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

International Tables for Crystallography (2006). Vol. D, ch. 2.2, pp. 297-298

Section 2.2.6. Space-group symmetry

K. Schwarza*

aInstitut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.6. Space-group symmetry

| top | pdf |

2.2.6.1. Representations and bases of the space group

| top | pdf |

The effect of a space-group operation [\{p|{\bf w}\}] on a Bloch function, labelled by [{\bf k}], is to transform it into a Bloch function that corresponds to a vector [p{\bf k}],[\{p|{\bf w}\}\psi_{{\bf k}}=\psi_{p{\bf k}},\eqno(2.2.6.1)]which can be proven by using the multiplication rule of Seitz operators (2.2.3.12)[link] and the definition of a Bloch state (2.2.4.17)[link].

A special case is the inversion operator, which leads to [\{i|{\bf E}\}\psi_{{\bf k}}=\psi_{-{\bf k}}.\eqno(2.2.6.2)]The Bloch functions [\psi_{{\bf k}}] and [\psi_{p{\bf k}}], where p is any operation of the point group P, belong to the same basis for a representation of the space group G. [ \langle \psi_{{\bf k}}|= \langle \psi_{p{\bf k}}|\hbox{ for all }p\in P\hbox{ for all }p{\bf k}\in {\rm BZ}.\eqno(2.2.6.3)]The same [p{\bf k}] cannot appear in two different bases, thus the two bases [\psi_{{\bf k}}] and [\psi_{{\bf k}^{\prime}}] are either identical or have no [{\bf k}] in common.

Irreducible representations of T are labelled by the N distinct [{\bf k}] vectors in the BZ, which separate in disjoint bases of G (with no [{\bf k}] vector in common). If a [{\bf k}] vector falls on the BZ edge, application of the point-group operation p can lead to an equivalent [{\bf k}^{\prime}] vector that differs from the original by [{\bf K}] (a vector of the reciprocal lattice). The set of all mutually inequivalent [{\bf k}] vectors of [p{\bf k}] ([p\in P]) define the star of the k vector ([S_{{\bf k}}]) (see also Section 1.2.3.3[link] of the present volume).

The set of all operations that leave a [{\bf k}] vector invariant (or transform it into an equivalent [{\bf k+K}]) forms the group [G_{{\bf k}}] of the [{\bf k}] vector. Application of q, an element of [G_{{\bf k}}], to a Bloch function (Section 2.2.8[link]) gives [q\psi_{{\bf k}}^{j}({\bf r})=\psi_{{\bf k}}^{j^{\prime}} ({\bf r})\hbox{ for }q\in G_{{\bf k}},\eqno(2.2.6.4)]where the band index j (described below) may change to [j^{\prime}]. The Bloch factor stays constant under the operation of q and thus the periodic cell function [u_{{\bf k}}^{j}({\bf r})] must show this symmetry, namely [qu_{{\bf k}}^{j}({\bf r})=u_{{\bf k}}^{j^{\prime}}({\bf r})\hbox{ for }q\in G_{{\bf k}}.\eqno(2.2.6.5)]For example, a [p_{x}]-like orbital may be transformed into a [p_{y}]-like orbital if the two are degenerate, as in a tetragonal lattice.

A star of [{\bf k}] determines an irreducible basis, provided that the functions of the star are symmetrized with respect to the irreducible representation of the group of [{\bf k}] vectors, which are called small representations. The basis functions for the irreducible representations are given according to Seitz (1937[link]) by [ \langle s\psi_{{\bf k}}^{j}|,\hbox{ where }s\in S_{{\bf k}},]written as a row vector [ \langle |] with [j=1,\ldots,n], where n is the dimension of the irreducible representation of [S_{{\bf k}}] with the order [\left| S_{{\bf k}}\right| ]. Such a basis consists of [n\left| S_{{\bf k} }\right| ] functions and forms an [n\left| S_{{\bf k}}\right|]-dimensional irreducible representation of the space group. The degeneracies of these representations come from the star of [{\bf k}] (not crucial for band calculations except for determining the weight of the [{\bf k}] vector) and the degeneracy from [G_{{\bf k}}]. The latter is essential for characterizing the energy bands and using the compatibility relations (Bouckaert et al., 1930[link]; Bradley & Cracknell, 1972[link]).

2.2.6.2. Energy bands

| top | pdf |

Each irreducible representation of the space group, labelled by [{\bf k}], denotes an energy [E^{j}({\bf k})], where [{\bf k}] varies quasi-continuously over the BZ and the superscript j numbers the band states. The quantization of [{\bf k}] according to (2.2.4.13)[link] and (2.2.4.15)[link] can be done in arbitrary fine steps by choosing corresponding periodic boundary conditions (see Section 2.2.4.2[link]). Since [{\bf k}] and [{\bf k+K}] belong to the same Bloch state, the energy is periodic in reciprocal space: [E^{j}({\bf k})=E^{j}({\bf k+K}).\eqno(2.2.6.6)]Therefore it is sufficient to consider [{\bf k}] vectors within the first BZ. For a given [{\bf k}], two bands will not have the same energy unless there is a multidimensional small representation in the group of [{\bf k}] or the bands belong to different irreducible representations and thus can have an accidental degeneracy. Consequently, this can not occur for a general [{\bf k}] vector (without symmetry).

References

Bouckaert, L. P., Smoluchowski, R. & Wigner, E. (1930). Theory of Brillouin zones and symmetry properties of wavefunctions in crystals. Phys. Rev. 50, 58–67.
Bradley, C. J. & Cracknell, A. P. (1972). The mathematical theory of symmetry in solids. Oxford: Clarendon Press.
Seitz, F. (1937). On the reduction of space groups. Ann. Math. 37, 17–28.








































to end of page
to top of page