Assessing temperature effects on multipole contributions and angular dependence in core-level spectroscopies

This study aims at assessing the thermal nuclei motion effects on the multipole transition channels involved in two core-level spectroscopies, x-ray absorption spectroscopy (XAS) and x-ray Raman scattering (XRS). Temperature effects on the $1s\ensuremath{\rightarrow}s$ monopole, $1s\ensuremath{\rightarrow}p$ dipole, and $1s\ensuremath{\rightarrow}d$ quadrupole transitions are investigated using two reference systems for which we present original experimental data: $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{Al}}_{2}{\mathrm{O}}_{3}$ at the Al $K$ edge probed by XRS at room temperature and rutile $\mathrm{Ti}{\mathrm{O}}_{2}$ at the Ti $K$ pre-edge probed by XAS at temperatures ranging from 6 to 700 K. Through the latter, this work enlightens the part of the pre-edge peak enhancement due to temperature in the $K$ pre-edge region of $3d$ transition metal, which is known to be routinely used to determine the concentration, valence or symmetry of the probed element in a given sample. Nuclear thermal fluctuations are taken into account using a method based on density functional theory that consists in averaging spectra over atomic configurations, generated within the harmonic approximation and obeying quantum statistics at finite temperature. Since only a finite number of such configurations are used, the numerically averaged spectra generally lose the symmetry of the equilibrium crystal positions. In this paper, we demonstrate that the physical average has to be symmetric and propose a method to restore the physical angular dependence of the spectra. The approach is successfully applied to investigate the angular dependent XAS spectra in rutile as a function of temperature. The two systems under study allow to draw general conclusions regarding the effect of nuclear quantum fluctuations on the different transition channels available to both core-level spectroscopies.