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

International Tables for Crystallography (2006). Vol. A, ch. 9.1, pp. 743-745

Section 9.1.6. Classifications

H. Burzlaffa and H. Zimmermannb*

aUniversität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@knot.uni-erlangen.de

9.1.6. Classifications

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By means of the above-mentioned lattice properties, it is possible to classify lattices according to various criteria. Lattices can be subdivided with respect to their topological types of domains, resulting in two classes in two dimensions and five classes in three dimensions. They are called Voronoi types (see Table 9.1.6.1[link]). If the classification involves topological and symmetry properties of the domains, 24 Symmetrische Sorten (Delaunay, 1933[link]) are obtained in three dimensions and 5 in two dimensions. Other classifications consider either the centring type or the point group of the lattice.

Table 9.1.6.1| top | pdf |
Representations of the five types of Voronoi polyhedra

VIVIIVIIIVIVVV
[Scheme scheme20] [Scheme scheme21] [Scheme scheme22] [Scheme scheme23] [Scheme scheme24]

The most important classification takes into account both the lattice point-group symmetry and the centring mode (Bravais, 1866[link]). The resulting classes are called Bravais types of lattices or, for short, Bravais lattices. Two lattices belong to the same Bravais type if and only if they coincide both in their point-group symmetry and in the centring mode of their conventional cells. The Bravais lattice characterizes the translational subgroup of a space group. The number of Bravais lattices is 1 in one dimension, 5 in two dimensions, 14 in three dimensions and 64 in four dimensions. The Bravais lattices may be derived by topological (Delaunay, 1933[link]) or algebraic procedures (Burckhardt, 1966[link]; Neubüser et al., 1971[link]). It can be shown (Wondratschek et al., 1971[link]) that `all Bravais types of the same crystal family can be obtained from each other by the process of centring'. As a consequence, different Bravais types of the same [crystal] family (cf. Section 8.1.4[link] ) differ in their centring mode. Thus, the Bravais types may be described by a lower-case letter designating the crystal family and an upper-case letter designating the centring mode. The relations between the point groups of the lattices and the crystal families are shown in Table 9.1.4.1[link]. Since the hexagonal and rhombohedral Bravais types belong to the same crystal family, the rhombohedral lattice is described by hR, h indicating the family and R the centring type. This nomenclature was adopted for the 1969 reprint (IT 1969[link]) of IT (1952)[link] and for Structure Reports since 1975 (cf. Trotter, 1975[link]).

References

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]
International Tables for X-ray Crystallography (1969). Vol. I, 3rd ed., edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1969).]
Bravais, A. (1866). Etudes cristallographiques. Paris.
Burckhardt, J. (1966). Die Bewegungsgruppen der Kristallographie, 2nd ed., pp. 82–89. Basel: Birkhäuser.
Delaunay, B. N. (1933). Neuere Darstellung der geometrischen Kristallographie. Z. Kristallogr. 84, 109–149.
Neubüser, J., Wondratschek, H. & Bülow, R. (1971). On crystallography in higher dimensions. I. General definitions. Acta Cryst. A27, 517–520.
Trotter, J. (1975). Editor. Structure reports. 60-year structure index 1913–1973. A. Metals and inorganic compounds. Utrecht: Bohn, Scheltema & Holkema. [Now available from Kluwer Academic Publishers, Dordrecht, The Netherlands.]
Wondratschek, H., Bülow, R. & Neubüser, J. (1971). On crystallography in higher dimensions. III. Results in R4. Acta Cryst. A27, 523–535.








































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