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Angular dependence and powder average of resonant inelastic X-ray scattering

Myrtille O. J. Y Hunault, Timothy G. Burrow, Fabien Besnard, Amélie Juhin, Christian Brouder

Abstract

Resonant Inelastic X-ray scattering (RIXS) is a synchrotron-based spectroscopy that has seen growing interest across a range of scientific disciplines beyond fundamental physics. The interpretation of experimental RIXS data requires theoretical calculations based on the Kramers-Heisenberg formula. However, due to the dependence of RIXS on both the incident and scattered photon properties, a tractable treatment of the angular dependence in this formula has been lacking. In this work, within the electric dipole approximation, we determine the number of fundamental spectra contributing to the RIXS cross-section for all crystallographic point groups. We then derive a general expression for the RIXS cross-section of isotropic samples such as un-textured powders, homogeneous glasses or liquids, explicitly accounting for the polarization and propagation directions of both the incident and scattered photons. Simplified forms of the RIXS expressions are subsequently obtained for most common point groups. Finally, we demonstrate the applicability of our formalism through a case study of uranium 3d4f RIXS.

Angular dependence and powder average of resonant inelastic X-ray scattering

Abstract

Resonant Inelastic X-ray scattering (RIXS) is a synchrotron-based spectroscopy that has seen growing interest across a range of scientific disciplines beyond fundamental physics. The interpretation of experimental RIXS data requires theoretical calculations based on the Kramers-Heisenberg formula. However, due to the dependence of RIXS on both the incident and scattered photon properties, a tractable treatment of the angular dependence in this formula has been lacking. In this work, within the electric dipole approximation, we determine the number of fundamental spectra contributing to the RIXS cross-section for all crystallographic point groups. We then derive a general expression for the RIXS cross-section of isotropic samples such as un-textured powders, homogeneous glasses or liquids, explicitly accounting for the polarization and propagation directions of both the incident and scattered photons. Simplified forms of the RIXS expressions are subsequently obtained for most common point groups. Finally, we demonstrate the applicability of our formalism through a case study of uranium 3d4f RIXS.
Paper Structure (38 sections, 132 equations, 5 figures, 3 tables)

This paper contains 38 sections, 132 equations, 5 figures, 3 tables.

Table of Contents

  1. RIXS cross section and sample symmetry
  2. Group action
  3. Action of an isometry on the cross-section
  4. Action of a symmetry on the cross-section
  5. Symmetry reduction of the cross-section
  6. Alternative way to the fundamental spectra
  7. Definition of spherical parameters
  8. Circular dichroism and permutation symmetry
  9. From site to point group symmetry
  10. RIXS cross-section for various point groups
  11. Case $C_1$ or $C_i$
  12. Case $C_2$, $C_s$ or $C_{2h}$
  13. Case $D_2$ or $C_{2v}$ or $D_{2h}$
  14. Case $C_4$ or $S_4$ or $C_{4h}$
  15. Case $D_4$ or $C_{4v}$ or $D_{2d}$ or $D_{4h}$
  16. ...and 23 more sections

Figures (5)

  • Figure 1: Schematic showing an overview of experiment configuration. Parallel and perpendicular polarization vectors are denoted as $\pi$ and $\sigma$ respectively. The incident and scattered photon momentum are denoted as $\widehat{\mathbf{k}}_i$ and $\widehat{\mathbf{k}}_s$ respectively. The angle of scattered photon detection is given by $\varphi$.
  • Figure 2: Schematic of $3d4f$ RIXS, depicting the correlation between incident ($\omega$), emitted ($\omega_s$), and transferred ($\omega$ - $\omega_s$) energies, and the corresponding electronic configurations of the initial $\lvert I\rangle$, intermediate $\lvert N\rangle$, and final $\lvert F \rangle$ states.
  • Figure 3: U^VIO2Cl4^2- a) simulated HERFD cuts through $3d4f$ RIXS maps (c-e), inplot is same as (a) but with limited y axis. b) simulated RXES cuts through RIXS maps (c-e), inplot is same as (b) but with limited y axis. c-e) simulated uranium $3d4f$ RIXS maps at different crystal analyzer (CA) positions $\varphi$; 90°, 90°$\pm$30°, and back-scattering case of 0°. HERFD (red dash) and RXES (blue dash) cuts are represented as the diagonal and vertical cuts in the maps. White horizontal lines (c-e) and black vertical lines (a-b) correlate to the most prominent peaks present.
  • Figure 4: Sr3U^VIO6 a-b) simulated uranium $3d4f$ RIXS maps at a) $\varphi$=0° and b) $\varphi$=90° with respect to the incident beam, $\widehat{\mathbf{k}}_i$. c) Polar plot of features A-E. The angular axis is representative of $\varphi$ and the radial axis is the peak intensity. Example positions of crystal analyzers are indicated by black dots.
  • Figure 5: U^IVO2 a-b) simulated uranium $3d4f$ RIXS maps at a) $\varphi$=0° and b) $\varphi$=90° with respect to the incident beam, $\widehat{\mathbf{k}}_i$. c) Polar plot of features A-E. The angular axis is representative of $\varphi$ and the radial axis is the peak intensity. Example positions of crystal analyzers are indicated by black dots.