Consider the term $S\cdot J$ in matrix element $\mathcal{M}$, which is related to the cross section. S corresponds to a spin-1/2 particle. J is the angular momentum operator corresponding to a nucleus at rest. After taking average over initial spins and summing over final spins, one gets $$ \langle|\mathcal{M}|^2\rangle \propto \frac{J(J+1)}{4} $$ If I take average over initial spins as follows, $$ \langle S^2\rangle=\frac{1}{2}\left(\frac{1(\frac{1}{2}+1)}{2}+\frac{-1(\frac{-1}{2}+1)}{2}\right)=\frac{1}{4} $$ I get the required 1/4 and $J^2$ will give me $J(J+1)$. But is this the right way? I used $m_s$ instead of $S$.
Is there any formal way to arrive at the same expression using angular momentum algebra?
