Spin-orbit sum rule for electric multipole transitions in nonresonant inelastic x-ray scattering

The sum rule for the branching ratio of dipole-excited core-valence transitions in isotropic x-ray absorption spectroscopy is extended to electric multipole transitions. The derived sum rule not only shows that the branching ratio is linearly related to the expectation value of the angular part of the spin-orbit operator in the valence states, but also shows a strong dependence on the rank of the multipole. The spin-orbit dependence vanishes in the weighted sum over the branching ratios. The effect can be an important diagnostic tool for high-energy spectroscopies.