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. [\chi_{ij}({\bf r})] must be invariant against all the symmetry operations [g] of the given space group [G]:[\chi_{jk}({\bf r})=R^g_{jm}R^{gT}_{nk}\chi_{mn}({\bf r}^g), \eqno(1.11.2.1)]where [R^g_{jk}] is the matrix of the point operation (rotation or mirror reflection), [r_{j}^g=R^g_{kj}(r_{k}-a_{k}^g)], and [a_{k}^g] is the associated vector of translation. The index [T] indicates a transposed matrix, and summation over repeated indices is implied hereafter. To meet the above demand, it is obviously sufficient for [\chi_{ij}({\bf r})] to be invariant against all generators of the group [G].

There is a simple direct method for obtaining [\chi_{ij}({\bf r})] obeying equation (1.11.2.1)[link]: we can take an arbitrary second-rank tensor [\alpha_{ij}({\bf r})] and average it over all the symmetry operations [g]:[\chi_{jk}({\bf r})=N^{-1}\textstyle\sum\limits_{g\in G} R^g_{jm}R^{gT}_{nk}\alpha_{mn}({\bf r}^g), \eqno(1.11.2.2)]where [N] is the number of elements [g] in the group [G]. A small problem is that [N] is infinite for any space group, but this can be easily overcome if we take [\alpha_{ij}({\bf r})] as periodic and obeying the translation symmetry of the given Bravais lattice. Then the number [N] of the remaining symmetry operations becomes finite (an example of this approach is given in Section 1.11.2.3[link]).








































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