abg Decay Theory Previously looked at kinematics now

abg Decay Theory • Previously looked at kinematics now study dynamics (interesting bit). • QM tunnelling and a decays • Fermi theory of b decay and e. c. • g decays Tony Weidberg Nuclear Physics Lectures 1

a Decay Theory • Consider 232 Th Z=90 R=7. 6 fm E=34 Me. V • Energy of a Ea=4. 08 Me. V • Question: How does the a escape? • Answer: QM tunnelling Tony Weidberg Nuclear Physics Lectures 2

radial wave function in alpha decay i. II I i. I Exponential decay of y nucleus Tony Weidberg r barrier (negative KE) Nuclear Physics Lectures small flux of real α 3

QM Tunnelling V E 0 t • B. C. at x=0 and x=t for Kt>>1 and k~K gives for 1 D rectangular barrier thickness t gives T=|D|2=exp(-2 Kt) • Integrate over Coulomb barrier from r=R to r=t Tony Weidberg Nuclear Physics Lectures 4

a-decay DEsep≈6 Me. V per nucleon for heavy nuclei DEbind(42 a)=28. 3 Me. V > 4*6 Me. V Neutrons Tony Weidberg Protons Alphas Nuclear Physics Lectures 5

Tony Weidberg Nuclear Physics Lectures 6

Alpha Decay Rates • Gamow factor • Number of hits, on surface of nucleus radius R ~ v/2 R. Decay rate Tony Weidberg Nuclear Physics Lectures 7

Experimental Tests • Predict log decay rate proportional to (Ea)1/2 • Agrees ~ with data for e-e nuclei. • Angular momentum effects: – Additional barrier – Small compared to Coulomb but still generates large extra exponential suppression. Eg l=1, R=15 fm El~0. 05 Me. V cf for Z-90 Ec~17 Me. V. • Spin/parity DJ=L Tony Weidberg parity change=(-)L Nuclear Physics Lectures 8

Experimental Tests Half-life (s) 1018 10 -6 4 Tony Weidberg Energy E (Me. V) Nuclear Physics Lectures 9 9

Fermi b Decay. Theory • Consider simplest case: n decay. • At quark level: d u+W followed by decay of virtual W. d u u n d u d p e. W- Tony Weidberg Nuclear Physics Lectures ne ( ) 10

Fermi Theory • 4 point interaction (low energy approximation). Tony Weidberg Nuclear Physics Lectures 11

Fermi Theory • e distribution determined by phase space (neglect nuclear recoil energy) • Use FGR : phase space & M. E. decay rate Tony Weidberg Nuclear Physics Lectures 12

Kurie Plot Continuous spectrum neutrino End point gives limit on neutrino mass Tony Weidberg Intensity Coulomb correction Fermi function K(Z, p) (I(p)/p 2 K(Z, p))1/2 Tritium b decay 18 Electron energy (ke. V) Nuclear Physics Lectures Electron energy (ke. V) 13

Selection Rules • Fermi Transitions: – en couple to give 0 spin: DS=0 – “Allowed transitions” DL=0 DJ=0. • Gamow-Teller transitions: – en couple to give 1 unit of spin: DS=0 or ± 1. – “Allowed transitions” DL=0 DJ=0 or ± 1. • “Forbidden” transitions: – Higher order terms correspond to non-zero DL. Therefore suppressed depending on (q. r)2 L – Usual QM rules give: J=L+S Tony Weidberg Nuclear Physics Lectures 14

Electron Capture • Can compete with b+ decay. • For “allowed” transitions. • Only l=0. n=1 largest. Tony Weidberg Nuclear Physics Lectures 15

Electron Capture (2) • Density of states: • Fermi’s Golden Rule: Tony Weidberg Nuclear Physics Lectures 16

Anti-neutrino Discovery • Inverse Beta Decay • Same matrix elements. • Fermi Golden Rule: Tony Weidberg Nuclear Physics Lectures 17

Anti-neutrino Discovery (2) • Phase space factor • Neglect nuclear recoil. • Combine with FGR Tony Weidberg Nuclear Physics Lectures 18

The Experiment • For E~ 1 Me. V s~10 -47 cm 2 • Pauli prediction and Cowan and Reines. Liquid Scint. 1 GW Nuclear Reactor Tony Weidberg H 20+Cd. Cl 2 Shielding Nuclear Physics Lectures PMTs 19

Parity Definitions • Eigenvalues of parity are +/- 1. • If parity is conserved: [H, P]=0 eigenstates of H are eigenstates of parity. If parity violated can have states with mixed parity. • If Parity is conserved result of an experiment should be unchanged by parity operation. Tony Weidberg Nuclear Physics Lectures 20

Parity Conservation • If parity is conserved for reaction a+b c+d. • Nb absolute parity of states that can be produced from vacuum (e. g. photons) can be defined. For other particles we can define relative parity. e. g. define hp=+1, hn=+1 then can determine parity of other nuclei. • If parity is conserved <pseudo-scalar>=0 (see next transparency). Tony Weidberg Nuclear Physics Lectures 21

<Op> = 0 QED Tony Weidberg Nuclear Physics Lectures 22

Is Parity Conserved In Nature? • Feynman’s bet. • Yes in electromagnetic and strong interactions. • Big surprise was that parity is violated in weak interactions. Tony Weidberg Nuclear Physics Lectures 23

Mme. Wu’s Cool Experiment • Align spins of 60 Co with magnetic field. • Adiabatic demagnetisation to get T ~ 10 m. K • Measure angular distribution of electrons and photons relative to B field. • Clear forward-backward asymmetry Parity violation. Tony Weidberg Nuclear Physics Lectures 24

The Experiment Tony Weidberg Nuclear Physics Lectures 25

Improved Experiment q is angle wrt spin of 60 Co. Tony Weidberg Nuclear Physics Lectures 26

g decays • When do they occur? – Nuclei have excited states cf atoms. Don’t worry about details E, JP (need shell model to understand). – EM interaction << strong interaction – Low energy states E < 6 Me. V above ground state can’t decay by strong interaction EM. • Important in cascade decays a and b. • Practical consequences – Fission. Significant energy released in g decays. – Radiotherapy: g from Co 60 decays. – Medical imaging eg Tc. Tony Weidberg Nuclear Physics Lectures 27

Energy Levels for Mo and Tc b decay leaves Tc in excited state. Useful for medical imaging Tony Weidberg Nuclear Physics Lectures 28

g Decay Theory (Beyond Syllabus) • Most common decay mode for nuclear excited states (below threshold for break-up) is g decay. • Lifetimes vary from years to 10 -16 s. nb long lifetimes can easily be observed unlike in atomic. Why? • Angular momentum conservation in g decays. – intrinsic spin of g is 1 and orbital angular momentum integer J is integer. – Only integer changes in J of nucleus allowed. – QM addition of J: – Absolutely forbidden (why? ): 0 0 Tony Weidberg Nuclear Physics Lectures 29

g Decays • Electric transitions • Typically k~1 Me. V/c r~ 1 fm k. r~1/200 use multipole expansion. Lowest term is electric dipole transitions, L=1. • Parity change for electric dipole. Tony Weidberg Nuclear Physics Lectures 30

Forbidden Transitions • If electric dipole transitions forbidden by angular momentum or parity can have “forbidden” transitions, eg electric quadropole. • Rate suppressed cf dipole by ~ (k. r)2 • Magnetic transitions also possible: • Classically: E=-m. B • M 1 transition rate smaller than E 1 by ~ 10 -3. • Higher order magnetic transitions also possible. • Parity selection rules: – Electric: Dp=(-1)L – Magnetic: Dp=(-1)L+1 Tony Weidberg Nuclear Physics Lectures 31

Internal Conversion • 0 0 absolutely forbidden: • What happens to a 0+ excited state? • Decays by either: – Internal conversion: nucleus emits a virtual photon which kicks out an atomic electron. Requires overlap of the electron with the nucleus only l=0. Probability of electron overlap with nucleus increases as Z 3. For high Z can compete with other g decays. – Internal pair conversion: nucleus emits a virtual photon which converts to e+e- pair. Tony Weidberg Nuclear Physics Lectures 32