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Unified theory of orientation averaging in X-ray spectroscopies: understanding polarization dependence in a Cartesian tensor approach

Sihan Zhang, Oana Bunău, Marius Retegan, Pieter Glatzel

Abstract

X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) are powerful probes of electronic structure owing to their chemical and orbital selectivity. For powder samples, however, interpreting RIXS spectral intensities remains challenging as the measured signal is an average over all orientations. Existing theoretical treatments rely largely on spherical-tensor formalisms, which often involve complex derivations and case-specific analyses. Meanwhile, recent advances in quantum-chemistry methods have made the evaluation of transition tensors in Cartesian coordinates both accurate and straightforward. Here, we present a general theoretical framework that translates Cartesian transition tensors into physically meaningful, orientation-averaged intensities for powder samples. The formalism allows predicting angular and polarization dependences \textit{ab initio} for both XAS and RIXS and is extendable to other spectroscopies. The resulting predictions show excellent agreement with RIXS experimental data at the Ce L$_3$ edge.

Unified theory of orientation averaging in X-ray spectroscopies: understanding polarization dependence in a Cartesian tensor approach

Abstract

X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) are powerful probes of electronic structure owing to their chemical and orbital selectivity. For powder samples, however, interpreting RIXS spectral intensities remains challenging as the measured signal is an average over all orientations. Existing theoretical treatments rely largely on spherical-tensor formalisms, which often involve complex derivations and case-specific analyses. Meanwhile, recent advances in quantum-chemistry methods have made the evaluation of transition tensors in Cartesian coordinates both accurate and straightforward. Here, we present a general theoretical framework that translates Cartesian transition tensors into physically meaningful, orientation-averaged intensities for powder samples. The formalism allows predicting angular and polarization dependences \textit{ab initio} for both XAS and RIXS and is extendable to other spectroscopies. The resulting predictions show excellent agreement with RIXS experimental data at the Ce L edge.
Paper Structure (24 sections, 100 equations, 6 figures, 3 tables)

This paper contains 24 sections, 100 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model description and mathematical background
  3. XAS and RIXS cross sections
  4. Orientation averaging and tensor-integral reduction
  5. Orientation averages for X-ray spectroscopy
  6. XAS
  7. RIXS
  8. General formalism
  9. Projections onto polarization channels
  10. E1E1 RIXS
  11. E2E1 and E1E2 RIXS
  12. Scattering from spherically symmetric states
  13. Application
  14. Methods
  15. Experiment setup
  16. ...and 9 more sections

Figures (6)

  • Figure 1: Photon-in photon-out scattering geometry
  • Figure 2: Positioning of the analyzer crystals with respect to the sample (blue for the vtc measurement, red for ctc).
  • Figure 3: Experimental 2$p$3$d$ RIXS for the $\alpha = 118^\circ$ analyzer. Features at 4840 and 4822 eV emission energies correspond to the L$_3$M$_5$ and L$_3$M$_4$ lines respectively. Peaks are annotated according to the final state they are originating from. The displayed masks were used to obtain the CEE cuts in Fig. \ref{['fig:L3M45_cut']}.
  • Figure 4: CEE cuts for all analyzers obtained with the masks in Fig. \ref{['fig:2p3dRIXS']}. There is no significant angular dependence for L$_3$M$_5$ (top) and L$_3$M$_4$ RIXS (bottom).
  • Figure 5: Valence-to-core RIXS planes recorded for each analyzer are shown. The applied mask isolates the angular-dependent feature at an energy loss of 6.5 eV, with a bandwidth of 1 eV.
  • ...and 1 more figures