Electromagnetic multipole moments and transitions

where we have $$ < s\vert\vert \mathbf{s}\,\vert\vert s>\, = \sqrt{3/2}\, , $$ and $$ < k_{a}\vert\vert O(M1,l )\vert\vert k_{b}>\, = \sqrt{ \frac{3}{4\pi }}\, < j_{a}\vert\vert \mathbf{l}\,\vert\vert j_{b}> \delta _{n_{a},n_{b}} \; g^{l }_{q} \; \mu _{N} $$ $$ =\sqrt{ \frac{3}{4\pi }}\, (-1)^{l _{a}+j_{b}+3/2} \sqrt{(2j_{a}+1)(2j_{b}+1)}\, \left\{\begin{array}{ccc} {l _{a}}& {l _{b}} & {1}\\ {j_{b}}& {j_{a}}& {1/2}\end{array}\right\} $$ $$ \times\langle l _{a}\vert\vert \mathbf{l}\,\vert\vert l _{b}\rangle \delta _{n_{a},n_{b}}g^{l }_{q}\mu _{N} , $$ where $$ \langle l _{a}\vert\vert \mathbf{l}\,\vert\vert l _{b}\rangle = \delta _{l _{a},l _{b}} \sqrt{l _{a}(l_{a}+1)(2l _{a}+1)}. $$ Thus the \( M1 \) operator connects only those orbitals which have the same \( n \) and \( l \) values.