International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.11, p. 270

## Section 1.11.2.1. General symmetry restrictions

V. E. Dmitrienko,a* A. Kirfelb and E. N. Ovchinnikovac

aA. V. Shubnikov Institute of Crystallography, Leninsky pr. 59, Moscow 119333, Russia,bSteinmann Institut der Universität Bonn, Poppelsdorfer Schloss, Bonn, D-53115, Germany, and cFaculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia
Correspondence e-mail:  dmitrien@crys.ras.ru

#### 1.11.2.1. General symmetry restrictions

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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).