Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 9.7, pp. 904-906

Section Relation to sphere packing

A. J. C. Wilson,a V. L. Karenb and A. Mighellb

aSt John's College, Cambridge CB2 1TP, England, and bNIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Relation to sphere packing

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The effect of molecular symmetry cannot be ignored in overall statistical surveys as well as in structure prediction. However, in most structures, the molecular symmetry is low or it is not used in the packing. In about 90% of crystalline compounds, the molecules crystallize in low-symmetry space groups, so that a given molecule has a 12-point contact with neighbouring molecules. As 12 corresponds to the number of nearest neighbours in cubic and hexagonal closest packing of spheres, the periodic assembly of most molecular structures can be regarded as the closest packing of distorted spheres, where symmetry ensures the interlocking of complex shapes (Gavezzotti, 1994[link]).

For the relatively infrequent cases where high molecular symmetry is reflected in high crystal symmetry, the packing of molecules can be derived from the appropriate, though not necessarily the densest, packing of spheres. For example, ten-point, eight-point and six-point molecular contacts can be achieved, respectively, by tetragonal close packing (I4/mmm), by I-centred cubic packing ([Im\overline {3}m]), and by primitive cubic packing ([Pm\overline {3}m]). For a review and some derivations of the densest packing of equal spheres, see Chapter 9.1[link] and Patterson & Kasper (1959[link]), Coutanceau Clarke (1972[link]), and Smith (1973[link]); and for packing of clusters of unequal spheres, see Williams (1987[link]).

With spheres having infinite point symmetry [{\cal K}_{\infty h}], every sphere can be located on syntropic symmetry elements at special positions with high symmetry up to the symmetry of the lattice. The lattice translations, pertinent to the fully symmorphic space group, are then able to generate the entire crystal structure. When spheres are deformed, symmetry is removed and the non-lattice translations involved with antimorphic space groups (which must be subgroups of the sphere-packing groups) become necessary to ensure space filling with the repetitive patterns of complex molecular shapes. In a similar manner, other objects with infinite elements of symmetry (e.g. rods; Lidin, Jacob & Andersson, 1995[link]) can be subjected to a rigorous analysis of close packing.


Coutanceau Clarke, J. A. R. (1972). New periodic close packings of identical spheres. Nature (London), 240, 408–410.
Gavezzotti, A. (1994). Molecular packing and correlations between molecular and crystal properties. Structure correlation, Vol. 2, edited by H.-B. Bürgi & J. D. Dunitz, Chap. 12, pp. 509–542. Weinheim/New York/Basel/Cambridge/Tokyo: VCH Publishers.
Lidin, S., Jacob, M. & Andersson, S. (1995). A mathematical analysis of rod packings. J. Solid State Chem. 114, 36–41.
Patterson, A. L. & Kasper, J. S. (1959). Close packing. International tables for X-ray crystallography, Vol. II, Mathematical tables, pp. 342–354. Birmingham: Kynoch Press.
Smith, A. J. (1973). Periodic close packings of identical spheres. Nature (London) Phys. Sci. 246(149), 10–11.
Williams, D. E. G. (1987). Close packing of spheres. J. Chem. Phys. 87, 4207–4210.

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