International
Tables for Crystallography Volume I Xray absorption spectroscopy and related techniques Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2022 
International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870720007521 Inelastic Xray scattering^{a}ESRF – The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France,^{b}Institut 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 ^{c}Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy The first and secondorder terms of the Xray scattering cross section are given and an overview is provided of the excitations that can be studied using inelastic Xray scattering. The secondorder (resonant) term is then elaborated on and the role of the corehole lifetime broadening is explained. Approximations and the oneelectron transitions that describe the scattering process are briefly addressed in a simplified equation. Xray 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. 
The dramatic advances in synchrotronradiation instrumentation over the past decades have made the observation and detailed analysis of weak Xray scattering events possible. Such experiments, together with extensive theoretical work, have provided powerful tools for researchers in all fields of natural sciences, including solidstate physics, catalysis, energy applications, environmental sciences, cultural heritage and biology. Spectroscopy of inelastic Xray 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 Xray scattering from electrons.
The geometry of the Xray scattering process is shown in Fig. 1. An incoming photon with energy ħω_{in}, momentum k_{in} and polarization ɛ_{in} is scattered by the sample, resulting in an outgoing photon with energy ħω_{out}, momentum k_{out} and polarization ɛ_{out}. The scattering process is called inelastic if ħω_{in} ≠ ħω_{out} 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 lowestenergy ground state. This may be the case for thermal excitations (the inelastic scattering data will show antiStokes lines) or pumpandprobe experiments where atoms are excited optically shortly before the Xray probe (Wernet et al., 2015; Baron, 2015).

Geometry of photon scattering. The sample is centred at the origin and for φ = 0, ɛ_{in} points along x in the case of linear polarized incoming light. 
The angle between k_{in} and k_{out} 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 ħω_{in} − ħω_{out} and k_{in} − k_{out}, respectively. The probability of observing a scattered photon in the solid angle dΩ is given by the doubledifferential cross section F(ħω_{in}, ħω_{out}, k_{in}, k_{out}, ɛ_{in}, ɛ_{out}), the expression for which will be given below.
A quantummechanical treatment of the scattering of Xrays 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 Xray sources at storage rings, but may break down for Xray freeelectron lasers (Beye et al., 2013; Rohringer et al., 2012). The terms in the Hamiltonian describing the perturbation may interfere and the expressions for the doubledifferential 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 Xrays is several orders of magnitude weaker than the charge scattering (Blume, 1985). Neglecting such terms, the Hamiltonian describing the perturbation becomeswhere p_{j} 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, ω_{in} − ω_{out}) 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 Xray 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 sin^{2}φ + cos^{2}θcos^{2}φ 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 k_{in} and k_{out}: isotropic (for σscattering) or cos^{2}(θ) (for πscattering) 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 τ_{n} either via emission of a photon (radiative decay) or emission of an electron (nonradiative or Auger decay). The lifetime broadening (halfwidth at halfmaximum) Γ_{n} = ℏ/(2τ_{n}) does not appear in the secondorder 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 δfunction by a Lorentzian function (Kotani & Shin, 2001),
In the literature, one finds the use of the terms resonant inelastic Xray scattering (RIXS), resonance Raman scattering (RRS) and (resonant) Xray 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 Xray 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 Xray scattering (REXS), which is frequently used in solidstate 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 Xrays 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 Xrays, 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 Xray 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.
Fluorescencedetected 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 photonout transitions that is recorded in an experiment only depends on the probability of reaching the intermediate state. This assumption is a prerequisite for all fluorescencedetected 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 corelevel excitations as the lowenergy range. A wealth of excitations may be observed here ranging from charge transfer, plasmon, intraatomic (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 elementselective 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 Xray 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 solidstate systems (Nordgren & Rubensson, 2013; Rubensson et al., 2013).
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 Xrayinduced transitions (Tulkki & Aberg, 1982; Gel'mukhanov & Ågren, 1999; Kas et al., 2011; JiménezMier 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 solidstate systems), the atomic number of the Xrayabsorbing element and the observed transition. It may be possible to simulate the resonant scattering cross section neglecting interference effects and considering only oneelectron 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 singleelectron 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 Xray 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.
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 innershell excitation that acts in the intermediate state may result in lowenergy excitations. The energy required for this transition appears as an energyloss 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.
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 preedges 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 finalstate 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 oneelectron and a manybody 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 oneelectron diagram provides a qualitative description of the transitions, while the total energy axis in a manybody scheme observes energy conservation and is thus able to describe the transitions more quantitatively.
RIXS may be grouped into transitions where the final states exhibit a hole in a core level or the lowenergy 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. Valencetocore (vtc or v2t) dipole transitions in 3d transition metals gain intensity from mixing between the metal and ligand orbitals (see below).
The finalstate configurations after photoionization are similar for Xray photoelectron spectroscopy (XPS) and XES. For example, the finalstate configuration for Kβ lines, 3p^{5}ɛp, where ɛ represents a continuum electron, results in a spectrum that is similar to 3p XPS with final states 3p^{5}ɛ(s, d). The RIXS final states can be compared with absorption edges at lower energies. For example, 1s2p RIXS reaches the finalstate configurations 2p^{5}3d^{n+1} that are identical to Ledge absorption spectroscopy. Resonant excitation of the vtc Xray 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 3d^{n+1}L in vtc RIXS, where L represents a hole in a ligand orbital, are chargetransfer 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 twostep process (equation 3).
The RIXS spectral intensity is often conveniently shown as a function of two variables. This may be the emitted energy or the energy transfer (finalstate energy) and the incoming energy (Glatzel & Bergmann, 2005; Wernet et al., 2015). In solidstate 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 lowenergy feature, i.e. the energy changes as a function of momentum transfer, shows that the excitation relates to a longrange 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 intermediatestate 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 finalstate core hole that is stronger than the interaction with the deeper intermediatestate 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 `offdiagonal' 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.
The RIXS plane becomes considerably more complex when the energy transfer is tuned below all corehole excitations (see Fig. 2). These lowenergy excitations may resonate at different incident energies and thus spread in the RIXS plane along the absorption edge at different finalstate 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 timeconsuming 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 incidentenergy scans on an absorption feature are frequently used to record the weak lowenergy excitations (Fig. 2). A constant finalstate 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 lowenergy excitations are often sharper than can be achieved experimentally (Glatzel, 2022).
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 fluorescencedetected absorption spectroscopy using either total yield or a solidstate 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 Xray emission spectrum and the diagonal cut through the RIXS plane may strongly deviate from a fluorescencedetected spectrum (Hämäläinen et al., 1991; Carra et al., 1995; Lafuerza et al., 2020). Such scans are denoted partial fluorescence yield (PFY), high energy resolution fluorescencedetected (HERFD) or highresolution (HR) Xray absorption spectroscopy.
The intermediate and finalstate lifetime broadenings are indicated in Fig. 5. In most applications the intermediatestate lifetime broadening is larger than the finalstate lifetime broadening. A HERFD scan moves at 45° through the RIXS plane and the spectral line broadening is limited by the finalstate lifetime (de Groot et al., 2002). Fig. 6 shows an example of HERFDXAS spectra recorded on platinum compounds, with the full RIXS plane in the inset. No offdiagonal spectral intensity (apart from the lifetime broadening) is observed. In this case, a HERFD scan becomes, to a good approximation, an Xray absorption scan with reduced lifetime broadening (Van Bokhoven et al., 2006; Hayashi, 2011). An alternative method to obtain similar spectra is high energy resolution offresonance spectroscopy (HEROS), where the incident energy is tuned well below (∼20 eV) an absorption edge and the emission energy is scanned (Błachucki et al., 2014; Kavčič et al., 2013). HERFD and HEROS scans may be compared with true absorption spectra that have been deconvoluted (Loeffen et al., 1996; Fister et al., 2007; Juhin et al., 2016).
A corelevel vacancy can be created by particle (ion/electron) impact or photoexcitation. A radiative decay of a corelevel vacancy is generally referred to as a fluorescence line. The energy scheme in Fig. 4 shows that spectator resonant inelastic scattering and nonresonant Xray 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 chargeneutral. 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 Xray 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 chargetransfer excitation) or doubly ionized (Fig. 7; Glatzel et al., 2001; Chantler et al., 2010; Rovezzi & Glatzel, 2014). In the case of photoexcitation, Xray emission should thus be treated by the Xray 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 Xray fluorescence following photoionization where several intermediate states contribute to the fluorescence intensity. 
The fluorescence or (nonresonant) Xray emission energies from several intermediate states may result in either overlapping or well separated spectral features. The probability of reaching multipleionized states depends on the incoming Xray 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 Xray emission lines and we briefly explain the origin of the line splittings (Fig. 8). Table 1 shows the values for the twoelectron Slater integrals that are used to evaluate the intraatomic 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 valenceshell multiplet structure as captured in Tanabe–Sugano diagrams. This splitting cannot be directly observed in the K fluorescence lines. The 2p^{5} (Kα) final states are dominated by the 2p spin–orbit interaction ζ_{2p}, while ζ_{3p} becomes negligible for the 3p^{5} (Kβ) final states. The (p, d) interactions are larger for Kβ than Kα. In particular, the exchange integrals G_{pd} dominate the spectral shape for the 3p^{5}3d^{5} Kβ lines (Tsutsumi, 1959).

The fluorescence lines at energies higher than Kβ_{1,3} have been named Kβ′′ and Kβ_{2,5}. 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 Kβ_{2,5} lines has led to the nomenclature valencetocore (vtc, v2c) for the spectral features in this region.
Xray 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. Coretocore 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 Xrayabsorbing atom. The strength of this interaction can be derived from the values of the intraatomic Slater integrals (Table 1). These intraatomic interactions are modified by the valenceelectron configuration. A change in oxidation or spin state will alter the interaction, resulting in spectral changes of the coretocore lines. The Kβ lines are thus frequently used to detect the valenceshell spin state of a 3d transitionmetal 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, coretocore 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).
Valencetocore transitions give direct information on the valenceshell configuration. The valence electrons of the Xrayabsorbing 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 corehole 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 preedges 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 oneelectron approximation and neglecting the corehole 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 (JiménezMier et al., 1999; Smolentsev et al., 2011).
The scattering of photons in the Xray regime causes the exchange of sizeable momentum q = k_{in} − k_{out}, where q ≃ 2ksin(θ/2) (k = k_{in} ≃ k_{out}) 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 socalled crystal field (d–d or f–f) 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 solidstate physics and in particular to the investigation of strongly correlated electron systems, for example hightemperature 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 hightemperature superconductors, and iridates the lowtemperature phase is characterized by an antiferromagnetic state, which can be perturbed by means of spinflip excitations that propagate throughout the entire magnetic domain. The fact that RIXS can induce a single spinflip 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 transitionmetal 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 energyloss window. In addition, the small beam size of the Xrays 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 spinflip transition is forbidden and only spinconserving 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 lowtemperature phase of many transitionmetal 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 momentumresolved 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 qresolved 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 hightemperature 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 d–d or f–f, excitations. In systems with localized (usually 3d or 4f) orbitals, the energies of electronic states are mostly set by intraatomic 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 secondorder nature that characterizes the RIXS process, which can therefore be used to extract quantitative information on the above interactions. However, the qdependence 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).
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 polarizationdependent RIXS measurements have not been exploited to the best of their potential, especially on hard Xray RIXS beamlines, where measurements are usually performed at fixed scattering angles (often in forward scattering, at a 90° scattering angle or in backscattering). 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 angleresolved 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 (ɛ_{in}, k_{in}) and of the scattered beam (ɛ_{out}, k_{out}) 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 (ɛ_{in}, k_{in}, ɛ_{out}, k_{out}). 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 tensorbased expression of the RIXS cross section was obtained from a pure grouptheoretical treatment without any assumptions (Juhin et al., 2014) and is therefore equivalent to equation (3). It involves tensor products of two aranked 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 9jsymbols, 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 tensorbased 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 bruteforce treatment of equation (3), considering a RIXS spectrum as a fourthrank Cartesian tensor, would lead to the measurement of 3^{4} = 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 crosssection 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 (k_{in}, k_{out}) 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 S^{000}, S^{110} and S^{220} 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 k_{in} and k_{out},
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 Xrays) 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 Xray 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 Xray RIXS beamlines, simultaneous measurements of the energy and polarization state of scattered Xrays was first achieved using a graded multilayer, allowing the disentanglement of σ and πpolarized 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 Xrays, 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 KCuF_{3} (Ishii et al., 2011). More recently, by means of circularpolarization analysis using a diamond phase retarder positioned between the sample and Ge(440) analyzers, a large magnetic circulardichroic effect in the Fe Kα_{1} Xray emission (nonresonant) spectrum was measured in an iron single crystal (Inami, 2017). The generalization of such polarization and angledependent experimental studies performed with soft and hard Xrays, 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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