Chapter 2.2. Tensorial interactions of X-rays

Whereas the anisotropy of X-ray absorption has been known since at least 1932 (see the discussion in Brouder, 1990), the tensorial character of RXD was not discovered until 1982, when D. and L. Templeton unveiled a new world by finding polarization dependence in the RXD analysis of RbUO₂(3NO₃) at the L₃ edge of uranium (Templeton et al., 1982). Such findings were explained one year later by Dmitrienko (1983), who showed that the occurrence of Bragg-forbidden reflections is a consequence of the anisotropy of the scattering amplitude (at the time, it was termed the `anisotropic tensor of the susceptibility'). This in turn implied that for nonsymmorphic space groups some extinction rules are no longer valid around the edge. Just a few years later, Gibbs et al. (1988) discovered an effect of magnetic anisotropy in RXD spectra by measuring an enhancement effect of the magnetic signal in Bragg reflections across the L₃ edge of holmium, determined by the exchange interaction and associated with its helical magnetic structure. The theoretical analysis of these effects was due to Hannon et al. (1988), who interpreted the polarization dependence of the resonant scattering amplitude on the basis of electric dipole (E1) and electric quadrupole (E2) transitions from the 2p core levels to the 5d and 4f states, respectively.

As described below, these observations paved the way for a wide range of experiments at synchrotron-radiation facilities in which the energy was tuned across the different absorption edges and the polarization dependence of the X-ray scattering cross section served as a basic tool to identify tensorial (charge and magnetic) contributions.

The first half of the 1990s was characterized by two main streams of research which, while initially parallel, finally met at the dawn of the twenty-first century. On one side, in the domain of magnetism in absorption, the work of Carra, Thole and van der Laan (Thole et al., 1992; Carra et al., 1993) led to their celebrated `sum rules' for X-ray magnetic circular dichroism (XMCD). These rules are valid at the level of E1 X-ray absorption, where it is possible to simplify the radial matrix elements of the transition, thus obtaining the expectation value of the magnetic moment along the direction of the incoming X-ray beam. In parallel, Luo et al. (1993) developed a general theory for both elastic and inelastic X-ray scattering, introducing the spin–orbit operators for the valence shell involved in the resonant process. The overall interpretation was still limited to E1–E1 scattering and based on a simplified SO(2) symmetry, with a scattering amplitude divided into scalar (nonmagnetic), vector (magnetic) and second-rank spherical tensor parts (see Table 1 for a comparison with the present classification). In a different domain, Finkelstein et al. (1992) and Templeton & Templeton (1994) showed that a great deal of information about phase transitions could be extracted from RXD in the E2–E2 and E1–E2 channels, respectively.

Table 1 Properties of multiple expansion tensors under time-reversal and parity operators (Di Matteo et al., 2005) Tensor Type Multipole X(0)(E1 − E1) + + Charge Monopole X(0)(E2 − E2) + + Charge Monopole X(1)(E1 − E1) − + Magnetic Dipole X(1)(E2 − E2) − + Magnetic Dipole X(1+)(E1 − E2) + − Electric Dipole X(1−)(E1 − E2) − − Polar toroidal Dipole X(2)(E1 − E1) + + Electric Quadrupole X(2)(E2 − E2) + + Electric Quadrupole X(2+)(E1 − E2) + − Axial toroidal Quadrupole X(2−)(E1 − E2) − − Magnetic Quadrupole X(3)(E2 − E2) − + Magnetic Octupole X(3+)(E1 − E2) + − Electric Octupole X(3−)(E1 − E2) − − Polar toroidal Octupole X(4)(E2 − E2) + + Electric Hexadecapole …

These findings led to a common interpretation of RXD and XAFS phenomena in terms of E1, E2 transitions and their interference, revealing subtle anisotropic effects. For example, Carra & Thole (1994) showed that E2–E2 RXD could be interpreted using the same framework as in Luo et al. (1993). The analysis of E1–E2 phenomena followed soon after the discovery of anomalous magnetic resonances in V₂O₃ (Paolasini et al., 1999, 2001), boosted by the search for non­reciprocal and magneto-chiral dichroic effects in XAFS by Goulon et al. (2002) and Di Matteo et al. (2003). This kind of research led to the development of the multipole interpretation of XAFS and RXD spectroscopy that constitutes a fundamental tool for the investigation of time-reversal and parity-breaking order parameters (Lovesey et al., 2005; Di Matteo, 2012).

Since the beginning of the twenty-first century, the huge improvement in experimental energy resolutions has directed the attention of the international community to the study of inelastic X-ray scattering phenomena. Whereas RXD is a coherent elastic phenomenon (it is also known as REXS, resonant elastic X-ray scattering) characterized by the same energy for incident and scattered photons, the study of electronic and magnetic excitations only became a main focus of research with the development of the technique now known as resonant inelastic X-ray scattering (RIXS). Nowadays, the main focus of most RIXS experiments is on the energy dependence of electronic and magnetic excitations and tensorial analysis in this domain, which although developed theoretically (van Veenendaal et al., 1996; Borgatti et al., 2004; Ament et al., 2011; Juhin et al., 2014) has not yet been fully exploited experimentally. It is the aim of the next section to review the spectroscopic tensorial formalism common to REXS, RIXS and XAFS.

The Kramers–Heisenberg formula for the partial differential cross section per unit solid angle (Ω) and per unit energy of the scattered photon is where r₀ ≃ 2.52 × 10⁻¹⁵ m is the classical electron radius and ℏω_i and ℏω_s are the incident and scattered photon energies, respectively. Here, I labels the initial state of the sample, with energy E_I, and F the final state after the scattering process, with energy E_F. The amplitudes A_FI and B_FI correspond to interaction amplitudes quadratic and linear in the vector potential, respectively (see Chantler & Creagh, 2024). In the analysis that follows we will neglect the quadratic term: , where ɛ_i and ɛ_s are the incident and scattered photon polarizations and k_i and k_s are their wavevectors, respectively. Remember that A_II is the elastic contribution to the Bragg scattering and is only weakly energy-dependent. Therefore, it does not usually represent the main scattering contribution close to an absorption edge, although in X-ray magnetic diffraction it can interfere with the REXS term originating from B_II (Paolasini & de Bergevin, 2008).

The amplitude B_FI originates from the second-order expansion in terms that are linear in the vector potential. This second-order expansion in the matter–radiation interaction Hamiltonian contains two terms: a resonant term close to an absorption edge and a nonresonant term. The latter, although providing important contributions to nonresonant magnetic scattering (Blume, 1985, 1994), can be neglected close to absorption edges. For this reason, we write just the resonant contribution aswhere m is the electron mass and Γ_N is the core-hole inverse lifetime in the intermediate state N. In this subsection the operators are considered in the E1 approximation; , so that transition-matrix elements can be written as . Higher-order multipole expansion is treated in the next subsection.

Interestingly, from the resonant Kramers–Heisenberg amplitude (with equal incident and scattered polarization and wavevector parameters), we can also express the absorption intensity aswhere represents the imaginary part.

Analogously, we can derive the cross section for REXS asWe notice that in the case of REXS the neglected A² term should be restored in order to take the scalar Bragg scattering into account, as well as the structure and form factors (see, for example, Di Matteo, 2012).

The general expression for RIXS can be expressed (Ament et al., 2011) as

Equations (3), (4) and (5) lead to the theoretical descriptions of XAFS, REXS and RIXS, respectively. Their tensor properties follow directly from the expansion of the scalar products between the polarization vector and the electron position, which allows each of the previous equations to be written in the form , where X can be any of the REXS or RIXS amplitudes or the XAFS cross section and α, β = x, y, z. In the case of XAFS, however, we have just one polarization vector, , as the XAFS cross section is a first-order process. Such a decomposition corresponds to the separation of the amplitude/cross section into a term that describes the geometry of the probe (the polarization and, for higher multipoles, wavevectors of the incident and scattered electromagnetic fields), the tensors , and a term describing the dynamical susceptibility of the sample, the tensors X_αβ. The latter tensors can be explicitly written for RIXS as or and for REXS and for XAFS, respectively.

The tensor properties of the sample in the E1–E1 approximation, as determined by the electromagnetic incident (and scattered) signal, are reproduced by the term . As the latter tensor is coupled to X_αβ through a scalar product, there is a one-to-one correspondence between the dynamical tensor properties of X_αβ and the geometrical tensor properties of the electromagnetic probe . Both can be written in terms of their irreducible components [under the rotation group SO(3)] as (i) a scalar, (ii) a vector and (iii) a second-rank spherical tensor. In the case of XAFS (REXS), the time-reversal invariance of the corresponding intensity (amplitude) allows a direct interpretation of the dynamical tensor properties of the vector as an orbital and/or spin magnetic moment, depending on the absorption edge. Because of the scalar coupling between the sample properties (X_αβ) and the electromagnetic tensor, the magnetic moment, time-reversal odd part of X_αβ must be coupled to the time-reversal odd part of the electromagnetic tensor, which is for XAFS and for REXS. For both XAFS and REXS, the time-reversal even second-rank spherical tensor is associated with the electric quadrupole moment of the sample projected onto the absorbing ion, and is determined by the spherical coupling of the two polarization vectors to rank 2: . We remind the reader that these order parameters (as for the higher-order multipoles in the next subsection) do not necessarily refer to ground-state properties (Di Matteo et al., 2005).

Interestingly, analysis of forbidden Bragg reflections at the K edge of transition metals also provided evidence for charge and orbital multipoles related to the anisotropic environment of atoms in crystals or thermal motion (Kokubun & Dmitrienko, 2012). Indeed, the E1 transitions at the K edge from the 1s core level to 4p intermediate states are sensitive to the crystal field surrounding the resonant atom and, because of the strongly anisotropic environment, the resonant behaviour is sensitive to the local crystallographic distortion, as in manganites (Murakami et al., 1998) or KCuF₃ (Paolasini et al., 2002). For manganites, the anisotropic reflections originally attributed to the orbital ordering of 3d shells have been shown to be related to Jahn–Teller cooperative distortions (Benfatto et al., 1999). A different interpretation of the same reflections is usually performed in the soft X-ray regime (Wilkins et al., 2003; Mulders et al., 2006).

We can go beyond the dipole approximation by a full multipole expansion of the exponential term exp(ik_i,s · r) in the vector potential (see Chantler & Creagh, 2024). Replacing the well known expression for the transition operators in equation (2), we obtain the multipole terms of the electromagnetic field determined by the coefficients of the spherical harmonics . In the previous expression, j_l(k_i,sr) are spherical Bessel functions, k_i,s and k_i,s are the incident or scattered wavevector and its modulus, and and are the solid angles in k and r space. The order of the multipole is determined by 2^l+1, as for l = 0 the transition operator is the dipole (ɛ_{i, s} · r), for l = 1 the quadrupole (i/2)(k_i,s · r)(ɛ_i,s · r) etc. (the origin of the factor 1/2 in the quadrupole term, which is a quantum-mechanical effect, is well explained in Brouder, 1990). We notice here that often the exponential is approximated by a Taylor expansion of the kind where γ_i,s is the angle between k_i,s and r. When k_i,sr ≪ 1, because of the limit j_l(k_i,sr) ≃ (k_i,sr)^l/(2l + 1)!!, the two expansions differ only in the rearrangement of some terms (Brouder et al., 2008).

Multipoles of the electromagnetic field interact with the equivalent multipoles of the sample through scalar products, so that the tensor properties of the sample that can be analyzed by the field are determined by the order of its electromagnetic multipoles. However, practical calculations of this coupling can often be quite lengthy, as exemplified in Juhin et al. (2014) for RIXS.

The multipoles that can be detected by XAFS, REXS and RIXS have been described in several research articles (Di Matteo & Natoli, 2002; Carra et al., 2003; Di Matteo et al., 2005; Marri & Carra, 2004; Lovesey et al., 2007; Lovesey & Balcar, 2010; Ament et al., 2011). At the resonant (absorbing) j-ion, the scattering amplitudes f_j (the square of the absorption intensities) for REXS (XAFS) can be expressed as a scalar product of two irreducible spherical tensors, where is a rank-p spherical tensor depending only on the photon field and representing the experimental geometry associated with the incident and scattered photon polarizations and wavevectors. In the E1–E1 case, it corresponds to the Cartesian tensor T_αβ introduced above, expressed in spherical components.

The rank-p spherical tensor contains information about the excitation spectrum at the j-ion of the sample and describes its anisotropy and, for REXS and XAFS, its time-reversal properties. Remember that whereas for absorption techniques (XAFS and RIXS) the total signal is trivially the sum over all atoms j in the sample, special care with Bragg factors should be considered in the case of REXS amplitudes (which can even interfere with the Thomson amplitude), as described in the examples in Section 3.2. The rank p defines the order of the multipole in the electromagnetic field expansion and the projection m can take (2p + 1) values that satisfy (−p ≤ m ≤ p), as in usual angular-momentum theory. The behaviour under time-reversal () and spatial parity () symmetries of a selected set of these tensors is shown in Table 1. As detailed in various references (Di Matteo et al., 2005; Lovesey et al., 2005), tensorial interactions of X-rays leading to XAFS and REXS signals are strongly dependent on the crystal point group and space group, respectively.

The odd terms under the operator appear in XAFS when the space group of the sample breaks the inversion symmetry (in REXS, instead, the constraint is smoother: it is only necessary that the resonant ion breaks the local inversion symmetry). In this case an interference tensor with mixed E1–E2 character is allowed. Interestingly, magnetic E1–E2 terms, which are odd under both and operators, are associated with toroidal multipoles, which now represent an important subject in condensed matter for possible magneto-spintronics outcomes (Spaldin et al., 2008).

Each of the multipoles in Table 1 can be associated with a physical effect. For example, the parity-even rank-2 tensors in XAFS are responsible for linear dichroism and/or magnetic X-ray linear dichroism (XLD; van der Laan et al., 1986). Analogously, the parity-even rank-1 tensors in XAFS are related to XMCD (Carra et al., 1993) as well as to its dispersive analogue X-ray Faraday rotation (Collins, 1999).

The parity-even rank-4 tensors associated with the electric quadrupole (E2–E2) transitions give rise to time-reversal even pre-edge effects in rhombohedral crystals (Dräger et al., 1988; Finkelstein et al., 1992; Carra & Thole, 1994) determined by valence-orbital anisotropies. Conversely, parity-even rank-3 tensors associated with the same E2–E2 events are magnetic in origin and can be associated with the interaction with the magnetic octupole of the sample, as in CeFe₂ (Paolasini et al., 2008).

As stated above, E1–E2 tensors, which are odd under spatial parity, are related to inversion-symmetry breaking and therefore can be associated with ferroelectricity (non­magnetic, time-reversal even tensors) or with magneto­electricity (magnetic, time-reversal odd tensors). Among the latter, we refer to the polar toroidal dipole and octupole, which can be detected using X-ray magnetochiral dichroism (XMχD; Goulon et al., 2002; Di Matteo & Natoli, 2002; Sessoli et al., 2015). Also useful to disentangle parity-odd, magnetic multipoles is the technique of X-ray nonreciprocal directional dichroism described by Kubota et al. (2004), in which the polar toroidal dipole and octupole and magnetic quadrupole are measured, corresponding to the tensors X⁽¹⁻⁾(E1 − E2), X⁽³⁻⁾(E1 − E2) and X⁽²⁻⁾(E1 − E2), respectively.

Finally, experiments performed at the M edges of actinides have provided indirect evidence for higher-order magnetic and electric multipoles (Santini et al., 2009) through the measurement of ordered electric quadrupoles in UPd₃ (McMorrow et al., 2001), NpO₂ (Paixão et al., 2002) and UO₂ (Wilkins et al., 2006).

It often happens that the different multipoles at a given energy are entangled, depending on the symmetry properties of the resonant atoms. In order to clearly disentangle their contributions, X-ray polarization and wavevector analysis is fundamental. Compared with XAFS, REXS provides extra degrees of freedom determined by the scattered X-ray polarization and wavevector. The experimental method for REXS in the hard X-ray regime is based on characterization of the polarization of the incident and scattered X-rays (linear and circular) and on an appropriate choice of scattering geometry, combined with an azimuthal rotation of the sample about the scattering wavevector Q (Paolasini et al., 2007). The X-ray polarization analysis is realized by an appropriate choice of crystal analysers mounted on the diffraction arm, which are able to rotate about the scattered wavevector k_s and are set at a Bragg angle close to θ = 45°. The selection rule for Thomson scattering ensures determination of the Poincaré–Stokes parameters of the scattered beam (Paolasini, 2014). The incident beam polarization can be controlled by the diamond phase plates, and both linear (half-wave plate mode) and circular (quarter-wave plate mode) incident polarization can be selected (Scagnoli et al., 2009). This complete set of experimental conditions leads, in some favourable cases, to a full disentanglement of the character of resonant contributions at different forbidden lattice reflections (Mazzoli et al., 2007).

REXS and XAFS experiments based on the multipole interpretation have contributed to enriching our knowledge of the electronic and magnetic properties of materials and their symmetries, and also to quantifying their degree of anisotropy. One of the key differences between the two techniques is that the XAFS signal is constrained by the symmetry of the total point group of the crystal, whereas in REXS the Bragg factors provide an extra degree of freedom weighting the spatial phase factor and allowing the detection of multipoles that would otherwise be symmetry-forbidden by the point group of the crystal.

This is what happens, for example, in the case of V₂O₃ (Paolasini et al., 1999), the total magnetic space group of which contains both spatial parity and time-reversal symmetries, thereby not allowing their breakdown at the level of the crystal unit cell. Therefore, parity odd and time-reversal odd multipoles for the total unit cell are symmetry-forbidden and cannot be revealed by XAFS-based techniques (i.e. NR-XLD is not possible; Di Matteo et al., 2003). However, the symmetry does not forbid the presence of magnetoelectric multipoles (both odd and odd) at each atomic vanadium site. Although their sum on the total unit cell is zero, their separate contributions can be recovered with REXS with a clever choice of the Bragg phase factors, as shown in the literature (Paolasini et al., 2001; Di Matteo et al., 2003; Lovesey et al., 2007). The experimental findings associated with such an analysis are summarized in Fig. 1. The REXS spectrum in Fig. 1(b) is the equivalent, for V₂O₃, of the hexadecapole found for α-Fe₂O₃ in Finkelstein et al. (1992). The symmetry reduction determined by the metal–insulator transition in V₂O₃ (trigonal to monoclinic) manifests itself in Fig. 1(c), where the quadrupole term of E1–E1 origin enters the azimuth scan and interferes with the high-order multipoles visible in the narrow pre-edge region.

Figure 1 XANES, polarization-dependent energy scans and the corresponding azimuthal dependences of forbidden lattice reflections in chromium-doped V2O3 at the V K edge (Paolasini et al., 2001). (a) XANES spectrum. (b) Room-temperature (003)h reflection (pseudo-hexagonal setting) and (c) the equivalent low-temperature reflection (monoclinic setting). (d) Low-temperature, Bragg-forbidden reflection [antiferromagnetic, ] associated with the magnetic dipole and octupole. (e) Low-temperature, Bragg-forbidden reflection [antiferrotoroidal, ] associated with the magnetic quadrupole and polar toroidal dipole and octupole. (f) V2O3 magnetic structure (short arrows) and projection of the vanadium anapole moments (long arrows) along the (am, cm) monoclinic plane.

Finally, Figs. 1(d) and 1(e) show the magnetic multipoles X⁽¹⁾(E1 − E1), X⁽¹⁾(E2 − E2) and X⁽³⁾(E2 − E2) (Fig. 1d) and X⁽¹⁻⁾(E1 − E2), X⁽²⁻⁾(E1 − E2) and X⁽³⁻⁾(E1 − E2) (Fig. 1e). They can be disentangled by a full linear and circular polarization analysis using a lengthy procedure, as described in Fernández-Rodríguez et al. (2010). The study of these contributions by resonant X-ray diffraction is of fundamental importance in current developments in the electronic structure of materials with complex electronic properties, such as magnetoelectricity, piezoelectricity and ferroelectricity.

An analogous example of the detection of time-reversal odd, parity odd multipoles by means of XAFS-based techniques is represented by the discovery of an effect coupling both NR-XLD and XMχD in GaFeO₃ by Kubota et al. (2004). In this case, four absorption spectra are measured with incoming linear polarization along perpendicular axes (as in usual XLD), but with a magnetic field applied either parallel or antiparallel to the magnetic easy axis of the sample. The details of the calculations are reported in Di Matteo et al. (2005). Here, we just underline that as in the previous case for REXS, the tensors X⁽¹⁻⁾(E1 − E2), X⁽²⁻⁾(E1 − E2) and X⁽³⁻⁾(E1 − E2) have been measured using this procedure, as in this case they were allowed by the total point group of GaFeO₃.

The development of multi-element crystal analyser spectrometers at third-generation synchrotron sources has opened new opportunities to investigate the dynamical behaviour of the low-energy charge, spin, orbital and lattice elementary excitations in solids in detail. Most of these techniques deal with the dominant dipole E1 transitions but, at least theoretically, higher-order multipoles can also be observed in both elastic and inelastic channels.

The experimental development of polarization analysis detection and azimuthal polarimetry, in parallel with improvement of the energy resolution, will in the future allow the tensorial order parameters (multipoles) characterizing X-ray scattering to be properly disentangled.

Moreover, the advent of X-ray free-electron lasers (XFELs) has allowed nonlinearity to enter the X-ray domain. In this sense, the close analogy with what happens in the optical region with laser beams (second-harmonic generation, parametric down conversions etc.) might boost an analogous search for exotic (inversion-breaking and time-reversal-breaking) order parameters induced by the nonlinear, tensorial coupling of the X-ray laser beam and the sample.