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. 275276
Section 1.11.6.1. Tensor atomic factors: internal symmetry
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a
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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 
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.
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