1. 8-1 Beta Decay Neutrino Hypothesis Derivation of Spectral Shape Kurie Plots Beta Decay Rate Constant Selection Rules Transitions Majority of radioactive nuclei are outside range of alpha decay §Beta decay àSecond particle found from U decay *Negative particle *Distribution of energies *Need another particle to balance spin ØParent, daughter, and electron ØNeed to account for half integer spin

2. 8-2  -Decay Class includes any radioactive decay process in which A remains unchanged, but Z changes §  - decay, electron capture,  + decay §energetic conditions for decay: à  - decay: M Z  M Z+1 àElectron capture: M Z  M Z-1, à  + decay: M Z  M Z-1 +2m e For odd A, one  -stable nuclide; for even A, at most three  -stable nuclides §Information available from mass parabolas qualitative connection between half life and decay energy, although not as simple as case of  -decay §depends strongly on spin and parity changes in addition to available energy

3. 8-3  - decay: Electron capture:  + decay: Overview of Beta Decay

4. 8-4 The Neutrino Solved problems associated with  -decay §Zero charge à neutron -> proton + electron §Small mass àElectron goes up to Q value §Anti-particle àAccount for creation of electron particle §spin of ½ and obeys Fermi statistics àcouple the total final angular momentum to initial spin of ½ ħ Carries away the appropriate amount of energy and momentum in each  process to conserve these properties Nearly undetectable due to its small rest mass and magnetic moment §observed by inverse  processes Antineutrinos emitted in  - decay; neutrinos emitted in  + decay §indistinguishable properties, except in capture reactions “created” at the moment of emission §n  p +  - + §p  n +  + +

5. 8-5 Spin in Beta Decay Spins can be combined in two ways and still couple to the initial spin of the neutron. spins of the created particles  S   1 in a parallel alignment  S   0 in an anti-parallel alignment àcombine with S=1/2 of the neutron for a resultant vector of 1/2. two possible relative alignments of the "created" spins are Fermi (F) (S  =0) Gamow-Teller (GT) (S  =1) §a source will produce a mixture of relative spins decay of even-even nuclei with N=Z (mirror nuclei) §neutron and protons are in the same orbitals à 0+ to 0+ decay can only take place by a Fermi transition heavy nuclei with protons and neutrons in very different orbitals §GT is main mode complex nuclei §rate of decay will depend on the overlap of the wave functions of ground state of the parent and the state of the daughter. §final state in the daughter depends on the decay mode. decay constant can be calculated if wave functions are known §observed rate gives some indication of the quantum mechanical overlap of the initial and final state wave functions.

6. 8-6 Energetics Beta decay §electron can be combined with the positive ion to create a neutral atom àrelease of very small binding energy àuse neutral atoms to calculate the Q value *assuming that the mass of the antineutrino is very small Consider beta decay of 14 C § 14 C  14 N + + β - +antineutrino + energy àEnergy = mass 14 C – mass 14 N Positron decay §2 extra electrons (daughter less Z, emission of positron) Electron Capture

7. 8-7 Q value calculation Find Q value for the Beta decay of 24 Na §1 amu = 931.5 MeV §M ( 24 Na)-M( 24 Mg) à23.990962782-23.985041699 à 0.005921 amu *5.5154 MeV §From mass excess à-8.4181 - -13.9336 à5.5155 MeV Q value for the EC of 22 Na §M ( 22 Na)-M( 22 Ne) §21.994436425-21.991385113 §0.003051 amu à2.842297 MeV §From mass excess à-5.1824 - -8.0247 à2.8432 MeV

8. 8-8 Positrons Pair production-process that involves creation of a positron- electron pair by a photon §nucleus carries off some momentum and energy Positron-electron annihilation-the falling of an electron into a whole in the sea of electrons of negative energy with the simultaneous emission of the corresponding amount of energy in the form of radiation §accounts for short lifetime of positrons Annihilation radiation-energy carried off by two  quanta of opposite momentum §conserves momentum

9. 8-9 Decay energy probability P(p e )dp e probability electron with momentum p e +dp e  2 of allowed transitions   2 of disallowed transitions §allowed transitions are ones in which both electron and neutrino are emitted with zero orbital angular momentum Magnitudes of  (0)  and  M if  are independent of the division of energy between electron and neutrino M if orbital overlap of initial and final states g is Fermi constant determined by experiment Spectrum shape determined entirely by  e (0)  and dn/dE o §dn/dE o density of final states with electron momentum àcoulomb interaction between nucleus and emitted electron (  e (0)  ) can, for now, be neglected àDensity of final states determined from total energy W

10. 8-10 Fermi Golden Rule Consider allowed transition §No angular momentum change decay constant is given by Fermi's Golden Rule calculate §matrix element which couples the initial and final states §a phase space factor which describes the volume of phase space available for the outgoing leptons §Small system perturbation

11. 8-11 Kurie Plot Comparison of theory and experiment for momentum measurements

12. 8-12 Coulomb Correction Correction for Coulomb interaction between the nucleus and the emitted electron (  e (0)  ) §this interaction decelerates electrons and accelerates positrons àelectron spectra contains more low-energy particles than predicted by statistical considerations àpositron spectra contains fewer Treated as perturbation on  e (0) and spectrum multiplied by the Fermi function

13. 8-13 Beta decay spectra

14. 8-14 Comparative Half Lives f o t 1/2 is called the comparative half life of a transition §may be thought of as the half life corrected for differences in Z and W o Approximations:

15. 8-15 Beta decays

16. 8-16 Capture-to-positron ratio ( ) measures competition between electron capture and positron emission §ratio increases with decreasing decay energy K/L ratio §when energetically possible, capture of K(1s) electrons predominates over capture of electrons with higher principal quantum number §at decay energies below the binding energy of K electrons, EC is possible only from the L(2s+2p), M(3s,3p,3d),... Useful Ratios

17. 8-17 Extranuclear Effects of EC If K-shell vacancy is filled by L electron, difference in binding energies emitted as x-ray or used in internal photoelectric process §Auger electrons are additional extranuclear electrons from L, M, or another shell emitted with kinetic energy equal to characteristic x-ray energy minus its own binding energy Fluorescence yield is the fraction of vacancies in a shell that is filled with accompanying x-ray emission §important in measuring disintegration rates of EC nuclides àradiations most frequently detected are the x- rays

18. 8-18 Selection Rules Allowed transitions are ones in which the electron and neutrino carry away no orbital angular momentum §largest transition probability for a given energy release If the electron and neutrino do not carry off angular momentum, spins of initial and final nucleus differ by no more than h/2  and parities must be the same If electron and neutrino emitted with intrinsic spins antiparallel, nuclear spin change (  I )is zero §singlet If electron and neutrino spins are parallel,  I may be +1, 0, - 1 §triplet §0  0 transitions are forbidden

19. 8-19 Selection Rules All transitions between states of  I=0 or 1 with no change in parity have the allowed spectrum shape Not all these transitions have similar f o t values §transitions with low f o t values are “favored” or “superallowed” àfound among  emitters of low Z and between mirror nuclei (one contains n neutrons and n+1 protons, the other n+1 neutrons and n protons) §our assumption of approximately equal  M if  2 values for all transitions with  I=0,  1 without parity change was erroneous

20. 8-20 Forbidden Transitions When the transition from initial to final nucleus cannot take place by emission of s-wave electron and neutrino §orbital angular momenta other than zero l value associated with given transition deduced from indirect evidence §ft values, spectrum shapes If l is odd, initial and final nucleus have opposite parities If l is even, parities must be the same Emission of electron and nucleus in singlet state requires  I  l Triple-state emission allows  I  l+1

21. 8-21 Transitions