Nuclear Decays Unstable nuclei can change N,Z.A to a nuclei at a lower energy (mass) If there is a mass difference such that energy is released, pretty much all decays occur but with very different lifetimes. have band of stable particles and band of “natural” radioactive particles (mostly means long lifetimes). Nuclei outside these bands are produced in labs and in Supernovas nuclei can be formed in excited states and emit a gamma while cascading down. P461 - nuclear decays

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 decays

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 decays

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 decays

Lifetimes just one channel with N(t) = total number at time t multiple possible decays. Calculate each (the “partial” widths) and then add up Measure lifetime. long-lived (t>10-8sec). Have a certain number and count the decays P461 - nuclear decays

Lifetimes Measure lifetime. medium-lived (t>10-13sec). Decay point separated from production point. Measure path length. Slope gives lifetime short-lived (10-23 < t <10-16 sec). Measure invariant mass of decay products. If have all  mass of initial. Width of mass distributions (its width) related to lifetime by Heisenberg uncertainty. 100 10 1 Dx P461 - nuclear decays

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 decays

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 dr is a thin barrier P461 - nuclear decays

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 0.002 for each change in L (increases potential barrier) P461 - nuclear decays

Alpha decay-Energy levels may need to have orbital angular momentum if sub-shell changes (for odd n/p nuclei) Z= 83-92 1h(9/2) N=127-136 2g(9/2) Z=93-100 2f(7/2) N=137-142 3d(5/2) so if f(7/2)  h(9/2) need L>0 but parity change if L=1  L=2,4 or d(5/2)  g(9/2) need L>1. No parity change L=2,4 not for even-even nuclei (I=0). suppression of about 0.002 for each change in L (increases potential barrier) s 0 p 1 d 2 f 3 g 4 h 5 P461 - nuclear decays

Parity + Angular Momentum Conservation in Alpha decay X  Y + a. 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 P461 - nuclear decays

Energy vs A Alpha decay P461 - nuclear decays

Lifetime vs Energy in Alpha Decays Perlman, Ghiorso, Seaborg, Physics Review 75, 1096 (1949) 10 log10 half-life in years -10 5 7 Alpha Energy MeV P461 - nuclear decays