International
Tables for
Crystallography
Volume I
X-ray absorption spectroscopy and related techniques
Edited by C. T. Chantler, F. Boscherini and B. Bunker

International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870720007521

## Inelastic X-ray scattering

Pieter Glatzel,a* Amélie Juhinb* and Marco Morettic*

aESRF – The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France,bInstitut de Minéralogie, de Physique des Matériaux et de Cosmochimie (IMPMC), UMR7590, CNRS, Sorbonne Université, Muséum National d'Histoire Naturelle, 4 Place Jussieu, 75252 Paris CEDEX 05, France, and cDipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Correspondence e-mail:  glatzel@esrf.fr, amelie.juhin@sorbonne-universite.fr, marco.moretti@polimi.it

The first- and second-order terms of the X-ray scattering cross section are given and an overview is provided of the excitations that can be studied using inelastic X-ray scattering. The second-order (resonant) term is then elaborated on and the role of the core-hole lifetime broadening is explained. Approximations and the one-electron transitions that describe the scattering process are briefly addressed in a simplified equation. X-ray emission (fluorescence) and outline momentum, angular and polarization dependence of the resonant scattering process are discussed.

Keywords: Kramers–Heisenberg equation; resonance; momentum dependence; angular dependence; polarization dependence.

### 1. Introduction

The dramatic advances in synchrotron-radiation instrumentation over the past decades have made the observation and detailed analysis of weak X-ray scattering events possible. Such experiments, together with extensive theoretical work, have provided powerful tools for researchers in all fields of natural sciences, including solid-state physics, catalysis, energy applications, environmental sciences, cultural heritage and biology. Spectroscopy of inelastic X-ray scattering requires an energy bandwidth of the incoming beam and of the spectrometer that is sufficiently small to resolve excited states. The nature of the excitation determines the required energy resolution. This chapter addresses inelastic X-ray scattering from electrons.

### 2. Basic equations

The geometry of the X-ray scattering process is shown in Fig. 1. An incoming photon with energy ħωin, momentum kin and polarization ɛin is scattered by the sample, resulting in an outgoing photon with energy ħωout, momentum kout and polarization ɛout. The scattering process is called inelastic if ħωin ≠ ħωout and elastic otherwise. While the energy of the scattered photon is in most cases smaller than the incoming photon energy, it may be larger if the initial state of the transition is not the lowest-energy ground state. This may be the case for thermal excitations (the inelastic scattering data will show anti-Stokes lines) or pump-and-probe experiments where atoms are excited optically shortly before the X-ray probe (Wernet et al., 2015; Baron, 2015).

 Geometry of photon scattering. The sample is centred at the origin and for φ = 0, ɛin points along x in the case of linear polarized incoming light.

The angle between kin and kout is the scattering angle θ. Literature on diffraction will use the notation 2θ for the scattering angle as it is twice the Bragg angle. Here, all scattering processes of light in matter are considered. The energy and momentum transfer are given by ħωin − ħωout and kinkout, respectively. The probability of observing a scattered photon in the solid angle dΩ is given by the double-differential cross section Fωin, ħωout, kin, kout, ɛin, ɛout), the expression for which will be given below.

A quantum-mechanical treatment of the scattering of X-rays in matter leads to the generalized Kramers–Heisenberg equation (Schuelke, 2007; Sakurai, 1967; Kotani & Shin, 2001; Ågren & Gel'Mukhanov, 2000). The quantized electromagnetic field is described by the vector potential A(r), the interaction of which with the electrons in the sample is assumed to be sufficiently weak to treat it within perturbation theory. This condition is fulfilled to a good approximation for X-ray sources at storage rings, but may break down for X-ray free-electron lasers (Beye et al., 2013; Rohringer et al., 2012). The terms in the Hamiltonian describing the perturbation may interfere and the expressions for the double-differential scattering cross sections given in the following always assume that one term dominates over all others, which are therefore neglected. The interaction of the electron spin with the magnetic field of the incoming X-rays is several orders of magnitude weaker than the charge scattering (Blume, 1985). Neglecting such terms, the Hamiltonian describing the perturbation becomeswhere pj is the momentum of the electron and the sum runs over all electrons (Rueff & Shukla, 2010). The term quadratic in A(r) contributes to first order and the term for nonresonant (NR) scattering readsS(q, ωinωout) is denoted the dynamic structure factor (Schuelke, 2007). Equation (2) describes elastic and inelastic scattering processes including Thomson, Compton and Bragg scattering (without the magnetic contributions). This term dominates at incoming light energies below or far above absorption edges (i.e. where photoelectric absorption is minimized) and at appropriately chosen scattering angles φ and θ that maximize nonresonant scattering. Equation (2) also describes high energy resolution nonresonant inelastic X-ray scattering (Baron, 2015; Krisch & Sette, 2017) and non­resonant Raman scattering. Compton scattering can be exploited to study the momentum density distribution of the electron system (Cooper et al., 2004). We note that the use of the expression Thomson scattering' is ambivalent in the literature. It may be used to refer to all processes described by equation (2) or only elastic scattering by free electrons where the dynamic structure factor equals 1.

The square of the dot product between the incoming and outgoing polarization leads to an angular dependence sin2φ + cos2θcos2φ for linear polarized incoming light (Fig. 1). This yields the two special cases where the incoming light polarization is either perpendicular (σ, senkrecht) or parallel (π) to the scattering plane as defined by kin and kout: isotropic (for σ-scattering) or cos2(θ) (for π-scattering) angular dependence, respectively.

The term p · A(r) contributes in second order to the perturbation terms. Keeping only the terms that give an important contribution when the incoming energy is tuned close to or above an absorption edge, one obtains for the scattering cross section

Only one ground state |g〉 is considered. The sums run over all intermediate states |n〉 and final states |f〉, as well as all electrons in the scattering system (j, j′). Equation (3) describes a resonant scattering process within the approximations given above. It is often referred to as the Kramers–Heisenberg equation or the resonant term of the generalized Kramers–Heisenberg equation. The equation evokes the following picture. The absorption of a photon results in the excitation of the system from ground state |g〉 to the intermediate state |n〉 with a core hole in the photoabsorbing atom. The intermediate state decays with a lifetime τn either via emission of a photon (radiative decay) or emission of an electron (nonradiative or Auger decay). The lifetime broadening (half-width at half-maximum) Γn = ℏ/(2τn) does not appear in the second-order term of the perturbation theory treatment. It is added to account for the finite lifetime of the intermediate state (Sakurai, 1967). Interference may occur for transitions between a ground and a final state that can be realized via different intermediate states.

A finite lifetime of the final state in equations (2) and (3) can be considered by replacing the δ-function by a Lorentzian function (Kotani & Shin, 2001),

In the literature, one finds the use of the terms resonant inelastic X-ray scattering (RIXS), resonance Raman scattering (RRS) and (resonant) X-ray emission spectroscopy [(R)XES] to denote processes that are described by equation (3). This chapter uses the acronym RIXS' for all scattering processes described by equation (3) except when the incoming energy is tuned well above an absorption edge, where the absorption cross section only varies slowly as a function of the incoming energy and the interaction between the photoexcited electron and the remaining ion is weak. In this case the term XES or X-ray fluorescence is used (but the process is described by equation 3).

If the ground state and final state are identical in equation (3) we describe resonant elastic X-ray scattering (REXS), which is frequently used in solid-state systems where the intensity of a diffraction peak or diffuse scattering is observed in an appropriate scattering geometry while the incoming light is tuned to the vicinity of an absorption edge (Ishihara, 2017). Combing REXS with an instrument for energy analysis of the scattered X-rays greatly reduces the signal from unwanted events and opens new paths for analysis (Ghiringhelli et al., 2012; Hwan Chun et al., 2015). We note that in both RIXS and REXS experiments it is possible to analyse the polarization of the scattered X-rays, which provides important information on the scattering process and hence the type of observed excitations (Braicovich et al., 2014; Detlefs et al., 2012). The terms resonant' and nonresonant' may be defined according to whether the scattering cross section given by equation (2) or equation (3), respectively, dominates. Following this definition, any XES process following photoionization is a resonant scattering process. This definition is, however, at odds with the general use of resonance' as an excitation with incoming energy close to an absorption edge. Conversely, an X-ray Raman scattering process described by equation (2) may be used to study an absorption edge and thus a resonant' excitation. Consequently, the definition of the term resonant' depends on the theoretical framework as well as personal preferences, and the term must be used with care.

Fluorescence-detected absorption spectroscopy is always described by equation (3) and thus does not directly measure the linear absorption coefficient μ(E) (Glatzel & Juhin, 2014). The cross section in equation (3) is proportional to the partial photoexcitation cross section [i.e. the term dominating μ(E) in the vicinity of an absorption edge] if the signal arising from photon-out transitions that is recorded in an experiment only depends on the probability of reaching the intermediate state. This assumption is a prerequisite for all fluorescence-detected absorption spectroscopic studies that aim at recording μ(E). It may not hold if, for example, the fluorescence yield varies across an absorption edge. Strong deviations between μ(E) and the fluorescence yield can be observed, for example, at the L edges of 3d transition metals (Kurian et al., 2012).

Fig. 2 provides some examples of excited final states that are frequently observed in (R)IXS. We refer to the energy range below core-level excitations as the low-energy range. A wealth of excitations may be observed here ranging from charge transfer, plasmon, intra-atomic (for example d–d, f–f and crystal field), spin–orbit, charge, spin or orbital order to phonons and vibrations. The scattering process in equation (3) is always element-selective if the intermediate state exhibits a hole in a core level. This is the defining characteristic of RIXS and distinguishes it from other techniques that are used to study similar excited states. The ability to observe excitations at low energies depends on the experimental resolution (Glatzel, 2022). IXS experimental setups in the hard X-ray range can reach an energy bandwidth of around 1 meV to explore the dynamic structure factor. Recently, a spectrographic approach to IXS was realized with an achieved energy bandwidth of 100 µeV (Chumakov et al., 2019). This allows, for example, mapping of phonon dispersion (Baron, 2015), but energy bandwidths of tens of meV also allow the observation of vibrations in molecules and liquids and phonons in solid-state systems (Nordgren & Rubensson, 2013; Rubensson et al., 2013).

 Some examples of excitations typically studied in inelastic X-ray scattering. The energy axis is the energy transfer or final-state energy in an (R)IXS experiment. Compton scattering (not shown here) shows a width and a maximum intensity at an energy that depends on the scattering angle.

### 3. Approximations to the resonant Kramers–Heisenberg equation

Approximations are necessary in order to evaluate equation (3). Such approximations are based on the theoretical approach that is chosen to describe the electronic structure and the X-ray-induced transitions (Tulkki & Aberg, 1982; Gel'mukhanov & Ågren, 1999; Kas et al., 2011; Jiménez-Mier et al., 1999; Ament et al., 2011; Geondzhian & Gilmore, 2018; de Groot & Kotani, 2008). The applicability of most approximations depends on the properties of the system under study (for example molecules versus solid-state systems), the atomic number of the X-ray-absorbing element and the observed transition. It may be possible to simulate the resonant scattering cross section neglecting interference effects and considering only one-electron transitions. An approach that allows evaluation of the full multiplet structure and interference effects may be more suitable in some cases (de Groot, 2005; de Groot & Kotani, 2008; van Veenendaal, 2015; Pollock et al., 2014; Haverkort et al., 2012; Wernet et al., 2015; Josefsson et al., 2012). It may be necessary to use different theoretical approaches for different energy regions of the same absorption edge (Glatzel, Weng et al., 2013).

Within an approximation it is possible to further classify the scattering processes. Considering single-electron transitions one may distinguish between a spectator and a participator scattering process (Fig. 3). The photoexcited electron remains in the receiving orbital above the Fermi energy as a spectator, while an electron from an orbital below the Fermi energy fills the core hole in the decay process. This process may also be referred to as resonant XES. In this case, similar to XES after photoionization well above the absorption edge, the X-ray emission process is independent of the incident polarization (Kotani et al., 2012). The direct RIXS transition (Ament et al., 2011) is largely equivalent to a spectator scattering process, but some authors distinguish between the two depending on the character of the valence band (Kotani et al., 2012) from which the electron decays to fill the core hole.

 Resonant inelastic scattering processes within a one-electron picture. A participator scattering process is also called indirect RIXS. A spectator decay is largely equivalent to a direct RIXS process.

A participator or indirect scattering process is characterized by a photoexcited electron that returns to the core hole. This process alone would result in elastic scattering with an angular dependence of the intensity on the scattering angle given by the incoming polarization. The perturbation of the electrons induced by the inner-shell excitation that acts in the intermediate state may result in low-energy excitations. The energy required for this transition appears as an energy-loss feature in the RIXS spectrum. Some authors use RIXS' only for indirect scattering processes and (resonant) XES' otherwise.

A theoretical treatment of spectator/direct and participator/indirect RIXS leads to different selection rules, intensities and angular dependence for the scattering cross sections (Ament et al., 2011; Kotani et al., 2012). The two processes have been characterized for nickel complexes (van Veenendaal et al., 2011). A distinction between the two processes may not be possible when they occur over similar energies and the approximations leading to the distinction break down.

### 4. Nomenclature and transition schemes

The nomenclature for denoting a RIXS process is ambivalent in the literature. Some authors use the shells that carry the electron hole in the intermediate and final state for direct RIXS, while the character of the orbital into which the photoexcitation occurs is not defined. At the K edge of 3d transition metals one may use the terms 1s2p and 1s3p RIXS to describe direct RIXS with excitation into the K absorption pre-edges that have mainly metal 3d character (de Groot & Kotani, 2008), i.e. resonantly excited Kα and Kβ lines. Indirect/participator RIXS at the K shell of 3d transition metals corresponds to 1s–4p excitation and decay (Ament et al., 2011). At the L edges of 3d (5d) transition metals the term RIXS is used when the transitions 2p to 3d (5d) and 3d (5d) to 2p are observed and final-state energies between 0 and a few eV are studied (Moretti Sala, Boseggia et al., 2014; Moretti Sala, Ohgushi et al., 2014; Calder et al., 2016).

Fig. 4 shows important transitions for a 3d transition metal using a one-electron and a many-body scheme assuming a spectator model. The photoexcited electron is elevated either to the continuum or to an unoccupied orbital close to the Fermi energy, i.e. the electron is resonantly excited. A one-electron diagram provides a qualitative description of the transitions, while the total energy axis in a many-body scheme observes energy conservation and is thus able to describe the transitions more quantitatively.

 One-electron (top) and total energy description of resonant (r) and nonresonant K emission in a 3d transition metal with ground-state configuration 3dn. Only spectator transitions are considered. Atomic shells are used for a simplified description. The resonant excitation is a quadrupole 1s to 3d transition. ɛ denotes an electron in a continuum level. L denotes a hole in a ligand orbital. The fine structure due to spin–orbit and multiplet interactions is neglected. Transitions from the valence shell are denoted valence-to-core (vtc or v2t) transitions.

RIXS may be grouped into transitions where the final states exhibit a hole in a core level or the low-energy regime, which in the present case exhibits a hole in the valence shell. The transitions between core levels are 2p (Kα) and 3p (Kβ) to 1s in Fig. 4. Valence-to-core (vtc or v2t) dipole transitions in 3d transition metals gain intensity from mixing between the metal and ligand orbitals (see below).

The final-state configurations after photoionization are similar for X-ray photoelectron spectroscopy (XPS) and XES. For example, the final-state configuration for Kβ lines, 3p5ɛp, where ɛ represents a continuum electron, results in a spectrum that is similar to 3p XPS with final states 3p5ɛ(s, d). The RIXS final states can be compared with absorption edges at lower energies. For example, 1s2p RIXS reaches the final-state configurations 2p53dn+1 that are identical to L-edge absorption spectroscopy. Resonant excitation of the vtc X-ray emission lines result in net excitations within the valence orbitals that have energies of a few eV and can be compared with UV–Vis spectroscopy. In fact, the final states 3dn+1L in vtc RIXS, where L represents a hole in a ligand orbital, are charge-transfer excitations that may gain spectral intensity via spectator or participator transitions. This energy range may also show d–d or f–f excitations (van Veenendaal et al., 2011; Kotani et al., 2012; Huotari, Pylkkänen et al., 2008; Ghiringhelli et al., 2009). The spectral differences between XES and photoelectron spectroscopy as well as between RIXS and absorption spectroscopy arise from different selection rules, as the final states in XES and RIXS are reached in a two-step process (equation 3).

### 5. The RIXS plane and line scans

The RIXS spectral intensity is often conveniently shown as a function of two variables. This may be the emitted energy or the energy transfer (final-state energy) and the incoming energy (Glatzel & Bergmann, 2005; Wernet et al., 2015). In solid-state physics the experiments are often performed on single crystals and the scattering intensity is plotted versus the energy transfer and the momentum transfer (Schmitt et al., 2014). The dispersion of a low-energy feature, i.e. the energy changes as a function of momentum transfer, shows that the excitation relates to a long-range order in the system and provides important information with respect to the spin, orbital and charge degrees of freedom (Ament et al., 2011; Haverkort, 2010; Schuelke, 2007; Le Tacon et al., 2013).

Fig. 5 shows the energy scheme, RIXS plane and line scans for a hypothetical system with two discrete and one broad (continuous) intermediate states, where the dependence on the momentum transfer is neglected. In the case of a spectator process where the intermediate-state core hole is replaced by a shallower core hole in the final state [for example 1s2p or 1s3p RIXS in 3d transition metals or 2p3d RIXS in 4d (5d) transition metals or rare earths], the final states will have a distribution similar to the intermediate states with additional splitting. The splitting, which is similar between intermediate and final states, arises for example from crystal field effects and electron–electron interactions within the valence shell. The additional splitting may arise from an inter­action of the valence electrons with the shallow final-state core hole that is stronger than the interaction with the deeper intermediate-state core hole. This stronger, more complex interaction may result in a richer multiplet structure. In the energy diagram in Fig. 5 this is indicated by a dashed line in the final states. In the RIXS plane, this additional splitting appears as off-diagonal' features, i.e. spectral intensity that does not occur with the main spectral intensity extending along a diagonal line in the RIXS plane and that becomes a fluorescence line well above the absorption edge.

 Bottom left: model RIXS plane based on the energy scheme shown in the inset on the bottom right. No experimental broadening has been applied. The energy scheme shows isolated (resonant) and continuum excitations. The lifetime broadenings are indicated in the RIXS plane. The dashed line shows a cut through the RIXS plane, which results in a high energy resolution fluorescence-detected (HERFD) XAS scan (top right, compared with an absorption scan). Cuts through the RIXS plane are shown for constant energy transfer (top left) and constant incident energy (bottom right).

The RIXS plane becomes considerably more complex when the energy transfer is tuned below all core-hole excitations (see Fig. 2). These low-energy excitations may resonate at different incident energies and thus spread in the RIXS plane along the absorption edge at different final-state energies (or energy transfers). Excitations that cannot be resolved experimentally may manifest themselves as intensity variations of the elastic peak.

It is desirable to record a full RIXS plane, as shown in Figs. 5, 6 and 9. This may be very time-consuming and often only line scans can be measured. Line scans can be recorded at fixed incident energy, fixed emission energy or constant energy transfer. Constant incident-energy scans on an absorption feature are frequently used to record the weak low-energy excitations (Fig. 2). A constant final-state scan is recorded by scanning the incoming and emitted energy simultaneously. Such a scan decomposes an absorption spectrum according to the final states that the intermediate states decay into. The experimental energy bandwidth in the emission detection is ideally below the natural width of the emission line. This is readily achieved in RIXS experiments with a core hole in the final state, but low-energy excitations are often sharper than can be achieved experimentally (Glatzel, 2022).

 Comparison between total fluorescence yield and HERFD XAS recorded simultaneously on K2PtIICl4 (solid line) and K2PtIVCl6 (dashed line). The inset shows the RIXS plane covering the L3 (2p3/2) absorption and Lα1 (2p3/2→3d5/2) emission.

A diagonal cut through the RIXS plane, as shown in Fig. 5, is experimentally easily achieved by fixing the emission energy while the incoming energy is scanned. It is reminiscent of fluorescence-detected absorption spectroscopy using either total yield or a solid-state detector with an energy bandwidth about two orders of magnitude larger than the natural linewidth. A sufficiently small energy bandwidth, as used in RIXS, resolves the fine structure in the X-ray emission spectrum and the diagonal cut through the RIXS plane may strongly deviate from a fluorescence-detected spectrum (Hämäläinen et al., 1991; Carra et al., 1995; Lafuerza et al., 2020). Such scans are denoted partial fluorescence yield (PFY), high energy resolution fluorescence-detected (HERFD) or high-resolution (HR) X-ray absorption spectroscopy.

### 6. X-ray fluorescence as an inelastic scattering process

A core-level vacancy can be created by particle (ion/electron) impact or photoexcitation. A radiative decay of a core-level vacancy is generally referred to as a fluorescence line. The energy scheme in Fig. 4 shows that spectator resonant inelastic scattering and nonresonant X-ray emission (fluorescence) following photoexcitation are closely related, but with important differences. The photoexcited electron in resonant scattering does not leave the atom and the scattering process is thus charge-neutral. Furthermore, different intermediate states may interfere in the scattering process, which influences the spectral intensities in the RIXS plane (equation 3). As suggested by some authors (Kotani & Shin, 2001; Glatzel et al., 2001), it is convenient and rigorous to also use the resonant Kramers–Heisenberg equation for nonresonant X-ray emission following photoionization. The photoelectron is carried along in the description of the intermediate and final states but does not interact with the bound electrons. This is important as photoionization may lead to multiple excited states that are either singly (for example charge-transfer excitation) or doubly ionized (Fig. 7; Glatzel et al., 2001; Chantler et al., 2010; Rovezzi & Glatzel, 2014). In the case of photoexcitation, X-ray emission should thus be treated by the X-ray scattering formalism as described by the resonant Kramers–Heisenberg equation. This means that the photoionization process must be explicitly calculated for a full treatment. It is, however, reasonable to assume that no interference occurs between intermediate states if the photoelectron is excited well above an absorption edge.

 Simplified energy scheme for X-ray fluorescence following photoionization where several intermediate states contribute to the fluorescence intensity.

The fluorescence or (nonresonant) X-ray emission energies from several intermediate states may result in either overlapping or well separated spectral features. The probability of reaching multiple-ionized states depends on the incoming X-ray energy (Huotari, Hämäläinen et al., 2008; Kavčič et al., 2009; Glatzel et al., 2003). We furthermore note that some authors have identified radiative Auger emission as a mechanism contributing to the fluorescence lines in 3d transition metals (Limandri et al., 2018).

The K fluorescence lines in 3d transition metals are the most frequently measured X-ray emission lines and we briefly explain the origin of the line splittings (Fig. 8). Table 1 shows the values for the two-electron Slater integrals that are used to evaluate the intra-atomic electron–electron interactions. The values for the (3d, 3d) interactions relate to the Racah parameters B and C and, together with the crystal field splitting, give rise to the valence-shell multiplet structure as captured in Tanabe–Sugano diagrams. This splitting cannot be directly observed in the K fluorescence lines. The 2p5 (Kα) final states are dominated by the 2p spin–orbit interaction ζ2p, while ζ3p becomes negligible for the 3p5 (Kβ) final states. The (p, d) interactions are larger for Kβ than Kα. In particular, the exchange integrals Gpd dominate the spectral shape for the 3p53d5 Kβ lines (Tsutsumi, 1959).

 Table 1 Atomic values (with 80% scaling) in eV for the Slater integrals in the 1s13d5, 2p53d5 (Kβ) and 3p53d5 (Kα) final states calculated using Cowan's code (Cowan, 1981)
1s13d52p53d53p53d5
ξd 0.06 0.06 0.05
ξp n/a 6.85 0.80
9.70 9.77 9.18
6.07 6.12 5.75
n/a 5.59 9.92
0.05 4.14 12.31
n/a 2.36 7.50
 X-ray emission lines following 1s core-hole creation in Fe2O3. The relative intensities between the Kα and Kβ lines were scaled according to theoretical values for the radiative transition probabilities. The insets show a magnification of the Kβ spectral region.

The fluorescence lines at energies higher than Kβ1,3 have been named Kβ′′ and Kβ2,5. They gain spectral intensities from occupied delocalized orbitals that have orbital momentum p character with respect to the photoexcited metal ion. Electrons in orbitals with metal d character can only contribute via electric quadrupole transitions and this intensity has been found to be negligible (Smolentsev et al., 2009; Lee et al., 2010; Gallo & Glatzel, 2014). The delocalized character of the orbitals that contribute to the Kβ′′ and Kβ2,5 lines has led to the nomenclature valence-to-core (vtc, v2c) for the spectral features in this region.

### 7. Chemical sensitivity in RIXS

X-ray absorption spectroscopy probes unoccupied levels, while XES arises from transitions between occupied orbitals. The mechanisms that give rise to the chemical sensitivity depend on the observed transitions. Core-to-core XES requires the hole (or unpaired electron) in the final state of the transition to strongly interact with the valence electrons in order to contain information about the chemical state of the X-ray-absorbing atom. The strength of this interaction can be derived from the values of the intra-atomic Slater integrals (Table 1). These intra-atomic interactions are modified by the valence-electron configuration. A change in oxidation or spin state will alter the interaction, resulting in spectral changes of the core-to-core lines. The Kβ lines are thus frequently used to detect the valence-shell spin state of a 3d transition-metal ion. Here, it is important to note that the mixing of atomic orbitals between metal and ligand atoms (covalence) also affects the interaction of the valence electrons with the core levels. Hence, core-to-core XES is sensitive to changes in covalence (Glatzel & Bergmann, 2005; Pollock et al., 2014). A different mechanism is a modification of the screening of the nuclear charge, i.e. a change in the effective nuclear charge, as experienced by the core electrons upon oxidation and reduction. This mechanism has been identified, for example, for sulfur (Alonso Mori et al., 2009).

Valence-to-core transitions give direct information on the valence-shell configuration. The valence electrons of the X-ray-absorbing atom may be directly involved in the transitions, as for example in sulfur or phosphorus (Mori et al., 2010; Stein et al., 2018). In the case of 3d transition metals the vtc lines following 1s core-hole creation reflect the occupied p density of states (Meisel et al., 1989; Smolentsev et al., 2009; Lee et al., 2010; Gallo & Glatzel, 2014). These emission lines may be used to identify the number and the type of ligands (Lancaster et al., 2011; Bergmann et al., 1999; Safonova et al., 2006; Safonov et al., 2006).

RIXS combines the sensitivity of XAS with XES. 1s2p RIXS at the K absorption pre-edges reveal spectral details that impose additional constraints on theoretical modelling (Glatzel et al., 2004; Glatzel, Schroeder et al., 2013; de Groot et al., 2005; Fig. 9). A one-electron approximation and neglecting the core-hole effect leads to a simplification of the Kramers–Heisenberg equation, and the RIXS process that directly involves the valence electrons can be described as a convolution of the unoccupied with the occupied density of states (Jiménez-Mier et al., 1999; Smolentsev et al., 2011).

 Contour plots of Mn 1s2p (Kα1) RIXS planes of (a) MnIIO, (b) , (c) and (d) . The energy axes are identical for all spectra. The intensity is normalized to the maximum in the pre-edge region (Glatzel, Schroeder et al., 2013).

### 8. Momentum dependence in RIXS

The scattering of photons in the X-ray regime causes the exchange of sizeable momentum q = kinkout, where |q| ≃ 2|k|sin(θ/2) (|k| = |kin| ≃ |kout|) is large enough to span large parts or all of the Brillouin zone. RIXS measurements on single crystals allow alignment of the momentum transfer along specific crystallographic directions. The momentum dependence of RIXS excitations can therefore be used to investigate their nature and infer important microscopic interactions. Typically, dispersive features are associated with collective modes involving multiple lattice sites, for example phonons, magnons and orbitons, while the lack of dispersion suggests the localized, or excitonic, character of an excitation, for example in the case of the so-called crystal field (dd or ff) transitions. The dispersive or nondispersive character is intrinsic to an excitation and does not depend on the details of the (direct or indirect) RIXS process that leads to the corresponding final state. The observation of dispersive quasiparticles using RIXS has made fundamental contributions in solid-state physics and in particular to the investigation of strongly correlated electron systems, for example high-temperature superconducting cuprates and spin–orbit coupled Mott insulating iridates.

The most prominent example of dispersive excitations studied by RIXS are certainly magnons: in insulating cuprates, the parent compounds of high-temperature superconductors, and iridates the low-temperature phase is characterized by an antiferromagnetic state, which can be perturbed by means of spin-flip excitations that propagate throughout the entire magnetic domain. The fact that RIXS can induce a single spin-flip transition in cuprates was anticipated theoretically by Ament and coworkers. It arises from a direct RIXS process and requires that the intermediate state is strongly spin–orbit coupled (Ament et al., 2009; Haverkort, 2010; Moretti Sala, Boseggia et al., 2014), so it typically occurs at the L (or M) edge of transition metals. The first experimental evidence of magnons was published by Braicovich et al. (2010) and led to the widespread use of RIXS to investigate magnetic excitations in 3d, 4d and 5d transition-metal oxides in a complementary way to inelastic neutron scattering (INS). RIXS is limited by a poorer energy resolution compared with INS, but can easily cover a larger energy-loss window. In addition, the small beam size of the X-rays facilitates the investigation of small amounts of material, including tiny single crystals and very thin (down to a single monolayer) films, and of materials that are unsuitable for neutron scattering. In particular, RIXS opened up the possibility of identifying magnetic modes in superconducting cuprates (paramagnons) and studying their momentum dependence accurately (Le Tacon et al., 2011; Dean et al., 2013), ultimately providing important insights into the microscopic interactions governing magnetism in these systems (Guarise et al., 2010; Peng et al., 2017). Moreover, unlike any other technique, magnetic excitations can also be studied under conditions of extreme high pressure (Rossi et al., 2019).

It should be mentioned that magnetic excitations are also visible via indirect RIXS processes. However, when not assisted by spin–orbit coupling, the spin-flip transition is forbidden and only spin-conserving magnetic excitations can be observed; bimagnons, a continuum formed by the combination of two interacting magnons, were first observed at the Cu K edge (Hill et al., 2008) and subsequently also at the O K edge (Bisogni et al., 2012).

Also direct is the RIXS process that leads to the creation of orbitons, i.e. collective excitations of an orbitally ordered state characterizing the low-temperature phase of many transition-metal oxides with strong electron correlation. Orbital flips can be produced via the RIXS process and can eventually propagate with characteristic dispersion relations that depend on the details of the microscopic orbital interactions (Schlappa et al., 2012).

Another important application of momentum-resolved RIXS is the study of phonons. In this case it is not the dispersion relation of phonons that is investigated, because this is often determined at an earlier stage by means of other experimental techniques, but rather their intensity throughout reciprocal space. Unlike magnons and orbitons, the RIXS process leading to the creation of a lattice vibration is indirect and contains the effect of the interaction between a specific phonon mode and the extra charge injected in the valence band for the duration of the RIXS intermediate state. In this respect, RIXS can be seen as a q-resolved probe of the electron–phonon coupling (Moser et al., 2015; Geondzhian & Gilmore, 2018; Rossi et al., 2019), a quantity that plays an important role in the physics of many materials, including high-temperature superconductors, but that can hardly be assessed by most experimental techniques.

Finally, as an example of excitations showing no (or little) dispersion we consider crystal field, typically dd or ff, excitations. In systems with localized (usually 3d or 4f) orbitals, the energies of electronic states are mostly set by intra-atomic interactions, including crystalline electric field, spin–orbit coupling and electron–electron correlation, that form multiplets. Transitions between different states are allowed by virtue of the second-order nature that characterizes the RIXS process, which can therefore be used to extract quantitative information on the above interactions. However, the q-dependence of crystal field excitations usually provides little information; rather, they show pronounced polarization effects that can be exploited to identify different final states (Moretti Sala, Boseggia et al., 2014; Amorese et al., 2018).

From an experimental point of view, the study of the momentum (and polarization) dependence of excitations requires the possibility of varying the scattering geometry in order to (i) change the magnitude |q| and (ii) align the direction of q relative to the sample orientation. To this end, modern RIXS spectrometers are able to rotate the scattering arm and have been equipped with manipulators possessing all of the degrees of freedom necessary to properly orient the sample (Moretti Sala et al., 2018; Brookes et al., 2018).

### 9. Angular and polarization dependence in RIXS

The richness of RIXS lies in the large number of possible spectra that can be obtained by varying the energy, direction and polarization state of the incident and scattered beams. This wealth of possibilities makes it difficult in practice to know whether a specific set of experiments measures all of the potential information. To date, the possibilities offered by angular- and polarization-dependent RIXS measurements have not been exploited to the best of their potential, especially on hard X-ray RIXS beamlines, where measurements are usually performed at fixed scattering angles (often in forward scattering, at a 90° scattering angle or in back-scattering). However, with such measurements one can expect a significant gain in information on the electronic structure, in a way comparable to that provided, for example, by angle-resolved photoemission spectroscopy (ARPES) measurements in comparison to conventional photoemission (XPS).

The idea consists of expressing a RIXS spectrum obtained for a given wavevector and polarization vector of the incident beam (ɛin, kin) and of the scattered beam (ɛout, kout) as a linear combination of terms which are `fundamental spectra' (i.e. independent of the incident and scattered beam properties) multiplied by a polynomial in (ɛin, kin, ɛout, kout). This task is not trivial because the Kramers–Heisenberg cross section in the form of equation (3) couples beam and sample properties inside the same matrix element, and accounts for incident and scattered beam properties through different matrix elements. Decoupling and recoupling them in a different way can be performed by using spherical tensors to express transition operators, polarization and wavevectors, which is very efficient because tensors can easily be transformed by any symmetry operation. Tensor formalism has been used with great success for XPS of localized magnetic systems (Thole & van der Laan, 1991, 1994; van der Laan & Thole, 1993, 1995a,b) and in XAS (Schillé et al., 1993; Thole et al., 1994; Brouder et al., 2008; Elnaggar et al., 2021). Such a formulation is therefore particularly interesting (i) to predict specific experimental configurations aiming at the observation of specific sample properties and (ii) to determine the smallest number of measurements (or calculations) that are needed to capture the full spectral information.

A spherical tensor-based expression of the RIXS cross section was obtained from a pure group-theoretical treatment without any assumptions (Juhin et al., 2014) and is therefore equivalent to equation (3). It involves tensor products of two a-ranked tensors, i.e. one tensor that concerns only the experimental conditions, γ(a), and one tensor that concerns only the sample, S(a). Experimental and sample tensors are coupled by 9j-symbols, ensuring that variables in the scattering process are coupled in a correct (physical) way. Each experimental tensor γ(a) is written as the product of one tensor describing only the properties of the incident beam and one tensor describing only the properties of the scattered beam, where the polarization part is fully separated from the wavevector part. Such a formulation is very powerful. The nature of the transitions involved (electric dipole or quadrupole) and the polarization state of the light (linear, circular, detected or not in the emission) determine the possible values for the rank of the experimental tensor: this allows the direct simplification of the RIXS cross section by easily identifying terms with zero contribution.

Sample tensors S(a) gather all of the physical information on the material and are independent of any system of coordinates. Their angular components (α = −a, …, a) are the fundamental spectra from which any experimental spectrum can be expressed. The number of sample tensors (possible values for a) and of nonzero tensor components is determined by the nature of the transitions involved and the symmetry of the sample.

Let us now illustrate this tensor-based expression on a common experimental case. We consider an isotropic sample (i.e. disordered molecules, a liquid, a polycrystal or a powder with no preferred orientation and no remanent magnetization) that is measured with electric dipole transitions in both excitation and emission, using a detector that does not analyze the polarization state of the scattered beam. The brute-force treatment of equation (3), considering a RIXS spectrum as a fourth-rank Cartesian tensor, would lead to the measurement of 34 = 81 different components in order to recover the full spectral information. By considering spherical tensors and imposing the isotropy condition (i.e. the only possible value for a is 0), the number of fundamental spectra is reduced to three, which implies a spectacular gain in measurement time. In this case the RIXS cross-section is simply expressed as the combination of three fundamental spectra, the relative weights of which depend on the relative orientation of the incident polarization vector and the scattered wavevector,

Only three measurements are necessary to record all of the spectral information. For example, in horizontal scattering geometry (Fig. 10), by fixing the scattering angle (kin, kout) at 90° and changing the incident polarization direction [linear horizontal (LH), linear vertical (LV) or linear at 45° (L45)],

 Experimental configurations in 90° scattering geometry using LH and LV polarization. L45 corresponds to a linear polarization vector along (LH + LV).

The three fundamental spectra S000, S110 and S220 can be determined from proper linear combinations of these three measurements.

In the case of electric quadrupole transitions in the absorption followed by electric dipole transitions in the emission (again, for an isotropic sample with no detection of the scattered polarization), the RIXS cross section depends on the relative orientation of the incident polarization vector and the scattered wavevector, but now also on the relative orientation of kin and kout,

Only three fundamental spectra are needed to recover the complete polarization and angular dependence of the cross section by choosing three measurements yielding linearly independent cross sections.

So far, most RIXS studies have focused on the dependence on energy and momentum (mostly with soft X-rays) and the polarization dependence has been overlooked. Nevertheless, this aspect should be more easily investigated in the future thanks to the technical developments that have recently been achieved in detection. On hard X-ray beamlines, the number of analyser crystals has significantly increased, implying that they are positioned at different scattering angles with respect to the sample, which is likely to yield spectral differences in the RIXS spectra according to the formula above (Kotani et al., 2012; Kotani & Shin, 2001; Matsubara et al., 2002). Secondly, analysis of the polarization of the emitted beam has remained a technical challenge until recently. On soft X-ray RIXS beamlines, simultaneous measurements of the energy and polarization state of scattered X-rays was first achieved using a graded multilayer, allowing the disentanglement of σ- and π-polarized components in the scattered beam (Braicovich et al., 2014). Such a polarimeter was used to show the collective nature of spin excitations in superconducting cuprates (Minola et al., 2015). For hard X-rays, an initial device was proposed a few years ago based on a pyrolytic graphite crystal placed in front of spherically bent Ge(800) crystals. Using the (006) reflection for the Cu K edge, with a polarization extinction ratio of 0.94, a clear polarization dependence was successfully observed in the d–d excitations of KCuF3 (Ishii et al., 2011). More recently, by means of circular-polarization analysis using a diamond phase retarder positioned between the sample and Ge(440) analyzers, a large magnetic circular-dichroic effect in the Fe Kα1 X-ray emission (non­resonant) spectrum was measured in an iron single crystal (Inami, 2017). The generalization of such polarization and angle-dependent experimental studies performed with soft and hard X-rays, together with calculations, could definitely contribute to a better understanding of the polarization and angular dependence in RIXS and check the applicability of the RIXS sum rules (Borgatti et al., 2004).

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