Case of a centre of symmetry

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.1, p. 15

Section 1.1.4.6.2.2. Case of a centre of symmetry

A. Authier^a ^*

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

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All the diagonal components are in this case equal to −1. One thus has:

(i) Tensors of even rank, $[t\hskip1pt^{ij\ldots}= (-1)^{2p}t\hskip1pt^{ij\ldots}]$ . The components are not affected by the presence of the centre of symmetry. The reduction of tensors of even rank is therefore the same in a centred group and in its noncentred subgroups, that is in any of the 11 Laue classes: $[\matrix{&\bar{{\bf 1}} \quad 1\hfill\cr &{\bf 2/{\bi m}} \quad 2,\ m \hfill\cr &{\bi mmm} \quad 222,\ 2mm \hfill\cr &\bar{{\bf 3}} \quad 3\hfill\cr &{\bar{\bf 3}}{\bi m} \quad 32, \ 3m\hfill\cr &{\bf 4/{\bi m}} \quad {\bar 4}, \ 4 \hfill\cr &{\bf 4/{\bi m m}} \quad {\bar 4}2m, \ 422, \ 4mm\hfill\cr &{\bf 6/{\bi m}} \quad {\bar 6}, \ 6 \hfill\cr &{\bf 6/{\bi m m}} \quad {\bar 6}2m, \ 622, \ 6mm\hfill\cr &{\bi m}{\bar{\bf 3}} \quad 23\hfill\cr &{\bi m}{\bar{\bf 3}}{\bi m} \quad 432, \ {\bar 4}32.\hfill\cr}]$ If a tensor is invariant with respect to two elements of symmetry, it is invariant with respect to their product. It is then sufficient to make the reduction for the generating elements of the group and (since this concerns a tensor of even rank) for the 11 Laue classes.
(ii) Tensors of odd rank, $[t\hskip1pt^{ij\ldots}= (-1)^{2p+1}t\hskip1pt^{ij\ldots}]$ . All the components are equal to zero. The physical properties represented by tensors of rank 3, such as piezoelectricity, piezomagnetism, nonlinear optics, for instance, will therefore not be present in a centrosymmetric crystal.

International Tables for Crystallography (2006). Vol. D. ch. 1.1, p. 15