Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.5, p. 162   | 1 | 2 |

Section 1.5.2. Introduction

M. I. Aroyoa* and H. Wondratschekb

aFaculty of Physics, University of Sofia, bulv. J. Boucher 5, 1164 Sofia, Bulgaria , and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:

1.5.2. Introduction

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This new chapter on representations widens the scope of the general topics of reciprocal space treated in this volume.

Space-group representations play a growing role in physical applications of crystal symmetry. They are treated in a number of papers and books but comparison of the terms and the listed data is difficult. The main reason for this is the lack of standards in the classification and nomenclature of representations. As a result, the reader is confronted with different numbers of types and barely comparable notations used by the different authors, see e.g. Stokes & Hatch (1988)[link], Table 7.

The k vectors, which can be described as vectors in reciprocal space, play a decisive role in the description and classification of space-group representations. Their symmetry properties are determined by the so-called reciprocal-space group [{\cal G}^{*}] which is always isomorphic to a symmorphic space group [{\cal G}_{0}]. The different symmetry types of k vectors correspond to the different kinds of point orbits in the symmorphic space groups [{\cal G}_{0}]. The classification of point orbits into Wyckoff positions in International Tables for Crystallography Volume A (IT A) (2005)[link] can be used directly to classify the irreducible representations of a space group, abbreviated irreps; the Wyckoff positions of the symmorphic space groups [{\cal G}_{0}] form a basis for a natural classification of the irreps. This was first discovered by Wintgen (1941)[link]. Similar results have been obtained independently by Raghavacharyulu (1961)[link], who introduced the term reciprocal-space group. In this chapter a classification of irreps is provided which is based on Wintgen's idea.

Although this idea is now more than 50 years old, it has been utilized only rarely and has not yet found proper recognition in the literature and in the existing tables of space-group irreps. Slater (1962)[link] described the correspondence between the special k vectors of the Brillouin zone and the Wyckoff positions of space group [Pm\bar{3}m]. Similarly, Jan (1972)[link] compared Wyckoff positions with points of the Brillouin zone when describing the symmetry [Pm\bar{3}] of the Fermi surface for the pyrite structure. However, the widespread tables of Miller & Love (1967)[link], Zak et al. (1969)[link], Bradley & Cracknell (1972)[link] (abbreviated as BC), Cracknell et al. (1979)[link] (abbreviated as CDML), and Kovalev (1986)[link] have not made use of this kind of classification and its possibilities, and the existing tables are unnecessarily complicated, cf. Boyle (1986)[link].

In addition, historical reasons have obscured the classification of irreps and impeded their application. The first considerations of irreps dealt only with space groups of translation lattices (Bouckaert et al., 1936[link]). Later, other space groups were taken into consideration as well. Instead of treating these (lower) symmetries as such, their irreps were derived and classified by starting from the irreps of lattice space groups and proceeding to those of lower symmetry. This procedure has two consequences:

  • (1) those k vectors that are special in a lattice space group are also correspondingly listed in the low-symmetry space group even if they have lost their special properties due to the symmetry reduction;

  • (2) during the symmetry reduction unnecessary new types of k vectors and symbols for them are introduced.

The use of the reciprocal-space group [{\cal G}^{*}] avoids both these detours.

In this chapter we consider in more detail the reciprocal-space-group approach and show that widely used crystallographic conventions can be adopted for the classification of space-group representations. Some basic concepts are developed in Section 1.5.3[link]. Possible conventions are discussed in Section 1.5.4[link]. The consequences and advantages of this approach are demonstrated and discussed using examples in Section 1.5.5[link].


International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.
Bouckaert, L. P., Smoluchowski, R. & Wigner, E. P. (1936). Theory of Brillouin zones and symmetry properties of wave functions in crystals. Phys. Rev. 50, 58–67.
Boyle, L. L. (1986). The classification of space group representations. In Proceedings of the 14th international colloquium on group-theoretical methods in physics, pp. 405–408. Singapore: World Scientific.
Bradley, C. J. & Cracknell, A. P. (1972). The mathematical theory of symmetry in solids: representation theory for point groups and space groups. Oxford: Clarendon Press.
Cracknell, A. P., Davies, B. L., Miller, S. C. & Love, W. F. (1979). Kronecker product tables, Vol. 1, General introduction and tables of irreducible representations of space groups. New York: IFI/Plenum.
Jan, J.-P. (1972). Space groups for Fermi surfaces. Can. J. Phys. 50, 925–927.
Kovalev, O. V. (1986). Irreducible and induced representations and co-representations of Fedorov groups. Moscow: Nauka. (In Russian.)
Miller, S. C. & Love, W. F. (1967). Tables of irreducible representations of space groups and co-representations of magnetic space groups. Boulder: Pruett Press.
Raghavacharyulu, I. V. V. (1961). Representations of space groups. Can. J. Phys. 39, 830–840.
Slater, L. S. (1962). Quantum theory of molecules and solids, Vol. 2. Amsterdam: McGraw-Hill.
Stokes, H. T. & Hatch, D. M. (1988). Isotropy subgroups of the 230 crystallographic space groups. Singapore: World Scientific.
Wintgen, G. (1941). Zur Darstellungstheorie der Raumgruppen. Math. Ann. 118, 195–215. (In German.)
Zak, J., Casher, A., Glück, M. & Gur, Y. (1969). The irreducible representations of space groups. New York: Benjamin.

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