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. 2.2, pp. 297-298
Section 2.2.6. Space-group symmetry^{a}Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria |
The effect of a space-group operation on a Bloch function, labelled by , is to transform it into a Bloch function that corresponds to a vector ,which can be proven by using the multiplication rule of Seitz operators (2.2.3.12) and the definition of a Bloch state (2.2.4.17).
A special case is the inversion operator, which leads to The Bloch functions and , where p is any operation of the point group P, belong to the same basis for a representation of the space group G. The same cannot appear in two different bases, thus the two bases and are either identical or have no in common.
Irreducible representations of T are labelled by the N distinct vectors in the BZ, which separate in disjoint bases of G (with no vector in common). If a vector falls on the BZ edge, application of the point-group operation p can lead to an equivalent vector that differs from the original by (a vector of the reciprocal lattice). The set of all mutually inequivalent vectors of () define the star of the k vector () (see also Section 1.2.3.3 of the present volume).
The set of all operations that leave a vector invariant (or transform it into an equivalent ) forms the group of the vector. Application of q, an element of , to a Bloch function (Section 2.2.8) gives where the band index j (described below) may change to . The Bloch factor stays constant under the operation of q and thus the periodic cell function must show this symmetry, namely For example, a -like orbital may be transformed into a -like orbital if the two are degenerate, as in a tetragonal lattice.
A star of determines an irreducible basis, provided that the functions of the star are symmetrized with respect to the irreducible representation of the group of vectors, which are called small representations. The basis functions for the irreducible representations are given according to Seitz (1937) by written as a row vector with , where n is the dimension of the irreducible representation of with the order . Such a basis consists of functions and forms an -dimensional irreducible representation of the space group. The degeneracies of these representations come from the star of (not crucial for band calculations except for determining the weight of the vector) and the degeneracy from . The latter is essential for characterizing the energy bands and using the compatibility relations (Bouckaert et al., 1930; Bradley & Cracknell, 1972).
Each irreducible representation of the space group, labelled by , denotes an energy , where varies quasi-continuously over the BZ and the superscript j numbers the band states. The quantization of according to (2.2.4.13) and (2.2.4.15) can be done in arbitrary fine steps by choosing corresponding periodic boundary conditions (see Section 2.2.4.2). Since and belong to the same Bloch state, the energy is periodic in reciprocal space: Therefore it is sufficient to consider vectors within the first BZ. For a given , two bands will not have the same energy unless there is a multidimensional small representation in the group of or the bands belong to different irreducible representations and thus can have an accidental degeneracy. Consequently, this can not occur for a general 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.