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

International Tables for Crystallography (2006). Vol. A, ch. 2.1, pp. 14-16

Section 2.1.3. Conventional coordinate systems and cells

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:

2.1.3. Conventional coordinate systems and cells

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A plane group or space group usually is described by means of a crystallographic coordinate system, consisting of a crystallographic basis (basis vectors are lattice vectors) and a crystallographic origin (origin at a centre of symmetry or at a point of high site symmetry). The choice of such a coordinate system is not mandatory since in principle a crystal structure can be referred to any coordinate system; cf. Section 8.1.4[link] .

The selection of a crystallographic coordinate system is not unique. Conventionally, a right-handed set of basis vectors is taken such that the symmetry of the plane or space group is displayed best. With this convention, which is followed in the present volume, the specific restrictions imposed on the cell parameters by each crystal family become particularly simple. They are listed in columns 6 and 7 of Table[link]. If within these restrictions the smallest cell is chosen, a conventional (crystallographic) basis results. Together with the selection of an appropriate conventional (crystallographic) origin (cf. Sections 2.2.2[link] and 2.2.7[link] ), such a basis defines a conventional (crystallographic) coordinate system and a conventional cell. The conventional cell of a point lattice or a space group, obtained in this way, turns out to be either primitive or to exhibit one of the centring types listed in Chapter 1.2[link] . The centring type of a conventional cell is transferred to the lattice which is described by this cell; hence, we speak of primitive, face-centred, body-centred etc. lattices. Similarly, the cell parameters are often called lattice parameters; cf. Section 8.3.1[link] and Chapter 9.1[link] for further details.

In the triclinic, monoclinic and orthorhombic crystal systems, additional conventions (for instance cell reduction or metrical conventions based on the lengths of the cell edges) are needed to determine the choice and the labelling of the axes. Reduced bases are treated in Chapters 9.1[link] and 9.2[link] , orthorhombic settings in Section[link] , and monoclinic settings and cell choices in Section 2.2.16[link] .

In this volume, all space groups within a crystal family are referred to the same kind of conventional coordinate system with the exception of the hexagonal crystal family in three dimensions. Here, two kinds of coordinate systems are used, the hexagonal and the rhombohedral systems. In accordance with common crystallographic practice, all space groups based on the hexagonal Bravais lattice hP (18 trigonal and 27 hexagonal space groups) are described only with a hexagonal coordinate system (primitive cell),2 whereas the seven space groups based on the rhombohedral Bravais lattice hR are treated in two versions, one referred to `hexagonal axes' (triple obverse cell) and one to `rhombohedral axes' (primitive cell); cf. Chapter 1.2[link] . In practice, hexagonal axes are preferred because they are easier to visualize.

Note: For convenience, the relations between the cell parameters a, c of the triple hexagonal cell and the cell parameters [a'], [\alpha'] of the primitive rhombohedral cell (cf. Table[link]) are listed:[a = a' \sqrt{2} \sqrt{1 - \cos \alpha'} = 2a' \sin {\alpha' \over 2}] [\eqalign{c &= a' \sqrt{3} \sqrt{1 + 2 \cos \alpha'}\phantom{\Big(_{2_{2\over2}}}\cr {c \over a} &= \sqrt{{3 \over 2}} \sqrt{{1 + 2 \cos \alpha' \over 1 - \cos \alpha'}} = \sqrt{{9 \over 4 \sin^{2} (\alpha'/2)} - 3}\phantom{\Big(_{2_{2\over2}}}\cr a' &= {\textstyle{1 \over 3}} \sqrt{3a^{2} + c^{2}}\phantom{\Big(_{2\over2}^{2\over2}}\cr \sin {\alpha' \over 2} &= {3 \over 2\sqrt{3 + (c^{2}/a^{2})}} \hbox{ or } \cos \alpha' = {(c^{2}/a^{2}) - {3 \over 2} \over (c^{2}/a^{2}) + 3}.\phantom{\bigg)^{2\over2}}}]

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