Nonresonant inelastic x-ray scattering from actinides and rare earths

Abstract

Calculated multiplet spectra are presented for the electric-multipole transitions in nonresonant inelastic x-ray scattering (NIXS) at the ${O}_{4,5}$ edges of actinides and the ${N}_{4,5}$ edges of rare earths. The multiplet structure is characteristic for the $f$ count as well as for the angular momentum coupling. As a remarkable peculiarity, the calculations for the actinide ${O}_{4,5}$ edge show that the higher multipole spectra resemble $jj$ coupling, whereas the dipole spectrum is close to $LS$ coupling. The exchange integral ${G}^{1}(5d,5f)$ is responsible for a shift of $\ensuremath{\sim}$15 eV of the dipole spectrum to higher energy. This high energy is conserved by a sum rule for the average energy of the accessible final states. In the higher multipole spectra, where the allowed final states are different, the splitting by the $5d$ spin-orbit interaction exceeds that of the electrostatic interaction. This leads to a distinct spin-orbit-split doublet structure, with a broad first peak and a narrower second peak. According to the spin-orbit sum rule, the $5f$ spin-orbit interaction per hole is linearly related to the core-level branching ratio, where the proportionality factor is opposite in sign for the $k=3$ and 5 spectra. Compared to the $k=3$ spectrum, the $k=5$ spectrum has overall a lower intensity in the low-energy region but a higher intensity in the high-energy region, especially for the heavier elements in the series. The NIXS at the rare-earth ${N}_{4,5}$ edge is quite different since the electrostatic and core spin-orbit splitting in the $k=3$ and 5 spectra are of similar size. The pre-edge region of the rare-earth spectra, where the linewidth is narrow, shows a rich multiplet structure that depends strongly on the initial-state $J$ value.