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/S1574870722001550

Tensorial interactions of X-rays

Luigi Paolasinia* and Sergio Di Matteob

aESRF – The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France, and bInstitut de Physique de Rennes CNRS UMR 6251, Université de Rennes, Rennes, France
Correspondence e-mail:  paolasini@esrf.fr

The tensorial interactions of X-rays in crystals are reviewed here, including magnetic effects, both in X-ray absorption and X-ray scattering. After a brief historical description of the development of theoretical concepts and experimental work in the last two decades of the twentieth century and the first decade of the twenty-first century, their analysis is detailed. The basic theoretical formulae are provided with the main information about their origin and a necessary link to the original literature in order to analyze them in more depth. Two examples of the experimental detection of higher-order magnetoelectric multipoles are provided: in V2O3, using resonant elastic scattering techniques, and in GaFeO3, using X-ray absorption-based techniques.

Keywords: general interactions; tensorial interactions; scattering; magnetic scattering.

1. Introduction

X-ray absorption/emission spectroscopy and resonant X-ray scattering close to an absorption edge provide a powerful probe of the anisotropies of electronic and magnetic degrees of freedom in crystals due to their tensorial character. These processes are controlled by electric dipole (E1) and electric quadrupole (E2) transitions, and have been used to determine the properties of valence electron states in transition metals, rare earths and actinides (Paolasini & de Bergevin, 2008[link]). Through the angular and polarization dependence of X-rays, both techniques are sensitive to the orientation of the un­occupied electronic orbitals and to their density of states. In the case of resonant X-ray diffraction (RXD), the tensorial character of atomic scattering amplitudes can break the conventional extinction rules for forbidden Bragg reflections related to glide planes and/or screw-axis symmetries. As a result, from analysis of the resonant reflections of polarized X-rays tuned across an absorption edge, we obtain information about the anisotropic part of the electronic and magnetic density of states, projected onto the resonant atom, and their structural order. A quantitative measurement of the anisotropies in magnetic and orbital degrees of freedom of electrons in solids is provided by the atomic and crystal multipoles which, due to their tensorial character and peculiar coupling with the X-ray polarization, can be directly measured and disentangled by an appropriate choice of the experimental setup (azimuthal and polarization dependence of incoming and scattered X-rays). As detailed below, the X-ray absorption fine-structure (XAFS) cross section is proportional to the imaginary part of the forward X-ray scattering amplitude (scattering vector Q = 0). This is at the origin of the close analogy between the theoretical interpretations of the tensorial characters of XAFS and RXD. Experimentally, of course, the two techniques are quite distinct.

In this chapter, after a brief historical introduction, we analyse the tensorial characters of XAFS and RXD, with particular emphasis on the theoretical description of high-order electromagnetic multipoles (Section 2[link]) and their experimental detection using X-ray scattering and absorption techniques (Section 3[link]).

1.1. Historical overview

Whereas the anisotropy of X-ray absorption has been known since at least 1932 (see the discussion in Brouder, 1990[link]), 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[link]). Such findings were explained one year later by Dmitrienko (1983[link]), 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[link]) 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[link]), 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[link]; Carra et al., 1993[link]) 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[link]) 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[link] for a comparison with the present classification). In a different domain, Finkelstein et al. (1992[link]) and Templeton & Templeton (1994[link]) 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 [{\hat T}] and parity [{\hat P}] operators (Di Matteo et al., 2005[link])

Tensor[{\hat T}][{\hat P}]TypeMultipole
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[link]) showed that E2–E2 RXD could be interpreted using the same framework as in Luo et al. (1993[link]). The analysis of E1–E2 phenomena followed soon after the discovery of anomalous magnetic resonances in V2O3 (Paolasini et al., 1999[link], 2001[link]), boosted by the search for non­reciprocal and magneto-chiral dichroic effects in XAFS by Goulon et al. (2002[link]) and Di Matteo et al. (2003[link]). 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[link]; Di Matteo, 2012[link]).

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[link]; Borgatti et al., 2004[link]; Ament et al., 2011[link]; Juhin et al., 2014[link]) 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.

2. X-ray tensorial interaction with matter: absorption and scattering

The tensor properties of the X-ray absorption cross section close to an absorption edge are a well established tool to investigate the electronic and magnetic symmetries of absorbing atoms in different areas of condensed matter, especially through dichroic techniques, either magnetic or non­magnetic, as detailed in Section 2.2[link]. The theoretical derivation of the angular and polarization dependence of both absorption and scattering cross sections can be obtained from the Kramers–Heisenberg formula, which, after some formal manipulation, can be expressed in terms of a multipole expansion in the incoming (and scattered) electromagnetic field. Of course, the tensor properties also appear in the analogous Cartesian expansion of the matter–radiation interaction and the two approaches can be linked with the usual Cartesian/spherical basis transformations (see, for example, Varshalovich et al., 1988[link]). As a reminder, the determination of tensorial symmetries through the multipole expansion of the charge and the spin currents of electrons in the valence states is nowadays an important issue in understanding several physical properties of materials, including magnetic, structural and transport properties (Lovesey et al., 2005[link]; Spaldin et al., 2008[link]; Santini et al., 2009[link]). This procedure, besides showing the tensor properties of the matter–radiation interaction, also underlines the necessity for a common theoretical background for REXS, RIXS and XAFS (Ament et al., 2011[link]).

2.1. Anisotropy in matter–radiation interaction: the case of X-ray scattering and absorption

The Kramers–Heisenberg formula for the partial differential cross section per unit solid angle (Ω) and per unit energy of the scattered photon is [{{d\sigma} \over {d\Omega d\omega_{\rm s}}} = r_{0}^{2}{{\omega_{\rm s}} \over {\omega_{\rm i}}}\textstyle \sum \limits_{\rm F} |A_{\rm FI}+B_{\rm FI}|^{2}\delta(E_{\rm F}+\hbar\omega_{\rm s}-E_{\rm I}-\hbar\omega_{\rm i}), \eqno (1)]where r0 ≃ 2.52 × 10−15 m is the classical electron radius and ℏωi and ℏωs are the incident and scattered photon energies, respectively. Here, I labels the initial state of the sample, with energy EI, and F the final state after the scattering process, with energy EF. The amplitudes AFI and BFI correspond to interaction amplitudes quadratic and linear in the vector potential, respectively (see Chantler & Creagh, 2022[link]). In the analysis that follows we will neglect the quadratic term: [A_{\rm FI} = \boldvarepsilon^{*}_{\rm s}\cdot\boldvarepsilon_{\rm i}\langle{\rm F}|\exp[i({\bf k}_{\rm i} -{\bf k}_{\rm s})\cdot {\bf r}]|{\rm I}\rangle], where ɛi and ɛs are the incident and scattered photon polarizations and ki and ks are their wavevectors, respectively. Remember that AII 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 BII (Paolasini & de Bergevin, 2008[link]).

The amplitude BFI 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[link], 1994[link]), can be neglected close to absorption edges. For this reason, we write just the resonant contribution as[B_{\rm FI}^{({\rm i,s})} = {{1} \over {m}}{\textstyle\sum\limits_{N}}{{\langle {\rm F}|{\hat O}_{\rm s}^{\dagger}|N \rangle\langle N|{\hat O}_{\rm i}|{\rm I}\rangle} \over {\hbar\omega_{\rm i}-(E_{N}-E_{\rm I})+i\Gamma _{N}}}, \eqno (2)]where m is the electron mass and ΓN is the core-hole inverse lifetime in the intermediate state N. In this subsection the operators [{\hat O}_{\rm i,s}] are considered in the E1 approximation; [{\hat O}_{\rm i,s} = \boldvarepsilon_{\rm i,s}\cdot{\bf p}], so that transition-matrix elements can be written as [\langle N|{\hat O}_{\rm i,s}|{\rm I}\rangle = (m/i\hbar)(E_{\rm I}-E_{N})\langle N| \boldvarepsilon_{\rm i,s}\cdot{\bf r}|{\rm I}\rangle]. 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 as[I_{\rm XAFS}^{({\rm i})} = -{{4\pi r_{0}c} \over {\omega_{\rm i}}}\Im\{B_{\rm II}^{(\rm i,i)}\}, \eqno (3)]where [\Im] represents the imaginary part.

Analogously, we can derive the cross section for REXS as[I_{\rm REXS}^{({\rm i,s})} = r_{0}^{2}|B_{\rm II}^{\rm (i,s)}|^{2}. \eqno (4)]We 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[link]).

The general expression for RIXS can be expressed (Ament et al., 2011[link]) as[I_{\rm RIXS}^{\rm (i,s)} = r_{0}^{2}{{\omega_{\rm s}} \over {\omega_{\rm i}}}\textstyle\sum\limits_{\rm F}|B_{\rm FI}^{\rm (i, s)}|^{2}. \eqno (5)]

Equations (3[link]), (4[link]) and (5[link]) 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 [X = \textstyle\sum_{\alpha,\beta}(\varepsilon^{*}_{\rm s})_{\alpha}(\varepsilon_{\rm i})_{\beta}X_{\alpha\beta}], where X can be any of the REXS or RIXS amplitudes or the XAFS cross section and αβ = x, y, z. In the case of XAFS, however, we have just one polarization vector, [\boldvarepsilon^{*}_{\rm s}\equiv\boldvarepsilon^{*}_{\rm i}], 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 [T_{\alpha\beta}\equiv(\varepsilon^{*}_{\rm s})_{\alpha}(\varepsilon_{\rm i})_{\beta}], and a term describing the dynamical susceptibility of the sample, the tensors Xαβ. The latter tensors can be explicitly written for RIXS as [X_{\alpha\beta}^{({\rm FI})} = {{1} \over {m}}{\textstyle \sum \limits_{N}} {{\langle {\rm F}|r_{\alpha}|N\rangle \langle N|r_{\beta}|{\rm I}\rangle} \over {\hbar\omega_{\rm i}-(E_{N}-E_{\rm I})+i\Gamma_{N}}}, \eqno (6)]or [X_{\alpha\beta}^{\rm REXS} = X_{\alpha\beta}^{({\rm II})}] and [X_{\alpha\beta}^{\rm XAFS} = -(4\pi r_{0}c/ \omega_{\rm i})\Im\{X_{\alpha\beta}^{({\rm II})}\}] 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 [(\varepsilon^{*}_{\rm s})_{\alpha}(\varepsilon_{\rm i})_{\beta}]. As the latter tensor is coupled to Xαβ through a scalar product, there is a one-to-one correspondence between the dynamical tensor properties of Xαβ and the geometrical tensor properties of the electromagnetic probe [(\varepsilon^{*}_{\rm s})_{\alpha}(\varepsilon_{\rm i})_{\beta}]. 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 (Xαβ) and the electromagnetic tensor, the magnetic moment, time-reversal odd part of Xαβ must be coupled to the time-reversal odd part of the electromagnetic tensor, which is [\boldvarepsilon^{*}_{\rm i}\times\boldvarepsilon_{\rm i}] for XAFS and [\boldvarepsilon^{*}_{\rm s}\times\boldvarepsilon_{\rm i}] 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: [\{{\varepsilon^{*}_{\rm s}},{\varepsilon^{*}_{\rm i}}\}^{(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[link]).

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[link]). 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[link]) or KCuF3 (Paolasini et al., 2002[link]). 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[link]). A different interpretation of the same reflections is usually performed in the soft X-ray regime (Wilkins et al., 2003[link]; Mulders et al., 2006[link]).

2.2. Beyond the dipole approximation: tensor symmetries and classification of high-order multipoles

We can go beyond the dipole approximation by a full multipole expansion of the exponential term exp(iki,s · r) in the vector potential (see Chantler & Creagh, 2022[link]). Replacing the well known expression [\exp(i{\bf k}_{\rm i,s}\cdot{\bf r}) = \textstyle\sum\limits_{l}i^{l}(2l+1)j_{l}(k_{\rm i,s}r)\sum \limits_{m}(-)^{m}Y_{l,-m}({\hat k}_{\rm i,s})Y_{l,m}({\hat r})]for the transition operators [{\hat O}_{\rm i,s}] in equation (2[link]), we obtain the multipole terms of the electromagnetic field determined by the coefficients of the spherical harmonics [Y_{l,m}({\hat r})]. In the previous expression, jl(ki,sr) are spherical Bessel functions, ki,s and ki,s are the incident or scattered wavevector and its modulus, and [{\hat k}] and [{\hat r}] are the solid angles in k and r space. The order of the multipole is determined by 2l+1, as for l = 0 the transition operator is the dipole (ɛi, s · r), for l = 1 the quadrupole (i/2)(ki,s · r)(ɛi,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[link]). We notice here that often the exponential is approximated by a Taylor expansion of the kind [\exp(i{\bf k}_{\rm i,s}\cdot{\bf r}) = {\textstyle\sum\limits_{n}}i^{n}{{(k_{\rm i,s}r)^{n}} \over {n!}}(\cos \gamma_{\rm i,s})^{n}, ]where γi,s is the angle between ki,s and r. When ki,sr ≪ 1, because of the limit jl(ki,sr) ≃ (ki,sr)l/(2l + 1)!!, the two expansions differ only in the rearrangement of some terms (Brouder et al., 2008[link]).

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[link]) for RIXS.

The multipoles that can be detected by XAFS, REXS and RIXS have been described in several research articles (Di Matteo & Natoli, 2002[link]; Carra et al., 2003[link]; Di Matteo et al., 2005[link]; Marri & Carra, 2004[link]; Lovesey et al., 2007[link]; Lovesey & Balcar, 2010[link]; Ament et al., 2011[link]). At the resonant (absorbing) j-ion, the scattering amplitudes fj (the square of the absorption intensities) for REXS (XAFS) can be expressed as a scalar product of two irreducible spherical tensors, [f_{j} = \textstyle\sum\limits_{p,m}(-1)^{m}T^{(p)}_{-m}\,X^{(p)}_{m}(j\semi\omega), \eqno (7)]where [T^{(p)}_{-m}] 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 Tαβ introduced above, expressed in spherical components.

The rank-p spherical tensor [X^{(p)}_{m}(j\semi \omega)] 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 [link]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 (−pmp), as in usual angular-momentum theory. The behaviour under time-reversal ([{\hat T}]) and spatial parity ([{\hat P}]) symmetries of a selected set of these tensors is shown in Table 1[link]. As detailed in various references (Di Matteo et al., 2005[link]; Lovesey et al., 2005[link]), 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 [{\hat P}] 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 [{\hat P}] and [{\hat T}] operators, are associated with toroidal multipoles, which now represent an important subject in condensed matter for possible magneto-spintronics outcomes (Spaldin et al., 2008[link]).

Each of the multipoles in Table 1[link] 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[link]). Analogously, the parity-even rank-1 tensors in XAFS are related to XMCD (Carra et al., 1993[link]) as well as to its dispersive analogue X-ray Faraday rotation (Collins, 1999[link]).

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[link]; Finkelstein et al., 1992[link]; Carra & Thole, 1994[link]) 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[link]).

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[link]; Di Matteo & Natoli, 2002[link]; Sessoli et al., 2015[link]). Also useful to disentangle parity-odd, magnetic multipoles is the technique of X-ray nonreciprocal directional dichroism described by Kubota et al. (2004[link]), 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[link]) through the measurement of ordered electric quadrupoles in UPd3 (McMorrow et al., 2001[link]), NpO2 (Paixão et al., 2002[link]) and UO2 (Wilkins et al., 2006[link]).

3. Experimental detection of high-order multipoles

The identification of high-order multipoles is enhanced by the experimental capability to discriminate the tensorial character of the multipoles classified in Table 1[link]. With this aim, a clever analysis in terms of incoming and outgoing polarizations and wavevectors is critical to determine their tensor rank and their symmetry properties under the action of parity [{\hat P}] and time-reversal [{\hat T}] operators. The experimental geometrical parameters are contained in the [T^{(p)}_{-m}] term in equation (7)[link], in which both the polarizations and the wavevectors of the incident and scattered beam couple with the tensors [X^{(p)}_{m}(j\semi\omega)] associated with the electronic symmetries of the sample, including magnetic tensors.

3.1. X-ray polarization and wavevector analysis

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[link]). 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 ks and are set at a Bragg angle close to θ = 45°. The selection rule [(\boldvarepsilon^{*}_{\rm s}\cdot\boldvarepsilon_{\rm i})] for Thomson scattering ensures determination of the Poincaré–Stokes parameters of the scattered beam (Paolasini, 2014[link]). 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[link]). 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[link]).

3.2. Two examples: REXS and XAFS cases

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[link]), the total magnetic space group of which contains both spatial parity [{\hat P}] and time-reversal [{\hat T}] 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[link]). However, the symmetry does not forbid the presence of magnetoelectric multipoles (both [{\hat P}] odd and [{\hat T}] 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[link]; Di Matteo et al., 2003[link]; Lovesey et al., 2007[link]). The experimental findings associated with such an analysis are summarized in Fig. 1[link]. The REXS spectrum in Fig. 1[link](b) is the equivalent, for V2O3, of the hexadecapole found for α-Fe2O3 in Finkelstein et al. (1992[link]). The symmetry reduction determined by the metal–insulator transition in V2O3 (trigonal to monoclinic) manifests itself in Fig. 1[link](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]

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[link]). (a) XANES spectrum. (b) Room-temperature (003)h reflection (pseudo-hexagonal setting) and (c) the equivalent low-temperature [(10\bar{1})_{m}] reflection (monoclinic setting). (d) Low-temperature, Bragg-forbidden reflection [antiferromagnetic, [(2\bar{2}1)_{m}]] associated with the magnetic dipole and octupole. (e) Low-temperature, Bragg-forbidden reflection [antiferrotoroidal, [(3\bar{1}1)_{m}]] 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[link](d) and 1[link](e) show the magnetic multipoles X(1)(E1 − E1), X(1)(E2 − E2) and X(3)(E2 − E2) (Fig. 1[link]d) and X(1−)(E1 − E2), X(2−)(E1 − E2) and X(3−)(E1 − E2) (Fig. 1[link]e). They can be disentangled by a full linear and circular polarization analysis using a lengthy procedure, as described in Fernández-Rodríguez et al. (2010[link]). 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[link]). 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[link]). 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.

3.3. Future trends

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.

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