Tables for
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2011). Vol. A1, ch. 1.6, p. 44

Section 1.6.2. The symmetry principle in crystal chemistry

Ulrich Müllera*

aFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:

1.6.2. The symmetry principle in crystal chemistry

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The usefulness of symmetry relations is intimately related to the symmetry principle in crystal chemistry. This is an old principle based on experience which has been worded during its long history in rather different ways. Bärnighausen (1980[link]) summarized it in the following way:

  • (1) In crystal structures the arrangement of atoms reveals a pronounced tendency towards the highest possible symmetry.

  • (2) Counteracting factors due to special properties of the atoms or atom aggregates may prevent the attainment of the highest possible symmetry. However, in most cases the deviations from the ideal symmetry are only small (key word: pseudosymmetry).

  • (3) During phase transitions and solid-state reactions which result in products of lower symmetry, the higher symmetry of the starting material is often indirectly preserved by the formation of oriented domains.

Aspect (1) is due to the tendency of atoms of the same kind to occupy equivalent positions, as stated by Brunner (1971[link]). This has physical reasons: depending on chemical composition, the kind of chemical bonding, electron configuration of the atoms, relative sizes of the atoms, pressure, temperature etc., there exists one energetically most favourable surrounding for atoms of a given species which all of these atoms strive to attain.

Aspect (2) of the symmetry principle is exploited in the following sections. Factors that counteract the attainment of the highest symmetry include: (1) stereochemically active lone electron pairs; (2) Jahn–Teller distortions; (3) covalent bonds; (4) Peierls distortions; (5) ordered occupation of originally equivalent sites by different atomic species (substitution derivatives); (6) partial occupation of voids in a packing of atoms; (7) partial vacation of atomic positions; (8) freezing (condensation) of lattice vibrations (soft modes) giving rise to phase transitions; and (9) ordering of atoms in a disordered structure.

Aspect (3) of the symmetry principle has its origin in an observation by Bernal (1938[link]). He noted that in the solid state reaction [{\rm Mn(OH)}_2 \rightarrow{\rm MnOOH}\rightarrow{\rm MnO}_2] the starting and the product crystal had the same orientation. Such reactions are called topotactic reactions after Lotgering (1959[link]) (for a more exact definition see Giovanoli & Leuenberger, 1969[link]). In a paper by Bernal & Mackay (1965[link]) we find the sentence: `One of the controlling factors of topotactic reactions is, of course, symmetry. This can be treated at various levels of sophistication, ranging from Lyubarskii's to ours, where we find that the simple concept of Buridan's ass illumines most cases.' According to the metaphor of Buridan (French philosopher, died circa 1358), the ass starves to death between two equal and equidistant bundles of hay because it cannot decide between them. Referred to crystals, such an asinine behaviour would correspond to an absence of phase transitions or solid-state reactions if there are two or more energetically equivalent orientations of the domains of the product. Crystals, of course, do not behave like the ass; they take both.


Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Commun. Math. Chem. 9, 139–175.
Bernal, J. D. (1938). Conduction in solids and diffusion and chemical change in solids. Geometrical factors in reactions involving solids. Trans. Faraday Soc. 34, 834–839.
Bernal, J. D. & Mackay, A. L. (1965). Topotaxy. Tschermaks mineralog. petrogr. Mitt., Reihe 3 (now Mineral. Petrol.), 10, 331–340.
Brunner, G. O. (1971). An unconventional view of the `closest sphere packings'. Acta Cryst. A27, 388–390.
Giovanoli, D. & Leuenberger, U. (1969). Über die Oxidation von Manganoxidhydroxid. Helv. Chim. Acta, 52, 2333–2347.
Lotgering, F. K. (1959). Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures – I. J. Inorg. Nucl. Chem. 9, 113–123.

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