General symmetry restrictions

Dmitrienko, V. E.; Kirfel, A.; Ovchinnikova, E. N.

doi:10.1107/97809553602060000910

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

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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. Kirfel^b and E. N. Ovchinnikova^c

^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
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): 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).

References

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