Cross-section for any final spin state

When the spin of the scattering nucleus is zero (S = 0, b+ = b — b), we get the trivial result 

However, if the target is a proton (S = 1/2), the situation is more complicated and in the following sections, two cases corresponding to particular but realistic initial states will be studied. 

When the incident beam is non-polarized, if the orientation of the nucleus spin is at random, each state has the same weight. Under such conditions, the average differential cross-section is 

Let us calculate this expression the number of states with total spin is 2S + 2. Thus, 

The number of states with total spin (S — 1/2) is 2S. Thus, 

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