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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
a
A. 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 |
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).
