International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 9.1, pp. 742743
Section 9.1.4. Special bases for lattices^{a}Universität Erlangen–Nürnberg, RobertKochStrasse 4a, D91080 Uttenreuth, Germany, and ^{b}Institut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D91054 Erlangen, Germany 
Different procedures are in use to select specific bases of lattices. The reduction procedures employ metrical properties to develop a sequence of basis transformations which lead to a reduced basis and reduced cell (see Chapter 9.3 ).
Another possibility is to make use of the symmetry properties of lattices. This procedure, with the aid of standardization rules, leads to the conventional crystallographic basis and cell. In addition to translational symmetry, a lattice possesses pointgroup symmetry. No crystal can have higher pointgroup symmetry than the point group of its lattice, which is called holohedry. The seven point groups of lattices in three dimensions and the four in two dimensions form the basis for the classification of lattices (Table 9.1.4.1). It may be shown by an algebraic approach (Burckhardt, 1966) or a topological one (Delaunay, 1933) that the arrangement of the symmetry elements with respect to the lattice vectors is not arbitrary but well determined. Taking as basis vectors lattice vectors along important symmetry directions and choosing the origin in a lattice point simplifies the description of the lattice symmetry operations (cf. Chapter 12.1 ). Note that such a basis is not necessarily a (primitive) basis of the lattice (see below). The choice of a basis controlled by symmetry is not always unique; in the monoclinic system, for example, one vector can be taken parallel to the symmetry direction but the other two vectors, perpendicular to it, are not uniquely determined by symmetry.
^{†}The symbols for crystal families were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985).

The choice of conventions for standardizing the setting of a lattice depends on the purpose for which it is used. The several sets of conventions rest on two conflicting principles: symmetry considerations and metric considerations. The following rules (i) to (vii) defining a conventional basis are taken from Donnay (1943; Donnay & Ondik, 1973); they deal with the conventions based on symmetry:
The metric parameters of the conventional basis are called lattice parameters. For the purpose of identification, additional metric rules have to be employed to make the labelling unique; they can be found in the introduction to Crystal Data (Donnay & Ondik, 1973).
When the above rules have been applied, it may occur that not all lattice points can be described by integral coordinates. In such cases, the unit cell contains two, three or four lattice points. The additional points may be regarded as centrings of the conventional cell. They have simple rational coordinates. For a conventional basis, the number of lattice points per cell is 1, 2, 3 or 4 (see Tables 9.1.7.1 and 9.1.7.2).
In two dimensions, only two centring types are needed:
In three dimensions, the following centring types are used: [see rule (iiib) above].
In orthorhombic and monoclinic lattices, some differently centred cells can be transformed into each other without violating the symmetry conditions for the choice of the basis vectors. In these cases, the different centred cells belong to the same centring mode. In the orthorhombic case, the three types of onefacecentred cells belong to the same centring mode because the symbol of the cell depends on the labelling of the basis vectors; C is usually preferred to A and B in the standard setting; the centring mode is designated S (seitenflächenzentriert). In the monoclinic case (bunique setting), A, I and C can be transformed into each other without changing the symmetry direction. C is used for the standard setting (cf. Section 2.2.3 ); it represents the centring mode S. The vectors a, c are conventionally chosen as short as the Ccentring allows so that they need not be the shortest two vectors in their net plane and need not fulfil the inequalities (9.1.4.1).
In some situations, the Icentring of the monoclinic conventional cell may be more advantageous. If the vectors a, c are the shortest ones leading to the centring I, they obey the inequalities (9.1.4.1).
References
International Tables for Xray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]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.
Donnay, J. D. H. (1943). Rules for the conventional orientation of crystals. Am. Mineral. 28, 313–328, 470.
Donnay, J. D. H. & Ondik, H. M. (1973). Editors. Crystal data, Vol. 2, 3rd ed. Introduction, p. 2. Washington: National Bureau of Standards.