Chapter 1.5 Crystallographic viewpoints in the classification of space-group representations

Reciprocal space

Second Online Edition (2010)

Part 1. General relationships and techniques

  1. M. I. Aroyo1,
  2. H. Wondratschek2

Published Online: 1 JUN 2010

DOI: 10.1107/97809553602060000762

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Aroyo, M. I. and Wondratschek, H. 2010. Crystallographic viewpoints in the classification of space-group representations. International Tables for Crystallography. B:1:1.5:175–192.

Author Information

  1. 1

    Departamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain

  2. 2

    Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany

Publication History

  1. Published Online: 1 JUN 2010


The k vectors are vectors in reciprocal space and play an important role in the description of space-group representations. Chapter 1.5 deals with the classification of these k vectors with special regard to crystallographic points of view. In 1941, Wintgen found that the k vectors of any space group can be classified in a natural way analogous to the classification of the Wyckoff positions of the symmorphic space groups. This is possible by introducing the so-called reciprocal-space group, which is isomorphic to a symmorphic space group. The symmetry types of k vectors correspond to the Wyckoff positions of this symmorphic space group and the tables of the Wyckoff positions in Volume A of International Tables for Crystallography present the classification directly. In this chapter, the basic concepts of representations of space groups are defined and the reciprocal-space group is introduced. The sometimes complicated Brillouin zone and its representation domain may be replaced by the often much simpler conventional unit cell of the reciprocal lattice and its asymmetric unit. The different k vectors of the same symmetry type are characterized by parameters which correspond to the coordinates of the representative points of the Wyckoff positions. The ranges of these parameters are chosen in such a way that each k-vector orbit is listed exactly once in the k-vector table of the space group. The Wintgen classification is applied in several examples (space groups equation image, equation image, equation image, equation image, equation image, equation image, equation image, equation image, equation image and equation image) and compared with the usual classification. For the third edition of this volume, the examples in this chapter are taken from the database of the Bilbao Crystallographic Server (


  • space groups;
  • irreducible representations;
  • Wintgen positions;
  • Brillouin zones;
  • reciprocal-space groups;
  • crystallographic orbits;
  • k vectors;
  • star of k;
  • space-group representations