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-decay theory

-decay theory. The decay rate. Fermi’s Golden Rule. transition (decay) rate (c). density of final states (b). transition matrix element (a). Turn off any Coulomb interactions. The decay rate (a). Fermi’s Golden Rule. V  = weak interaction potential. u = nuclear states.

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-decay theory

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  1. -decay theory

  2. The decay rate Fermi’s Golden Rule transition (decay) rate(c) density of final states(b) transition matrix element(a) Turn off any Coulomb interactions

  3. The decay rate (a) Fermi’s Golden Rule V = weak interaction potential u = nuclear states   = lepton () states Integral over nuclear volume

  4. uD uP   uP  W  uD The decay rate (a) “Four-fermion” (contact) interaction (W) Intermediate vector boson Interaction range

  5. The decay rate (a) Assume:Short range interaction contact interaction g=weak interaction coupling constant Assume:, are weakly interacting“free particles” in nucleus Approximate leptons as plane waves

  6. The decay rate (a) Assume:We can expand lepton wave functions and simplify And similarly for the neutrino wave function. Test the approximation --- deBroglie  >> Rtherefore, lepton ,   constant over nuclear volume.(We will revisit this assumption later!)

  7. The decay rate (a) Therefore -- the matrix element simplifies to -- Mfi is the nuclear matrix element; overlap of uD and uP Remember the assumptions we have made!!

  8. The decay rate Fermi’s Golden Rule transition (decay) rate(c) density of final states(b) transition matrix element(a)

  9. The decay rate (b) Fermi’s Golden Rule Quantization of particles in a fixed volume (V)  discrete momentum/energy states (phase space) -- Number of states dN in space-volume V, and momentum-volume 4p2dp

  10. The decay rate (b) Do not observe ; therefore remove -dependence -- Assume At fixed Ee

  11. The decay rate (b) Density of final states Fermi’s Golden Rule Differential rate

  12. The decay rate Fundamental (uniform) interaction strength Determines spectral shape! Differential decay rate Overlap of initial and final nuclear wave functions; largest when uP uD  a number

  13. Ef(Q) Q-value for decay Definition of Ef

  14. d(pe) c.f. Fig. 9.2

  15. d(Ee)

  16. d(Te) c.f. Fig. 9.2

  17. Consider assumptions Coulomb Effects -- Look at data for differential rates - c.f.,Fig. 9.3 Calculate corrections for Coulomb effects on  or  Fermi FunctionF(Z’,pe) or F(Z’,Te) ve velocity of electronfar from nucleus

  18. Consider assumptions Lepton wavefunctions -- In some cases, the lowest order term possible in the expansion is not1, but one of the higher order terms! • More complicated matrix element; impacts rate! • Additional momentum dependence to the differential rate spectrum; changes the spectrum shape!

  19. Consider assumptions Lepton wavefunctions -- “Allowed term” “First forbidden term” “Second forbidden term” etc….

  20. Consider assumptions Lepton wavefunctions -- Change in spectral shape from higher order terms  “Shape Factor” S(pe,p)

  21. The decay rate Nuclear matrix element Fermi function Shape correction Density of final states

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