International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 3.1, p. 347
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The phases considered being crystalline, their invariance groups, G or F, coincide with crystallographic space groups. Let us only recall here that each of these groups of infinite order is constituted by elements of the form where R is a point-symmetry operation and t a translation. The symmetry operations R generate the point group of the crystal. On the other hand, among the translations t there is a subset forming an infinite group of `primitive' translations T generating the three-dimensional Bravais lattice of the crystal.
For a space group G, there is an infinite set of unequivalent irreducible representations. An introduction to their properties can be found in Chapter 1.2 as well as in a number of textbooks. They cannot be tabulated in a synthetic manner as the better-known representations of finite groups. They have to be constructed starting from simpler representations. Namely, each representation is labelled by a double index.