Tables for
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.6, p. 111

Section Restrictions on space groups

U. Shmuelia and H. D. Flackb Restrictions on space groups

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The values of certain chemical and physical properties of a bulk compound, or its crystals, have implications for the assignment of the space group of a crystal structure. In the chemical domain, notably in proteins and small-molecule natural products, information concerning the enantiomeric purity of the bulk compound or of its individual crystals is most useful. Further, all physical properties of a crystal are limited by the point group of the crystal structure in ways that depend on the individual nature of the physical property.

It is very well established that the crystal structure of an enantiomerically pure compound will be chiral (see Flack, 2003[link]). By an enantiomerically pure compound one means a compound whose molecules are all chiral and all these molecules possess the same chirality. The space group of a chiral crystal structure will only contain the following types of symmetry operation: translations, pure rotations and screw rotations. Inversion in a point, mirror reflection or rotoinversion do not occur in the space group of a chiral crystal structure. Taking all this together means that the crystal structure of an enantiomerically pure compound will show one of 65 space groups (known as the Sohncke space groups), all noncentrosymmetric, containing only translations, rotations and screw rotations. As a consequence, the point group of a chiral crystal structure is limited to the 11 point groups containing only pure rotations (i.e. 1, 2, 222, 4, 422, 3, 32, 6, 622, 23 and 432). Particular attention must be paid as to whether a measurement of enantiomeric purity of a compound applies to the bulk material or to the single crystal used for the diffraction experiment. Clearly, a compound whose bulk is enantiomerically pure will produce crystals which are enantiomerically pure. The converse is not necessarily true (i.e. enantiomerically pure crystals do not necessarily come from an enantiomerically pure bulk). For example, a bulk compound which is a racemate (i.e. an enantiomeric mixture containing 50% each of the opposite enantiomers) may produce either (a) crystals of the racemic compound (i.e. crystals containing 50% each of the opposite enantiomers) or (b) a racemic conglomerate (i.e. a mixture of enantiomerically pure crystals in a proportion of 50% of each pure enantiomer) or (c) some other rarer crystallization modes. Consequently, as part of a single-crystal structure analysis, it is highly recommended to make a measurement of the enantiomeric purity of the single crystal used for the diffraction experiment.

Much information on methods of establishing the enantiomeric purity of a compound can be found in a special issue of Chirality devoted to the determination of absolute configuration (Allenmark et al., 2007[link]). Measurements in the fluid state of optical activity, optical rotatory dispersion (ORD), circular dichroism (CD) and enantioselective chromatography are of prime importance. Many of these are sufficiently sensitive to be applicable not only to the bulk compound but also to the single crystal used for the diffraction experiment taken into solution. CD may also be applied in the solid state.

Many physical properties of a crystalline solid are anisotropic and the symmetry of a physical property of a crystal is limited both by the point-group symmetry of the crystal and by symmetries inherent to the physical property under study. For further information on this topic see Part 1[link] of Volume D (Authier et al., 2014[link]). Unfortunately, many of these physical properties are intrinsically centrosymmetric, so few of them are of use in distinguishing between the subgroups of a Laue group, a common problem in space-group determination. In Chapter 3.2 of the present volume, Hahn & Klapper show to which point groups a crystal must belong to be capable of displaying some of the principal physical properties of crystals (Table[link] ). Measurement of morphology, pyroelectricity, piezoelectricity, second harmonic generation and optical activity of a crystalline sample can be of use.


Allenmark, S., Gawronski, J. & Berova, N. (2007). Editors. Chirality, 20, 605–759.
Authier, A., Borovik-Romanov, A. S., Boulanger, B., Cox, K. G., Dmitrienko, V. E., Ephraïm, M., Glazer, A. M., Grimmer, H., Janner, A., Jannsen, T., Kenzelmann, M., Kirfel, A., Kuhs, W. F., Küppers, H., Mahan, G. D., Ovchinnikova, E. N., Thiers, A., Zarembowitch, A. & Zyss, J. (2014). International Tables for Crystallography, Volume D, Physical Properties of Crystals, edited by A. Authier, 2nd edition, Part 1. Chichester: Wiley.
Flack, H. D. (2003). Chiral and achiral crystal structures. Helv. Chim. Acta, 86, 905–921.

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