Beta decay - wikipedia Beta decay Jump to : navigation , search β decay in an atomic nucleus ( the accompanying antineutrino is omitted ) . The inset shows beta decay of a free neutron . In both processes , an intermediate virtual boson is not shown . 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Bohr N. Bohr Chadwick Cockcroft Ir. Curie Fr. Curie Pi. Curie Skłodowska - Curie Davisson Fermi Hahn Jensen Lawrence Mayer Meitner Oliphant Oppenheimer Proca Purcell Rabi Rutherford Soddy Strassmann Szilárd Teller Thomson Walton Wigner In nuclear physics , beta decay ( β - decay ) is a type of radioactive decay in which a beta ray ( fast energetic electron or positron ) and a neutrino are emitted from an atomic nucleus . For example , beta decay of a neutron transforms it into a proton by the emission of an electron , or conversely a proton is converted into a neutron by the emission of a positron ( positron emission ) , thus changing the nuclide type . Neither the beta particle nor its associated neutrino exist within the nucleus prior to beta decay , but are created in the decay process . By this process , unstable atoms obtain a more stable ratio of protons to neutrons . The probability of a nuclide decaying due to beta and other forms of decay is determined by its nuclear binding energy . The binding energies of all existing nuclides form what is called the nuclear band or valley of stability . For either electron or positron emission to be energetically possible , the energy release ( see below ) or Q value must be positive . Beta decay is a consequence of the weak force , which is characterized by relatively lengthy decay times . Nucleons are composed of up or down quarks , and the weak force allows a quark to change type by the exchange of a W boson and the creation of an electron / antineutrino or positron / neutrino pair . For example , a neutron , composed of two down quarks and an up quark , decays to a proton composed of a down quark and two up quarks . Decay times for many nuclides that are subject to beta decay can be thousands of years . Electron capture is sometimes included as a type of beta decay , because the basic nuclear process , mediated by the weak force , is the same . In electron capture , an inner atomic electron is captured by a proton in the nucleus , transforming it into a neutron , and an electron neutrino is released . Contents ( hide ) 1 Description 2 History 2.1 Discovery and initial characterization 2.2 Neutrinos 2.3 β decay and electron capture 2.4 Non-conservation of parity 3 β decay 4 β decay 5 Electron capture ( K - capture ) 6 Nuclear transmutation 6.1 Competition of beta decay types 6.2 Stability of naturally occurring nuclides 7 Conservation rules for beta decay 7.1 Baryon number is conserved 7.2 Lepton number is conserved 7.3 Angular momentum 8 Energy release 8.1 β decay 8.2 β decay 8.3 Electron capture 9 Beta emission spectrum 9.1 Fermi function 9.2 Kurie plot 10 Helicity ( polarization ) of neutrinos , electrons and positrons emitted in beta decay 11 Types of beta decay transitions 11.1 Fermi transitions 11.2 Gamow - Teller transitions 11.3 Forbidden transitions 12 Rare decay modes 12.1 Bound - state β decay 12.2 Double beta decay 13 See also 14 References 15 Bibliography 16 External links Description ( edit ) The two types of beta decay are known as beta minus and beta plus . In beta minus ( β ) decay , a neutron is converted to a proton and the process creates an electron and an electron antineutrino ; while in beta plus ( β ) decay , a proton is converted to a neutron and the process creates a positron and an electron neutrino . β decay is also known as positron emission . Beta decay conserves a quantum number known as the lepton number , or the number of electrons and their associated neutrinos ( other leptons are the muon and tau particles ) . These particles have lepton number + 1 , while their antiparticles have lepton number − 1 . Since a proton or neutron has lepton number zero , β decay ( a positron , or antielectron ) must be accompanied with an electron neutrino , while β decay ( an electron ) must be accompanied by an electron antineutrino . An example of electron emission ( β decay ) is the decay of carbon - 14 into nitrogen - 14 with a half - life of about 5,730 years : C → + + ν In this form of decay , the original element becomes a new chemical element in a process known as nuclear transmutation . This new element has an unchanged mass number A , but an atomic number Z that is increased by one . As in all nuclear decays , the decaying element ( in this case ) is known as the parent nuclide while the resulting element ( in this case ) is known as the daughter nuclide . An example of positron emission is the decay of magnesium - 23 into sodium - 23 with a half - life of about 11.3 s : Mg → Na + + ν β decay also results in nuclear transmutation , with the resulting element having an atomic number that is decreased by one . A beta spectrum , showing a typical division of energy between electron and antineutrino The beta spectrum , or distribution of energy values for the beta particles , is continuous . The total energy of the decay process is divided between the electron , the antineutrino , and the recoiling nuclide . In the figure to the right , an example of an electron with 0.40 MeV energy from the beta decay of Bi is shown . In this example , the total decay energy is 1.16 MeV , so the antineutrino has the remaining energy : 1.16 - 0.40 = 0.76 MeV . An electron at the far right of the curve would have the maximum possible kinetic energy , leaving the energy of the neutrino to be only its small rest mass . History ( edit ) This section needs additional citations for verification . Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed . ( September 2014 ) ( Learn how and when to remove this template message ) Discovery and initial characterization ( edit ) Radioactivity was discovered in 1896 by Henri Becquerel in uranium , and subsequently observed by Marie and Pierre Curie in thorium and in the new elements polonium and radium . In 1899 , Ernest Rutherford separated radioactive emissions into two types : alpha and beta ( now beta minus ) , based on penetration of objects and ability to cause ionization . Alpha rays could be stopped by thin sheets of paper or aluminium , whereas beta rays could penetrate several millimetres of aluminium . In 1900 , Paul Villard identified a still more penetrating type of radiation , which Rutherford identified as a fundamentally new type in 1903 and termed gamma rays . Alpha , beta , and gamma are the first three letters of the Greek alphabet . In 1900 , Becquerel measured the mass - to - charge ratio ( m / e ) for beta particles by the method of J.J. Thomson used to study cathode rays and identify the electron . He found that m / e for a beta particle is the same as for Thomson 's electron , and therefore suggested that the beta particle is in fact an electron . In 1901 , Rutherford and Frederick Soddy showed that alpha and beta radioactivity involves the transmutation of atoms into atoms of other chemical elements . In 1913 , after the products of more radioactive decays were known , Soddy and Kazimierz Fajans independently proposed their radioactive displacement law , which states that beta ( i.e. , β ) emission from one element produces another element one place to the right in the periodic table , while alpha emission produces an element two places to the left . Neutrinos ( edit ) The study of beta decay provided the first physical evidence for the existence of the neutrino . In both alpha and gamma decay , the resulting particle has a narrow energy distribution , since the particle carries the energy from the difference between the initial and final nuclear states . However , the kinetic energy distribution , or spectrum , of beta particles measured by Lise Meitner and Otto Hahn in 1911 and by Jean Danysz in 1913 showed multiple lines on a diffuse background . These measurements offered the first hint that beta particles have a continuous spectrum . In 1914 , James Chadwick used a magnetic spectrometer with one of Hans Geiger 's new counters to make more accurate measurements which showed that the spectrum was continuous . The distribution of beta particle energies was in apparent contradiction to the law of conservation of energy . If beta decay were simply electron emission as assumed at the time , then the energy of the emitted electron should have a particular , well - defined value . For beta decay , however , the observed broad distribution of energies suggested that energy is lost in the beta decay process . This spectrum was puzzling for many years . A second problem is related to the conservation of angular momentum . Molecular band spectra showed that the nuclear spin of nitrogen - 14 is 1 ( i.e. equal to the reduced Planck constant ) and more generally that the spin is integral for nuclei of even mass number and half - integral for nuclei of odd mass number . This was later explained by the proton - neutron model of the nucleus . Beta decay leaves the mass number unchanged , so the change of nuclear spin must be an integer . However , the electron spin is 1 / 2 , hence angular momentum would not be conserved if beta decay were simply electron emission . From 1920 -- 1927 , Charles Drummond Ellis ( along with Chadwick and colleagues ) further established that the beta decay spectrum is continuous . In 1933 , Ellis and Nevill Mott obtained strong evidence that the beta spectrum has an effective upper bound in energy . Niels Bohr had suggested that the beta spectrum could be explained if conservation of energy was true only in a statistical sense , thus this principle might be violated in any given decay . However , the upper bound in beta energies determined by Ellis and Mott ruled out that notion . Now , the problem of how to account for the variability of energy in known beta decay products , as well as for conservation of momentum and angular momentum in the process , became acute . In a famous letter written in 1930 , Wolfgang Pauli attempted to resolve the beta - particle energy conundrum by suggesting that , in addition to electrons and protons , atomic nuclei also contained an extremely light neutral particle , which he called the neutron . He suggested that this `` neutron '' was also emitted during beta decay ( thus accounting for the known missing energy , momentum , and angular momentum ) , but it had simply not yet been observed . In 1931 , Enrico Fermi renamed Pauli 's `` neutron '' the `` neutrino '' ( roughly ' little neutral one ' in Italian ) . In 1934 , Fermi published his landmark theory for beta decay , where he applied the principles of quantum mechanics to matter particles , supposing that they can be created and annihilated , just as the light quanta in atomic transitions . Thus , according to Fermi , neutrinos are created in the beta - decay process , rather than contained in the nucleus ; the same happens to electrons . The neutrino interaction with matter was so weak that detecting it proved a severe experimental challenge . Further indirect evidence of the existence of the neutrino was obtained by observing the recoil of nuclei that emitted such a particle after absorbing an electron . Neutrinos were finally detected directly in 1956 by Clyde Cowan and Frederick Reines in the Cowan -- Reines neutrino experiment . The properties of neutrinos were ( with a few minor modifications ) as predicted by Pauli and Fermi . β Decay and electron capture ( edit ) In 1934 , Frédéric and Irène Joliot - Curie bombarded aluminium with alpha particles to effect the nuclear reaction He + Al → + n , and observed that the product isotope emits a positron identical to those found in cosmic rays ( discovered by Carl David Anderson in 1932 ) . This was the first example of β decay ( positron emission ) , which they termed artificial radioactivity since is a short - lived nuclide which does not exist in nature . In recognition of their discovery the couple were awarded the Nobel Prize in Chemistry in 1935 . The theory of electron capture was first discussed by Gian - Carlo Wick in a 1934 paper , and then developed by Hideki Yukawa and others . K - electron capture was first observed in 1937 by Luis Alvarez , in the nuclide V. Alvarez went on to study electron capture in Ga and other nuclides . Non-conservation of parity ( edit ) In 1956 , Tsung - Dao Lee and Chen Ning Yang noticed that there was no evidence that parity was conserved in weak interactions , and so they postulated that this symmetry may not be preserved by the weak force . They sketched the design for an experiment for testing conservation of parity in the laboratory . Later that year , Chien - Shiung Wu and coworkers conducted the Wu experiment showing an asymmetrical beta decay of cobalt - 60 at cold temperatures that proved that parity is not conserved in beta decay . This surprising result overturned long - held assumptions about parity and the weak force . In recognition of their theoretical work , Lee and Yang were awarded the Nobel Prize for Physics in 1957 . Β decay ( edit ) The leading - order Feynman diagram for β decay of a neutron into a proton , electron , and electron antineutrino via an intermediate boson . In β decay , the weak interaction converts an atomic nucleus into a nucleus with atomic number increased by one , while emitting an electron ( ) and an electron antineutrino ( ν ) . β decay generally occurs in neutron - rich nuclei . The generic equation is : X → X ' + + ν where A and Z are the mass number and atomic number of the decaying nucleus , and X and X ' are the initial and final elements , respectively . Another example is when the free neutron ( n ) decays by β decay into a proton ( p ) : n → p + + ν . At the fundamental level ( as depicted in the Feynman diagram on the right ) , this is caused by the conversion of the negatively charged ( − 1 / 3 e ) down quark to the positively charged ( + 2 / 3 e ) up quark by emission of a boson ; the boson subsequently decays into an electron and an electron antineutrino : d → u + + ν . Β decay ( edit ) Main article : Positron emission The leading - order Feynman diagram for β decay of a proton into a neutron , positron , and electron neutrino via an intermediate boson . In β decay , or `` positron emission '' , the weak interaction converts an atomic nucleus into a nucleus with atomic number decreased by one , while emitting a positron ( ) and an electron neutrino ( ν ) . β decay generally occurs in proton - rich nuclei . The generic equation is : X → X ' + + ν This may be considered as the decay of a proton inside the nucleus to a neutron p → n + + ν However , β decay can not occur in an isolated proton because it requires energy due to the mass of the neutron being greater than the mass of the proton . β decay can only happen inside nuclei when the daughter nucleus has a greater binding energy ( and therefore a lower total energy ) than the mother nucleus . The difference between these energies goes into the reaction of converting a proton into a neutron , a positron and a neutrino and into the kinetic energy of these particles . This process is opposite to negative beta decay , in that the weak interaction converts a proton into a neutron by converting an up quark into a down quark resulting in the emission of a or the absorption of a . Electron capture ( K - capture ) ( edit ) Main article : Electron capture In all cases where β decay ( positron emission ) of a nucleus is allowed energetically , so too is electron capture allowed . This is a process during which a nucleus captures one of its atomic electrons , resulting in the emission of a neutrino : X + → X ' + ν An example of electron capture is one of the decay modes of krypton - 81 into bromine - 81 : Kr + → Br + ν All emitted neutrinos are of the same energy . In proton - rich nuclei where the energy difference between the initial and final states is less than 2m c , β decay is not energetically possible , and electron capture is the sole decay mode . If the captured electron comes from the innermost shell of the atom , the K - shell , which has the highest probability to interact with the nucleus , the process is called K - capture . If it comes from the L - shell , the process is called L - capture , etc . Electron capture is a competing ( simultaneous ) decay process for all nuclei that can undergo β decay . The converse , however , is not true : electron capture is the only type of decay that is allowed in proton - rich nuclides that do not have sufficient energy to emit a positron and neutrino . Nuclear transmutation ( edit ) Main article : Nuclear drip line If the proton and neutron are part of an atomic nucleus , the above described decay processes transmute one chemical element into another . For example : Cs → Ba + + ν ( beta minus decay ) Na → Ne + + ν ( beta plus decay ) Na + → Ne + ν ( electron capture ) Beta decay does not change the number ( A ) of nucleons in the nucleus , but changes only its charge Z . Thus the set of all nuclides with the same A can be introduced ; these isobaric nuclides may turn into each other via beta decay . For a given A there is one that is most stable . It is said to be beta stable , because it presents a local minima of the mass excess : if such a nucleus has ( A , Z ) numbers , the neighbour nuclei ( A , Z − 1 ) and ( A , Z + 1 ) have higher mass excess and can beta decay into ( A , Z ) , but not vice versa . For all odd mass numbers A , there is only one known beta - stable isobar . For even A , there are up to three different beta - stable isobars experimentally known ; for example , Zr , Mo , and Ru are all beta - stable . There are about 355 known beta - decay stable nuclides . Competition of beta decay types ( edit ) Usually , unstable nuclides are clearly either `` neutron rich '' or `` proton rich '' , with the former undergoing beta decay and the latter undergoing electron capture ( or more rarely , due to the higher energy requirements , positron decay ) . However , in a few cases of odd - proton , odd - neutron radionuclides , it may be energetically favorable for the radionuclide to decay to an even - proton , even - neutron isobar either by undergoing beta - positive or beta - negative decay . An often - cited example is the single isotope Cu ( 29 protons , 35 neutrons ) , which illustrates three types of beta decay in competition . Copper - 64 has a half - life of about 12.7 hours . This isotope has one unpaired proton and one unpaired neutron , so either the proton or the neutron can decay . This particular nuclide ( though not all nuclides in this situation ) is almost equally likely to decay through proton decay by positron emission ( 18 % ) or electron capture ( 43 % ) to Ni , as it is through neutron decay by electron emission ( 39 % ) to Zn . Stability of naturally occurring nuclides ( edit ) Most naturally occurring nuclides on earth are beta stable . Those that are not have half - lives ranging from under a second to periods of time significantly greater than the age of the universe . One common example of a long - lived isotope is the odd - proton odd - neutron nuclide , which undergoes all three types of beta decay ( β , β and electron capture ) with a half - life of 7016402990552000000 ♠ 1.277 × 10 years . Conservation rules for beta decay ( edit ) Baryon number is conserved ( edit ) B = n q − n q  ̄ 3 ( \ displaystyle B = ( \ frac ( n_ ( q ) - n_ ( \ bar ( q ) ) ) ( 3 ) ) ) where n q ( \ displaystyle n_ ( q ) ) is the number of constituent quarks , and n q  ̄ ( \ displaystyle n_ ( \ overline ( q ) ) ) is the number of constituent antiquarks . Beta decay just changes neutron to proton or , in the case of positive beta decay ( electron capture ) proton to neutron so the number of individual quarks do n't change . It is only the baryon flavor that changes , here labelled as the isospin . Up and down quarks have total isospin I = 1 2 ( \ displaystyle I = ( \ frac ( 1 ) ( 2 ) ) ) and isospin projections I z = ( 1 2 up quark − 1 2 down quark ( \ displaystyle I_ ( \ text ( z ) ) = ( \ begin ( cases ) ( \ frac ( 1 ) ( 2 ) ) & ( \ text ( up quark ) ) \ \ - ( \ frac ( 1 ) ( 2 ) ) & ( \ text ( down quark ) ) \ end ( cases ) ) ) All other quarks have I = 0 . In general I z = 1 2 ( n u − n d ) ( \ displaystyle I_ ( \ text ( z ) ) = ( \ frac ( 1 ) ( 2 ) ) ( n_ ( \ text ( u ) ) - n_ ( \ text ( d ) ) ) ) Lepton number is conserved ( edit ) L ≡ n l − n l  ̄ ( \ displaystyle L \ equiv n_ ( \ ell ) - n_ ( \ bar ( \ ell ) ) ) so all leptons have assigned a value of + 1 , antileptons − 1 , and non-leptonic particles 0 . n → p + e − + ν  ̄ e L : 0 = 0 + 1 − 1 ( \ displaystyle ( \ begin ( matrix ) & ( \ text ( n ) ) & \ rightarrow & ( \ text ( p ) ) & + & ( \ text ( e ) ) ^ ( - ) & + & ( \ bar ( \ nu ) ) _ ( \ text ( e ) ) \ \ L : &0& = &0& + &1& - &1 \ end ( matrix ) ) ) Angular momentum ( edit ) For allowed decays , the net orbital angular momentum is zero , hence only spin quantum numbers are considered . The electron and antineutrino are fermions , spin - 1 / 2 objects , therefore they may couple to total S = 1 ( \ displaystyle S = 1 ) ( parallel ) or S = 0 ( \ displaystyle S = 0 ) ( anti-parallel ) . For forbidden decays , orbital angular momentum must also be taken into consideration . Energy release ( edit ) The Q value is defined as the total energy released in a given nuclear decay . In beta decay , Q is therefore also the sum of the kinetic energies of the emitted beta particle , neutrino , and recoiling nucleus . ( Because of the large mass of the nucleus compared to that of the beta particle and neutrino , the kinetic energy of the recoiling nucleus can generally be neglected . ) Beta particles can therefore be emitted with any kinetic energy ranging from 0 to Q. A typical Q is around 1 MeV , but can range from a few keV to a few tens of MeV . Since the rest mass of the electron is 511 keV , the most energetic beta particles are ultrarelativistic , with speeds very close to the speed of light . Β decay ( edit ) Consider the generic equation for beta decay X → X ' + + ν . The Q value for this decay is Q = ( m N ( X Z A ) − m N ( X Z + 1 A ′ ) − m e − m ν  ̄ e ) c 2 ( \ displaystyle Q = \ left ( m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) - m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( ( \ mathit ( Z ) ) + 1 ) X ' ) ) \ right ) - m_ ( e ) - m_ ( ( \ overline ( \ nu ) ) _ ( e ) ) \ right ) c ^ ( 2 ) ) , where m N ( X Z A ) ( \ displaystyle m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) ) is the mass of the nucleus of the X atom , m e ( \ displaystyle m_ ( e ) ) is the mass of the electron , and m ν  ̄ e ( \ displaystyle m_ ( ( \ overline ( \ nu ) ) _ ( e ) ) ) is the mass of the electron antineutrino . In other words , the total energy released is the mass energy of the initial nucleus , minus the mass energy of the final nucleus , electron , and antineutrino . The mass of the nucleus m is related to the standard atomic mass m by m ( X Z A ) c 2 = m N ( X Z A ) c 2 + Z m e c 2 − ∑ i = 1 Z B i ( \ displaystyle m \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) c ^ ( 2 ) = m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) c ^ ( 2 ) + Zm_ ( e ) c ^ ( 2 ) - \ sum _ ( i = 1 ) ^ ( Z ) B_ ( i ) ) . That is , the total atomic mass is the mass of the nucleus , plus the mass of the electrons , minus the sum of all electron binding energies B for the atom . This equation is rearranged to find m N ( X Z A ) ( \ displaystyle m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) ) , and m N ( X Z + 1 A ′ ) ( \ displaystyle m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( ( \ mathit ( Z ) ) + 1 ) X ' ) ) \ right ) ) is found similarly . Substituting these nuclear masses into the Q - value equation , while neglecting the nearly - zero antineutrino mass and the difference in electron binding energies , which is very small for high - Z atoms , we have Q = ( m ( X Z A ) − m ( X Z + 1 A ′ ) ) c 2 ( \ displaystyle Q = \ left ( m \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) - m \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( ( \ mathit ( Z ) ) + 1 ) X ' ) ) \ right ) \ right ) c ^ ( 2 ) ) This energy is carried away as kinetic energy by the electron and neutrino . Because the reaction will proceed only when the Q - value is positive , β decay can occur when the mass of atom X is greater than the mass of atom X ' . Β decay ( edit ) The equations for β decay are similar , with the generic equation X → X ' + + ν giving Q = ( m N ( X Z A ) − m N ( X Z − 1 A ′ ) − m e − m ν e ) c 2 ( \ displaystyle Q = \ left ( m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) - m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( ( \ mathit ( Z ) ) - 1 ) X ' ) ) \ right ) - m_ ( e ) - m_ ( \ nu _ ( e ) ) \ right ) c ^ ( 2 ) ) . However , in this equation , the electron masses do not cancel , and we are left with Q = ( m ( X Z A ) − m ( X Z − 1 A ′ ) − 2 m e ) c 2 ( \ displaystyle Q = \ left ( m \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) - m \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( ( \ mathit ( Z ) ) - 1 ) X ' ) ) \ right ) - 2m_ ( e ) \ right ) c ^ ( 2 ) ) Because the reaction will proceed only when the Q - value is positive , β decay can occur when the mass of atom X exceeds that of X ' by at least twice the mass of the electron . Electron capture ( edit ) The analogous calculation for electron capture must take into account the binding energy of the electrons . This is because the atom will be left in an excited state after capturing the electron , and the binding energy of the captured innermost electron is significant . Using the generic equation for electron capture X + → X ' + ν we have Q = ( m N ( X Z A ) + m e − m N ( X Z − 1 A ′ ) − m ν e ) c 2 ( \ displaystyle Q = \ left ( m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) + m_ ( e ) - m_ ( N ) \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( ( \ mathit ( Z ) ) - 1 ) X ' ) ) \ right ) - m_ ( \ nu _ ( e ) ) \ right ) c ^ ( 2 ) ) , which simplifies to Q = ( m ( X Z A ) − m ( X Z − 1 A ′ ) ) c 2 − B n ( \ displaystyle Q = \ left ( m \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( \ mathit ( Z ) ) X ) ) \ right ) - m \ left ( ( \ ce ( ^ ( \ mathit ( A ) ) _ ( ( \ mathit ( Z ) ) - 1 ) X ' ) ) \ right ) \ right ) c ^ ( 2 ) - B_ ( n ) ) , where B is the binding energy of the captured electron . Because the binding energy of the electron is much less than the mass of the electron , nuclei that can undergo β decay can always also undergo electron capture , but the reverse is not true . Beta emission spectrum ( edit ) Beta spectrum of Bi . E = Q = 1.16 MeV is the maximum energy Beta decay can be considered as a perturbation as described in quantum mechanics , and thus Fermi 's Golden Rule can be applied . This leads to an expression for the kinetic energy spectrum N ( T ) of emitted betas as follows : N ( T ) = C L ( T ) F ( Z , T ) p E ( Q − T ) 2 ( \ displaystyle N ( T ) = C_ ( L ) ( T ) F ( Z , T ) pE ( Q-T ) ^ ( 2 ) ) where T is the kinetic energy , C is a shape function that depends on the forbiddenness of the decay ( it is constant for allowed decays ) , F ( Z , T ) is the Fermi Function ( see below ) with Z the charge of the final - state nucleus , E = T + mc is the total energy , p = √ ( E / c ) − ( mc ) is the momentum , and Q is the Q value of the decay . The kinetic energy of the emitted neutrino is given approximately by Q minus the kinetic energy of the beta . As an example , the beta decay spectrum of Bi ( originally called RaE ) is shown to the right . Fermi function ( edit ) The Fermi function that appears in the beta spectrum formula accounts for the Coulomb attraction / repulsion between the emitted beta and the final state nucleus . Approximating the associated wavefunctions to be spherically symmetric , the Fermi function can be analytically calculated to be : F ( Z , T ) = 2 ( 1 + S ) Γ ( 1 + 2 S ) 2 ( 2 p ρ ) 2 S − 2 e π η Γ ( S + i η ) 2 , ( \ displaystyle F ( Z , T ) = ( \ frac ( 2 ( 1 + S ) ) ( \ Gamma ( 1 + 2S ) ^ ( 2 ) ) ) ( 2p \ rho ) ^ ( 2S - 2 ) e ^ ( \ pi \ eta ) \ Gamma ( S + i \ eta ) ^ ( 2 ) , ) where S = √ 1 − α Z ( α is the fine - structure constant ) , η = ± αZE / pc ( + for electrons , − for positrons ) , ρ = r / ħ ( r is the radius of the final state nucleus ) , and Γ is the Gamma function . For non-relativistic betas ( Q ≪ m c ) , this expression can be approximated by : F ( Z , T ) ≈ 2 π η 1 − e − 2 π η . ( \ displaystyle F ( Z , T ) \ approx ( \ frac ( 2 \ pi \ eta ) ( 1 - e ^ ( - 2 \ pi \ eta ) ) ) . ) Other approximations can be found in the literature . Kurie plot ( edit ) A Kurie plot ( also known as a Fermi -- Kurie plot ) is a graph used in studying beta decay developed by Franz N.D. Kurie , in which the square root of the number of beta particles whose momenta ( or energy ) lie within a certain narrow range , divided by the Fermi function , is plotted against beta - particle energy . It is a straight line for allowed transitions and some forbidden transitions , in accord with the Fermi beta - decay theory . The energy - axis ( x-axis ) intercept of a Kurie plot corresponds to the maximum energy imparted to the electron / positron ( the decay 's Q - value ) . With a Kurie plot one can find the limit on the effective mass of a neutrino . Helicity ( polarization ) of neutrinos , electrons and positrons emitted in beta decay ( edit ) After the discovery of parity non-conservation ( see History ) , it was found that , in beta decay , electrons are emitted mostly with negative helicity , i.e. , they move , naively speaking , like left - handed screws driven into a material ( they have negative longitudinal polarization ) . Conversely , positrons have mostly positive helicity , i.e. , they move like right - handed screws . Neutrinos ( emitted in positron decay ) have positive helicity , while antineutrinos ( emitted in electron decay ) have negative helicity . The higher the energy of the particles , the higher their polarization . Types of beta decay transitions ( edit ) Main article : Beta decay transition Beta decays can be classified according to the angular momentum ( L - value ) and total spin ( S - value ) of the emitted radiation . Since total angular momentum must be conserved , including orbital and spin angular momentum , beta decay occurs by a variety of quantum state transitions to various nuclear angular momentum or spin states , known as `` Fermi '' or `` Gamow - Teller '' transitions . When beta decay particles carry no angular momentum ( L = 0 ) , the decay is referred to as `` allowed '' , otherwise it is `` forbidden '' . Other decay modes , which are rare , are known as bound state decay and double beta decay . Fermi transitions ( edit ) A Fermi transition is a beta decay in which the spins of the emitted electron ( positron ) and anti-neutrino ( neutrino ) couple to total spin S = 0 ( \ displaystyle S = 0 ) , leading to an angular momentum change Δ J = 0 ( \ displaystyle \ Delta J = 0 ) between the initial and final states of the nucleus ( assuming an allowed transition ) . In the non-relativistic limit , the nuclear part of the operator for a Fermi transition is given by O F = G V ∑ a τ ^ a ± ( \ displaystyle ( \ mathcal ( O ) ) _ ( F ) = G_ ( V ) \ sum _ ( a ) ( \ hat ( \ tau ) ) _ ( a \ pm ) ) with G V ( \ displaystyle G_ ( V ) ) the weak vector coupling constant , τ ± ( \ displaystyle \ tau _ ( \ pm ) ) the isospin raising and lowering operators , and a ( \ displaystyle a ) running over all protons and neutrons in the nucleus . Gamow - Teller transitions ( edit ) A Gamow - Teller transition is a beta decay in which the spins of the emitted electron ( positron ) and anti-neutrino ( neutrino ) couple to total spin S = 1 ( \ displaystyle S = 1 ) , leading to an angular momentum change Δ J = 0 , ± 1 ( \ displaystyle \ Delta J = 0 , \ pm 1 ) between the initial and final states of the nucleus ( assuming an allowed transition ) . In this case , the nuclear part of the operator is given by O G T = G A ∑ a σ ^ a τ ^ a ± ( \ displaystyle ( \ mathcal ( O ) ) _ ( GT ) = G_ ( A ) \ sum _ ( a ) ( \ hat ( \ sigma ) ) _ ( a ) ( \ hat ( \ tau ) ) _ ( a \ pm ) ) with G A ( \ displaystyle G_ ( A ) ) the weak axial - vector coupling constant , and σ ( \ displaystyle \ sigma ) the spin Pauli matrices , which can produce a spin - flip in the decaying nucleon . Forbidden transitions ( edit ) When L > 0 , the decay is referred to as `` forbidden '' . Nuclear selection rules require high L - values to be accompanied by changes in nuclear spin ( J ) and parity ( π ) . The selection rules for the Lth forbidden transitions are : Δ J = L − 1 , L , L + 1 ; Δ π = ( − 1 ) L , ( \ displaystyle \ Delta J = L - 1 , L , L + 1 ; \ Delta \ pi = ( - 1 ) ^ ( L ) , ) where Δπ = 1 or − 1 corresponds to no parity change or parity change , respectively . The special case of a transition between isobaric analogue states , where the structure of the final state is very similar to the structure of the initial state , is referred to as `` superallowed '' for beta decay , and proceeds very quickly . The following table lists the ΔJ and Δπ values for the first few values of L : Forbiddenness ΔJ Δπ Superallowed 0 no Allowed 0 , 1 no First forbidden 0 , 1 , 2 yes Second forbidden 1 , 2 , 3 no Third forbidden 2 , 3 , 4 yes Rare decay modes ( edit ) Bound - state β decay ( edit ) A very small minority of free neutron decays ( about four per million ) are so - called `` two - body decays '' , in which the proton , electron and antineutrino are produced , but the electron fails to gain the 13.6 eV energy necessary to escape the proton , and therefore simply remains bound to it , as a neutral hydrogen atom . In this type of beta decay , in essence all of the neutron decay energy is carried off by the antineutrino . For fully ionized atoms ( bare nuclei ) , it is possible in likewise manner for electrons to fail to escape the atom , and to be emitted from the nucleus into low - lying atomic bound states ( orbitals ) . This can not occur for neutral atoms with low - lying bound states which are already filled by electrons . The phenomenon in fully ionized atoms was first observed for Dy in 1992 by Jung et al. of the Darmstadt Heavy - Ion Research group . Although neutral Dy is a stable isotope , the fully ionized Dy undergoes β decay into the K and L shells with a half - life of 47 days . Another possibility is that a fully ionized atom undergoes greatly accelerated β decay , as observed for Re by Bosch et al. , also at Darmstadt . Neutral Re does undergo β decay with a half - life of 42 × 10 years , but for fully ionized Re this is shortened by a factor of 10 to only 32.9 years . For comparison the variation of decay rates of other nuclear processes due to chemical environment is less than 1 % . Double beta decay ( edit ) Main article : Double beta decay Some nuclei can undergo double beta decay ( ββ decay ) where the charge of the nucleus changes by two units . Double beta decay is difficult to study , as the process has an extremely long half - life . In nuclei for which both β decay and ββ decay are possible , the rarer ββ decay process is effectively impossible to observe . However , in nuclei where β decay is forbidden but ββ decay is allowed , the process can be seen and a half - life measured . Thus , ββ decay is usually studied only for beta stable nuclei . Like single beta decay , double beta decay does not change A ; thus , at least one of the nuclides with some given A has to be stable with regard to both single and double beta decay . `` Ordinary '' double beta decay results in the emission of two electrons and two antineutrinos . 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Question: this type of nuclear decay effectively turns a proton into a neutron