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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 1.2, p. 4
https://doi.org/10.1107/97809553602060000501

Chapter 1.2. Printed symbols for conventional centring types

Th. Hahna*

a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

This chapter lists the printed symbols used for the centring types of lattices and cells throughout this volume. The list is accompanied by notes and cross-references to recent IUCr nomenclature reports.

Keywords: symbols; centred cell; centred lattice; crystallography.

1.2.1. Printed symbols for the conventional centring types of one-, two- and three-dimensional cells

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For `reflection conditions', see Tables 2.2.13.1[link] and 2.2.13.3[link] . For the new centring symbol S, see Note (iii) below.

Printed symbol Centring type of cell Number of lattice points per cell Coordinates of lattice points within cell
One dimension
[{\scr p}] Primitive 1 0
Two dimensions
p Primitive 1 0, 0
c Centred 2 0, 0; [{1 \over 2}], [{1 \over 2}]
h † Hexagonally centred 3 0, 0; [{2 \over 3}], [{1 \over 3}]; [{1 \over 3}], [{2 \over 3}]
Three dimensions
P Primitive 1 0, 0, 0
C C -face centred 2 0, 0, 0; [{1 \over 2}], [{1 \over 2}], 0
A A -face centred 2 0, 0, 0; 0, [{1 \over 2}], [{1 \over 2}]
B B -face centred 2 0, 0, 0; [{1 \over 2}], 0, [{1 \over 2}]
I Body centred 2 0, 0, 0; [{1 \over 2}], [{1 \over 2}], [{1 \over 2}]
F All-face centred 4 0, 0, 0; [{1 \over 2}], [{1 \over 2}], 0; 0, [{1 \over 2}], [{1 \over 2}]; [{1 \over 2}], 0, [{1 \over 2}]
R ‡ [\cases{\hbox{Rhombohedrally centred}\cr \hbox{(description with `hexagonal axes')}\cr \hbox{Primitive}\cr \hbox{(description with `rhombohedral axes')}\cr}] 3 [\!\openup1pt{\cases {0,{\hbox to 1pt{}} 0,{\hbox to 1pt{}} 0{\hbox{; }}{\hbox to 2pt{}}{2 \over 3},{\hbox to -1.5pt{}} {1 \over 3},{\hbox to -1.5pt{}} {1 \over 3}{\hbox{; }}{\hbox to 1pt{}}{1 \over 3},{\hbox to -1pt{}} {2 \over 3},{\hbox to -1pt{}} {2 \over 3} \hbox{ (`obverse setting')}\cr 0,{\hbox to 1pt{}} 0,{\hbox to 1pt{}} 0{\hbox{; }}{\hbox to 1.5pt{}}{1 \over 3},{\hbox to -1pt{}} {2 \over 3},{\hbox to -1.5pt{}} {1 \over 3}{\hbox{; }}{\hbox to 1pt{}}{2 \over 3}, {1 \over 3},{\hbox to -1.5pt{}} {2 \over 3} \hbox{ (`reverse setting')}\cr}}]
1 0, 0, 0
H § Hexagonally centred 3 0, 0, 0; [{2 \over 3}], [{1 \over 3}], 0; [{1 \over 3}], [{2 \over 3}], 0
† The two-dimensional triple hexagonal cell h is an alternative description of the hexagonal plane net, as illustrated in Fig. 5.1.3.8[link] . It is not used for systematic plane-group description in this volume; it is introduced, however, in the sub- and supergroup entries of the plane-group tables (Part 6[link] ). Plane-group symbols for the h cell are listed in Chapter 4.2[link] . Transformation matrices are contained in Table 5.1.3.1[link] .
‡In the space-group tables (Part 7[link] ), as well as in IT (1935)[link] and IT (1952)[link], the seven rhombohedral R space groups are presented with two descriptions, one based on hexagonal axes (triple cell), one on rhombohedral axes (primitive cell). In the present volume, as well as in IT (1952)[link], the obverse setting of the triple hexagonal cell R is used. Note that in IT (1935)[link] the reverse setting was employed. The two settings are related by a rotation of the hexagonal cell with respect to the rhombohedral lattice around a threefold axis, involving a rotation angle of 60°, 180° or 300° (cf. Fig. 5.1.3.6[link] ). Further details may be found in Chapter 2.1[link] , Section 4.3.5[link] and Chapter 9.1[link] . Transformation matrices are contained in Table 5.1.3.1[link] .
§The triple hexagonal cell H is an alternative description of the hexagonal Bravais lattice, as illustrated in Fig. 5.1.3.8[link] . It was used for systematic space-group description in IT (1935)[link], but replaced by P in IT (1952)[link]. In the space-group tables of this volume (Part 7[link] ), it is only used in the sub- and supergroup entries (cf. Section 2.2.15[link] ). Space-group symbols for the H cell are listed in Section 4.3.5[link] . Transformation matrices are contained in Table 5.1.3.1[link] .

1.2.2. Notes on centred cells

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  • (i) The centring type of a cell may change with a change of the basis vectors; in particular, a primitive cell may become a centred cell and vice versa. Examples of relevant transformation matrices are contained in Table 5.1.3.1[link] .

  • (ii) Section 1.2.1[link] contains only those conventional centring symbols which occur in the Hermann–Mauguin space-group symbols. There exist, of course, further kinds of centred cells which are unconventional; an interesting example is provided by the triple rhombohedral D cell, described in Section 4.3.5.3[link] .

  • (iii) For the use of the letter S as a new general, setting-independent `centring symbol' for monoclinic and orthorhombic Bravais lattices see Chapter 2.1[link] , especially Table 2.1.2.1[link] , and de Wolff et al. (1985[link]).

  • (iv) Symbols for crystal families and Bravais lattices in one, two and three dimensions are listed in Table 2.1.2.1[link] and are explained in the Nomenclature Report by de Wolff et al. (1985[link]).

References

First citation Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). I. Band, edited by C. Hermann. Berlin: Borntraeger. [Reprint with corrections: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]Google Scholar
 First citation International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]Google Scholar
 First citation Wolff, P. M. de, Belov, N. V., Bertaut, E. F., Buerger, M. J., Donnay, J. D. H., Fischer, W., Hahn, Th., Koptsik, V. A., Mackay, A. L., Wondratschek, H., Wilson, A. J. C. & Abrahams, S. C. (1985). Nomenclature for crystal families, Bravais-lattice types and arithmetic classes. Report of the International Union of Crystallography Ad-hoc Committee on the Nomenclature of Symmetry. Acta Cryst. A41, 278–280.Google Scholar








































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