The reduced single-particle matrix elements are given by $$ \langle k_{a},p\vert\vert \sigma t_{-}\vert\vert k_{b},n\rangle=\langle k_{a},n\vert\vert \sigma t_{+}\vert\vert k_{b},p\rangle= 2\langle k_{a}\vert\vert \mathbf{s}\vert\vert k_{b}\rangle, $$ where the matrix elements of \( \mathbf{s} \) are given by $$ \langle k_{a}\vert\vert \mathbf{s}\vert\vert k_{b}\rangle=\langle j_{a}\vert\vert \mathbf{s}\vert\vert j_{b}\rangle \delta_{n_{a},n_{b}} $$ $$ =(-1)^{l_{a}+j_{a}+3/2} \sqrt{(2j_{a}+1)(2j_{b}+1)}\left\{\begin{array}{ccc} {1/2}& {1/2}& {1} \\ {j_{b}}& {j_{a}}& {l_{a}}\end{array}\right\} \langle s\vert\vert \mathbf{s}\,\vert\vert s\rangle \delta _{\ell _{a},\ell _{b}} \delta_{n_{a},n_{b}} , $$ with $$ \langle s\vert\vert \mathbf{s}\vert\vert s\rangle= \sqrt{3/2}. $$