Tables for
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2011). Vol. A1, ch. 1.1, pp. 2-3   | 1 | 2 |

Section 1.1.2. Symmetry and crystal-structure determination

Mois I. Aroyo,a* Ulrich Müllerb and Hans Wondratschekc

aDepartamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain,bFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany, and cInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:

1.1.2. Symmetry and crystal-structure determination

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In 1895, Wilhelm Röntgen, then at the University of Würzburg, discovered what he called X-rays. Medical applications emerged the following year, but it was 17 years later that Max von Laue suggested at a scientific discussion in Munich that a crystal should be able to act as a diffraction grating for X-rays. Two young physicists, Walther Friedrich and Paul Knipping, successfully performed the experiment in May 1912 with a crystal of copper sulfate. In von Laue's opinion, the experiment was so important that he later publicly donated one third of his 1914 Nobel prize to Friedrich and Knipping. The discovery immediately aroused the curiosity of father William Henry Bragg and son William Lawrence Bragg. The son's experiments in Cambridge, initially with NaCl and KCl, led to the development of the Bragg equation and to the first crystal-structure determinations of diamond and simple inorganic materials. Since then, the determination of crystal structures has been an ever-growing enterprise.

The diffraction of X-rays by crystals is partly determined by the space group and partly by the relative arrangement of the atoms, i.e. by the atomic coordinates and the lattice parameters. The presentations by Fedorov and Schoenflies of the 230 space groups were not yet appropriate for use in structure determinations with X-rays. The breakthrough came with the fundamental book of Paul Niggli (1919[link]), who described the space groups geometrically by symmetry elements and point positions and provided the first tables of what are now called Wyckoff positions. Niggli emphasized the importance of the multiplicity and site symmetry of the positions and demonstrated with examples the meaning of the reflection conditions.

Niggli's book pointed the way. The publication of related tables by Ralph W. G. Wyckoff (1922[link]) included diagrams of the unit cells with special positions and symmetry elements. Additional tables by Astbury & Yardley (1924[link]) listed `abnormal spacings' for the space groups, i.e. the reflection conditions. These tables made the concepts and the data of geometric crystallography widely available and were the basis for the series Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935[link]) (abbreviated as IT 35), International Tables for X-ray Crystallography, Vol. I (1952, 1965, 1969[link]) (abbreviated as IT 52) and International Tables for Crystallography, Vol. A (1983[link] and subsequent editions 1987, 1992, 1995, 2002, 2005[link]) (abbreviated as IT A).

Group–subgroup relations were used in the original derivation of the space groups and plane groups. However, in the first decades of crystal-structure determinations, the derivation of geometric data (atomic coordinates) was of prime importance and the group-theoretical information in the publications was small, although implicitly present. With the growing number of crystal structures determined, however, it became essential to understand the rules and laws of crystal chemistry, to classify the incomprehensible set of structures into crystal-structure types, to develop methods for ordering the structure types in a systematic way, to show relations among them and to find common underlying principles.

To this end, different approaches were presented over time. By 1926, the number of crystal structures was already large enough for Viktor Moritz Goldschmidt to formulate basic principles of packing of atoms and ions in inorganic solids (Goldschmidt, 1926[link]). Shortly afterwards, Linus Pauling (1928[link], 1929[link]) formulated his famous rules about ionic radii, valence bonds, coordination polyhedra and the joining of these polyhedra. Later, Wilhelm Biltz (1934[link]) focused attention on the volume requirements of atoms. Many other important factors determining crystal structures, such as chemical bonding, molecular shape, valence-electron concentration, electronic band structures, crystal-orbital overlap populations and others, have been the subject of subsequent studies. Each one of these aspects can serve as an ordering principle in crystal chemistry, giving insights from a different point of view.

For the aspects mentioned above, symmetry considerations are only secondary tools or even unimportant, and group–subgroup relations hardly play a role. Although symmetry is indispensable for the description of a specific crystal structure, for a long time crystal symmetry and the group–subgroup relations involved did not attract much attention as possible tools for working out the relations between crystal structures. This has even been the case for most textbooks on solid-state chemistry and physics. The lack of symmetry considerations is almost a characteristic feature of many of these books. There is a reason for this astonishing fact: the necessary group-theoretical material only became available in a useful form in 1965, namely as a listing of the maximal subgroups of all space groups by Neubüser & Wondratschek. However, for another 18 years this material was only distributed among interested scientists before it was finally included in the 1983[link] edition of IT A. And yet, even in the printing of 2005[link], the listing of the subgroups in Volume A is incomplete. It is this present Volume A1 which contains the complete listing.


Astbury, W. T. & Yardley, K. (1924). Tabulated data for the examination of the 230 space-groups by homogeneous X-rays. Philos. Trans. R. Soc. London, 224, 221–257.
Biltz, W. (1934). Raumchemie der festen Stoffe. Leipzig: L. Voss.
Goldschmidt, V. M. (1926). Untersuchungen über den Bau und Eigenschaften von Krystallen. Skr. Nor. Vidensk. Akad. Oslo Mat.-Nat. Kl. 1926 No. 2 and 1927 No. 8.
International Tables for Crystallography (1983). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 1st ed. Dordrecht: Kluwer Academic Publishers.
International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Dordrecht: Kluwer Academic Publishers.
International Tables for X-ray Crystallography (1952, 1965, 1969). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Bd. Edited by C. Hermann. Berlin: Borntraeger. (In German, English and French.)
Niggli, P. (1919). Geometrische Kristallographie des Diskontinuums. Leipzig: Gebrüder Borntraeger. Reprint (1973) Wiesbaden: Dr M. Saendig.
Pauling, L. (1928). The coordination theory of the structure of ionic compounds. Probleme der modernen Physik (Sommerfeld-Festschrift), p. 11. Leipzig: S. Hirzel.
Pauling, L. (1929). The principles determining the structures of complex ionic crystals. J. Am. Chem. Soc. 51, 1010–1026.
Wyckoff, R. W. G. (1922). The Analytical Expression of the Results of the Theory of Space Groups. Washington: Carnegie Institution.

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