Space-group types

Wondratschek, H.

doi:10.1107/97809553602060000515

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. 8.2, pp. 726-727

Section 8.2.2. Space-group types

H. Wondratschek^a ^*

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

8.2.2. Space-group types

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The finest commonly used classification of three-dimensional space groups, i.e. the one resulting in the highest number of classes, is the classification into the 230 (crystallographic) space-group types.¹ The word `type' is preferred here to the word `class', since in crystallography `class' is already used in the sense of `crystal class', cf. Sections 8.2.3 and 8.2.4. The classification of space groups into space-group types reveals the common symmetry properties of all space groups belonging to one type. Such common properties of the space groups can be considered as `properties of the space-group types'.

The practising crystallographer usually assumes the 230 space-group types to be known and to be described in this volume by representative data such as figures and tables. To the experimentally determined space group of a particular crystal structure, e.g. of pyrite FeS₂, the corresponding space-group type No. 205 $[(Pa\bar{3} \equiv T_{h}^{6})]$ of International Tables is assigned. Two space groups, e.g. those of FeS₂ and CO₂, belong to the same space-group type if their symmetries correspond to the same entry in International Tables.

The rigorous definition of the classification of space groups into space-group types can be given in a more geometric or a more algebraic way. Here matrix algebra will be followed, by which primarily the classification into the 219 so-called affine space-group types is obtained.² For this classification, each space group is referred to a primitive basis and an origin. In this case, the matrices $[{\bi W}_{j}]$ of the symmetry operations consist of integral coefficients and $[\det ({\bi W}_{j}) = \pm 1]$ holds. Two space groups $[{\cal G}]$ and $[{\cal G}']$ are then represented by their $[(n + 1) \times (n + 1)]$ matrix groups $[\specialfonts\{{\bbsf W} \}]$ and $[\specialfonts\{{\bbsf W} '\}]$ . These two matrix groups are now compared.

Definition: The space groups $[{\cal G}]$ and $[{\cal G}']$ belong to the same space-group type if, for each primitive basis and each origin of $[{\cal G}]$ , a primitive basis and an origin of $[{\cal G}']$ can be found so that the matrix groups $[\specialfonts\{{\bbsf W}\}]$ and $[\specialfonts\{{\bbsf W}'\}]$ are identical. In terms of matrices, this can be expressed by the following definition:

Definition: The space groups $[{\cal G}]$ and $[{\cal G}']$ belong to the same space-group type if an $[(n + 1) \times (n + 1)]$ matrix $[\specialfonts{\bbsf P}]$ exists, for which the matrix part P is an integral matrix with $[\det\! ({\bi P}) = \pm 1]$ and the column part p consists of real numbers, such that $[\specialfonts\{{\bbsf W}'\} = {\bbsf P}^{-1} \{{\bbsf W} \} {\bbsf P} \eqno(8.2.2.1)]$ holds. The matrix part P of $[\specialfonts{\bbsf P}]$ describes the transition from the primitive basis of $[{\cal G}]$ to the primitive basis of $[{\cal G}']$ . The column part p of $[\specialfonts{\bbsf P}]$ expresses the (possibly) different origin choices for the descriptions of $[{\cal G}]$ and $[{\cal G}']$ .

Equation (8.2.2.1) is an equivalence relation for space groups. The corresponding classes are called affine space-group types. By this definition, one obtains 17 plane-group types for $[E^{2}]$ and 219 space-group types for $[E^{3}]$ , see Fig. 8.2.1.1. Listed in the space-group tables are 17 plane-group types for $[E^{2}]$ and 230 space-group types for $[E^{3}]$ . Obviously, the equivalence definition of the space-group tables differs somewhat from the one used above. In practical crystallography, one wants to distinguish between right- and left-handed screws and does not want to change from a right-handed to a left-handed coordinate system. In order to avoid such transformations, the matrix P of equation (8.2.2.1) is restricted by the additional condition $[\det\! ({\bi P}) = + 1]$ . Using matrices $[\specialfonts{\bbsf P}]$ with $[\det\! ({\bi P}) = + 1]$ only, 11 space-group types of $[E^{3}]$ split into pairs, which are the so-called pairs of enantiomorphic space-group types. The Hermann–Mauguin and Schoenflies symbols (in parentheses) of the pairs of enantiomorphic space-group types are $[P4_{1}\hbox{--}P4_{3}]$ $[ (C_{4}^{2}\hbox{--} C_{4}^{4})]$ , $[P4_{1}22\hbox{--} P4_{3}22]$ $[(D_{4}^{3}\hbox{--} D_{4}^{7})]$ , $[P4_{1}2_{1}2\hbox{--} P4_{3}2_{1}2]$ $[(D_{4}^{4}\hbox{--} D_{4}^{8})]$ , $[P3_{1}\hbox{--} P3_{2}]$ $[(C_{3}^{2}\hbox{--} C_{3}^{3})]$ , $[P3_{1}21\hbox{--} P3_{2}21]$ $[(D_{3}^{4}\hbox{--} D_{3}^{6})]$ , $[P3_{1}12\hbox{--} P3_{2}12]$ $[(D_{3}^{3}\hbox{--} D_{3}^{5})]$ , $[P6_{1}\hbox{--} P6_{5}]$ $[(C_{6}^{2}\hbox{--} C_{6}^{3})]$ , $[P6_{2}\hbox{--} P6_{4}]$ $[(C_{6}^{4}\hbox{--} C_{6}^{5})]$ , $[P6_{1}22\hbox{--} P6_{5}22]$ $[(D_{6}^{2}\hbox{--} D_{6}^{3})]$ , $[P6_{2}22\hbox{--} P6_{4}22]$ $[(D_{6}^{4}\hbox{--} D_{6}^{5})]$ and $[P4_{1}32\hbox{--} P4_{3}32]$ $[(O^{7}\hbox{--} O^{6})]$ . In order to distinguish between the two definitions of space-group types, the first is called the classification into the 219 affine space-group types and the second the classification into the 230 crystallographic or positive affine or proper affine space-group types, see Fig. 8.2.1.1. Both classifications are useful.

In Section 8.1.6 , symmorphic space groups were defined. It can be shown (with either definition of space-group type) that all space groups of a space-group type are symmorphic if one of these space groups is symmorphic. Therefore, it is also possible to speak of types of symmorphic and non-symmorphic space groups. In $[E^{3}]$ , symmorphic space groups do not occur in enantiomorphic pairs. This does happen, however, in $[E^{4}]$ .

The so-called space-group symbols are really symbols of `crystallographic space-group types'. There are several different kinds of symbols (for details see Part 12 ). The numbers denoting the crystallographic space-group types and the Schoenflies symbols are unambiguous but contain little information. The Hermann–Mauguin symbols depend on the choice of the coordinate system but they are much more informative than the other notations.

References

International Tables for Crystallography (2006). Vol. A. ch. 8.2, pp. 726-727