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

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- Solution c

Solution

Unpolarized projectile and target spins supply

(2 J_{x} + 1) (2 J_{X} + 1)

equally weighted spin orientations. An isolated compound-nucleus level of spin

J

has

2 J + 1

degenerate magnetic substates. Sum over these resonance substates and average over the entrance spins to obtain

g = \frac{2 J + 1}{( 2 J _{x} + 1 ) ( 2 J _{X} + 1 )} .

This 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.

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