Subgroups and supergroups of space groups

Wondratschek, H.

doi:10.1107/97809553602060000516

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

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International Tables for Crystallography (2006). Vol. A. ch. 8.3, pp. 734-736

Section 8.3.3. Subgroups and supergroups of space groups

H. Wondratschek^a ^*

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

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), 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 . 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.

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).

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 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) and (ii), e.g. by reducing the order of the point group and by thinning out the lattice of translations.

Subgroups of the first kind (i) are called translationengleiche or t subgroups⁷ because the set $[{\cal T}]$ of all (pure) translations is retained. In case (ii), 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), 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) of subgroups is more difficult to survey than kinds (i) and (ii). Fortunately, a theorem of Hermann states that the maximal subgroups of a space group $[{\cal G}]$ are of type (i) or (ii).

Theorem of Hermann (1929). 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) can never occur among the maximal subgroups. They can, however, be derived by a stepwise process of linking maximal subgroups of types (i) and (ii), 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 . The possible t subgroups were first listed by Hermann (1935); corrections have been reported by Ascher et al. (1969).

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 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 .

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, 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 . 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). 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.

References

International Tables for Crystallography (2004). Vol. A1, Symmetry relations between space groups, edited by H. Wondratschek & U. Müller. Dordrecht: Kluwer Academic Publishers.Google Scholar

Ascher, E., Gramlich, V. & Wondratschek, H. (1969). Korrekturen zu den Angaben `Untergruppen' in den Raumgruppen der Internationalen Tabellen zur Bestimmung von Kristallstrukturen (1935), Band 1. Acta Cryst. B25, 2154–2156.Google Scholar

Hermann, C. (1929). Zur systematischen Strukturtheorie. IV. Untergruppen. Z. Kristallogr. 69, 533–555.Google Scholar

Hermann, C. (1935). Internationale Tabellen zur Bestimmung von Kristallstrukturen, Band 1. Berlin: Borntraeger.Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 8.3, pp. 734-736