P461 - decays II1 Beta Decays Beta decays are proton -> neutrons or neutron -> proton transitions involve W exchange and are weak interaction the last reaction is electron capture where one of the atomic electrons overlaps the nuclei. Same matrix element (essentially) bit different kinematics the semi-empirical mass formula gives a minimum for any A. If mass difference between neighbors is large enough, decay will occur

P461 - decays II2 Beta Decays - Q Values Determine Q of reactions by looking at mass difference 1 MeV more Q in EC than beta+ emission. More phase space BUT need electron wavefunction overlap with nucleus..

P461 - decays II3 Beta+ vs Electron Capture Fewer beta+ emitters than beta- in “natural” nuclei (but many in “artificial” important in Positron Emission Tomography - PET) sometimes both beta+ and EC for some nuclei. Different widths sometimes only EC allowed monoenergetic neutrino. E=.87 MeV. Important reaction in the Sun. Note EC rate different in Sun as it is a plasma and not atoms

P461 - decays II4 Beta Decay - 3 Body The neutrino is needed to conserve angular momentum (Z,A) -> (Z+1,A) for A=even have either Z,N even-even -> odd-odd or odd-odd->even-even p,n both spin 1/2 and so for even-even or odd-odd nuclei I=0,1,2,3……. But electron has spin 1/2 I(integer -> I(integer+ + 1/2(electron) doesn’t conserve J need spin 1/2 neutrino. Also observed that electron spectrum is continuous indicative of >2 body decay Pauli/Fermi understood this in 1930s electron neutrino discovered 1953 (Reines and Cowan) muon neutrino discovered 1962 (Schwartz +Lederman/Steinberger) tau neutrino discovered 2000 at Fermilab

P461 - decays II5 3 Body Kinematics While 3 body the nuclei are very heavy and easy approximation is that electron and neutrino split available Q (nuclei has similar momentum) maximum electron energy when E(nu)=0 example

P461 - decays II6 Beta decay rate Start from Fermi Golden Rule first approximation (Fermi). Beta=constant=strength of weak force Rule 1: parity of nucleus can’t change (integral of odd*even=0) Rule 2: as antineutrino and electron are spin 1/2 they add to either 0 or 1. Gives either

P461 - decays II7 Beta decay rate II Orbital angular momentum suppression of for each value of (in matrix element calculation) look at density of states factor. Want # quantum states per energy interval we know from quantum statistics that each particle (actually each spin state) has 3 body decay but recoil nucleus is so heavy it doesn’t contribute

P461 - decays II8 Beta decay rate III Conservation of energy allows one to integrate over the neutrino (there is a delta function) this gives a distribution in electron momentum/energy which one then integrates over. (end point depends on neutrino mass) F is a function which depends on Q. It is almost loqrithmic

P461 - decays II9 Beta decay rate IV FT is “just kinematics” measuring FT can study nuclear wavefunctions M’ and strength of the weak force at low energies lower values of FT are when M’ approaches 1 beta decays also occur for particles electron is now relativistic and E=pc. The integral is now easy to do. For massive particles (with decay masses small), Emax = M/2 and so rate goes as fifth power of mass