X-ray $M_{4,5}$ resonant Raman scattering from La metal with a final $4p$ hole:  Calculations with $4p-4d-4f$ configuration interaction in the final state and comparison to experiments

We consider the x-ray resonant Raman scattering (RRS) in La in the whole $M_{4,5}$ region ending with a state with a $4p$ hole, along the sequence ${3d}^{10}{4f}^0\to {3d}^9{4f}^1\to {3d}^{10}{4p}^5{4f}^1.$ The final state configuration mixes with that with two $4d$ holes, i.e., ${3d}^{10}{4d}^8{4f}^{n+2\mathrm{}}$ having almost the same energy. Thus RRS must be described by introducing final-state configuration interaction (CI) between states with one $4p$ hole and with two $4d$ holes. This approach allows detailed experimental data on La metal to be interpreted on the basis of a purely ionic approach. It is shown that the inclusion of CI is crucial and has very clear effects. The calculations with the Kramers-Heisenberg formula describe all measured spectral features appearing in the strict Raman regime, i.e., dispersing with the incident photon energy. In the experiment also a nondispersive component is present when the excitation energy is greater than about 2 eV above the $M_5$ peak. The shape and position of this component is well accounted for by a model based on all possible partitions of the excitation energy between localized and extended states. However the intensity of the nondispersive component is greater in the measurements, suggesting a rearrangement in the intermediate excited state. The comparison of ionic calculations with the metal measurements is legitimate, as shown by the comparison between the measurements on La metal and on ${\mathrm{LaF}}_3$ with $M_5$ excitation giving the same spectrum within the experimental accuracy. Moreover the experiment shows that the final lifetime broadening is much greater in the final states corresponding to lower outgoing photon energies than in the states corresponding to higher outgoing photon energies.