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. 279280
Section 1.11.6.6. Tensor atomic factors (spherical tensor representation)^{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 
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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