Introduction

local susceptibilitiesThe 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 parametersatomic displacementparameters (ADPs) (Chapter Atomic displacement parametersW. F. Kuhs1.9 ) or electric field gradient tensorselectric field gradient (EFG) (Section ElectronsK. Schwarz2.2.15 ) 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 reflectionsforbidden reflections excited at radiation energies close to X-ray absorption edgesX-ray absorption edges of atoms, and reflections caused by magnetic scatteringmagnetic scattering.

We begin with the X-ray dielectric susceptibilitydielectric susceptibilityX-ray, which expresses the response of crystalline matter to an incident X-ray wave characterized by its energy (frequency), polarizationpolarization 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 X-ray diffraction theories presented in The structure factorP. CoppensChapters 1.2 and Dynamical theory of X-ray diffractionA. Authier5.1 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 X-ray susceptibility is assumed to comply with the periodicity of the crystalline matter.

It is important that the dielectric susceptibilitydielectric susceptibility is (i) a local crystal property and (ii) a tensor physical property, because it relates the polarization vectorspolarization vector 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 X-ray 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 energy-dependent dispersiondispersion corrections as considered in Section X-raysU. W. ArndtD. C. CreaghR. D. DeslattesJ. H. HubbellP. IndelicatoE. G. Kessler JrE. Lindroth4.2.6 of International Tables for Crystallography Volume C (2004). As a result of atomic wavefunction distortions caused by neighbouring atoms, these scalar dispersion correctionsdispersion 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 anisotropyanisotropy 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 X-raysU. W. ArndtD. C. CreaghR. D. DeslattesJ. H. HubbellP. IndelicatoE. G. Kessler JrE. Lindroth4.2.6 of Volume C (2004).

However, even if the anisotropyanisotropy of the atomic factor is small, it can be crucial for some effects, for instance the excitation of so-called `forbiddenforbidden reflections' 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 extinctionsextinctionsextinctionssystematic extinctions are due to (i) the translation symmetry of the non-primitive Bravais latticesBravais 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 factorscattering factor (as the Fourier transform of an independent atom/ion with spherically symmetric electron-density distribution) is an element-specific scalar that depends only on the scattering-vector 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 electron-density distribution caused by chemical bondingchemical bonding and/or (ii) atomic vibrations (Dawson, 1975) if the atom in question occupies a special site.

In contrast, an anisotropyanisotropy of the atomic factor affects all reflections and can therefore violate general extinction rulesextinction rules related to glide planes and/or screw axes, i.e. symmetry elements with translation components, in nonsymmorphic space groups. Even a very small X-ray anisotropyanisotropyX-ray can be quantitatively studied with this type of forbidden reflectionsforbidden 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 scatteringanomalous scattering was first observed in an NaBrO₃ crystal (Templeton & Templeton, 1985, 1986) and then studied for Cu₂O (Eichhorn & Kirfel, 1988), TiO₂ and MnF₂ (Kirfel & Petcov, 1991), and for many other compounds with different crystal symmetries. Within the dipole approximation, a systematic compilation of `forbiddenforbidden reflections' reflection properties for all relevant space groups up to tetragonal symmetry and an application to partial-structure 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 X-ray studies) in diffraction of Mössbauer radiationMössbauer 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 diffractionneutron diffraction, in crystals with local anisotropyanisotropy of the magnetic susceptibilitymagnetic susceptibility (Gukasov & Brown, 2010). All these latter findings are, however, beyond the scope of this chapter.

X-ray polarizationpolarization phenomena similar to those in visible optics and spectroscopy (birefringencebirefringence, linear and circular dichroismcircular dichroism, the Faraday rotationFaraday 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 X-rayspolarized X-rays (a historical overview can be found in Rogalev et al., 2006). In particular, for non-magnetic media, X-ray natural circular dichroismcircular 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 X-ray absorption spectroscopiesX-ray absorption spectroscopy using polarized X-rays have been developed for magnetic materials; examples are XMCD (X-ray magnetic circular dichroismX-ray magnetic circular dichroism) (Schütz et al., 1987; Thole et al., 1992; Carra et al., 1993) and XMLD (X-ray magnetic linear dichroismmagnetic linear dichroism) (Thole et al., 1986; van der Laan et al., 1986; Arenholz et al., 2006; van der Laan et al., 2008). X-ray magnetochiral dichroismmagnetochiral dichroism (XMD) was discovered by Goulon et al. (2002) and is used as a probe of toroidal moment in solids. Sum rules connecting X-ray 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 X-ray spectroscopies and are widely used for applications. These types of X-ray absorption spectroscopies are not considered here, as this chapter is mainly devoted to X-ray tensorial properties observed in single-crystal diffractionX-ray diffraction.

Symmetry restrictions on local tensorial susceptibility and forbidden reflectionslocal susceptibilitiesforbidden reflections

Several different approaches can be used to determine the local susceptibilitylocal susceptibilities with appropriate symmetry. For illustration, we start with the simple but very important case of a symmetric tensor of rank 2symmetric tensorsrank 2 defined in the Cartesian system, (in this case, we do not distinguish covariantcovariant and contravariantcontravariant components, see Chapter Introduction to the properties of tensorsA. Authier1.1 ). From the physical point of view, such tensors appear in the dipole–dipole approximation (see Section 1.11.4).

Polarization properties and azimuthal dependencepolarization

There are two important properties that distinguish forbidden reflectionsforbidden reflections from conventional (`allowed') ones: non-trivial polarization effects and strong azimuthal dependence of intensity (and sometimes also of polarization) corresponding to the symmetry of the direction of the scattering vectorscattering 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 reciprocal-lattice vectorreciprocal latticevector, 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 factorstructure factors 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 factorstructure factors is a tensor of rank 2, then the reflection intensity obtained with incident and reflected radiation with polarization vectorspolarization vector, respectively, and (prepared and analysed by a corresponding polarizerpolarizer and analyseranalyser) 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 orthogonalityorthogonality 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 factorstructure factors is a direct product of two vectors, then the polarizationpolarization of the diffracted beam does not depend on the incident polarization.

The polarization analysis of forbidden reflectionsforbidden reflections frequently uses the linear polarization vectorspolarization vector and . Vector is perpendicular to the scattering plane, whereas the vectors and are in the scattering plane so that and form right-hand triads. Note that the components of the polarization vectors, etc., change with the azimuthal angle if the crystal is rotated about the scattering vectorscattering vector.

In special cases, circular polarizationscircular polarization are very useful and sometimes even indispensable, because they enable us to distinguish right- and left-hand crystals or to unravel interferences between magnetic and electric scattering (see below).

If the incident radiation is - or -polarized or non-polarized, 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 X-rayspolarized X-rays 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 screw-axis forbidden reflectionsforbidden 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 type-I forbidden reflectionsforbidden 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 vectorscattering 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) diffractionRenninger diffraction. Such measurements could also be useful for determining the phases of the complex and above.

The polarization properties of type-II reflections are quite distinct from those of type-I 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 type-I reflections, the intensities of the type-II reflections are different for - and -polarized incident beams. What is more interesting is that type-II reflections are `chiral', i.e. their intensities differ for right-handcircularly polarized radiationright-handed and left-handcircularly polarized radiationleft-handed circularly polarized incident radiation. As an example, we take the type-II back-reflections () 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 (right-hand if and left-hand if ) are reflected and that the back-reflected radiation has the same circular polarizationcircular polarization in both cases. For opposite polarization, the reflection is absent. Thus, under these circumstances, the crystal may be regarded as a circular polarizerpolarizer or analyseranalyser. If , the eigen-polarizations 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 chiralitychirality of type-II reflections can be used to distinguish enantiomorphous crystalsenantiomorphous crystals. Although this was suggested many years ago, its potential was only recently proved by experiments, first on α-quartzquartzalpha- (low-temperature), SiO₂, and berliniteberlinite (AlPO₄), AlPO₄ (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 X-ray diffraction methodsX-ray diffraction for absolute structure determination fail in elemental crystals.

The non-trivial polarizationpolarization and azimuthal properties discussed above are, in most cases, determined by symmetry, and they are used as evidence confirming the origin of the forbidden reflectionsforbidden reflections. They are also used for obtaining detailed information about anisotropyanisotropy of local susceptibilitylocal susceptibilities 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 magnetitemagnetite (Fe₃O₄), Fe₃O₄ – 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).

Physical mechanisms for the anisotropy of atomic X-ray susceptibilityX-ray susceptibilityanisotropy

Conventional non-resonant Thomson scatteringThomson scattering in condensed matter is the result of the interaction of the electric fieldelectric 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 X-ray diffractionX-ray 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 X-ray–electron interactions by the Pauli equation (Berestetskii et al., 1982), which is a low-energy approximation to the Dirac equation (typical X-ray 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 HamiltonianHamiltonian:where is the momentum of the pth electronmomentum of the 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 vectorspolarization vector, and and are the photon annihilation and creation operators.

Considering X-ray scatteringX-ray scattering by different atoms in solids as independent processes [in Section The structure factorP. Coppens1.2.4 of International Tables for Crystallography Volume B, this is called `the isolated-atom approximation in X-ray diffractionX-ray 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 non-resonant Thomson scatteringThomson scattering and is the energy width of the excited state . The second line gives non-resonant magnetic scatteringmagnetic scatteringnon-resonant with the spin and orbital terms given by the rank-3 tensors (1.11.5.2) and (1.11.5.1), respectively. Compared to the second-to-last 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 non-resonant terms, in particular at photon energies far from resonance. The third term gives the dispersion correctionsdispersion corrections also addressed as resonant scatteringresonant scattering, magnetic and non-magnetic. 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 factorscattering factor looks likewhere is the ordinary Thomson (non-resonant) 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 scatteringresonant 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 non-resonant scatteringmagnetic scatteringnon-resonant, which is also anisotropic (see the next section).

Non-resonant magnetic scatteringmagnetic scatteringnon-resonant

Far from resonance (), the non-resonant parts of the scattering factorscattering 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 re-emission of radiation. The non-resonant magnetic term is small compared to the charge (Thomson) scatteringThomson scattering owing (a) to small numbers of unpaired (magnetic) electrons and (b) to the factor of about 0.02 for a typical X-ray energy . This is the reason why it is so difficult to observe non-resonant magnetic scattering with conventional X-ray sources (de Bergevin & Brunel, 1972, 1981; Brunel & de Bergevin, 1981), in contrast to the nowadays normal use of synchrotron radiation.

Non-resonant magnetic scatteringmagnetic scatteringnon-resonant yields polarizationpolarization 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 scatteringneutron 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 Levi-Civita symbol).

Being convoluted with polarization vectorspolarization vector (Blume, 1985; Lovesey & Collins, 1996; Paolasini, 2012), the non-resonant 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 factorscattering factor are significantly different. Consequently, the two contributions can be separated by analysing the polarizationpolarization of the scattered radiation with the help of an analyseranalyser crystal (Gibbs et al., 1988). Usually the incident (synchrotron) radiation is σ-polarized, i.e. the polarization vectorpolarization vector is perpendicular to the scattering plane. If due to the orientation of the analysing crystal only the σ-polarized 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.

Resonant atomic factors: multipole expansionresonant scattering factor

Strong enhancement of resonant scatteringresonant 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–dipoledipole–dipole interaction contribution, is the dipole–quadrupoledipole–quadrupole interaction contribution and is the quadrupole–quadrupolequadrupole–quadrupole interaction 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 X-ray absorption edgesX-ray absorption edges because the products are small for the typical sizes of the inner shells involved. In resonant X-ray 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 dispersiondispersion effects similar to those in visible optics.

Glossary

Table 1.11.2.1| top | pdf | The indices of the screw-axis/glide-plane forbidden reflections () and independent components of their tensorial structure factors forbidden reflections Other components: , , . The direction of the z axis is selected along the corresponding screw axes. The last column lists different types of polarization properties defined in Section 1.11.3. Screw axis or glide plane Type 0 0 I II II 0 0 I 0 0 II 0 0 II 0 0 I 0 0 II 0 0 I 0 0 II 0 0 0 0   0 0 II 0 0 0 0   0 0 II 0 0 I 0 0 II 0 0 0 0   0 0 II

The indices of the screw-axis/glide-plane forbidden reflections () and independent components of their tensorial structure factors forbidden reflections The indices of the screw-axis/glide-plane forbidden reflections () and independent components of their tensorial structure factors forbidden reflections

Table 1.11.2.2| top | pdf | The indices of the forbidden reflections and corresponding tensors of structure factors for the cubic space groups ()forbidden reflectionsstructure factors Space group Indices of reflections Expressions for and additional restrictions (1.11.2.23) (1.11.2.6); for (1.11.2.6); for (1.11.2.6); for (1.11.2.6) (1.11.2.24) (1.11.2.24) (1.11.2.23);   (1.11.2.24) (1.11.2.23);   (1.11.2.24) (1.11.2.24) (1.11.2.22); for , for (1.11.2.22); for (1.11.2.22); for , for (1.11.2.22); for   (1.11.2.6); for (1.11.2.22); for (1.11.2.6); for (1.11.2.22); for (1.11.2.6); for (1.11.2.6); for   (1.11.2.22); for (1.11.2.6); for   (1.11.2.22); : , for

The indices of the forbidden reflections and corresponding tensors of structure factors for the cubic space groups ()forbidden reflectionsstructure factors The indices of the forbidden reflections and corresponding tensors of structure factors for the cubic space groups ()forbidden reflectionsstructure factors

Table 1.11.6.1| top | pdf | Coefficients corresponding to various kinds of tensor symmetry with respect to space inversion , rotations , and time reversal electric fieldmagnetic fieldtoroidal moment Tensor type Example Transformation type Even Strain 1 1 1 1 Electric Electric field 1 −1 1 −1 Magnetic Magnetic field 1 1 −1 −1 Magnetoelectric Toroidal moment 1 −1 −1 1

Coefficients corresponding to various kinds of tensor symmetry with respect to space inversion , rotations , and time reversal electric fieldmagnetic fieldtoroidal moment Coefficients corresponding to various kinds of tensor symmetry with respect to space inversion , rotations , and time reversal electric fieldmagnetic fieldtoroidal moment

Table 1.11.6.2| top | pdf | Identification of properties under time inversion and space inversion of tensors associated with multipole expansion After Di Matteo et al. (2005) and Paolasini (2012). Rank of tensor Resonant process T P Type Multipole 0 E1E1 + + charge monopole 0 E2E2 + + charge monopole 1 E1E1 − + magnetic dipole 1 E2E2 − + magnetic dipole 1 E1E2 + − electric dipole 1 E1E2 − − polar toroidal dipole 2 E1E1 + + electric quadrupole 2 E2E2 + + electric quadrupole 2 E1E2 + − axial toroidal quadrupole 2 E1E2 − − magnetic quadrupole 3 E2E2 − + magnetic octupole 3 E1E2 + − electric octupole 3 E1E2 − − polar toroidal octupole 4 E2E2 + + electric hexadecapole

Identification of properties under time inversion and space inversion of tensors associated with multipole expansion Identification of properties under time inversion and space inversion of tensors associated with multipole expansion