International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 8.3, pp. 732-740
https://doi.org/10.1107/97809553602060000516

Chapter 8.3. Special topics on space groups

H. Wondratscheka*

aInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

Part 8 provides the theoretical background to the data in the tables and diagrams of Volume A. Chapter 8.3 is devoted to special topics on space groups. The reader finds sections on the conventional crystallographic coordinate systems, on crystallographic point orbits and their distribution into Wyckoff positions according to their site symmetries, on sub- and supergroups of space groups, on the space-group generators used in Volume A and on normalizers of space groups.

8.3.1. Coordinate systems in crystallography

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The matrices W and the columns w of crystallographic symmetry operations [\hbox{\sf W}] depend on the choice of the coordinate system. A suitable choice is essential if W and w are to be obtained in a convenient form.

Example

In a space group I4mm, the matrix part of a clockwise fourfold rotation around the c axis is described by the W matrix [4^{-}\ 00z: \pmatrix{\phantom{0\;}0 &1 &0\hfill\cr - 1 &0 &0\hfill\cr \phantom{0\;}0 &0 &1\hfill\cr}] if referred to the conventional crystallographic basis a, b, c. Correspondingly, the matrix [m\ 0yz: \pmatrix{- 1 &0 &0\hfill\cr \phantom{0}0 &1 &0\hfill\cr \phantom{0}0 &0 &1\hfill\cr}] represents a reflection in a plane parallel to b and c. These matrices are easy to handle and their geometrical significance is evident. Referred to the primitive basis [{\bf a}'], [{\bf b}'], [{\bf c}'], defined by [{\bf a}' = {1 \over 2}( - {\bf a} + {\bf b} + {\bf c})], [{\bf b}' = {1 \over 2}({\bf a} - {\bf b} + {\bf c})], [{\bf c}' = {1 \over 2}({\bf a} + {\bf b} - {\bf c})], the matrices representing the same symmetry operations would be [4^{-}: \pmatrix{1 &\phantom{0}0 &- 1\hfill\cr 1 &\phantom{0}0 &\phantom{0\;}0\hfill\cr 1 &- 1 &\phantom{0\;}0\hfill\cr}; \quad m: \pmatrix{1 &\phantom{0}0 &\phantom{0\;}0\hfill\cr 1 &\phantom{0}0 &- 1\hfill\cr 1 &- 1 &\phantom{0\;}0\hfill\cr}.] These matrices are more complicated to work with, and their geometrical significance is less obvious.

The conventional coordinate systems obey rules concerning the vector bases and the origins.

  • (i) In all cases, the conventional coordinate bases are chosen such that the matrices W only consist of the integers 0, [+]1 and −1, that they are reduced as much as possible, and that they are of simplest form, i.e. contain six or at least five zeros for three dimensions and two or at least one zeros for two dimensions. This fact can be expressed in geometric terms by stating that `symmetry directions (Blickrichtungen) are chosen as coordinate axes' (axes of rotation, screw rotation or rotoinversion, normals of reflection or glide planes); cf. Section 2.2.4[link] and Chapter 9.1[link] . Shortest translation vectors compatible with these conditions are chosen as basis vectors. In many cases, the conventional vector basis is not a primitive but rather a nonprimitive crystallographic basis, i.e. there are lattice vectors with fractional coefficients. The centring type of the conventional cell and thus the lattice type can be recognized from the first letter of the Hermann–Mauguin symbol.

    Example

    The letter P for [E^{3}] (or p for [E^{2}]), taken from `primitive', indicates that a primitive basis is being used conventionally for describing the crystal structure and its symmetry operations. In this case, the vector lattice L consists of all vectors [{\bf u} = u_{1}{\bf a}_{1} + \ldots + u_{n}{\bf a}_{n}] with integral coefficients [u_{i}], but contains no other vectors. If the Hermann–Mauguin symbol starts with a `C' in [E^{3}] or with a `c' in [E^{2}], in addition to all such vectors u all vectors [{\bf u} + {1 \over 2}({\bf a} + {\bf b})] also belong to L. The letters A, B, I, F and R are used for the conventional bases of the other types of lattices, cf. Section 1.2.1[link] .

    In a number of cases, the symmetry of the space group determines the conventional vector basis uniquely; in other cases, metrical criteria, e.g. the length of basis vectors, may be used to define a conventional vector basis.

  • (ii) The choice of the conventional origin in the space-group tables of this volume has been dealt with by Burzlaff & Zimmermann (1980)[link]. In general, the origin is a point of highest site symmetry, i.e. as many symmetry operations [\hbox{\sf W}_{j}] as possible leave the origin fixed, and thus have [{\bi w}_{j} = {\bi o}]. Special reasons may justify exceptions from this rule, for example for space groups [I2_{1}2_{1}2_{1} \equiv D_{2}^{9}] (No. 24), [P4_{3}32 \equiv O^{6}] (No. 212), [P4_{1}32 \equiv O^{7}] (No. 213), [I4_{1}32 \equiv O^{8}] (No. 214) and [I\bar{4}3d \equiv T_{d}^{6}] (No. 220); cf. Section 2.2.7[link] . If in a centrosymmetric space group a centre of inversion is not a point of highest site symmetry, the space group is described twice, first with the origin in a point of highest site symmetry, and second with the origin in a centre of inversion, e.g. at 222 and at [\bar{1}] for space group [Pnnn \equiv D_{2h}^{2}] (No. 48); cf. Section 2.2.1[link] .1 For space groups with low site symmetries, the origin is chosen so as to minimize the number of nonzero coefficients of the [w_{j}], e.g. on a twofold screw axis for space group [P2_{1} \equiv C_{2}^{2}] (No. 4).

A change of the coordinate system, i.e. referring the crystal pattern and its symmetry operations [\hbox{\sf W}] to a new coordinate system, results in new coordinates [\specialfonts{\bbsf x}'] and new matrices [\specialfonts{\bbsf{W}}']; cf. Section 5.1.3[link] .

8.3.2. (Wyckoff) positions, site symmetries and crystallographic orbits

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The concept of positions and their site symmetries is fundamental for the determination and description of crystal structures. Let, for instance, [P\bar{1}] be the space group of a crystal structure with tetrahedral [AX_{4}] and triangular [BY_{3}] groups. Then the atoms A and B cannot be located at centres of inversion, as the symmetry of tetrahedra and triangles is incompatible with site symmetry [\bar{1}]. If the space group is [P2/m], again the points with site symmetry [2/m] cannot be the loci of A or B, but points with site symmetries 2, m or 1 can.

The relations between `site symmetry' and `positions' can be formulated in a rather general way.

Definition: The set of all symmetry operations of a space group [{\cal G}] that leave a point X invariant forms a finite group, the site-symmetry group [{\cal S}(X)] of X with respect to [{\cal G}].2

With regard to the symmetry operations of a space group [{\cal G}], two kinds of points are to be distinguished. A point X is called a point of general position with respect to a space group [{\cal G}] if there is no symmetry operation of [{\cal G}] (apart from the identity operation) that leaves X fixed, i.e. if [{\cal S}(X) = {\cal I}]. A point X is called a point of special position with respect to a space group [{\cal G}] if there is at least one other symmetry operation of [{\cal G}], in addition to the identity operation, that leaves X fixed, i.e. if [{\cal S}(X) \gt {\cal I}].

The subdivision of the set of all points into two classes, those of general and those of special position with respect to a space group [{\cal G}], constitutes only a very coarse classification. A finer classification is obtained as follows.

Definition: A Wyckoff position [\hbox{\sf W}_{{\cal G}}] (for short, position; in German, Punktlage) consists of all points X for which the site-symmetry groups [{\cal S}(X)] are conjugate subgroups3 of [{\cal G}].

For practical purposes, each Wyckoff position of a space group is labelled by a letter which is called the Wyckoff letter (Wyckoff notation in earlier editions of these Tables). Wyckoff positions without variable parameters (e.g. [0, 0, 0; 0, 0, {1 \over 2}; \ldots]) and with variable parameters (e.g. [x, y, z; x, 0, {1 \over 4}; \ldots]) have to be distinguished.

The number of different Wyckoff positions of each space group is finite, the maximal numbers being nine for plane groups (realized in p2mm) and 27 for space groups (realized in Pmmm).

A finer classification of the points of [E^{n}] with respect to [{\cal G}], which always results in an infinite number of classes, is the subdivision of all points into sets of symmetrically equivalent points. In the following, these sets will be called crystallographic orbits according to the following definition.

Definition: The set of all points that are symmetrically equivalent to a point X with respect to a space group [{\cal G}] is called the crystallographic orbit of X with respect to [{\cal G}].

Example

Described in a conventional coordinate system, the crystallographic orbit of a point X of general position with respect to a plane group p2 consists of the points [x, y]; [\bar{x},\bar{y}]; [x + 1, y]; [\bar{x} + 1, \bar{y}]; [x, y + 1]; [\bar{x},\bar{y} + 1]; [x - 1, y]; [\bar{x} - 1,\bar{y}]; [x, y - 1]; [\bar{x},\bar{y} - 1]; [x + 1,y + 1]; [\ldots] etc.

Crystallographic orbits are infinite sets of points due to the infinite number of translations in each space group. Any one of its points may represent the whole crystallographic orbit, i.e. may be the generating point X of a crystallographic orbit.4

Because the site-symmetry groups of different points of the same crystallographic orbit are conjugate subgroups of [{\cal G}], a crystallographic orbit consists either of points of general position or of points of special position only. Therefore, one can speak of `crystallographic orbits of general position' or general crystallographic orbits and of `crystallographic orbits of special position' or special crystallographic orbits with respect to [{\cal G}]. Because all points of a crystallographic orbit belong to the same Wyckoff position of [{\cal G}], one also can speak of Wyckoff positions of crystallographic orbits.5

The points of each general crystallographic orbit of a space group [{\cal G}] are in a one-to-one correspondence with the symmetry operations of [{\cal G}]. Starting with the generating point X (to which the identity operation corresponds), to each point [\tilde{X}] of the crystallographic orbit belongs exactly one symmetry operation [\hbox{\sf W}] of [{\cal G}] such that [\tilde{X}] is the image of X under [\hbox{\sf W}]. This one-to-one correspondence is the reason why the `coordinates' listed for the general position in the space-group tables may be interpreted in two different ways, either as the coordinates of the image points of X under [{\cal G}] or as a short-hand notation for the pairs (W, w) of the symmetry operations [\hbox{\sf W}] of [{\cal G}]; cf. Sections 8.1.6[link] and 11.1.1[link] . Such a one-to-one correspondence does not exist for the special crystallographic orbits, where each point corresponds to a complete coset of a left coset decomposition of [{\cal G}] with respect to the site-symmetry group [{\cal S}(X)] of X. Thus, the data listed for the special positions are to be understood only as the coordinates of the image points of X under [{\cal G}].

Space groups with no special crystallographic orbits are called fixed-point-free space groups. The following types of fixed-point-free space groups occur: p1 and pg in [E^{2}]; [P_{1} \equiv C_{1}^{1}] (No. 1), [P2_{1} \equiv C_{2}^{2}] (No. 4), [Pc \equiv C_{s}^{2}] (No. 7), [Cc \equiv C_{s}^{4}] (No. 9), [P2_{1}2_{1}2_{1} \equiv D_{2}^{4}] (No. 19), [Pca2_{1} \equiv C_{2v}^{5}] (No. 29), [Pna2_{1} \equiv C_{2v}^{9}] (No. 33), [P4_{1} \equiv C_{4}^{2}] (No. 76), [P4_{3} \equiv C_{4}^{4}] (No. 78), [P3_{1} \equiv C_{3}^{2}] (No. 144), [P3_{2} \equiv C_{3}^{3}] (No. 145), [P6_{1} \equiv C_{6}^{2}] (No. 169) and [P6_{5} \equiv C_{6}^{3}] (No. 170) in [E^{3}].

Though the classification of the points of space [E^{n}] into Wyckoff positions [\hbox{\sf W}_{{\cal G}}] of a space group [{\cal G}] is unique, the labelling of the Wyckoff positions by Wyckoff letters (Wyckoff notation) is not.

Example

In a space group [P\bar{1}] there are eight classes of centres of inversion [\bar{1}], represented in the space-group tables by [0,0,0]; [0,0,{1 \over 2}]; [0,{1 \over 2},0; \ldots; {1 \over 2},{1 \over 2},{1 \over 2}]. The site-symmetry groups [\{\hbox{\sf I}, \bar{1}\}] within each class are `symmetrically equivalent', i.e. they are conjugate subgroups of [P\bar{1}]. The groups [\{\hbox{\sf I}, \bar{1}\}] of different classes, however, are not `symmetrically equivalent' with respect to [P\bar{1}]. Each class is labelled by one of the Wyckoff letters [a,b,\ldots, h]. This letter depends on the choice of origin and on the choice of coordinate axes. Cyclic permutation of the labels of the basis vectors a, b, c, for instance, induces a cyclic permutation of Wyckoff positions b–c–d and e–f–g; origin shift from 0, 0, 0 to the point [{1 \over 2},0,0] results in an exchange of Wyckoff letters in the pairs a–d, b–f, c–e and g–h. Even if the coordinate axes are determined by some extra condition, e.g. [a \leq b \leq c], there exist no rules for fixing the origin in [P\bar{1}] when describing a crystal structure. The eight classes of centres of inversion of [P\bar{1}] are well established but none of them is inherently distinguished from the others.

The example shows that the different Wyckoff positions of a space group [{\cal G}] may permute under an isomorphic mapping of [{\cal G}] onto itself, i.e. under an automorphism of [{\cal G}]. Accordingly, it is useful to collect into one set all those Wyckoff positions of a space group [{\cal G}] that may be permuted by automorphisms of [{\cal G}]. These sets are called `Wyckoff sets'. The Wyckoff letters belonging to the different Wyckoff positions of the same Wyckoff set are listed by Koch & Fischer (1975)[link]; changes in Wyckoff letters caused by changes of the coordinate system have been listed by Boyle & Lawrenson (1973[link], 1978[link]).

To introduce `Wyckoff sets' more formally, it is advantageous to use the concept of normalizers; cf. Ledermann (1976)[link]. The affine normalizer [{\cal N}]6 of a space group [{\cal G}] in the group [{\cal A}] of all affine mappings is the set of those affine mappings which map [{\cal G}] onto itself. The space group [{\cal G}] is a normal subgroup of [{\cal N}], [{\cal N}] itself is a subgroup of [{\cal A}]. The mappings of [{\cal N}] which are not symmetry operations of [{\cal G}] may transfer one Wyckoff position of [{\cal G}] onto another Wyckoff position.

Definition: Let [{\cal N}] be the normalizer of a space group [{\cal G}] in the group of all affine mappings. A Wyckoff set with respect to [{\cal G}] consists of all points X for which the site-symmetry groups are conjugate subgroups of [{\cal N}].

The difference between Wyckoff positions and Wyckoff sets of [{\cal G}] may be explained as follows. Any Wyckoff position of [{\cal G}] is transformed onto itself by all elements of [{\cal G}] , but not necessarily by the elements of the (larger) group [{\cal N}]. Any Wyckoff set, however, is transformed onto itself even by those elements of [{\cal N}] which are not contained in [{\cal G}].

Remark: A Wyckoff set of [{\cal G}] is a set of points. Obviously, with each point X it contains all points of the crystallographic orbit of X and all points of the Wyckoff position of X. Accordingly, one can speak not only of `Wyckoff sets of points', but also of `Wyckoff sets of crystallographic orbits' and `Wyckoff sets of Wyckoff positions' of [{\cal G}]. Wyckoff sets of crystallographic orbits have been used in the definition of lattice complexes (Gitterkomplexe), under the name Konfigurationslage, by Fischer & Koch (1974)[link]; cf. Part 14[link] .

The concepts `crystallographic orbit', `Wyckoff position' and `Wyckoff set' have so far been defined for individual space groups only. It is no problem, but is of little practical interest, to transfer the concept of `crystallographic orbit' to space-group types. It would be, on the other hand, of great interest to transfer `Wyckoff positions' from individual space groups to space-group types. As mentioned above, however, such a step is not unique. For this reason, the concept of `Wyckoff set' has been introduced to replace `Wyckoff positions'. Different space groups of the same space-group type have corresponding Wyckoff sets, and one can define `types of Wyckoff sets' (consisting of individual Wyckoff sets) in the same way that `types of space groups' (consisting of individual space groups) were defined in Section 8.2.2[link] .

Definition: Let the space groups [{\cal G}] and [{\cal G}'] belong to the same space-group type. The Wyckoff sets [\hbox{\sf K}] of [{\cal G}] and [\hbox{\sf K}'] of [{\cal G}'] belong to the same type of Wyckoff sets if the affine mappings which transform [{\cal G}] onto [{\cal G}'] also transform [\hbox{\sf K}] onto [\hbox{\sf K}'].

Types of Wyckoff sets have been used by Fischer & Koch (1974)[link], under the name Klasse von Konfigurationslagen, when defining lattice complexes. There are 1128 types of Wyckoff sets of the 219 (affine) space-group types and 51 types of Wyckoff sets of the 17 plane-group types [Koch & Fischer (1975)[link] and Chapter 14.1[link] ].

8.3.3. Subgroups and supergroups of space groups

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Relations between crystal structures imply relations between their space groups, which can often be expressed by group–subgroup relations. These group–subgroup relations may be recognized from relations between the lattices and between the point groups of the crystal structures.

Example

The crystal structures of silicon, Si, and sphalerite, ZnS, belong to space-group types [Fd\bar{3}m \equiv O_{h}^{7}] (No. 227) and [F\bar{4}3m \equiv T_{d}^{2}] (No. 216) with lattice constants [a_{\rm Si} = 5.43] and [a_{\rm ZnS} = 5.41\;\hbox{\AA}]. The structure of sphalerite is obtained from that of silicon by replacing alternately half of the Si atoms by Zn and half by S, and by adjusting the lattice constant. The strong connection between the two crystal structures is reflected in the relation between their space groups: the space group of sphalerite is a subgroup (of index 2) of that of silicon (ignoring the small difference in lattice constants).

Data on sub- and supergroups of the space groups are useful for the discussion of structural relations and phase transitions. It must be kept in mind, however, that group–subgroup relations only describe symmetry relations. It is important, therefore, to ascertain that the consequential relations between the atomic coordinates of the particles of the crystal structures also hold, before a structural relation can be deduced from a symmetry relation.

Examples

NaCl and CaF2 belong to the same space-group type [Fm\bar{3}m \equiv O_{h}^{5}] (No. 225) and have lattice constants [a = 5.64] and [a = 5.46\;\hbox{\AA}], respectively. The ions, however, occupy unrelated positions and so the symmetry relation does not express a structural relation. Pyrite, FeS2, and solid carbon dioxide, CO2, belong to the same space-group type [Pa\bar{3} \equiv T_{h}^{6}] (No. 205). They have lattice constants [a = 5.42] and [a = 5.55\;\hbox{\AA}], respectively, and the particles occupy analogous Wyckoff positions. Nevertheless, the structures of these compounds are not related because the positional parameters [x = 0.386] of S in FeS2 and [{x = 0.11}] of O in CO2 differ so much that the coordinations of corresponding atoms are dissimilar.

To formulate group–subgroup relations some definitions are necessary:

Definitions: A set [\{\hbox{\sf H}_{i}\}] of symmetry operations [\hbox{\sf H}_{i}] of a space group [{\cal G}] is called a subgroup [{\cal H}] of [{\cal G}] if [\{\hbox{\sf H}_{i}\}] obeys the group conditions, i.e. is a symmetry group. The subgroup [{\cal H}] is called a proper subgroup of [{\cal G}] if there are symmetry operations of [{\cal G}] not contained in [{\cal H}]. A subgroup [{\cal H}] of a space group [{\cal G}] is called a maximal subgroup of [{\cal G}] if there is no proper subgroup [{\cal M}] of [{\cal G}] such that [{\cal H}] is a proper subgroup of [{\cal M}], i.e. [{\cal G} \gt {\cal M} \gt {\cal H}].

Examples:

Maximal subgroups [{\cal H}] of a space group P1 with lattice vectors a, b, c are, among others, subgroups P1 for which [{\bf a}'' = p{\bf a}], [{\bf b}'' = {\bf b}], [{\bf c}'' = {\bf c}], p prime. If p is not a prime number, e.g. [p = q \cdot r], the subgroup [{\cal H}] is not maximal, because a proper subgroup [{\cal M}] exists with [{\bf a}' = q{\bf a}], [{\bf b}' = {\bf b}], [{\bf c}' = {\bf c}]. [{\cal M}] again has [{\cal H}] as a proper subgroup with [{\bf a}'' = r{\bf a}', {\bf b}'' = {\bf b}',{\bf c}'' = {\bf c}'].

[P2_{1}/c] has maximal subgroups [P2_{1}], Pc and [P\bar{1}] with the same unit cell, whereas P1 is obviously not a maximal subgroup of [P2_{1}/c].

A three-dimensional space group may have subgroups with no translations (site-symmetry groups; cf. Section 8.3.2[link]), with one- or two-dimensional lattices of translations (line groups, frieze groups, rod groups, plane groups and layer groups), or with a three-dimensional lattice of translations (space groups). The number of subgroups of a space group is always infinite.

In this section, only those subgroups of a space group will be considered which are also space groups. This includes all maximal subgroups because a maximal subgroup of a space group is itself a space group. To simplify the discussion, we suppose the set of all maximal subgroups of every space group to be known. In this case, any subgroup [{\cal H}] of a given space group [{\cal G}] may be obtained via a chain of maximal subgroups [{\cal H}_{1}, {\cal H}_{2}, \ldots, {\cal H}_{r-1}, {\cal H}_{r} = {\cal H}] such that [{\cal G} = {\cal H}_{0} \gt {\cal H}_{1} \gt {\cal H}_{2} \gt \ldots \gt {\cal H}_{r-1} \gt {\cal H}_{r} = {\cal H}] where [{\cal H}_{j}] is a maximal subgroup of [{\cal H}_{j-1}] of index [[i_{j}]], with [j = 1, \ldots, r]; for the term `index' see below and Section 8.1.6[link] . There may be many such chains between [{\cal G}] and [{\cal H}]. On the other hand, all subgroups of [{\cal G}] of a given index [i] are obtained if all chains are constructed for which [[i_{1}] * [i_{2}] * \ldots * [i_{r}] = [i]] holds.

For example, [P2/c \gt P2 \gt P1], [P2/c \gt P\bar{1} \gt P1], [P2/c \gt Pc \gt P1] are all possible chains of maximal subgroups for [P2/c] if the original translations are retained completely. The seven subgroups of index [4] with the same translations as the original space group [P6_{3}/mcm] are obtained via the 21 different chains of Fig. 8.3.3.1[link].

[Figure 8.3.3.1]

Figure 8.3.3.1 | top | pdf |

Space group [P6_{3}/mcm] with t subgroups of index [2] and [4]. All 21 possible subgroup chains are displayed by lines.

Not only the number of all subgroups but even the number of all maximal subgroups of a given space group is infinite. This infinite number, however, only occurs for a certain kind of subgroup and can be reduced as described below. It is thus useful to consider the different kinds of subgroups of a space group in a way introduced by Hermann (1929)[link].

It should be kept in mind that all group–subgroup relations considered here are relations between individual space groups but they are valid for all space groups of a space-group type, as the following example shows. A particular space group P2 has a subgroup P1 which is obtained from P2 by retaining all translations but eliminating all rotations and combinations of rotations with translations. For every space group of space-group type P2 such a subgroup P1 exists. Thus the relationship exists, in an extended sense, for the two space-group types involved. One can, therefore, list these relationships by means of the symbols of the space-group types.

For every subgroup [{\cal H}] of a space group [{\cal G}], a `right coset decomposition' of [{\cal G}] relative to [{\cal H}] can be defined as [{\cal G} = {\cal H} + {\cal H} \hbox{\sf G}_{2} + \ldots + {\cal H} \hbox{\sf G}_{i}.] The elements [\hbox{\sf G}_{2},\ldots, \hbox{\sf G}_{i}] of [{\cal G}] are such that [\hbox{\sf G}_{j}] is contained only in the coset [{\cal H} \hbox{\sf G}_{j}]. The integer [i], i.e. the number of cosets, is called the index of [{\cal H}] in [{\cal G}]; cf. the footnote[link] to Section 8.1.6.

The index [i] of a subgroup has a geometric significance. It determines the `dilution' of symmetry operations of [{\cal H}] compared with those of [{\cal G}]. This dilution can occur in essentially three different ways:

  • (i) by reducing the order of the point group, i.e. by eliminating all symmetry operations of some kind. The example [P2 \rightarrow P1] mentioned above is of this type.

  • (ii) by loss of translations, i.e. by `thinning out' the lattice of translations. For the space group P121 mentioned above this may happen in different ways:

    • (a) by suppressing all translations of the kind [(2u + 1){\bf a} + v{\bf b} + w{\bf c}], u, v, w integral (new basis [{\bf a}' = 2{\bf a}], [{\bf b}' = {\bf b}], [{\bf c}' = {\bf c}]), and, hence, by eliminating half of the twofold axes, or

    • (b1) by [{\bf b}' = 2{\bf b}], i.e. by thinning out the translations parallel to the twofold axes, or

    • (b2) again by [{\bf b}' = 2{\bf b}] but replacing the twofold rotation axes by twofold screw axes.

  • (iii) by combination of (i)[link] and (ii)[link], e.g. by reducing the order of the point group and by thinning out the lattice of translations.

Subgroups of the first kind (i)[link] are called translationengleiche or t subgroups7 because the set [{\cal T}] of all (pure) translations is retained. In case (ii)[link], the point group [{\cal P}] and thus the crystal class of the space group is unchanged. These subgroups are called klassengleiche or k subgroups. In the general case (iii)[link], both the translation subgroup [{\cal T}] of [{\cal G}] and the point group [{\cal P}] are changed; the subgroup has lost translations and belongs to a crystal class of lower order.

Obviously the third kind (iii)[link] of subgroups is more difficult to survey than kinds (i)[link] and (ii)[link]. Fortunately, a theorem of Hermann states that the maximal subgroups of a space group [{\cal G}] are of type (i)[link] or (ii)[link].

Theorem of Hermann (1929)[link]. A maximal subgroup of a space group [{\cal G}] is either a t subgroup or a k subgroup of [{\cal G}].

According to this theorem, subgroups of kind (iii)[link] can never occur among the maximal subgroups. They can, however, be derived by a stepwise process of linking maximal subgroups of types (i)[link] and (ii)[link], as has been shown by the chains discussed above.

8.3.3.1. Translationengleiche or t subgroups of a space group

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The `point group' [{\cal P}] of a given space group [{\cal G}] is a finite group. Hence, the number of subgroups and consequently the number of maximal subgroups of [{\cal P}] is finite. There exist, therefore, only a finite number of maximal t subgroups of [{\cal G}]. All maximal t subgroups of every space group [{\cal G}] are listed in the space-group tables of this volume; cf. Section 2.2.15[link] . The possible t subgroups were first listed by Hermann (1935)[link]; corrections have been reported by Ascher et al. (1969)[link].

8.3.3.2. Klassengleiche or k subgroups of a space group

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Every space group [{\cal G}] has an infinite number of maximal k subgroups. For dimensions 1, 2 and 3, however, it can be shown that the number of maximal k subgroups is finite, if subgroups belonging to the same affine space-group type as [{\cal G}] are excluded. The number of maximal subgroups of [{\cal G}] belonging to the same affine space-group type as [{\cal G}] is always infinite. These subgroups are called maximal isomorphic subgroups. In Part 13[link] isomorphic subgroups are treated in detail. In the space-group tables, only data on the isomorphic subgroups of lowest index are listed. The way in which the isomorphic and non-isomorphic k subgroups are listed in the space-group tables is described in Section 2.2.15[link] .

Remark: Enantiomorphic space groups have an infinite number of maximal isomorphic subgroups of the same type and an infinite number of maximal isomorphic subgroups of the enantiomorphic type.

Example

All k subgroups [{\cal G}'] of a given space group [{\cal G} \equiv P3_{1}], with basis vectors [{\bf a}' = {\bf a},{\bf b}' = {\bf b},{\bf c}' = p{\bf c}, p] being any prime number except 3, are maximal isomorphic subgroups. They belong to space-group type [P3_{1}] if [p = 3r + 1, r] any integer. They belong to the enantiomorphic space-group type [P3_{2}] if [p = 3r + 2].

Even though in the space-group tables some kinds of maximal subgroups are listed completely whereas others are listed only partly, it must be emphasized that in principle there is no difference in importance between t, non-isomorphic k and isomorphic k sub-groups. Roughly speaking, a group–subgroup relation is `strong' if the index [i] of the subgroup is low. All maximal t and maximal non-isomorphic k subgroups have indices less than four in [E^{2}] and five in [E^{3}], index four already being rather exceptional. Maximal isomorphic k subgroups of arbitrarily high index exist for every space group.

8.3.3.3. Supergroups

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Sometimes a space group [{\cal H}] is known and the possible space groups [{\cal G}], of which [{\cal H}] is a subgroup, are of interest.

Definition: A space group [{\cal R}] is called a minimal supergroup of a space group [{\cal G}] if [{\cal G}] is a maximal subgroup of [{\cal R}].

Examples

In Fig. 8.3.3.1[link], the space group [P6_{3}/mcm] is a minimal supergroup of [P\bar{6}c2, \ldots, P\bar{3}c1;\ P\bar{6}c2] is a minimal supergroup of [P\bar{6}, P3c1] and P312; etc.

If [{\cal G}] is a maximal t subgroup of [{\cal R}] then [{\cal R}] is a minimal t supergroup of [{\cal G}]. If [{\cal G}] is a maximal k subgroup of [{\cal R}] then [{\cal R}] is a minimal k supergroup of [{\cal G}]. Finally, if [{\cal G}] is a maximal isomorphic subgroup of [{\cal R}], then [{\cal R}] is a minimal isomorphic supergroup of [{\cal G}]. Data on minimal t and minimal non-isomorphic k supergroups are listed in the space-group tables; cf. Section 2.2.15[link] . Data on minimal isomorphic supergroups are not listed because they can be derived easily from the corresponding subgroup relations.

The complete data on maximal subgroups of plane and space groups are listed in Volume A1 of International Tables for Crystallography (2004)[link]. For each space group, all maximal subgroups of index [2], [3] and [4] are listed individually. The infinitely many maximal isomorphic subgroups are listed as members of a few (infinite) series. The main parameter in these series is the index p, [p^{2}] or [p^{3}], where p runs through the infinite number of primes.

8.3.4. Sequence of space-group types

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The sequence of space-group entries in the space-group tables follows that introduced by Schoenflies (1891)[link] and is thus established historically. Within each geometric crystal class, Schoenflies has numbered the space-group types in an obscure way. As early as 1919, Niggli (1919)[link] considered this Schoenflies sequence to be unsatisfactory and suggested that another sequence might be more appropriate. Fedorov (1891)[link] used a different sequence in order to distinguish between symmorphic, hemisymmorphic and asymmorphic space groups.

The basis of the Schoenflies symbols and thus of the Schoenflies listing is the geometric crystal class. For the present Tables, a sequence might have been preferred in which, in addition, space-group types belonging to the same arithmetic crystal class were grouped together. It was decided, however, that the long-established sequence in the earlier editions of International Tables should not be changed.

In Table 8.3.4.1[link], those geometric crystal classes are listed in which the Schoenflies sequence separates space groups belonging to the same arithmetic crystal class. The space groups are rearranged in such a way that space groups of the same arithmetic crystal class are grouped together. The arithmetic crystal classes are separated by rules spanning the first three columns of the table and the geometric crystal classes are separated by rules spanning the full width of the table. In all cases not listed in Table 8.3.4.1[link], the Schoenflies sequence, as used in these Tables, does not break up arithmetic crystal classes. Nevertheless, some rearrangement would be desirable in other arithmetic crystal classes too. For example, the symmorphic space group should always be the first entry of each arithmetic crystal class.

Table 8.3.4.1| top | pdf |
Listing of space-group types according to their geometric and arithmetic crystal classes

No.Hermann–Mauguin symbolSchoenflies symbolGeometric crystal class
10 [P2/m] [C_{2h}^{1}] [2/m]
11 [P2_{1}/m] [C_{2h}^{2}]
13 [P2/c] [C_{2h}^{4}]
14 [P2_{1}/c] [C_{2h}^{5}]
12 [C2/m] [C_{2h}^{3}]
15 [C2/c] [C_{2h}^{6}]
149 P312 [D_{3}^{1}] 32
151 [P3_{1}12] [D_{3}^{3}]
153 [P3_{2}12] [D_{3}^{5}]
150 P321 [D_{3}^{2}]
152 [P3_{1}21] [D_{3}^{4}]
154 [P3_{2}21] [D_{3}^{6}]
155 R32 [D_{3}^{7}]
156 P3m1 [C_{3v}^{1}] 3m
158 P3c1 [C_{3v}^{3}]
157 P31m [C_{3v}^{2}]
159 P31c [C_{3v}^{4}]
160 R3m [C_{3v}^{5}]
161 R3c [C_{3v}^{6}]
195 P23 [T^{1}] 23
198 [P2_{1}3] [T^{4}]
196 F23 [T^{2}]
197 I23 [T^{3}]
199 [I2_{1}3] [T^{5}]
200 [Pm\bar{3}] [T_{h}^{1}] [m\bar{3}]
201 [Pn\bar{3}] [T_{h}^{2}]
205 [Pa\bar{3}] [T_{h}^{6}]
202 [Fm\bar{3}] [T_{h}^{3}]
203 [Fd\bar{3}] [T_{h}^{4}]
204 [Im\bar{3}] [T_{h}^{5}]
206 [Ia\bar{3}] [T_{h}^{7}]
207 P432 [O^{1}] 432
208 [P4_{2}32] [O^{2}]
213 [P4_{1}32] [O^{7}]
212 [P4_{3}32] [O^{6}]
209 F432 [O^{3}]
210 [F4_{1}32] [O^{4}]
211 I432 [O^{5}]
214 [I4_{1}32] [O^{8}]
215 [P\bar{4}3m] [T_{d}^{1}] [\bar{4}3m]
218 [P\bar{4}3n] [T_{d}^{4}]
216 [F\bar{4}3m] [T_{d}^{2}]
219 [F\bar{4}3c] [T_{d}^{5}]
217 [I\bar{4}3m] [T_{d}^{3}]
220 [I\bar{4}3d] [T_{d}^{6}]

8.3.5. Space-group generators

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In group theory, a set of generators of a group is a set of group elements such that each group element may be obtained as an ordered product of the generators. For space groups of one, two and three dimensions, generators may always be chosen and ordered in such a way that each symmetry operation [\hbox{\sf W}] can be written as the product of powers of h generators [\hbox{\sf G}_{j}\ (j = 1, 2, \ldots, h)]. Thus,[\hbox{\sf W} = \hbox{\sf G}_{h}^{k_{h}} * \hbox{\sf G}_{h-1}^{k_{h-1}} * \ldots * \hbox{\sf G}_{3}^{k_{3}} * \hbox{\sf G}_{2}^{k_{2}} * \hbox{\sf G}_{1},] where the powers [k_{j}] are positive or negative integers (including zero).

Description of a group by means of generators has the advantage of compactness. For instance, the 48 symmetry operations in point group [m\bar{3}m] can be described by two generators. Different choices of generators are possible. For the present Tables, generators and generating procedures have been chosen such as to make the entries in the blocks General position (cf. Section 2.2.11[link] ) and Symmetry operations (cf. Section 2.2.9[link] ) as transparent as possible. Space groups of the same crystal class are generated in the same way (for sequence chosen, see Table 8.3.5.1)[link], and the aim has been to accentuate important subgroups of space groups as much as possible. Accordingly, a process of generation in the form of a `composition series' has been adopted, see Ledermann (1976)[link]. The generator [\hbox{\sf G}_{1}] is defined as the identity operation, represented by (1) x, y, z. [\hbox{\sf G}_{2}], [\hbox{\sf G}_{3}] and [\hbox{\sf G}_{4}] are the translations with translation vectors a, b and c, respectively. Thus, the coefficients [k_{2}], [k_{3}] and [k_{4}] may have any integral value. If centring translations exist, they are generated by translations [\hbox{\sf G}_{5}] (and [\hbox{\sf G}_{6}] in the case of an F lattice) with translation vectors d (and e). For a C lattice, for example, d is given by [{\bf d} = {1 \over 2}({\bf a} + {\bf b})]. The exponents [k_{5}] (and [k_{6}]) are restricted to the following values:

Table 8.3.5.1| top | pdf |
Sequence of generators for the crystal classes

The space-group generators differ from those listed here by their glide or screw components. The generator 1 is omitted, except for crystal class 1. The subscript of a symbol denotes the characteristic direction of that operation, where necessary. The subscripts z, y, 110, [1\bar{1}0], [10\bar{1}] and 111 refer to the directions [001], [010], [110], [[1\bar{1}0]], [[10\bar{1}]] and [111], respectively. For mirror reflections m, the `direction of m' refers to the normal to the mirror plane. The subscripts may be likewise interpreted as Miller indices of that plane.

Hermann–Mauguin symbol of crystal classGenerators [\hbox{\sf G}_{i}] (sequence left to right)
1 1
[\bar{1}] [\bar{1}]
2 2
m m
[2/m] [2,\bar{1}]
222 [2_{z},2_{y}]
mm2 [2_{z},m_{y}]
mmm [2_{z},2_{y},\bar{1}]
4 [2_{z},4]
[\bar{4}] [2_{z},\bar{4}]
[4/m] [2_{z},4,\bar{1}]
422 [2_{z},4,2_{y}]
4mm [2_{z},4,m_{y}]
[\bar{4}2m] [2_{z},\bar{4},2_{y}]
[\bar{4}m2] [2_{z},\bar{4},m_{y}]
[4/mmm] [2_{z},4,2_{y},\bar{1}]
3 3
[\bar{3}] [3,\bar{1}]
321 [3,2_{110}]
(rhombohedral coordinates [3_{111},2_{10\bar{1}}])
312 [3,2_{1\bar{1}0}]
3m1 [3,m_{110}]
(rhombohedral coordinates [3_{111},m_{10\bar{1}}])
31m [3,m_{1\bar{1}0}]
[\bar{3}m1] [3,2_{110},\bar{1}]
(rhombohedral coordinates [3_{111},2_{10\bar{1}},\bar{1}])
[\bar{3}1m] [3,2_{1\bar{1}0},\bar{1}]
6 [3,2_{z}]
[\bar{6}] [3,m_{z}]
[6/m] [3,2_{z},\bar{1}]
622 [3,2_{z},2_{110}]
6mm [3,2_{z},m_{110}]
[\bar{6}m2] [3,m_{z},m_{110}]
[\bar{6}2m] [3,m_{z},2_{110}]
[6/mmm] [3,2_{z},2_{110},\bar{1}]
23 [2_{z},2_{y},3_{111}]
[m\bar{3}] [2_{z},2_{y},3_{111},\bar{1}]
432 [2_{z},2_{y},3_{111},2_{110}]
[\bar{4}3m] [2_{z},2_{y},3_{111},m_{1\bar{1}0}]
[m\bar{3}m] [2_{z},2_{y},3_{111},2_{110},\bar{1}]

Lattice letter A, B, C, I: [k_{5} = 0] or 1.

Lattice letter R (hexagonal axes): [k_{5} = 0], 1 or 2.

Lattice letter F: [k_{5} = 0] or 1; [k_{6} = 0] or 1.

As a consequence, any translation [\hbox{\sf T}] of [{\cal G}] with translation vector [{\bf t} = k_{2}{\bf a} + k_{3}{\bf b} + k_{4}{\bf c}(+ k_{5}{\bf d} + k_{6}{\bf e})] can be obtained as a product [\hbox{\sf T} = (\hbox{\sf G}_{6})^{k_{6}} * (\hbox{\sf G}_{5})^{k_{5}} * \hbox{\sf G}_{4}^{k_{4}} * \hbox{\sf G}_{3}^{k_{3}} * \hbox{\sf G}_{2}^{k_{2}} * \hbox{\sf G}_{1},] where [k_{2},\ldots,k_{6}] are integers determined by [\hbox{\sf T}]. [\hbox{\sf G}_{6}] and [\hbox{\sf G}_{5}] are enclosed between parentheses because they are effective only in centred lattices.

The remaining generators generate those symmetry operations that are not translations. They are chosen in such a way that only terms [\hbox{\sf G}_{j}] or [\hbox{\sf G}_{j}^{2}] occur. For further specific rules, see below.

The process of generating the entries of the space-group tables may be demonstrated by the example of Table 8.3.5.2[link], where [{\cal G}_{j}] denotes the group generated by [\hbox{\sf G}_{1}, \hbox{\sf G}_{2},\ldots, \hbox{\sf G}_{j}]. For [j \geq 5], the next generator [\hbox{\sf G}_{j+1}] has always been taken as soon as [\hbox{\sf G}_{j}^{k_{j}} \in {\cal G}_{j-1}], because in this case no new symmetry operation would be generated by [\hbox{\sf G}_{j}^{k_{j}}]. The generating process is terminated when there is no further generator. In the present example, [\hbox{\sf G}_{7}] completes the generation: [{\cal G}_{7} \equiv P6_{1}22].

Table 8.3.5.2| top | pdf |
Example of a space-group generation [{\cal G}: {P6_{1}22 \equiv D_{6}^{2}}] (No. 178)

 CoordinatesSymmetry operations
[\hbox{\sf G}_{1}] (1) [x, y, z]; Identity I
[\!\matrix{\hbox{\sf G}_{2}\hfill\cr \hbox{\sf G}_{3}\hfill\cr \hbox{\sf G}_{4}\hfill\cr}] [\!\left.\matrix{t(100)\hfill\cr t(010)\hfill\cr t(001)\hfill\cr}\right\}] [\!\left\{\matrix{\hbox{These are the generating translations.}\hfill\cr {\cal G}_{4} \hbox{ is the group } {\cal T} \hbox{ of all translations}\hfill\cr \hbox{of } P6_{1}22\hfill\cr}\right.]
[ \hbox{\sf G}_{5}] [(2)\ \bar{y},x - y, z + {1 \over 3};] Threefold screw rotation
[ \hbox{\sf G}_{5}^{2}] [(3)\ \bar{x} + y, \bar{x}, z + {2 \over 3};] Threefold screw rotation
[ \hbox{\sf G}_{5}^{3} = t(001):] Now the space group [{\cal G}_{5} \equiv P3_{1}] has been generated  
[ \hbox{\sf G}_{6}] [(4)\ \bar{x},\bar{y}, z + {1 \over 2};] Twofold screw rotation
[ \hbox{\sf G}_{6}* \hbox{\sf G}_{5}] [(5)\ y,\bar{x} + y,z + {5 \over 6};] Sixfold screw rotation
[ \hbox{\sf G}_{6}* \hbox{\sf G}_{5}^{2}] [x - y,x,z + {7 \over 6} \sim (6)\ x - y,x, z + {1 \over 6};] Sixfold screw rotation
[ \hbox{\sf G}_{6}^{2} = t(001):] Now the space group [{\cal G}_{6} \equiv P6_{1}] has been generated  
[ \hbox{\sf G}_{7}] [(7)\ y,x,\bar{z} + {1 \over 3};] Twofold rotation, direction of axis [110]
[ \hbox{\sf G}_{7}* \hbox{\sf G}_{5}] [(8)\ x - y,\bar{y},\bar{z};] Twofold rotation, axis [100]
[ \hbox{\sf G}_{7}* \hbox{\sf G}_{5}^{2}] [\bar{x},\bar{x} + y, \bar{z} - {1 \over 3} \sim (9)\ \bar{x},\bar{x} + y, \bar{z} + {2 \over 3};] Twofold rotation, axis [010]
[ \hbox{\sf G}_{7}* \hbox{\sf G}_{6}] [\bar{y},\bar{x},\bar{z} - {1 \over 6} \sim (10)\ \bar{y},\bar{x},\bar{z} + {5 \over 6};] Twofold rotation, axis [[\bar{1}10]]
[ \hbox{\sf G}_{7}* \hbox{\sf G}_{6}* \hbox{\sf G}_{5}] [\bar{x} + y,y, \bar{z} - {1 \over 2} \sim (11)\ \bar{x} + y, y,\bar{z} + {1 \over 2};] Twofold rotation, axis [120]
[ \hbox{\sf G}_{7}* \hbox{\sf G}_{6}* \hbox{\sf G}_{5}^{2}] [x,x - y, \bar{z} - {5 \over 6} \sim (12)\ x,x - y,\bar{z} + {1 \over 6};] Twofold rotation, axis [210]
[ \hbox{\sf G}_{7}^{2} = \hbox{\sf I}] [{\cal G}_{7} \sim P6_{1}22]  

8.3.5.1. Selected order for non-translational generators

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For the non-translational generators, the following sequence has been adopted:

  • (a) In all centrosymmetric space groups, an inversion (if possible at the origin O) has been selected as the last generator.

  • (b) Rotations precede symmetry operations of the second kind. In crystal classes [\bar{4}2m\hbox{--}\bar{4}m2] and [\bar{6}2m\hbox{--}\bar{6}m2], as an exception, [\bar{4}] and [\bar{6}] are generated first in order to take into account the conventional choice of origin in the fixed points of [\bar{4}] and [\bar{6}].

  • (c) The non-translational generators of space groups with C, A, B, F, I or R symbols are those of the corresponding space group with a P symbol, if possible. For instance, the generators of [I2_{1}2_{1}2_{1}] are those of [P2_{1}2_{1}2_{1}] and the generators of Ibca are those of Pbca, apart from the centring translations.

    Exceptions: I4cm and [I4/mcm] are generated via P4cc and [P4/mcc], because P4cm and [P4/mcm] do not exist. In space groups with d glides (except [I\bar{4}2d]) and also in [I4_{1}/a], the corresponding rotation subgroup has been generated first. The generators of this subgroup are the same as those of the corresponding space group with a lattice symbol P.

    Example

    [F4_{1}/d\bar{3}2/m: P4_{1}32 \rightarrow F4_{1}32 \rightarrow F4_{1}/d\bar{3}2/m].

  • (d) In some cases, rule (c) could not be followed without breaking rule (a), e.g. in Cmme. In such cases, the generators are chosen to correspond to the Hermann–Mauguin symbol as far as possible. For instance, the generators (apart from centring) of Cmme and Imma are those of Pmmb, which is a non-standard setting of Pmma. (Combination of the generators of Pmma with the C- or I-centring translation results in non-standard settings of Cmme and Imma.)

For the space groups with lattice symbol P, the generation procedure has given the same triplets (except for their sequence) as in IT (1952)[link]. In non-P space groups, the triplets listed sometimes differ from those of IT (1952)[link] by a centring translation.

8.3.6. Normalizers of space groups

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The concept of normalizers, well known to mathematicians since the nineteenth century, is finding more and more applications in crystallography. Normalizers play an important role in the general theory of space groups of n-dimensional space. By the so-called Zassenhaus algorithm, one can determine the space-group types of n-dimensional space, provided the arithmetic crystal classes and for each arithmetic crystal class a representative integral [(n \times n)]-matrix group are known. The crucial step is then to determine for these matrix groups their normalizers in GL(n, Z). This was done for [n = 4] by Brown et al. (1978)[link] in the derivation of the 4894 space-group types. Now, the program package Carat solves this problem and was used, among others, for the enumeration of the 28 927 922 affine space-group types for six-dimensional space, see Table 8.1.1.1[link] .

Crystallographers have been applying normalizers in their practical work for some time without realizing this fact, and only in the last decades have they become aware of the importance of normalizers. Normalizers first seem to have been derived visually, see Hirshfeld (1968)[link]. A derivation of the normalizers of the space groups using matrix methods is found in a paper by Boisen et al. (1990)[link].

For the practical application of normalizers in crystallographic problems, see Part 15[link] , which also contains detailed lists of normalizers of the point groups and space groups, as well as of space groups with special metrics. In this section, a short elementary introduction to normalizers will be presented.

In Section 8.1.6[link] , the elements of a space group [{\cal G}] have been divided into classes with respect to the subgroup [{\cal T}] of all translations of [{\cal G}]; these classes have been called cosets. `Coset decomposition' can be performed for any pair `group [{\cal G}] and subgroup [{\cal H}]'. The subgroup [{\cal H}] is called normal if the decomposition of [{\cal G}] into right and left cosets results in the same cosets.

The decomposition of the elements of a group [{\cal G}] into `conjugacy classes' is equally important in crystallography. These classes are defined as follows:

Definition: The elements [\hbox{\sf A}] and [\hbox{\sf B}] of a group [{\cal G}] are said to be conjugate in [{\cal G}], if there exists an element [{\sf G} \in {\cal G}] such that [{\sf B} = {\sf G}^{-1} \hbox{\sf A}{\sf G}].

Example

The symmetry group 4mm of a square consists of the symmetry operations 1, 2, 4, [4^{-1}], [m_{x}], [m_{y}], [m_{d}] and [m_{d'}], see Fig. 8.3.6.1[link]. Vertex 1 is left invariant by 1 and [m_{d'}], vertex 2 by 1 and [m_{d}]. The operations [m_{d}] and [m_{d'}] are conjugate, because [m_{d'} = 4^{-1}m_{d}]4 holds, as are [m_{x}] and [m_{y}] [(m_{y} = 4^{-1}m_{x}4)], or 4 and [4^{-1}\ (4^{-1} =] [m_{x}^{-1}4m_{x})]. The operations 1 and 2 have no conjugates.

[Figure 8.3.6.1]

Figure 8.3.6.1 | top | pdf |

Square with `symmetry elements' [m_{x}], [m_{y}], [m_{d}], [m_{d'}] (mirror lines) and ♦ (fourfold rotation point). The vertices are numbered 1, 2, 3 and 4.

As proved in mathematical textbooks, e.g. Ledermann (1976)[link], conjugacy indeed subdivides a group into classes of elements. The unit element 1 always forms a conjugacy class for itself, as does any element that commutes with every other element of the group. For finite groups, the number of elements in a conjugacy class is a factor of the group order. In infinite groups, such as space groups, conjugacy classes may contain an infinite number of elements.

Conjugacy can be transferred from elements to groups. Let [{\cal H}] be a subgroup of [{\cal G}]. Then another subgroup [{\cal H}'] of [{\cal G}] is said to be conjugate to [{\cal H}] in [{\cal G}], if there exists an element [{\sf G} \in {\cal G}], such that [{\cal H}' = {\sf G}^{-1} {\cal H}{\sf G}]. In this way, the set of all subgroups [{\cal H}_{i}] of a group [{\cal G}] is divided into classes of conjugate subgroups or conjugacy classes of subgroups. Conjugacy classes may contain different numbers of subgroups but, for finite groups, the number of subgroups in each class is always a factor of the order of the group. Conjugacy classes which contain only one subgroup are of special interest; they are called normal subgroups. There are always two trivial normal subgroups of a group [{\cal G}]: the group [{\cal G}] itself and the group [{\cal I}] consisting of the unit element 1 only. For space groups, the group [{\cal T}] of all translations of the group always forms a normal subgroup of [{\cal G}].

In the above-mentioned example of the square, the subgroups [\{1,m_{x}\}] and [\{1,m_{y}\}] as well as [\{1,m_{d}\}] and [\{1,m_{d'}\}] each form conjugate pairs, whereas the subgroups [\{1,2,4,4^{-1}\}], [\{1,2,m_{x},m_{y}\}], [\{1,2,m_{d},m_{d'}\}] and [\{1,2\}] are normal subgroups.

The characterization of normal subgroups, [{\cal H}] must obey the condition [{\cal H} = {\sf G}^{-1}{\cal H}{\sf G}] for all elements [{\sf G} \in {\cal G}], is identical with the one used in Section 8.1.6[link] . This relation can also be expressed as [{\sf G}{\cal H} = {\cal H}{\sf G}] which means that the right and left cosets coincide. Thus each subgroup [{\cal H}] of index [2] is normal, because there is only one coset in addition to [{\cal H}] itself which is then necessarily the right as well as the left coset.

If [{\cal H}] is not a normal subgroup of [{\cal G}], then [{\cal H} = {\sf G}^{-1} {\cal H} {\sf G}] cannot hold for all [{\sf G} \in {\cal G}], because there is at least one subgroup [{\cal H}'] of [{\cal G}] which is conjugate to [{\cal H}]. This situation leads to the introduction of the normalizer [{\cal N}_{{\cal G}}({\cal H})] of a subgroup [{\cal H}] of [{\cal G}].

Definition: The set of all elements [{\sf G} \in {\cal G}], for which [{\sf G}^{-1} {\cal H} {\sf G} = {\cal H}] holds, is called the normalizer [{\cal N}_{{\cal G}}({\cal H})] of [{\cal H}] in [{\cal G}].

The normalizer [{\cal N}_{{\cal G}}({\cal H})] is always a group and thus a subgroup of [{\cal G}] which obeys the relation [{\cal G} \geq {\cal N}_{{\cal G}}({\cal H})  \, {\underline{\triangleright}} \,\,{\cal H}]. The symbol [{\underline{\triangleright}}] means that [{\cal H}] is a normal subgroup of [{\cal N}_{{\cal G}}({\cal H})]. The subgroup [{\cal H}] is normal in [{\cal G}] if its normalizer [{\cal N}_{{\cal G}}({\cal H}) = {\cal G}] coincides with [{\cal G}]. Otherwise, other subgroups conjugate to [{\cal H}] exist. To determine the number of conjugate subgroups of [{\cal H}], one decomposes [{\cal G}] into cosets relative to [{\cal N}_{{\cal G}}({\cal H})]. The elements of each such coset transform [{\cal H}] into a conjugate subgroup [{\cal H}'], such that the number of conjugates (including [{\cal H}] itself) equals the index of [{\cal N}_{{\cal G}}({\cal H})] in [{\cal G}].

Examples

  • (1) The normalizer [{\cal N}_{4mm}(\{1, m_{d}\})] of the subgroup [\{1, m_{d}\}], see Fig. 8.3.6.1[link], consists of the elements [\{1, 2, m_{d}, m_{d'}\}]; its index in the full group 4mm of the square is [2]. Therefore, there are two conjugate subgroups [\{1, m_{d}\}] and [\{1, m_{d'}\}]. All operations of the normalizer map the group [\{1, m_{d}\}] onto itself and thus describe the `symmetry in 4mm of this symmetry group'.

  • (2) Under space-group type [Fm\bar{3}m] (No. 225), four subgroups of type [Pm\bar{3}m] (No. 221) of index [4] are listed; four subgroups of type [Pn\bar{3}m] (No. 224) are similarly listed. Are these subgroups conjugate in the original space group? Clearly, a space group of type [Pm\bar{3}m] cannot be conjugate to one of the different type [Pn\bar{3}m] because there exists no affine mapping transforming one into the other. On the other hand, one verifies that the four subgroups of type [Pm\bar{3}m] are conjugate relative to [Fm\bar{3}m]: the transforming elements are the (centring) translations 000, [{1 \over 2}{1 \over 2}0, {1 \over 2}0{1 \over 2}, 0{1 \over 2}{1 \over 2}]. Similarly, the four subgroups of type [Pn\bar{3}m] are conjugate in [Fm\bar{3}m] by the same (centring) translations. The normalizers [{\cal N}_{Fm\bar{3}m}(Pm\bar{3}m)] and [{\cal N}_{Fm\bar{3}m}(Pn\bar{3}m)] are the subgroups themselves; their indices in [Fm\bar{3}m] are [4], i.e. the numbers of conjugates are four.

Obviously, it would be impractical to list the normalizer for each type of group–subgroup pair. There are, however, some normalizers of outstanding importance from which, moreover, the normalizers determining the usual conjugacy relations can be obtained easily. Since space groups are groups of motions and space-group types are affine equivalence classes of space groups, cf. Sections 8.1.6[link] and 8.2.2[link] , the groups [{\cal E}] of all motions and [{\cal A}] of all affine mappings are groups of special significance for any space group. The normalizers of a space group relative to these two groups are considered now. Part 15[link] contains lists of these normalizers with detailed comments.

The normalizer of a space group [{\cal G}] in the group [{\cal A}] of all affine mappings is called the affine normalizer [{\cal N}\!_{{\cal A}}({\cal G})] of the space group [{\cal G}]. The affine normalizers of space groups of the same space-group type are affinely equivalent. One thus can speak of the `type of normalizers of a space-group type'. In many cases, these normalizers are either space groups or isomorphic to space groups, but they may also be other groups due to arbitrarily small translations (for polar space groups) and/or due to noncrystallographic point groups (for triclinic and monoclinic space groups).

Affine normalizers are of more theoretical interest. For example, they determine the occurrence of enantiomorphism of space groups, cf. Section 8.2.2[link] . A space-group type splits into a pair of enantiomorphic space-group types, if and only if its normalizers are contained in [{\cal A}^{+}], the group of all affine mappings with positive determinant.

The normalizer of a space group [{\cal G}] in the group [{\cal E}] of all motions (Euclidean group) is called the Euclidean normalizer [{\cal N}_{{\cal E}}({\cal G})] of the space group [{\cal G}]. For all trigonal, tetragonal, hexagonal and cubic space groups, [{\cal N}_{\!\cal E} = {\cal N}\!_{{\cal A}}] holds. In these cases, as well as in any context in which statements are valid for both normalizers, the abbreviated form [{\cal N}] is frequently used.

The group [{\cal T}({\cal N})] of all translations of [{\cal N}] is the same for both normalizers, [{\cal N}\!_{{\cal A}}({\cal G})] and [{\cal N}_{{\!\cal E}}({\cal G})], because any translation is a motion. It can be calculated easily: To be an element of [{\cal T}({\cal N})], the translation (I, t) has to satisfy the equation [({\bi I},{\bi t})^{-1} ({\bi W},{\bi w})({\bi I},{\bi t}) = ({\bi W}', {\bi w}') \in {\cal G}] for any operation (W, w) of [{\cal G}]. This results in [({\bi W}, {\bi w} + ({\bi W} - {\bi I}){\bi t}) = ({\bi W}', {\bi w}')] or [{\bi W}' = {\bi W}] and [{\bi w}' = {\bi w} + ({\bi W} - {\bi I}){\bi t}]. From this [({\bi W} - {\bi I}){\bi t} \in {\cal T}({\cal G})] follows, i.e. [({\bi W} - {\bi I}){\bi t}] must be a lattice translation of [{\cal G}]. To determine [{\cal T}({\cal N})], it is sufficient to apply this equation to the [{\bi W}_{i}] of the generators of [{\cal G}].

The conditions for the groups [{\cal T}({\cal N})] are the same for all space groups of the same arithmetic crystal class, because those space groups are generated by symmetry operations with the same matrix parts, and their lattices belong to the same centring type, if referred to conventional coordinate systems. The other elements of the normalizer are not obtained as easily.

In contrast to [{\cal N}\!_{{\cal A}}], the type of [{\cal N}_{{\!\cal E}}] is not a property of the space-group type, as the following example shows. The Euclidean normalizer of a space group P222 is an orthorhombic space group Pmmm if [a \neq b \neq c \neq a]. It is a tetragonal space group if accidentally [a = b] (or [b = c] or [c = a]), and it is even cubic if accidentally [a = b = c]. The listings in Part 15[link] also contain the normalizers for the case of lattices with accidental symmetries.

Acknowledgements

Part 8 is more than other parts of this volume the product of the combined efforts of many people. Most members of the IUCr Commission on International Tables made stimulating suggestions. Norman F. M. Henry, Cambridge, Theo Hahn, Aachen, and Aafje Looijenga-Vos, Groningen, have especially to be mentioned for their tireless efforts to find an intelligible presentation. Joachim Neubüser, Aachen, prepared the first draft for part 1 of the Pilot Issue (1972) under the title Mathematical Introduction to Symmetry. His article is the basis of the present text, to which again he made many valuable comments. J. Neubüser also stimulated the applications of normalizers in crystallography, outlined in Section 8.3.6[link] and Part 15[link] . L. Laurence Boyle, Canterbury, improved the English style and made constructive remarks.

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