Past exam of the particle and nuclear physics course of the University of Cambridge / 2012 / Paper 1 / 4 / c / Solution

Unpolarized projectile and target spins supply $(2J_x+1)(2J_X+1)$ equally weighted spin orientations. An isolated compound-nucleus level of spin $J$ has $2J+1$ degenerate magnetic substates. Sum over these resonance substates and average over the entrance spins to obtainThis resonance spin statistical factor is a statistical spin weight, not always a probability smaller than one: orbital angular momentum also participates in forming the resonance. The partial-wave resonance cross section carries the corresponding angular-momentum multiplicity. Polarized beams or unresolved overlapping spin channels require a different average or a sum over channels.