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. 271272
Section 1.11.2.3. Local tensorial susceptibility of cubic crystals^{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 
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
Belyakov, V. A. & Dmitrienko, V. E. (1985). The blue phase of liquid crystals. Sov. Phys. Usp. 28, 535–562.Google ScholarHahn, Th. (2005). Editor. International Tables for Crystallography, Volume A, SpaceGroup Symmetry, 5th ed. Heidelberg: Springer.Google Scholar