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. 275280
Section 1.11.6. Resonant atomic factors: multipole expansion^{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 
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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