Use Eq. (30.12) to evaluate the reduced matrix element of a spherical harmonic between states that are $l-s$ coupled to good total angular momentum $j$; thus obtain Eq. (30.13). The reduced matrix element of a spherical harmonic $\left\langle l\left|\boldsymbol{Y}_{L}\right| l^{\prime}\rightangle$ between states of good $l$ was evaluated previously in Eq. (6.74).

Data from Eq. 30.12

Data from Eq. 30.13