, it needed to be controlled. Many developments laid the groundwork of electron optics; see the paper by Chester J. Calbick for an overview of the early work. One significant step was the work of Heinrich Hertz in 1883 who made a cathode-ray tube with electrostatic and magnetic deflection, demonstrating manipulation of the direction of an electron beam. Others were focusing of electrons by an axial magnetic field by Emil Wiechert in 1899, improved oxide-coated cathodes which produced more electrons by Arthur Wehnelt in 1905 and the development of the electromagnetic lens in 1926 by Hans Busch.
Building an electron microscope involves combining these elements, similar to an optical microscope but with magnetic or electrostatic lenses instead of glass ones. To this day the issue of who invented the transmission electron microscope is controversial, as discussed by Thomas Mulvey and more recently by Yaping Tao. Extensive additional information can be found in the articles by Martin Freundlich, Reinhold Rüdenberg and Mulvey.
One effort was university based. In 1928, at the Technische Hochschule in Charlottenburg (now Technische Universität Berlin), Adolf Matthias (Professor of High Voltage Technology and Electrical Installations) appointed Max Knoll to lead a team of researchers to advance research on electron beams and cathode-ray oscilloscopes. The team consisted of several PhD students including Ernst Ruska. In 1931, Max Knoll and Ernst Ruska successfully generated magnified images of mesh grids placed over an anode aperture. The device, a replicate of which is shown in Figure 5, used two magnetic lenses to achieve higher magnifications, the first electron microscope. (Max Knoll died in 1969, so did not receive a share of the Nobel Prize in Physics in 1986.)
Apparently independent of this effort was work at Siemens-Schuckert by Reinhold Rudenberg. According to patent law (U.S.