Nuclear Radiation

Basics Empirically, it is found that --

Basics is -- is the probability per unit time per nucleus of a particular transition (e.g., decay, de-excitation, etc.)between two states. is called the “decay constant”.

Basics The mean lifetime for this process is  -

Basics The half-lifetime is

Basics Transition probability-- - from state A to state B  AB - from state A to state C  AC - from state A to state D  AD Total transition probability is --  T =  AB +  AC +  AD

The decay rate is then Note that the time dependence of the activity will depend on T for all of the individual transitions (decays), not on the partial decay constants.

Basics Partial lifetimes -- The lifetime of the parent gets shorter as more decay modes are possible.

Basics Number of species A that survive to time t Number of species B that have been formed by time t if species B is stable - This is the familiar “growth curve”.

Basics If species B is not stable, then -- i.e., the rate of change of number of species B is (rate of formation of B) - (rate of decay of B)

QM representation If the potential is time independent (V), then the solution of the SE gives - stationary states and discrete energy eigenvalues How do we deal with systems for which there is a transition between two states? Proposal: retain notion of stationary states from V; add a small (weak) perturbation in the potential to get V + V’

QM Solve SE with V to get stationary states  for the system. Then calculate the transition rate (probability per unit time) from Fermi’s Golden Rule - Transition matrix element  (E f ) is the density of final states

QM Fermi’s Golden Rule -  i is the initial state function for the system  f is final state function for the system:  (E f ) is determined by -- –Available states  f in final state nucleus –Available kinematic states  f for emitted particles

QM We need to match observation -- Full wave function for V Time-dependent probability in that state Corresponding wave function Time-independent nuclear state function

QM What are implications of this approach? But, the decay lifetime   s Therefore,  E ≤ MeV Nuclear level spacing  10 2 keV Therefore - discrete eigenstates still ~OK and  (E f ) does not depend on  E; only one  f Discrete eigenvalues no longer has exact meaning!

QM  (E f ) depends on the emitted particles and their kinematics. (c.f.,  -decay!) V’ is indicative of the weak-nuclear force -- or the weak interaction. (Specific form of V’ later…) is assumed to be small and therefore, is small so that  i becomes a mixture of  i and  f