Polarization dependency in Resonant Inelastic X-Ray Scattering

TL;DR

The paper introduces a polarization-sensitive framework for Resonant Inelastic X-Ray Scattering (RIXS) based on a fourth-rank Cartesian tensor $\chi^{(3)}$, which is contracted with a nine-component polarization basis $\hat{\mathbf{e}}=\{\boldsymbol{\epsilon}_{\mathrm{in}}\otimes\boldsymbol{\epsilon}_{\mathrm{out}}^{*}\}$ and represented in a spherical/cubic-harmonic basis. This approach cleanly separates experimental geometry from intrinsic material properties, allowing up to $81$ fundamental spectra in the most general dipole-dipole case, with symmetry reducing the number of independent components for any given point group. The authors develop a systematic scheme to count and couple tensor elements, derive how the tensor reorganizes under both non-magnetic and magnetic symmetries, and provide explicit treatments for powder averaging and Bragg-analyzer effects. The framework yields practical tools for experiment planning, data interpretation, and RIXS simulations, including explicit rules for averaging over outgoing polarization, powder orientations, and analyzer-induced polarization filtering. Overall, the work offers a general, symmetry-aware toolkit to predict and analyze the polarization dependence of RIXS spectra across diverse materials and experimental geometries.

Abstract

Resonant Inelastic X-Ray Scattering (RIXS) is a well-established tool for probing excitations in a wide range of materials. The measured spectra strongly depend on the scattering geometry, via its influence on the polarization of the incoming and outgoing light. By employing a tensor representation of the 4-point response function that governs the RIXS intensity, we disentangle the experimental geometry from the intrinsic material properties. In dipole-dipole RIXS processes and low-symmetry crystals, up to 81 linearly independent fundamental spectra can be measured as a function of light polarization. However, for crystals or molecules with symmetry, the number of independent fundamental spectra that define the RIXS tensor is significantly reduced. This work presents a systematic framework for determining the number of fundamental spectra and expressing the RIXS tensor in terms of these fundamental components. Given a specific experimental geometry, the measured spectrum can be represented as a linear combination of these fundamental spectra. To validate our approach, we performed calculations for different point group symmetries, both with and without an applied magnetic field. Within the same framework, we derived expressions for powder spectra in momentum-independent processes and spectra obtained using Bragg spectrometers. This formalism provides a valuable toolkit for optimizing experiment planning, data interpretation, and RIXS simulation.

Figures (4)

• Figure 1: Schematic representation of the RIXS experimental setup. The photon momentum is denoted as $\hbar\mathbf{k}$, while the transferred momentum to the sample is given by $\hbar\mathbf{q}$. The polarization basis vectors are defined as $\boldsymbol{\sigma}$, which is perpendicular to the scattering plane (vertical), and $\boldsymbol{\pi}$, which lies within the scattering plane (horizontal). The coordinate system is determined by the sample orientation, independent of the scattering plane.
• Figure 2: Imaginary part of the 2p$_{3/2}$3d RIXS tensor for systems with (a) SO(3), (b) $O_h$, (c) $D_{4h}$, and (d) $D_{2h}$ point group symmetry. Each tensor consists of 81 energy transfer over excitation energy intensity maps. Excitations energies are expressed relatively to the binding energy of Ni 2p$_{1/2}$ states. A symmetric logarithmic color scale is used to plot the values, with a linear regime between -0.01 and 0.01. The tensors are represented in a cubic harmonic basis, as defined in Eq. (\ref{['eq:sphericalkubiktensorbasis']}). The dark grey lines separate blocks with different angular momentum l.
• Figure 3: Imaginary part of the 2p$_{3/2}$3d RIXS tensor for systems with (a) SO(3), (b) $O_h$, (c) $D_{4h}$, and (d) $D_{2h}$ point group symmetry in the presence of a small but finite external magnetic field $B_{\mathrm{ext}} || \mathbf{z}$. Only the resulting single degenerate ground state is considered. Each tensor consists of 81 energy transfer over excitation energy intensity maps. Excitations energies are expressed relatively to the binding energy of Ni 2p$_{1/2}$ states. A symmetric logarithmic color scale is used to plot the values, with a linear regime between -0.01 and 0.01. The tensors are represented in a spherical harmonic basis, as defined in Eq. (\ref{['eq:sphericalkubiktensorbasis']}). Table \ref{['tab:branchingWithBext']} lists the branching rules for the unitary magnetic subgroups involved. The dark grey lines separate blocks with different angular momentum l.
• Figure 4: Schematic representation of a general RIXS geometry with scattering angle $2\theta$. After RIXS, the outgoing photons from the sample elastically scatter at a Bragg crystal or grating (green) before being detected. The RIXS scattering plane can differ from the Bragg plane. $\hbar\mathbf{k}$ represents the photon momentum. $\bm{\mathrm{\sigma}}$ and $\bm{\mathrm{\pi}}$ are the vertical (out-of-plane) and horizontal (in-plane) polarization basis vectors, respectively.