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ber is the same as neutron number see also nuclear drip line if the proton or neutron is part of an atomic nucleus the above described decay processes transmute one chemical element into another for example 137 55 cs 137 56 ba e ν e beta minus decay 22 11 na 22 10 ne e ν e beta plus decay 22 11 na e 22 10 ne ν e 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 minimum 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 124 50 sn 124 52 te and 124 54 xe are all beta stable there are about 350 known beta decay stable nuclides 34 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 35 three types of beta decay in competition are illustrated by the single isotope 64 29 cu 29 protons 35 neutrons which has a half life of about 12 7 hours 34 this isotope has one unpaired proton and one unpaired neutron so either the proton or the neutron can decay 36 this particular nuclide is almost equally likely to undergo proton decay by positron emission 18 or by electron capture 43 both forming 64 ni or neutron decay by electron emission 39 forming 64 zn 34 36 stability of naturally occurring nuclides edit most naturally occurring nuclides on earth are beta stable nuclides that are not beta stable 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 40 19 k which undergoes all three types of beta decay β β and electron capture with a half life of 1 248 10 9 years 34 conservation rules for beta decay edit baryon number is conserved edit b n q n q 3 displaystyle b frac n_ text q n_ bar text q 3 where n q displaystyle n_ text q is the number of constituent quarks and n q displaystyle n_ overline text 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 doesn 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 textstyle 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 1 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 in the case of 187 re the maximum speed of the beta particle is only 9 8 of the speed of light the following table gives some examples examples of beta decay energies isotope energy kev decay mode free neutron 0 782 33 β 00 3 h tritium 00 18 59 β 0 11 c 0 960 4 1982 4 β ε 0 14 c 0 156 475 β 0 20 f 5390 86 β 0 37 k 5125 48 6147 48 β ε 163 ho 000 2 555 ε 187 re 000 2 467 β 210 bi 1162 2 β tritium β decay is being used in the katrin experimental search which could have detected sterile neutrinos 37 β decay edit consider the generic equation for beta decay a z x a z 1 x e ν e 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_ text n left ce mathit a _ mathit z x right m_ text n left ce mathit a _ mathit z 1 x right m_ text e m_ overline nu _ text e right c 2 where m n x z a displaystyle m_ text n left ce mathit a _ mathit z x right is the mass of the nucleus of the a z x atom m e displaystyle m_ text e is the mass of the electron and m ν e displaystyle m_ overline nu _ text 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 n 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_ text n left ce mathit a _ mathit z x right c 2 zm_ text 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 i for the atom this equation is rearranged to find m n x z a displaystyle m_ text n left ce mathit a _ mathit z x right and m n x z 1 a displaystyle m_ text 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 antineutrino because the reaction will proceed only when the q value is positive β decay can occur when the mass of atom a z x is greater than the mass of atom a z 1 x 38 β decay edit the equations for β decay are similar with the generic equation a z x a z 1 x e ν e giving q m n x z a m n x z 1 a m e m ν e c 2 displaystyle q left m_ text n left ce mathit a _ mathit z x right m_ text n left ce mathit a _ mathit z 1 x right m_ text e m_ nu _ text 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_ text 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 a z x exceeds that of a z 1 x by at least twice the mass of the electron 38 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 a z x e a z 1 x ν e 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_ text n left ce mathit a _ mathit z x right m_ text e m_ text n left ce mathit a _ mathit z 1 x right m_ nu _ text 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_ text n where b n 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 38 beta emission spectrum edit beta spectrum of 210 bi e max 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 39 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 l 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 2 is the total energy p e c 2 m c 2 displaystyle p sqrt e c 2 mc 2 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 210 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 40 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 p is the final momentum γ the gamma function and if α is the fine structure constant and r n the radius of the final state nucleus s 1 α 2 z 2 displaystyle s sqrt 1 alpha 2 z 2 η z e 2 e ℏ c p displaystyle eta pm ze 2 e hbar cp for electrons for positrons and ρ r n ℏ displaystyle rho r_ text n hbar for non relativistic betas q m e c 2 this expression can be approximated by 41 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 42 43 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 momentum or energy lies within a certain narrow range divided by the fermi function is plotted against beta particle energy 44 45 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 46 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 47 conversely positrons have mostly positive helicity i e they move like right handed screws neutrinos emitted in positron decay have negative helicity while antineutrinos emitted in electron decay have positive helicity 48 the higher the velocity energy of the particles the higher their helicity polarization this reflects the absolute chirality selection in w boson decay citation needed 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 _ text f g_ text v sum _ a hat tau _ a pm with g v displaystyle g_ text 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 gt g a a σ a τ a displaystyle mathcal o _ text gt g_ text a sum _ a hat sigma _ a hat tau _ a pm with g a displaystyle g_ text 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 main article beta decay transition 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 l th 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 δ π 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 δ j and δ π values for the first few values of l forbiddenness δ j δ π 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 bound state β decay edit a very small minority of fre...
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