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. 270272
Section 1.11.2. Symmetry restrictions on local tensorial susceptibility and forbidden reflections^{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 
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.

References
Authier, A. (2005). Dynamical Theory of Xray Diffraction. Oxford University Press.Authier, A. (2008). In International Tables for Crystallography, Volume B, Reciprocal Space, edited by U. Shmueli, 3rd ed., pp. 626–646. Heidelberg: Springer.
Belyakov, V. A. & Dmitrienko, V. E. (1985). The blue phase of liquid crystals. Sov. Phys. Usp. 28, 535–562.
Dmitrienko, V. E. (1983). Forbidden reflections due to anisotropic Xray susceptibility of crystals. Acta Cryst. A39, 29–35.
Hahn, Th. (2005). Editor. International Tables for Crystallography, Volume A, SpaceGroup Symmetry, 5th ed. Heidelberg: Springer.