International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 2.1, p. 144

Table 2.1.1.2 

Th. Hahna and A. Looijenga-Vosb

Table 2.1.1.2| top | pdf |
Symbols for the conventional centring types of one-, two- and three-dimensional cells

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. 1.5.1.8[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 of International Tables for Crystallography, Vol. A1 (2010)[link], abbreviated as IT A1. Plane-group symbols for the h cell are listed in Section 1.5.4[link] . Transformation matrices are contained in Table 1.5.1.1[link] .
In the space-group tables (Chapter 2.3[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] and IT A (2002)[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. 1.5.1.6[link] ). Further details may be found in Section 1.5.4[link] and Chapter 3.1[link] . Transformation matrices are contained in Table 1.5.1.1[link] .
§The triple hexagonal cell H is an alternative description of the hexagonal Bravais lattice, as illustrated in Fig. 1.5.1.8[link] . It was used for systematic space-group description in IT (1935)[link], but replaced by P in IT (1952)[link]. It is used in the tables of maximal subgroups and minimal supergroups of the space groups in IT A1 (2010)[link]. Space-group symbols for the H cell are listed in Section 1.5.4[link] . Transformation matrices are contained in Table 1.5.1.1[link] .