Chapter 1.1. Historical introduction

The documentation concerning group-theoretical aspects of group–subgroup relations and the rising importance of these relations in three-dimensional crystallography is scattered widely in the literature. This short review, therefore, cannot be exhaustive and may even be unbalanced if the authors have missed essential sources. Not included here is the progress made in the general theory of crystallographic groups, e.g. in higher dimensions, which is connected with the names of the mathematicians and physicists Ascher, Brown, Janner, Janssen, Neubüser, Plesken, Zassenhaus and others. This volume A1 of International Tables for Crystallography, abbreviated IT A1, is concerned with objects belonging to the `classical' theory of crystallographic groups.

For a long time, the objective of crystallography, as implied by the word itself, was the description of crystals which were found in nature or which grew from solutions of salts or from melts. Crystals often display more-or-less planar faces which are symmetrically equivalent or can be made so by parallel shifting of the faces along their normals such that they form a regular body. The existence of characteristic angles between crystal faces was observed by Niels Stensen in 1669. The importance of the shape of crystals and their regularity was supported by observations on the cleavage of crystals. It was in particular this regularity which attracted mineralogists, chemists, physicists and eventually mathematicians, and led to the establishment of the laws by which this regularity was governed. The law of symmetry and the law of rational indices were formulated by René Just Haüy around 1800. These studies were restricted to the shape of macroscopic crystals and their physical properties, because only these were accessible to measurements until the beginning of the twentieth century.

Later in the nineteenth century, interest turned to `regular systems' of points, for which the arrangement of points around every point is the same. These were studied by Wiener (1863) and Sohncke (1879). For such sets, there are isometric mappings of any point onto any other point which are such that they map the whole set onto itself. The primary aim was the classification and listing of such regular systems, while the groups of isometries behind these systems were of secondary importance. These regular systems were at first the sets of crystal faces, of face normals of crystal faces and of directions in crystals which are equivalent with respect to the physical properties or the symmetry of the external shape of the crystal. The listing and the classification of these finite sets was based on the law of rational indices and resulted in the derivation of the 32 crystal classes of point groups by Moritz Ludwig Frankenheim in 1826, Johann Friedrich Christian Hessel in 1830 and Axel V. Gadolin in 1867, cited by Burckhardt (1988). The list of this classification contained crystal classes that had not been observed in any crystal and included group–subgroup relations implicitly.

Crystal cleavage led Haüy in 1784 to assume small parallel­epi­peds as building blocks of crystals. In 1824, Ludwig August Seeber explained certain physical properties of crystals by placing molecules at the vertices of the parallelepipeds. The concepts of unit cell and of translation symmetry were thus introduced implicitly.

The classification of the 14 underlying lattices of translations was completed by Auguste Bravais (1850). In the second half of the nineteenth century, interest turned to the derivation and classification of infinite regular systems of points or figures, in particular by Leonhard Sohncke and Evgraf Stepanovich Fedorov. Sohncke, Fedorov (1891), Arthur Schoenflies (1891) and later William Barlow (1894) turned to the investigation of the underlying groups, the space groups and plane groups. The derivation and classification of these groups were completed in the early 1890s. It was a plausible hypothesis that the structures of crystals were related to combinations of regular systems of atoms (Haag, 1887) and that the symmetry of a crystal structure should be a space group, but both conjectures were speculations at that time with no experimental proof. This also applies to the atom packings described by Sohncke and Barlow, such as a model of the NaCl structure (which they did not assign to any substance).

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), 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) included diagrams of the unit cells with special positions and symmetry elements. Additional tables by Astbury & Yardley (1924) 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) (abbreviated as IT 35), International Tables for X-ray Crystallography, Vol. I (1952, 1965, 1969) (abbreviated as IT 52) and International Tables for Crystallography, Vol. A (1983 and subsequent editions 1987, 1992, 1995, 2002) (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). Shortly afterwards, Linus Pauling (1928, 1929) formulated his famous rules about ionic radii, valence bonds, coordination polyhedra and the joining of these polyhedra. Later, Wilhelm Biltz (1934) 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 edition of IT A. And yet, even in the 2002 edition, the listing of the subgroups in Volume A is incomplete. It is this present Volume A1 which now contains the complete listing.

The systematic survey of group–subgroup relations of space groups started with the fundamental publication by Carl Hermann (1929). This paper is the last in a series of four publications, dealing with

In paper (IV), Hermann introduced two distinct kinds of subgroups. The translationengleiche subgroups of a space group have retained all translations of but belong to a crystal class of lower symmetry than the crystal class of (Hermann used the term zellengleiche instead of translationengleiche, see the footnote to Definition 1.2.6.2.1 in Chapter 1.2). The klassenglei­che subgroups are those which belong to the same crystal class as , but have lost translations compared with . General subgroups are those which have lost translations as well as crystal-class symmetry. Hermann proved the theorem, later called Hermann's theorem, that any general subgroup is a klassengleiche subgroup of a uniquely determined translationengleiche subgroup of . In particular, this implies that a maximal subgroup of is either a translationengleiche subgroup or a klassengleiche subgroup of .

Because of the strong relation (homomorphism) between a space group and its point group , the set of translationengleiche subgroups of is in a one-to-one correspondence with the set of subgroups of the point group . The crystallographic point groups are groups of maximal order 48 with well known group–subgroup relations and with not more than 96 subgroups. Thus, the maximal translationengleiche subgroups of any space group can be obtained easily by comparison with the subgroups of its point group . The kind of derivation of the space-group types by H. Heesch (1930) also gives access to translationengleiche subgroups. In IT 35, the types of the translationengleiche subgroups were listed for each space group [for a list of corrections to these data, see Ascher et al. (1969)]. A graph of the group–subgroup relations between the crystallographic point groups can also be found in IT 35; the corresponding graphs for the space groups were published by Ascher (1968). In these lists and graphs the subgroups are given only by their types, not individually.

The group–subgroup relations between the space groups were first applied in Vol. 1 of Strukturbericht (1931). In this volume, a crystal structure is described by the coordinates of the atoms, but the space-group symmetry is stated not only for spherical particles but also for molecules or ions with lower symmetry. Such particles may reduce the site symmetry and with it the space-group symmetry to that of a subgroup. In addition, the symmetry reduction that occurs if the particles are combined into larger structural units is stated. The listing of these detailed data was discontinued both in the later volumes of Strukturbericht and in the series Structure Reports. Meanwhile, experience had shown that there is no point in assuming a lower symmetry of the crystal structure if the geometrical arrangement of the centres of the particles does not indicate it.

With time, not only the classification of the crystal structures but also a growing number of investigations of (continuous) phase transitions increased the demand for data on subgroups of space groups. Therefore, when the Executive Committee of the International Union of Crystallography decided to publish a new series of International Tables for Crystallography, an extension of the subgroup data was planned. Stimulated and strongly supported by the mathematician J. Neubüser, the systematic derivation of the subgroups of the plane groups and the space groups began. The listing was restricted to the maximal subgroups of each space group, because any subgroup of a space group can be obtained by a chain of maximal subgroups.

The derivation by Neubüser & Wondratschek started in 1965 with the translationengleiche subgroups of the space groups, because the complete set of these (maximally 96) subgroups could be calculated by computer. All klassengleiche subgroups of indices 2, 3, 4, 6, 8 and 9 were also obtained by computer. As the index of a maximal non-isomorphic subgroup of a space group is restricted to 2, 3 or 4, all maximal non-isomorphic subgroups of all space groups were contained in the computer outputs. First results and their application to relations between crystal structures are found in Neubüser & Wondratschek (1966). In the early tables, the subgroups were only listed by their types. For International Tables, an extended list of maximal non-isomorphic subgroups was prepared. For each space group the maximal translationengleiche subgroups and those maximal klassengleiche subgroups for which the reduction of the translations could be described as `loss of centring translations' of a centred lattice are listed individually. For the other maximal klassengleiche subgroups, i.e. those for which the conventional unit cell of the subgroup is larger than that of the original space group, the description by type was retained, because the individual subgroups of this kind were not completely known in 1983. The deficiency of such a description becomes clear if one realizes that a listed subgroup type may represent 1, 2, 3, 4 or even 8 individual subgroups.

In the present Volume A1, all maximal non-isomorphic subgroups are listed individually, in Chapter 2.2 for the plane groups and in Chapters 2.3 and 3.2 for the space groups. In addition, graphs for the translationengleiche subgroups (Chapter 2.4 ) and for the klassengleiche subgroups (Chapter 2.5 ) supplement the tables. After several rounds of checking by hand and after comparison with other listings, e.g. those by H. Zimmermann (unpublished) or by Neubüser and Eick (unpublished), intensive computer checking of the hand-typed data was carried out by F. Gähler as described in Chapter 1.4 .

The mathematician G. Nebe describes general viewpoints and new results in the theory of subgroups of space groups in Chapter 1.5 .

The maximal isomorphic subgroups are a special subset of the maximal klassengleiche subgroups. Maximal isomorphic subgroups are treated separately because each space group has an infinite number of maximal isomorphic subgroups and, in contrast to non-isomorphic subgroups, there is no limit for the index of a maximal isomorphic subgroup of .

An isomorphic subgroup of a space group seems to have first been described in a crystal–chemical relation when the crystal structure of Sb₂ZnO₆ (structure type of tapiolite, Ta₂FeO₆) was determined by Byström et al. (1941): `If no distinction is drawn between zinc and antimony, this structure appears as three cassiterite-like units stacked end-on-end' (Wyckoff, 1965). The space group of Sb₂ZnO₆ is a maximal isomorphic subgroup of index 3 with of the space group (, No. 136) of cassiterite SnO₂ (rutile type).

The first systematic study attempting to enumerate all isomorphic subgroups (not just maximal ones) for each space-group type was by Billiet (1973). However, the listing was incomplete and, moreover, in the case of enantiomorphic pairs of space-group types, only those with the same space-group symbol (called isosymbolic space groups) were taken into account.

Sayari (1976) derived the conventional bases for all maximal isomorphic subgroups of all plane groups. The general laws of number theory which underlie these results for plane-group types p4, p3 and p6 and space-group types derived from point groups 4, , , 3, , 6, and were published by Müller & Brelle (1995). Bertaut & Billiet (1979) suggested a new analytical approach for the derivation of all isomorphic subgroups of space and plane groups.

Because of the infinite number of maximal isomorphic subgroups, only a few representatives of lowest index are listed in IT A with their lattice relations but without origin specification, cf. IT A (2002), Section 2.2.15.2 . Part 13 of IT A (Billiet & Bertaut, 2005) is fully devoted to isomorphic subgroups, cf. also Billiet (1980) and Billiet & Sayari (1984).

In this volume, all maximal isomorphic subgroups are listed as members of infinite series, where each individual subgroup is specified by its index, its generators and the coordinates of its conventional origin as parameters.

The relations between a space group and its subgroups become more transparent if they are considered in connection with their normalizers in the affine group and the Euclidean group (Koch, 1984). Even the corresponding normalizers of Hermann's group play a role in these relations, cf. Wondratschek & Aroyo (2001).

In addition to subgroup data, supergroup data are listed in IT A. If is a maximal subgroup of , then is a minimal supergroup of . In IT A, the type of a space group is listed as a minimal non-isomorphic supergroup of if is listed as a maximal non-isomorphic subgroup of . Thus, for each space group one can find in the tables the types of those groups for which is listed as a maximal subgroup. The supergroup data of IT A1 are similarly only an inversion of the subgroup data.

Phase transitions . In 1937, Landau introduced the idea of the order parameter for the description of second-order phase transitions (Landau, 1937). Landau theory has turned out to be very useful in the understanding of phase transitions and related phenomena. Such a transition can only occur if there is a group–subgroup relation between the space groups of the two crystal structures. Often only the space group of one phase is known (usually the high-temperature phase) and subgroup relations help to eliminate many groups as candidates for the unknown space group of the other phase. Landau & Lifshitz (1980) examined the importance of group–subgroup relations further and formulated two theorems regarding the index of the group–subgroup pair. The significance of the subgroup data in second-order phase transitions was also pointed out by Ascher (1966, 1967), who formulated the maximal-subgroup rule: `The symmetry group of a phase that arises in a ferroelectric transition is a maximal polar subgroup of the group of the high-temperature phase.' There are analogous applications of the maximal-subgroup rule (with appropriate modifications) to other types of continuous transitions.

The group-theoretical aspects of Landau theory have been worked out in great detail with major contributions by Birman (1966a,b), Cracknell (1975), Stokes & Hatch (1988), Tolédano & Tolédano (1987) and many others. For example, Landau theory gives additional criteria based on thermodynamic arguments for second-order phase transitions. The general statements are reformulated into group-theoretical rules which permit a phase-transition analysis without the tedious algebraic treatment involving high-order polynomials. The necessity of having complete subgroup data for the space groups for the successful implementation of these rules was stated by Deonarine & Birman (1983): ` there is a need for tables yielding for each of the 230 three-dimensional space groups a complete lattice of decomposition of all its subgroups.' Domain-structure analysis (Chapter 3.4 in IT D; Janovec & Přívratská, 2003) and symmetry-mode analysis (Aroyo & Perez-Mato, 1998) are further aspects of phase-transition problems where group–subgroup relations between space groups play an essential role. Domain structures are also considered in Section 1.2.7 .

In treating successive phase transitions within Landau theory, Levanyuk & Sannikov (1971) introduced the idea of a hypothetical parent phase whose symmetry group is a supergroup of the observed (initial) space group. Moreover, the detection of pseudosymmetries is necessary for the prediction of higher-temperature phase transitions, cf. Kroumova et al. (2002) and references therein.

In a reconstructive phase transition, there is no group–subgroup relation between the symmetries of the two structures. Nevertheless, it has been pointed out that the `transition path' between the two structures may involve an intermediate unstable structure whose space group is a common subgroup of the space groups of the two phases (Sowa, 2001; Stokes & Hatch, 2002).

Overlooked symmetry . In recent times, the number of crystal-structure determinations with a wrongly assigned space group has been increasing (Baur & Kassner, 1992; Marsh et al., 2002). Frequently, space groups with too low a symmetry have been chosen. In most cases, the correct space group is a supergroup of the space group that has been assigned. A criterion for the correct assignment of the space group is given by Fischer & Koch (1983). Computer packages for treating the problem can be made more efficient if the possible supergroups are known.

Twinning can also lead to a wrong space-group assignment if it is not recognized, as a twinned crystal can feign a higher or lower symmetry. The true space group of the correct structure is usually a supergroup or subgroup of the space group that has been assumed (Nespolo & Ferraris, 2004).

Relations between crystal structures . Working out relations between different crystal structures with the aid of crystallographic group–subgroup relations was systematically developed by Bärnighausen (1980). The work became more widely known through a number of courses taught in Germany and Italy from 1976 to 1996 (in 1984 as a satellite meeting to the Congress of the International Union of Crystallography). For a script of the 1992 course, see Chapuis (1992). The basic ideas can also be found in the textbook by Müller (1993).

According to Bärnighausen, a family tree of group–subgroup relations is set up. At the top of the tree is the space group of a simple, highly symmetrical structure, called the aristotype by Megaw (1973) or the basic structure by Buerger (1947, 1951). The space groups of structures resulting from distortions or atomic substitutions (the hettotypes or derivative structures) are subgroups of the space group of the aristotype. Apart from many smaller Bärnighausen trees, some trees that interrelate large numbers of crystal structures have been published, cf. Section 1.3.1 . Such trees may even include structures as yet unknown, i.e. the symmetry relations can also serve to predict new structure types that are derived from the aristotype; in addition, the number of such structure types can be calculated for each space group of the tree (McLarnan, 1981a,b,c; Müller, 1992, 1998, 2003).

Setting up a Bärnighausen tree not only requires one to find the group–subgroup relations between the space groups involved. It also requires there to be an exact correspondence between the atomic positions of the crystal structures considered. For a given structure, each atomic position belongs to a certain Wyckoff position of the space group. Upon transition to a subgroup, the Wyckoff position will or will not split into different Wyckoff positions of the subgroup. With the growing number of applications of group–subgroup relations there had been an increasing demand for a list of the relations of the Wyckoff positions for every group–subgroup pair. These listings are accordingly presented in Part 3 of this volume.