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

Hahn, Th.

doi:10.1107/97809553602060000501

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

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International Tables for Crystallography (2006). Vol. A. ch. 1.2, p. 4

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

Th. Hahn^a ^*

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

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 and 2.2.13.3 . 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	$[\!\openup1pt{\cases {0,{\hbox to 1pt{}} 0,{\hbox to 1pt{}} 0{\hbox{; }}{\hbox to 2pt{}}{2 \over 3},{\hbox to -1.5pt{}} {1 \over 3},{\hbox to -1.5pt{}} {1 \over 3}{\hbox{; }}{\hbox to 1pt{}}{1 \over 3},{\hbox to -1pt{}} {2 \over 3},{\hbox to -1pt{}} {2 \over 3} \hbox{ (`obverse setting')}\cr 0,{\hbox to 1pt{}} 0,{\hbox to 1pt{}} 0{\hbox{; }}{\hbox to 1.5pt{}}{1 \over 3},{\hbox to -1pt{}} {2 \over 3},{\hbox to -1.5pt{}} {1 \over 3}{\hbox{; }}{\hbox to 1pt{}}{2 \over 3}, {1 \over 3},{\hbox to -1.5pt{}} {2 \over 3} \hbox{ (`reverse setting')}\cr}}]$
R ^‡		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

. 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

). Plane-group symbols for the h cell are listed in Chapter 4.2

. Transformation matrices are contained in Table 5.1.3.1

.
^‡In the space-group tables (Part 7

), as well as in IT (1935)

and IT (1952)

, 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)

, the obverse setting of the triple hexagonal cell R is used. Note that in IT (1935)

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

). Further details may be found in Chapter 2.1

, Section 4.3.5

and Chapter 9.1

. Transformation matrices are contained in Table 5.1.3.1

.
^§The triple hexagonal cell H is an alternative description of the hexagonal Bravais lattice, as illustrated in Fig. 5.1.3.8

. It was used for systematic space-group description in IT (1935)

, but replaced by P in IT (1952)

. In the space-group tables of this volume (Part 7

), it is only used in the sub- and supergroup entries (cf. Section 2.2.15

). Space-group symbols for the H cell are listed in Section 4.3.5

. Transformation matrices are contained in Table 5.1.3.1

References

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

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

International Tables for Crystallography (2006). Vol. A. ch. 1.2, p. 4