International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D. ch. 1.11, pp. 269283
https://doi.org/10.1107/97809553602060000910 Chapter 1.11. Tensorial properties of local crystal susceptibilities^{a}A. V. Shubnikov Institute of Crystallography, Leninsky pr. 59, Moscow 119333, Russia,^{b}Steinmann Institut der Universität Bonn, Poppelsdorfer Schloss, Bonn, D53115, Germany, and ^{c}Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia With the advent of synchrotron radiation, nonmagnetic and magnetic resonant Xray diffraction has become an important tool in modern materials research, e.g. on electronic states of systems. Polarized Xrays are sensitive to the local environments of resonant scattering atoms and their partial structures. At energies close to an absorption edge of an absorbing element, in particular, `forbidden' reflections can be excited, which would be extinct in absence of local anisotropic Xray susceptibility. Anisotropy of energydependent susceptibility is treated in terms of tensor atomic scattering factors, giving rise to tensor structure factors, so that the intensity and polarization of the scattered radiation depend not only on the energy and polarization of the incident radiation, but also on both the crystal symmetry and the site symmetry. Owing to the anisotropy, rotation of the crystal about the scattering vector (azimuthal rotation) becomes an additional important parameter of investigation. This chapter considers the treatment and potential of anisotropic resonant scattering, both in the absence and presence of magnetic scattering, and the impact of symmetry on local physical properties, particularly symmetry and physical phenomena that allow and restrict forbidden reflections as well as reflections caused by magnetic scattering. Keywords: Xray susceptibility; resonant diffraction; forbidden reflections; Xray magnetic scattering. 
The tensorial characteristics of macroscopic physical properties (as described in Chapters 1.3 , 1.4 and 1.6 –1.8 of this volume) are determined by the crystal point group, whereas the symmetry of local crystal properties, such as atomic displacement parameters (Chapter 1.9 ) or electric field gradient tensors (Section 2.2.15 ) are regulated by the crystal space group. In the present chapter, we consider further examples of the impact of symmetry on local physical properties, particularly both symmetry and physical phenomena that allow and restrict forbidden reflections excited at radiation energies close to Xray absorption edges of atoms, and reflections caused by magnetic scattering.
We begin with the Xray dielectric susceptibility, which expresses the response of crystalline matter to an incident Xray wave characterized by its energy (frequency), polarization and wavevector. The response is a polarization of the medium, finally resulting in a scattered wave with properties generally different from the initial ones. Thus, the dielectric susceptibility plays the role of a scattering amplitude, which relates the scattered wave to the incident wave. This is the basis of the different approaches to Xray diffraction theories presented in Chapters 1.2 and 5.1 of International Tables for Crystallography Volume B (2008). Here, we consider only elastic scattering, i.e. the energies of the incident and scattered waves are identical, and the Xray susceptibility is assumed to comply with the periodicity of the crystalline matter.
It is important that the dielectric susceptibility is (i) a local crystal property and (ii) a tensor physical property, because it relates the polarization vectors of the incident and scattered radiation. Consequently, the symmetry of the tensor is determined by the symmetry of the crystal space group, rather than by that of the point group as in conventional optics. In the vast majority of Xray applications, this tensor can reasonably be assumed to be given by the product of the unit tensor and a scalar susceptibility, which is proportional to the electron density plus exclusively energydependent dispersion corrections as considered in Section 4.2.6 of International Tables for Crystallography Volume C (2004). As a result of atomic wavefunction distortions caused by neighbouring atoms, these scalar dispersion corrections can also become anisotropic tensors, namely in the close vicinity (usually less than about 50 eV) of absorption edges of elements. For heavy elements, the anisotropy of the tensor atomic factor can exceed 20 e atom^{−1}. Appropriate references to detailed descriptions of the phenomenon can be found in Brouder (1990), Materlik et al. (1994) and in Section 4.2.6 of Volume C (2004).
However, even if the anisotropy of the atomic factor is small, it can be crucial for some effects, for instance the excitation of socalled `forbidden' reflections, which vanish in absence of anisotropy. Indeed, the crystal symmetry imposes strong restrictions on the indices of possible (`allowed') reflections. The systematic reflection conditions for the different space groups and for special atomic sites in the unit cell are listed in International Tables for Crystallography Volume A (Hahn, 2005). The resulting extinctions are due to (i) the translation symmetry of the nonprimitive Bravais lattices, (ii) the symmetry elements of the space group (glide planes and/or screw axes) and (iii) special sites. The first kind cannot be violated. The other extinctions are obtained if the atomic scattering factor (as the Fourier transform of an independent atom/ion with spherically symmetric electrondensity distribution) is an elementspecific scalar that depends only on the scatteringvector length and the dispersion corrections. Then the intensities of extinct reflections generally vanish. These reflections are `forbidden', but for different physical reasons not all of their intensities are necessarily strictly zero. Such reflections can appear owing to an asphericity of (i) an atomic electrondensity distribution caused by chemical bonding and/or (ii) atomic vibrations (Dawson, 1975) if the atom in question occupies a special site.
In contrast, an anisotropy of the atomic factor affects all reflections and can therefore violate general extinction rules related to glide planes and/or screw axes, i.e. symmetry elements with translation components, in nonsymmorphic space groups. Even a very small Xray anisotropy can be quantitatively studied with this type of forbidden reflections, and yield information about electronic states of crystals or partial structures of resonant scatterers. This was first recognized by Templeton & Templeton (1980), and a detailed theory was developed only a few years later (Dmitrienko, 1983, 1984). The excitation of forbidden reflections caused by anisotropic anomalous scattering was first observed in an NaBrO_{3} crystal (Templeton & Templeton, 1985, 1986) and then studied for Cu_{2}O (Eichhorn & Kirfel, 1988), TiO_{2} and MnF_{2} (Kirfel & Petcov, 1991), and for many other compounds with different crystal symmetries. Within the dipole approximation, a systematic compilation of `forbidden' reflection properties for all relevant space groups up to tetragonal symmetry and an application to partialstructure analysis followed (Kirfel et al., 1991; Kirfel & Petcov, 1992; Kirfel & Morgenroth, 1993; Morgenroth et al., 1994). Today, there are numerous surveys devoted to this well developed subject, and further details, applications and references can be found therein (Belyakov & Dmitrienko, 1989; Carra & Thole, 1994; Hodeau et al., 2001; Lovesey et al., 2005; Dmitrienko et al., 2005; Altarelli, 2006; Collins et al., 2007; Collins & Bombardi, 2010; Finkelstein & Dmitrienko, 2012). Forbidden reflections of the last type have also been observed (well before corresponding Xray studies) in diffraction of Mössbauer radiation (Belyakov & Aivazyan, 1969; Belyakov, 1975; Champeney, 1979) and, at optical wavelengths, in the blue phases of chiral liquid crystals (Belyakov & Dmitrienko, 1985; Wright & Mermin, 1989; Seideman, 1990; Crooker, 2001). Similar phenomena have also been reported to exist in chiral smectic liquid crystals (Gleeson & Hirst, 2006; Barois et al., 2012) and, considering neutron diffraction, in crystals with local anisotropy of the magnetic susceptibility (Gukasov & Brown, 2010). All these latter findings are, however, beyond the scope of this chapter.
Xray polarization phenomena similar to those in visible optics and spectroscopy (birefringence, linear and circular dichroism, the Faraday rotation) have been discussed since the beginning of the 20th century (Hart & Rodriques, 1981; Templeton & Templeton, 1980, 1982). Experimental studies and applications were mainly prompted by the development of synchrotrons and storage devices as sources of polarized Xrays (a historical overview can be found in Rogalev et al., 2006). In particular, for nonmagnetic media, Xray natural circular dichroism (XNCD) is used as a method for studying electronic states with mixed parity (Natoli et al., 1998; Goulon et al., 2003). Various kinds of Xray absorption spectroscopies using polarized Xrays have been developed for magnetic materials; examples are XMCD (Xray magnetic circular dichroism) (Schütz et al., 1987; Thole et al., 1992; Carra et al., 1993) and XMLD (Xray magnetic linear dichroism) (Thole et al., 1986; van der Laan et al., 1986; Arenholz et al., 2006; van der Laan et al., 2008). Xray magnetochiral dichroism (XMD) was discovered by Goulon et al. (2002) and is used as a probe of toroidal moment in solids. Sum rules connecting Xray spectral parameters with the physical properties of the medium have also been developed (Thole et al., 1992; Carra et al., 1993; Goulon et al., 2003) for various kinds of Xray spectroscopies and are widely used for applications. These types of Xray absorption spectroscopies are not considered here, as this chapter is mainly devoted to Xray tensorial properties observed in singlecrystal diffraction.
Several different approaches can be used to determine the local susceptibility with appropriate symmetry. For illustration, we start with the simple but very important case of a symmetric tensor of rank 2 defined in the Cartesian system, (in this case, we do not distinguish covariant and contravariant components, see Chapter 1.1 ). From the physical point of view, such tensors appear in the dipole–dipole approximation (see Section 1.11.4).
The most general expression for the tensor of susceptibility is exclusively restricted by the crystal symmetry, i.e. must be invariant against all the symmetry operations of the given space group :where is the matrix of the point operation (rotation or mirror reflection), , and is the associated vector of translation. The index indicates a transposed matrix, and summation over repeated indices is implied hereafter. To meet the above demand, it is obviously sufficient for to be invariant against all generators of the group .
There is a simple direct method for obtaining obeying equation (1.11.2.1): we can take an arbitrary secondrank tensor and average it over all the symmetry operations :where is the number of elements in the group . A small problem is that is infinite for any space group, but this can be easily overcome if we take as periodic and obeying the translation symmetry of the given Bravais lattice. Then the number of the remaining symmetry operations becomes finite (an example of this approach is given in Section 1.11.2.3).
In spite of its simplicity, equation (1.11.2.1) provides nontrivial restrictions on the tensorial structure factors of Bragg reflections. The sets of allowed reflections, listed in International Tables for Crystallography Volume A (Hahn, 2005) for all space groups and for all types of atom sites, are based on scalar Xray susceptibility. In this case, reflections can be forbidden (i.e. they have zero intensity) owing to glideplane and/or screwaxis symmetry operations. This is because the scalar atomic factors remain unchanged upon mirror reflection or rotation, so that the contributions from symmetryrelated atoms to the structure factors can cancel each other. In contrast, atomic tensors are sensitive to both mirror reflections and rotations, and, in general, the tensor atomic factors of symmetryrelated atoms have different orientations in space. As a result, forbidden reflections can in fact be excited just due to the anisotropy of susceptibility, so that the selection rules for possible reflections change.
It is easy to see how the most general tensor form of the structure factors can be deduced from equation (1.11.2.1). The structure factor of a reflection with reciprocallattice vector is proportional to the Fourier harmonics of the susceptibility. The corresponding relations (Authier, 2005, 2008) simply have to be rewritten in tensorial form:where is the classical electron radius, is the Xray wavelength and is the volume of the unit cell.
Considering first the glideplane forbidden reflections, there may, for instance, exist a glide plane perpendicular to the axis, i.e. any point is transformed by this plane into . The corresponding matrix of this symmetry operation changes the sign of ,and the translation vector into . Substituting (1.11.2.4) into (1.11.2.1) and exchanging the integration variables in (1.11.2.3), one obtains for the structure factors of reflections If is scalar, i.e. , then for odd , hence vanishes. This is the well known conventional extinction rule for a glide plane, see International Tables for Crystallography Volume A (Hahn, 2005). If, however, is a tensor, the mirror reflection changes the signs of the and tensor components [as is also obvious from equation (1.11.2.5)]. As a result, the and components should not vanish for and the tensor structure factor becomesIn general, the elements and are complex, and it should be emphasized from the symmetry point of view that they are different and arbitrary for different and . However, from the physical point of view, they can be readily expressed in terms of tensor atomic factors, where only those chemical elements are relevant whose absorptionedge energies are close to the incident radiation energy (see below).
It is also easy to see that for the nonforbidden (= allowed) reflections , the nonzero tensor elements are just those which vanish for the forbidden reflections:Here the result is mainly provided by the diagonal elements , but there is still an anisotropic part that contributes to the structure factor, as expressed by the offdiagonal element. In principle, the effect on the total intensity as well as the element itself can be assessed by careful measurements using polarized radiation.
For the screwaxis forbidden reflections, the most general form of the tensor structure factor can be found as before (Dmitrienko, 1983; see Table 1.11.2.1). Again, as in the case of the glide plane, for each forbidden reflection all components of the tensor structure factor are determined by at most two independent complex elements and . There may, however, exist further restrictions on these tensor elements if other symmetry operations of the crystal space group are taken into account. For example, although there are screw axes in space group , and reflections remain forbidden because the lattice is body centred, and this applies not only to the dipole–dipole approximation considered here, but also within any other multipole approximation.

In Table 1.11.2.1, resulting from the dipole–dipole approximation, some reflections still remain forbidden. For instance, in the case of a screw axis, there is no anisotropy of susceptibility in the plane due to the inevitable presence of the threefold rotation axis. For and axes, the reflections with also remain forbidden because only dipole–dipole interaction (of Xrays) is taken into account, whereas it can be shown that, for example, quadrupole interaction permits the excitation of these reflections.
Let us consider in more detail the local tensorial properties of cubic crystals. This case is particularly interesting because for cubic symmetry the secondrank tensor is isotropic, so that a global anisotropy is absent (but it exists for tensors of rank 4 and higher). Local anisotropy is of importance for some physical parameters, and it can be described by tensors depending periodically on the three space coordinates. This does not only concern Xray susceptibility, but can also, for instance, result from describing orientation distributions in chiral liquid crystals (Belyakov & Dmitrienko, 1985) or atomic displacements (Chapter 1.9 of this volume) and electric field gradients (Chapter 2.2 of this volume) in conventional crystals.
The symmetry element common to all cubic space groups is the threefold axis along the cube diagonal. The matrix of the symmetry operation isThis transformation results in the circular permutation , and from equation (1.11.2.1) it is easy to see that invariance of demands the general formwhere and are arbitrary functions with the periodicity of the corresponding Bravais lattice: for primitive lattices ( being arbitrary integers) plus in addition = for bodycentered lattices or = = = for facecentered lattices.
Depending on the space group, other symmetry elements can enforce further restrictions on and :
::: (1.11.2.10) and: (1.11.2.10) and: (1.11.2.10) and: (1.11.2.11) and (1.11.2.12).
: (1.11.2.10) and: (1.11.2.11) and: (1.11.2.11) and: (1.11.2.10) and: (1.11.2.10) and: (1.11.2.11) and: (1.11.2.10), (1.11.2.12) and (1.11.2.19).
: (1.11.2.10), (1.11.2.13) and (1.11.2.15).
: (1.11.2.10), (1.11.2.12) and (1.11.2.20).
: (1.11.2.10), (1.11.2.13) and (1.11.2.19).
: (1.11.2.10), (1.11.2.14) and (1.11.2.19).
: (1.11.2.10), (1.11.2.13) and (1.11.2.20).
: (1.11.2.11), (1.11.2.12) and (1.11.2.21).
For all , the sets of coordinates are chosen here as in International Tables for Crystallography Volume A (Hahn, 2005); the first one being adopted if Volume A offers two alternative origins. The expressions (1.11.2.10) or (1.11.2.11) appear for all space groups because all of them are supergroups of or .
The tensor structure factors of forbidden reflections can be further restricted by the cubic symmetry, see Table 1.11.2.2. For the glide plane , the tensor structure factor of reflections is given by (1.11.2.6), whereas for the diagonal glide plane , it is given byand additional restrictions on and can become effective for or . For forbidden reflections of the type, the tensor structure factor is eitherorsee Table 1.11.2.2.

There are two important properties that distinguish forbidden reflections from conventional (`allowed') ones: nontrivial polarization effects and strong azimuthal dependence of intensity (and sometimes also of polarization) corresponding to the symmetry of the direction of the scattering vector. The azimuthal dependence means that the intensity and polarization properties of the reflection can change when the crystal is rotated around the direction of the reciprocallattice vector, i.e. they change with the azimuthal angle of the incident wavevector k defined relative to the scattering vector. The polarization and azimuthal properties, both mainly determined by symmetry, are two of the most informative characteristics of forbidden reflections. A third one, energy dependence, is determined by physical interactions, electronic and/or magnetic, where the role of symmetry is indirect but nevertheless also important (e.g. in splitting of atomic levels etc., see Section 1.11.4).
In the kinematical theory, usually used for weak reflections, one obtains for unpolarized incident radiation the intensity of a conventional reflection as given bywhere is the Bragg angle, is the scalar structure factor of reflection , and is a scale factor, which depends on the incident beam intensity, the sample volume, the geometry of diffraction etc. (see International Tables for Crystallography Volume B ), and can be set to hereafter.
If the structure factor is a tensor of rank 2, then the reflection intensity obtained with incident and reflected radiation with polarization vectors, respectively, and (prepared and analysed by a corresponding polarizer and analyser) is given bywhere the star denotes the complex conjugate. The maximum of this expression is reached when is equal to the polarization of the diffracted beam. In general, the polarization of the diffracted secondary radiation, , depends on the incident beam polarization :where(the second term in this expression provides orthogonality between the polarization vector and the corresponding wavevector). If the polarization of the diffracted beam is not analysed, the total intensity of the diffracted beam is equal to . If the tensor structure factor is a direct product of two vectors, then the polarization of the diffracted beam does not depend on the incident polarization.
The polarization analysis of forbidden reflections frequently uses the linear polarization vectors and . Vector is perpendicular to the scattering plane, whereas the vectors and are in the scattering plane so that and form righthand triads. Note that the components of the polarization vectors, etc., change with the azimuthal angle if the crystal is rotated about the scattering vector.
In special cases, circular polarizations are very useful and sometimes even indispensable, because they enable us to distinguish right and lefthand crystals or to unravel interferences between magnetic and electric scattering (see below).
If the incident radiation is  or polarized or nonpolarized, then the total reflection intensities for these three cases are given by the following expressions:A more general approach uses the Stokes parameters for the description of partially polarized Xrays and the Müller matrices for the scattering process (see a survey by Detlefs et al., 2012). This issue will, however, not be discussed further since there is no principal difference to conventional optics.
Let us consider the polarization and azimuthal characteristics of screwaxis forbidden reflections listed in Table 1.11.2.1. These characteristics are rather different for two types of reflections: type I reflections are those for which , while all other reflections constitute the rest, type II.
The typeI forbidden reflections have the simplest polarization properties. From equations (1.11.3.5)–(1.11.3.7) and Table 1.11.2.1, one obtains and , where is given byfor a screw axis andfor , , and screw axes, where is the azimuthal angle of crystal rotation about the scattering vector . Thus, polarized incident radiation results in reflected radiation with polarization and vice versa; and unpolarized incident radiation gives unpolarized reflected radiation.
Note that there is no azimuthal dependence of intensity in (1.11.3.10). Nevertheless, the phase of the diffracted beams changes with azimuthal rotation, as might be observed via interference with another scattering process, for example, with multiple (Renninger) diffraction. Such measurements could also be useful for determining the phases of the complex and above.
The polarization properties of typeII reflections are quite distinct from those of typeI reflections. The intensities belonging to various polarization channels, i.e. combinations of primary and secondary beam polarizations (, etc.), exhibit different azimuthal symmetries for different screw axes.
For and screw axes, the azimuthal symmetry is threefold:where . The sign corresponds to in Table 1.11.2.1.
For , , and screw axes, the symmetry is fourfold:where and .
No azimuthal dependence exists for the screw axes , , and :
Unlike the typeI reflections, the intensities of the typeII reflections are different for  and polarized incident beams. What is more interesting is that typeII reflections are `chiral', i.e. their intensities differ for righthand and lefthand circularly polarized incident radiation. As an example, we take the typeII backreflections () for three and sixfold screw axes. We find from Table 1.11.2.1 and equations (1.11.3.1) and (1.11.3.3) that only the beams with definite circular polarization (righthand if and lefthand if ) are reflected and that the backreflected radiation has the same circular polarization in both cases. For opposite polarization, the reflection is absent. Thus, under these circumstances, the crystal may be regarded as a circular polarizer or analyser. If , the eigenpolarizations are elliptic and the axial ratio of the polarization ellipse is equal to for the sixfold screw axes (whereas for the three and fourfold screw axes, this ratio depends on the parameters and ).
The chirality of typeII reflections can be used to distinguish enantiomorphous crystals. Although this was suggested many years ago, its potential was only recently proved by experiments, first on αquartz, SiO_{2}, and berlinite, AlPO_{4} (Tanaka et al., 2008; Tanaka, Kojima et al., 2010), later for tellurium (Tanaka, Collins et al., 2010). All three candidates crystallize in the space groups or . The case of tellurium is particularly interesting because standard Xray diffraction methods for absolute structure determination fail in elemental crystals.
The nontrivial polarization and azimuthal properties discussed above are, in most cases, determined by symmetry, and they are used as evidence confirming the origin of the forbidden reflections. They are also used for obtaining detailed information about anisotropy of local susceptibility and, hence, about structural and electronic properties. For instance, careful analysis of polarization and azimuthal dependences allows one to distinguish between different scenarios of the Verwey phase transition in magnetite, Fe_{3}O_{4} – a longstanding and confusing problem (see Hagiwara et al., 1999; García et al., 2000; Renevier et al., 2001; García & Subías, 2004; Nazarenko et al., 2006; Subías et al., 2012).
Conventional nonresonant Thomson scattering in condensed matter is the result of the interaction of the electric field of the electromagnetic wave with the charged electron subsystem. However, there are also other mechanisms of interaction, e.g. interaction of electromagnetic waves with spin and orbital moments, which was first considered by Platzman & Tzoar (1970) for molecules and solids. They predicted the sensitivity of Xray diffraction to a magnetic structure of a crystal, as later observed in the pioneering works of de Bergevin & Brunel (de Bergevin & Brunel, 1972, 1981; Brunel & de Bergevin, 1981). It is reasonable to describe all Xray–electron interactions by the Pauli equation (Berestetskii et al., 1982), which is a lowenergy approximation to the Dirac equation (typical Xray energies are where m is the electron mass). The equation accounts for charge and spin interaction with the electromagnetic field of the wave, and spin–orbit interaction (Blume, 1985, 1994) using the following Hamiltonian:where is the momentum of the pth electron, and is the vector potential of the electromagnetic wave with wavevector and polarization .
Here and below + , where is a quantization volume, index labels two polarizations of each wave, are the polarizations vectors, and and are the photon annihilation and creation operators.
Considering Xray scattering by different atoms in solids as independent processes [in Section 1.2.4 of International Tables for Crystallography Volume B, this is called `the isolatedatom approximation in Xray diffraction'; the validity of this approximation has been discussed by Kolpakov et al. (1978)], the atomic scattering amplitude , which describes the scattering of a wave with wavevector and polarization into a wave with wavevector and polarization , can be written aswhere the tensor atomic factor depends not only on the wavevectors but also on the atomic environment, magnetic and orbital moments etc. From equation (1.11.4.1) and with the help of perturbation theory (Berestetskii et al., 1982), the atomic factor can be expressed aswhere the first line describes the nonresonant Thomson scattering and is the energy width of the excited state . The second line gives nonresonant magnetic scattering with the spin and orbital terms given by the rank3 tensors (1.11.5.2) and (1.11.5.1), respectively. Compared to the secondtolast line, where the energy denominator can be close to zero, the last line is usually neglected, but sometimes it has to be added to the nonresonant terms, in particular at photon energies far from resonance. The third term gives the dispersion corrections also addressed as resonant scattering, magnetic and nonmagnetic. In equation (1.11.4.3), and are the ground and excited states energies, respectively; is the probability that the incident state of the scatterer is occupied; and is the scattering vector (in the case of diffraction , where is the Bragg angle). The vector operator has the formThe second term in this equation is small and is frequently omitted.
In general, the total atomic scattering factor looks likewhere is the ordinary Thomson (nonresonant) factor, and are the isotropic corrections to the dispersion and absorption, which become stronger near absorption edges (), and and are the real and imaginary contributions accounting for resonant anisotropic scattering and are sensitive to the local symmetry of the resonant atom and its magnetism. In the latter case, one should add the tensor (–) describing magnetic nonresonant scattering, which is also anisotropic (see the next section).
Far from resonance (), the nonresonant parts of the scattering factor, and , described by the first two terms in (1.11.4.3) are the most important. In the classical approximation (Brunel & de Bergevin, 1981), there are four physical mechanisms (electric or magnetic, dipolar or quadrupolar) describing the interaction of an electron and its magnetic moment with an electromagnetic wave, causing the reemission of radiation. The nonresonant magnetic term is small compared to the charge (Thomson) scattering owing (a) to small numbers of unpaired (magnetic) electrons and (b) to the factor of about 0.02 for a typical Xray energy . This is the reason why it is so difficult to observe nonresonant magnetic scattering with conventional Xray sources (de Bergevin & Brunel, 1972, 1981; Brunel & de Bergevin, 1981), in contrast to the nowadays normal use of synchrotron radiation.
Nonresonant magnetic scattering yields polarization properties quite different from those obtained from charge scattering. Moreover, it can be divided into two parts, which are associated with the spin and orbital moments. In contrast to the case of neutron magnetic scattering, the polarization properties of these two parts are different, as described by the tensors (Blume, 1994)where is a completely antisymmetric unit tensor (the LeviCivita symbol).
Being convoluted with polarization vectors (Blume, 1985; Lovesey & Collins, 1996; Paolasini, 2012), the nonresonant magnetic term can be rewritten aswith vectors and given byAccording to (1.11.5.4) and (1.11.5.5), the polarization dependences of the spin and orbit contributions to the atomic scattering factor are significantly different. Consequently, the two contributions can be separated by analysing the polarization of the scattered radiation with the help of an analyser crystal (Gibbs et al., 1988). Usually the incident (synchrotron) radiation is σpolarized, i.e. the polarization vector is perpendicular to the scattering plane. If due to the orientation of the analysing crystal only the σpolarized part of the scattered radiation is recorded, we can see from (1.11.5.4) that the orbital contribution to the scattering atomic factor vanishes, whereas it differs from zero considering the scattering channel.
Strong enhancement of resonant scattering occurs when the energy of the incident radiation gets close to the energy of an electron transition from an inner shell to an empty state (be it localized or not) above the Fermi level. There are two widely used approaches for calculating resonant atomic amplitudes. One uses Cartesian, the other spherical (polar) coordinates, and both have their own advantages and disadvantages. Supposing in (1.11.4.3)and using the expression for the velocity matrix element (Berestetskii et al., 1982) , it is possible to present the resonant part of the atomic factor (1.11.4.3) aswhere , is a dimensionless tensor corresponding to the dipole–dipole contribution, is the dipole–quadrupole contribution and is the quadrupole–quadrupole term. All the tensors are complex and depend on the energy and the local properties of the medium. The expansion (1.11.6.1) over the wavevectors is possible near Xray absorption edges because the products are small for the typical sizes of the inner shells involved. In resonant Xray absorption and scattering, the contribution of the magnetic multipole transitions is usually much less than that of the electric multipole transitions. Nevertheless, the scattering amplitude corresponding to events has also been considered (Collins et al., 2007). The tensors and describe the spatial dispersion effects similar to those in visible optics.
Different types of tensors transform under the action of the extended orthogonal group (Sirotin & Shaskolskaya, 1982) aswhere the coefficients depend on the kind of tensor (see Table 1.11.6.1) and are coefficients describing proper rotations.

Various parts of the resonant scattering factor (1.11.6.3) possess different kinds of symmetry with respect to: (1) space inversion or parity, (2) rotations and (3) time reversal . Both dipole–dipole and quadrupole–quadrupole terms are parityeven, whereas the dipole–quadrupole term is parityodd. Thus, dipole–quadrupole events can exist only for atoms at positions without inversion symmetry.
It is convenient to separate the timereversible and timenonreversible terms in the contributions to the atomic tensor factor (1.11.6.3). The dipole–dipole contribution to the resonant atomic factor can be represented as a sum of an isotropic, a symmetric and an antisymmetric part, written as (Blume, 1994)where ,and and ; means the probability of the timereversed state . If, for example, has a magnetic quantum number m, then has a magnetic quantum number .
In nonmagnetic crystals, the probability of states with is the same, so that and ; in this case is symmetric under permutation of the the indices.
Similarly, the dipole–quadrupole atomic factor can be represented as (Blume, 1994)wherewith . In (1.11.6.10) the first plus () corresponds to the nonmagnetic case (time reversal) and the minus () corresponds to the timenonreversal magnetic term, while the second corresponds to the symmetric and antisymmetric parts of the atomic factor. We see that can contribute only to scattering, while can contribute to both resonant scattering and resonant Xray propagation. The latter term is a source of the socalled magnetochiral dichroism, first observed in Cr_{2}O_{3} (Goulon et al., 2002, 2003), and it can be associated with a toroidal moment in a medium possessing magnetoelectric properties. The symmetry properties of magnetoelectic tensors are described well by Sirotin & Shaskolskaya (1982), Nye (1985) and Cracknell (1975). Which magnetoelectric properties can be studied using Xray scattering are widely discussed by Marri & Carra (2004), Matsubara et al. (2005), Arima et al. (2005) and Lovesey et al. (2007).
It follows from (1.11.6.8) and (1.11.6.10) that and the dipole–quadrupole term can be represented as a sum of the symmetric and antisymmetric parts. From the physical point of view, it is useful to separate the dipole–quadrupole term into and , because only works in conventional optics where . The dipole–quadrupole terms are due to the hybridization of excited electronic states with different spacial parities, i.e. only for atomic sites without an inversion centre.
The pure quadrupole–quadrupole term in the tensor atomic factor is equal towith the fourthrank tensor given by
This fourthrank tensor has the following symmetries:
We can definewith , whereWe see that vanishes in timereversal invariant systems, which is true for nonmagnetic structures.
In timereversal invariant systems, equation (1.11.6.3) can be rewritten aswhere corresponds to the symmetric part of the dipole–dipole contribution, and mean the symmetric and antisymmetric parts of the thirdrank tensor describing the dipole–quadrupole term, and denotes a symmetric quadrupole–quadrupole contribution. From the physical point of view, it is useful to separate the dipole–quadrupole term into and , because in conventional optics, where , only is relevant.
The tensors contributing to the atomic factor in (1.11.6.16), , , , , are of different ranks and must obey the site symmetry of the atomic position. Generally, the tensors can be different, even for crystallographically equivalent positions, but all tensors of the same rank can be related to one of them, because all are connected through the symmetry operations of the crystal space group. In contrast, the scattering amplitude tensor does not necessarily comply with the point symmetry of the atomic position, because this symmetry is usually violated considering the arbitrary directions of the radiation wavevectors and .
Equation (1.11.6.16) is also frequently considered as a phenomenological expression of the tensor atomic factor where each tensor possesses internal symmetry (with respect to index permutations) and external symmetry (with respect to the atomic environment of the resonant atom). For instance, the tensor is symmetric, the rank3 tensor has a symmetric and a antisymmetric part, and the rank4 tensor is symmetric with respect to the permutation of each pair of indices. The external symmetry of coincides with the symmetry of the dielectric susceptibility tensor (Chapter 1.6 ). Correspondingly, the thirdrank tensors and are similar to the gyration susceptibility and electrooptic tensors (Chapter 1.6 ), and has the same tensor form as that for elastic constants (Chapter 1.3 ). The symmetry restrictions on these tensors (determining the number of independent elements and relationships between tensor elements) are very important and widely used in practical work on resonant Xray scattering. Since they can be found in Chapters 1.3 and 1.6 or in textbooks (Sirotin & Shaskolskaya, 1982; Nye, 1985), we do not discuss all possible symmetry cases in the following, but consider in the next section one specific example for Xray scattering when the symmetries of the tensors given by expression (1.11.6.3) do not coincide with the most general external symmetry that is dictated by the atomic environment.
It is fairly obvious from expressions (1.11.6.3) and (1.11.6.16) that in the nonmagnetic case the symmetric and antisymmetric thirdrank tensors, and , which describe the dipole–quadrupole contribution to the Xray scattering factor, are not independent: the antisymmetric part, which is also responsible for opticalactivity effects, can be expressed via the symmetric part (but not vice versa). Indeed, both of them can be described by a symmetric thirdrank tensor resulting from the secondorder Born approximation (1.11.6.3),whereFrom equation (1.11.6.17), one can infer that the symmetry restrictions for and are the same. Then it can be seen that can be expressed via .
For any symmetry, and have the same number of independent elements (with a maximum 18 for site symmetry 1). Thus, one can reverse equation (1.11.6.17) and express directly in terms of :
Using equations (1.11.6.18) and (1.11.6.20), one can express all nine elements of through :according to which the antisymmetric part of the dipole–quadrupole term is a linear function of the symmetric one [however, not vice versa: equations (1.11.6.21) cannot be reversed].
Note that the equations (1.11.6.21) impose an additional restriction on , which applies to all atomic site symmetries:This is, in fact, a well known result: the pseudoscalar part of vanishes in the dipole–quadrupole approximation used in equation (1.11.6.3). Thus, for point symmetry 1, has only eight independent elements rather than nine. This additional restriction works in all cases of higher symmetries provided the pseudoscalar part is allowed by the symmetry (i.e. point groups 2, 3, 4, 6, 222, 32, 422, 622, 23 and 432). All other symmetry restrictions on arise automatically from equation (1.11.6.21) taking into account the symmetry of [symmetry limitations on and for all crystallographic point groups can be found in Sirotin & Shaskolskaya (1982) and Nye (1985)].
Let us consider two examples, ZnO and anatase, TiO_{2}, where the dipole–dipole contributions to forbidden reflections vanish, whereas both the symmetric and antisymmetric dipolequadrupole terms are in principal allowed. In these crystals, the dipole–quadrupole terms have been measured by Goulon et al. (2007) and Kokubun et al. (2010).
In ZnO, crystallizing in the wurtzite structure, the 3m symmetry of the atomic positions imposes the following restrictions on :where , , , are energydependent complex tensor elements [keeping the notations by Sirotin & Shaskolskaya (1982), the x axis is normal to the mirror plane, the y axis is normal to the glide plane and the z axis corresponds to the c axis of ZnO]. If we suppose these restrictions for Zn at , then for the other Zn at , which is related to the first site by the glide plane, there is the following set of elements: . Therefore, the structure factors of the glideplane forbidden reflections are proportional to .
For the symmetric and antisymmetric parts one obtains from equations (1.11.6.17) and (1.11.6.18) the nonzero componentsand
Physically, we can expect that because survives even for tetrahedral symmetry , whereas is nonzero owing to a deviation from tetrahedral symmetry; in ZnO, the local coordinations of the Zn positions are only approximately tetrahedral.
In the anatase structure of TiO_{2}, the symmetry of the atomic positions imposes restrictions on the tensors [keeping the notations of Sirotin & Shaskolskaia (1982): the x and y axes are normal to the mirror planes, and the z axis is parallel to the c axis]:where and are energydependent complex parameters. If we apply these restrictions to the Ti atoms at and , then for the other two inversionrelated Ti atoms at and (centre ), the parameters are and .
For the symmetric and antisymmetric parts one obtains as nonvanishing componentsand
It is important to note that the symmetric part of the atomic factor can be affected by a contribution from thermalmotioninduced dipole–dipole terms. The latter terms are tensors of rank 3 proportional to the spatial derivatives , which take the same tensor form as but are not related to by equations (1.11.6.21). In ZnO, which was studied in detail by Collins et al. (2003), the thermalmotioninduced contribution is rather significant, while for anatase the situation is less clear.
Once the tensor atomic factors have been determined [either from phenomenological expressions like (1.11.6.16), according to the sitesymmetry restrictions, or from given microscopic expressions, e.g. (1.11.4.3)], tensor structure factors are obtained by summation over the contributions of all atoms in the unit cell, as in conventional diffraction theory:where the index t enumerates the crystallographically different types of scatterers (atoms belonging to the same or different chemical elements), the index u denotes the crystallographically equivalent positions; is a siteoccupancy factor, and is the Debye–Waller temperature factor. The tensors of the atomic factors, , , , , are, in general, different for crystallographically equivalent positions, that is for different u, and it is exactly this difference that enables the excitation of the resonant forbidden reflections.
Extinction rules and polarization properties for forbidden reflections are different for tensor structure factors of different ranks, a circumstance that may be used for experimental separation of different tensor contributions (for tensors of rank 2, information is given in Tables 1.11.2.1 and 1.11.2.2). In the harmonic approximation, anisotropies of the atomic thermal displacements (Debye–Waller factor) are also described by tensors of rank 2 or higher, but, owing to these, excitations of glideplane and screwaxis forbidden reflections are not possible.
Magnetic crystals possess different densities of states with opposite spin directions. During a multipole transition from the ground state to an excited state (or the reverse), the projection of an electron spin does not change, but the projection of the orbital moment varies. The consideration of all possible transitions allows for the formulation of the sum rules (Carra et al., 1993; Strange, 1994) that are widely used in Xray magnetic circular dichroism (XMCD). When measuring the differences of the absorption coefficients at the absorption edges of transition elements or at the M edges of rareearth elements (Erskine & Stern, 1975; Schütz et al., 1987; Chen et al., 1990), these rules allow separation of the spin and orbital contributions to the XMCD signal, and hence the study of the spin and orbital moments characterizing the ground state. In magnetic crystals, the tensors change their sign with time reversal because if and/or (Zeeman splitting in a magnetic field). That the antisymmetric parts of the tensors differ from zero follows from equations (1.11.6.7), (1.11.6.10) and (1.11.6.15).
Time reversal also changes the incident and scattered vectors corresponding to permutation of the Cartesian tensor indices. For dipole–dipole resonant events, the symmetric part does not vary with exchange of indices, hence it is time and parityeven. The antisymmetric part changes its sign upon permutation of the indices, so it is parityeven and timeodd, being associated with a magnetic moment (1.11.6.41). This part of the tensor is responsible for the existence of Xray magnetic circular dichroism (XMCD) and the appearance of the magnetic satellites in various kinds of magnetic structures.
If the rotation symmetry of a secondrank tensor is completely described by rotation about the magnetic moment m, then the antisymmetric secondrank tensor can be represented as , where is an antisymmetric thirdrank unit tensor and are the coordinates of the magnetic moment of the resonant atom. So, the scattering amplitude for the dipole–dipole transition can be given as, and are energydependent coefficients referring to the sth atom in the unit cell and is a unit vector along the magnetic moment. The third term in (1.11.6.41) is time nonreversal, and it is responsible for the magnetic linear dichroism (XMLD). This kind of Xray dichroism is also influenced by the crystal field (Thole et al., 1986; van der Laan et al., 1986).
The coefficients , and involved in (1.11.6.41) may be represented in terms of spherical harmonics. Using the relations (Berestetskii et al., 1982; Hannon et al., 1988)andfor , and , , respectively, one obtainswithwhere is the probability of the initial state , is that for the transition from state to a final state , and is the ratio of the partial line width of the excited state due to a pure radiative decay and the width due to all processes, both radiative and nonradiative (for example, the Auger decay).
Magnetic ordering is frequently accompanied by a local anisotropy in the crystal. In this case, both kinds of local anisotropies exist simultaneously and must be taken into account in, for example, XMLD (van der Laan et al., 1986) and XMχD (Goulon et al., 2002). In resonant Xray scattering experiments, simultaneous existence of forbidden reflections provided by spin and orbital ordering (Murakami et al., 1998) as well as magnetic and crystal anisotropy (Ji et al., 2003; Paolasini et al., 2002, 1999) have been observed. The explicit Cartesian form of the tensor atomic factor in the presence of both a magnetic moment and crystal anisotropy has been proposed by Blume (1994). When the symmetry of the atomic site is high enough, i.e. the atom lies on an norder axis (), then the tensors and can be represented asandwhere and depend on the energy, and is a unit vector along the symmetry axis under consideration. One can see that the atomic tensor factor is given by a sum of three terms: the first is due to the symmetry of the local crystal anisotropy, the second describes pure magnetic scattering, and the last (`combined') term is induced by interference between magnetic and nonmagnetic resonant scattering. This issue was first discussed by Blume (1994) and later in more detail by Ovchinnikova & Dmitrienko (1997, 2000). All the terms can give rise to forbidden reflections, i.e. sets of pure resonant forbidden magnetic and nonmagnetic reflections can be observed for the same crystal, see Ji et al. (2003) and Paolasini et al. (2002, 1999). Only reflections caused by the `combined' term (Ovchinnikova & Dmitrienko, 1997) have not been observed yet.
Neglecting the crystal field, an explicit form of the fourthrank tensors describing the quadrupole–quadrupole events in magnetic structures was proposed by Hannon et al. (1988) and Blume (1994):
Then, being convoluted with polarization vectors, the scattering amplitude of the quadrupole transition () can be written as a sum of 13 terms belonging to five orders of magnetic moments (Hannon et al., 1988; Blume, 1994). The final expression that gives the quadrupole contribution to the magnetic scattering amplitude in terms of individual spin components is rather complicated and can be found, for example, in Hill & McMorrow (1996). In the presence of both a magnetic moment and local crystal anisotropy, the fourthrank tensor describing events depends on both kinds of anisotropy and can include the `combined' part in explicit form, as found by Ovchinnikova & Dmitrienko (2000).
Another representation of the scattering amplitude is widely used in the scientific literature (Hannon et al., 1988; Luo et al., 1993; Carra et al., 1993; Lovesey & Collins, 1996) for the description of resonant multipole transitions. In order to obtain the scattering amplitude and intensity for a resonant process described by some set of spherical tensor components, the tensor that describes the atomic scattering must be contracted by a tensor of the same rank and inversion/timereversal symmetry which describes the Xray probe, so that the result would be a scalar. There are well known relations between the components of the atomic factor tensor, both in Cartesian and spherical representations. For the dipole–dipole transition, the resonant scattering amplitude can be written as (Hannon et al., 1988; Collins et al., 2007; Paolasini, 2012; Joly et al., 2012)where are the Cartesian tensor components, depends only on the incident and scattered radiation and the polarization vectors, and is associated with the tensor properties of the absorbing atom and can be represented in terms of a multipole expansion.
It is convenient to decompose each tensor into its irreducible parts. For example, an tensor containing nine Cartesian components can be represented as a sum of three irreducible tensors with ranks (one component), (three components) and (five components). This decomposition is unique.
It follows from (1.11.6.14) that the fourthrank tensor describing the quadrupole–quadrupole Xray scattering can also be divided into two parts: the timereversal part, , and the nontimereversal part, . Both can be explicitly represented by (1.11.6.3) and (1.11.6.2), in which all these tensors are parityeven. The explicit form of the fourthrank tensors is suitable for the analysis of possible effects in resonant Xray absorption and scattering. Nevertheless, sometimes the following representation of the scattering amplitude as a product of spherical tensors is preferable:
Here, the dipole–quadrupole tensor atomic factor given by (1.11.6.10) is represented by a sum over several tensors with different symmetries. All tensors are parityodd, but the tensors and are also nontimereversal. The scattering amplitude corresponding to the dipole–quadrupole resonant Xray scattering can be represented asThe explicit form of can be found in Marri & Carra (2004). Various parts of possess different symmetry with respect to the reversal of space and time .
The spherical representation of the tensor atomic factor allows one to analyse its various components, as they possess different symmetries with respect to rotations or space and time inversion. For each , is related to a specific term of the multipole expansion of the system. Multipole expansions of electric and magnetic fields generated by charges and permanent currents are widely used in characterizing the electromagnetic state of a physical system (Berestetskii et al., 1982). The transformation rules for electric and magnetic multipoles of both parities under space inversion and time reversal are of great importance for electromagnetic effects in crystals. The correspondence between the and electromagnetic multipoles is shown in Table 1.11.6.2. In this table, the properties of the tensors under time reversal and space inversion on one side are identified with multipole terms describing the physical system on the other. In fact, for any given tensor of rank there is one electromagnetic multipole of the same rank ( dipole, quadrupole, octupole, hexadecapole) and with the same and properties. Note that odd tensors have both odd (−) and even (+) terms for any , whereas even tensors (both and ) are odd for odd rank and even for even rank, respectively (Di Matteo et al., 2005).

An important contribution of Luo et al. (1993) and Carra et al. (1993) consisted of expressing the amplitude coefficients in terms of experimentally significant quantities, electron spin and orbital moments. This procedure is valid within the fastcollision approximation, when either the deviation from resonance, , or the width, , is large compared to the splitting of the excitedstate configuration. The approximation is expected to hold for the and edges of the rare earths and actinides, as well as for the and edges of the actinides. In this energy regime, the resonant factors can be summed independently, leaving amplitude coefficients that may be written in terms of multipole moment operators, which are themselves singleparticle operators summed over the valence electrons in the initial state.
Magnetic scattering has become a powerful method for understanding magnetic structures (Tonnere, 1996; Paolasini, 2012), particularly as it is suitable even for powder samples (Collins et al., 1995). Since the first studies (Gibbs et al., 1988), resonant magnetic Xray scattering has been observed at various edges of transition metals and rare earths. The studies include magnetics and multiferroics with commensurate and incommensurate modulation (Walker et al., 2009; Kim et al., 2011; Ishii et al., 2006; Partzsch et al., 2012; Lander, 2012; Beale et al., 2012; Lovesey et al., 2012; Mazzoli et al., 2007) as well as multik magnetic structures (Bernhoeft et al., 2012), and structures with orbital ordering (Murakami et al., 1998) and higherorder multipoles (Princep et al., 2011). It has also been shown that effects can be measured not only at the edges of magnetic atoms [K edges of transition metals, L edges of rareearth elements and M edges of actinides (Vettier, 2001, 2012)], but also at the edges of nonmagnetic atoms (Mannix et al., 2001; van Veenendaal, 2003).
Thus, magnetic and nonmagnetic resonant Xray diffraction clearly has the potential to be an important working tool in modern materials research. The advantage of polarized Xrays is their sensitivity to both the local atomic environments of resonant atoms and their partial structures. The knowledge of the local and global crystal symmetries and of the interplay of their effects is therefore of great value for a better understanding of structural, electronic and magnetic features of crystalline condensed matter.

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