Nature of the groups and of their irreducible representations

Tolédano, J.-C.

doi:10.1107/97809553602060000642

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. 3.1, p. 347

Section 3.1.2.5.1. Nature of the groups and of their irreducible representations

J.-C. Tolédano^d ^*

3.1.2.5.1. Nature of the groups and of their irreducible representations

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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 $[\{R |{\bf t}\}]$ 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.

(i) The first index is a k vector in reciprocal space, belonging to the first Brillouin zone of this space. The former vector defines a subgroup $[G({\bf k})]$ of G. This group is the set of elements $[\{R |{\bf t}\}]$ of G whose component R leaves k unmoved, or transforms it into an `equivalent' vector (i.e. differing from k by a reciprocal-lattice vector). The group $[G({\bf k})]$ has irreducible representations labelled $[\tau_m ({\bf k})]$ of dimension which are defined in available tables.
(ii) A representation of G can be denoted $[\Gamma_{{\bf k}, m}]$ . It can be constructed according to systematic rules on the basis of the knowledge of $[\tau_m ({\bf k})]$ . Its dimension is where r is the number of vectors in the `star' of k. This star is the set of vectors, unequivalent to k, having the same modulus as k and obtained from k by application of all the point-symmetry elements R of G.

References

International Tables for Crystallography (2006). Vol. D. ch. 3.1, p. 347