Electromagnetic multipole moments and transitions

The electromagnetic moment operator can be expressed in terms of the electromagnetic transition operators. By the parity selection rule of the moments are nonzero only for \( M1 \), \( E2 \), \( M3 \), \( E4,\ldots \). The most common are: $$ \mu =\sqrt{\frac{4\pi }{3}}\langle J,M=J\vert {\cal O}(M1)\vert J,M=J\rangle= \sqrt{\frac{4\pi }{3}}\left\{\begin{array}{ccc} J & 1 &J \\ -J & 0 & J \end{array}\right\} \langle J\vert \vert {\cal O}(M1)\vert \vert J\rangle, $$ and $$ Q = \sqrt{\frac{16\pi }{5}}\langle J,M=J\vert {\cal O}(E2)\vert J,M=J\rangle= \sqrt{\frac{16\pi }{5}}\, \left(\begin{array}{ccc} {J}& {2} & {J}\\ {-J} & {0}& {J}\end{array}\right)\langle J\vert\vert {\cal O}(E2)\vert\vert J\rangle. $$