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-decay theory
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The decay rate Fermi’s Golden Rule density of final states (b) transition (decay) rate (c) transition matrix element (a) Turn off any Coulomb interactions
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The decay rate (a) Fermi’s Golden Rule V = weak interaction potential u = nuclear states = lepton ( ) states Integral over nuclear volume
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The decay rate (a) uPuP uDuD “Four-fermion” (contact) interaction uPuP W uDuD (W) Intermediate vector boson Interaction range
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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
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The decay rate (a) Assume: We can expand lepton wave functions and simplify And similarly for the neutrino wave function. Test the approximation --- deBroglie >> R therefore, lepton , constant over nuclear volume. (We will revisit this assumption later!)
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The decay rate (a) Therefore -- the matrix element simplifies to -- M fi is the nuclear matrix element; overlap of u D and u P Remember the assumptions we have made!!
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The decay rate Fermi’s Golden Rule density of final states (b) transition (decay) rate (c) transition matrix element (a)
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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 p 2 dp
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The decay rate (b) Do not observe ; therefore remove -dependence -- At fixed E e Assume
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The decay rate (b) Fermi’s Golden Rule Differential rate Density of final states
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The decay rate Fundamental (uniform) interaction strength Differential decay rate Overlap of initial and final nuclear wave functions; largest when u P u D a number Determines spectral shape!
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E f (Q) Q-value for decay Definition of E f
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d (p e ) c.f. Fig. 9.2
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d (E e )
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d (T e ) c.f. Fig. 9.2
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Consider assumptions Look at data for differential rates - c.f., Fig. 9.3 Calculate corrections for Coulomb effects on or Fermi Function F(Z’,p e ) or F(Z’,T e ) Coulomb Effects -- v e velocity of electron far from nucleus
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Consider assumptions Lepton wavefunctions -- In some cases, the lowest order term possible in the expansion is not 1, 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!
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Consider assumptions Lepton wavefunctions -- “Allowed term” “First forbidden term” “Second forbidden term” etc….
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Consider assumptions Lepton wavefunctions -- Change in spectral shape from higher order terms “Shape Factor” S(p e,p )
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The decay rate Fermi function Shape correction Density of final states Nuclear matrix element
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