ΔS=±1Magnetic Multipole Radiative Transitions

The magnetic multipole transition probability is calculated in terms of the matrix elements of the magnetic multipole. The magnetic $\mathrm{jm}$ moment ${Q_{\mathrm{jm}}}^{\mathrm{}\left(\mathrm{mg}\right)\mathrm{}}$ is defined as $\left(\frac{e}{\mu }\right){\left[\frac{4\pi }{\mathrm{}(2j+1)\mathrm{}}\right]}^{\frac{1}{2}}\Sigma \stackrel{}{i}\left(\nabla {r_i}^j{Y}_{\mathrm{jm}}\right)[{\mathrm{}(j+1\mathrm{})\mathrm{}}^{-1\mathrm{}}{\mathrm{l}}_i+{\mathrm{s}}_i],$ where $e$ and $\mu$ are electron charge and electron mass, $r_i$, ${\mathrm{l}}_i$, and ${\mathrm{s}}_i$ are the coordinate, orbital angular momentum, and spin-angular momentum of the $i\mathrm{th}$ electron and $Y_{\mathrm{jm}}$ is the spherical harmonic. Magnetic quadrupole and octupole moments are explicitly given. It is shown that for the ${{}_{}{}^3{}_{}{}^{}\Sigma _u^{}{}_{}{}^{}}^{+}\leftrightarrow {{}_{}{}^1{}_{}{}^{}\Sigma _g^{}{}_{}{}^{}}^{+}$ transition of the hydrogen molecule, the magnetic quadrupole transition is more important than the conventional spin-orbit electric dipole transition. The magnetic octupole transition has the same order of magnitude as the spin-orbit magnetic dipole transition.