P461 - nuclear decays1 General Comments on Decays Use Fermi Golden rule (from perturbation theory) rate proportional to cross section or 1/lifetime the matrix element connects initial and final states where V contains the “physics” (EM vs strong vs weak coupling and selection rules) the density of states factor depends on the amount of energy available. Need to conserve momentum and energy “kinematics”. If large energy available then higher density factor and higher rate. Nonrelativistic (relativistic has 1/E also. PHYS584)

P461 - nuclear decays2 Simplified Phase Space Decay: A --> a + b + c ….. Q = available kinetic energy large Q -> large phase space -> higher rate larger number of final state products possibly means more phase space and higher rate as more variation in momentums. Except if all the mass of A is in the mass of final state particles 3 body has little less Q but has 4 times the rate of the 2 body (with essentially identical matrix elements)

P461 - nuclear decays3 Phase Space:Channels If there are multiple decay channels, each adds to “phase space”. That is one calculates the rate to each and then adds all of them up single nuclei can have an alpha decay and both beta+ and beta- decay. A particle can have hundreds of possible channels often one dominates or an underlying virtual particle dominates and then just dealing with its “decays” still need to do phase space for each….

P461 - nuclear decays4 Alpha decay Alpha particle is the He nucleus (2p+2n) ~all nuclei Z > 82 alpha decay. Pb(82,208) is doubly magic with Z=82 and N=126 the kinematics are simple as non-relativistic and alpha so much lighter than heavy nuclei really nuclear masses but can use atomic as number of electrons do not change

P461 - nuclear decays5 Alpha decay-Barrier penetration One of the first applications of QM was by Gamow who modeled alpha decay by assuming the alpha was moving inside the nucleus and had a probability to tunnel through the Coulomb barrier from 1D thin barrier (460) for particle with energy E hitting a barrier potential V and thickness gives Transmission = T now go to a Coulomb barrier V= A/r from the edge of the nucleus to edge of barrier and integrate- each dX is a thin barrier

P461 - nuclear decays6 Alpha decay-Barrier penetration Then have the alpha bouncing around inside the nucleus. It “strikes” the barrier with frequency the decay rate depends on barrier height and barrier thickness (both reduced for larger energy alpha) and the rate the alpha strikes the barrier larger the Q larger kinetic energy and very strong (exponential) dependence on this as alpha has A=4, one gets 4 different chains (4n, 4n+1, 4n+2, 4n+3). The nuclei in each chain are similar (odd/even, even/even, etc) but can have spin and parity changes at shell boundaries if angular momentum changes, then a suppression of about for each change in L (increases potential barrier)

P461 - nuclear decays7 Parity + Angular Momentum Conservation in Alpha decay X -> Y + alpha. The spin of the alpha = 0 but it can have non-zero angular momentum. Look at Parity P if parity X=Y then L=0,2…. If not equal L=1,3… to conserve both Parity and angular momentum