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

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

Chapter 1.2. Printed symbols for conventional centring types

Th. Hahna*

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail:

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.

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

| top | pdf |

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

Printed symbolCentring type of cellNumber of lattice points per cellCoordinates 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.[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[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.[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[link] .
§The triple hexagonal cell H is an alternative description of the hexagonal Bravais lattice, as illustrated in Fig.[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[link] .

1.2.2. Notes on centred cells

| top | pdf |

  • (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[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[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[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[link] and are explained in the Nomenclature Report by de Wolff et al. (1985[link]).


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).]
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]
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.

to end of page
to top of page