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, p. 270
Section 1.11.2.1. General symmetry restrictions
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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, D-53115, Germany, and ^{c}Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia |
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 second-rank 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).