International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 175-176   | 1 | 2 |

Section 1.5.2. Introduction

M. I. Aroyoa* and H. Wondratschekb

aDepartamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:  wmpararm@lg.ehu.es

1.5.2. Introduction

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This 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 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 symmetries 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 60 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\overline{3}m]. Similarly, Jan (1972)[link] compared Wyckoff positions with points of the Brillouin zone when describing the symmetry [Pm\overline{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 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 symbols of k vectors are introduced.

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

The relations between the special k vectors as listed by CDML and the Wyckoff positions of the space groups of IT A have been derived and displayed in figures and tables for a few space groups by Aroyo & Wondratschek (1995[link]). The k-vector classification scheme based on Wintgen's (1941[link]) reciprocal-space group approach has been applied meanwhile to all space groups. The compilation of Brillouin-zone figures and k-vector correlation tables for the 230 space groups constitutes the wavevector database of the Bilbao Crystallographic Server (1998[link]), a website of crystallographic databases and programs that can be used free of charge from any computer with a web browser via the Internet (Aroyo, Perez-Mato et al., 2006[link]; Aroyo, Kirov et al., 2006[link]). Simple retrieval tools give direct access to the figures and tables for any space group. The wavevector database available on the server forms the background of the description and classification of the space-group irreps calculated and applied by different programs of the server.

This chapter is a modification of Chapter 1.5 of the second edition of International Tables for Crystallography, Volume B, published in 2001. As in the previous edition, 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]. In contrast to Chapter 1.5 in the second edition of IT B, the consequences and advantages of the reciprocal-space group approach are demonstrated and discussed in Section 1.5.5[link] using examples from the database of the Bilbao Crystallographic Server (1998[link]).

References

Bilbao Crystallographic Server (1998). http://www.cryst.ehu.es/ .
International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.
Aroyo, M. I., Kirov, A., Capillas, C., Perez-Mato, J. M. & Wondratschek, H. (2006). Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups. Acta Cryst. A62, 115–128.
Aroyo, M. I., Perez-Mato, J. M., Capillas, C., Kroumova, E., Ivantchev, S., Madariaga, G., Kirov, A. & Wondratschek, H. (2006). Bilbao Crystallographic Server: I. Databases and crystallographic computing programs. Z. Kristallogr. 221, 15–27.
Aroyo, M. I. & Wondratschek, H. (1995). Crystallographic viewpoints in the classification of space-group representations. Z. Kristallogr. 210, 243–254.
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.
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.
Zak, J., Casher, A., Glück, M. & Gur, Y. (1969). The Irreducible Representations of Space Groups. New York: Benjamin.








































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