International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2011). Vol. A1, ch. 2.1, pp. 75-76   | 1 | 2 |

## Section 2.1.2.5.3. Space groups with a rhombohedral lattice

Hans Wondratscheka* and Mois I. Aroyob

aInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail:  wondra@physik.uni-karlsruhe.de

#### 2.1.2.5.3. Space groups with a rhombohedral lattice

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The seven trigonal space groups with a rhombohedral lattice are often called rhombohedral space groups. Their HM symbols begin with the lattice letter R and they are listed with both hexagonal axes and rhombohedral axes.

Rules

 (a) A rhombohedral subgroup of a rhombohedral space group is listed in the same setting as : if is referred to hexagonal axes, so is ; if is referred to rhombohedral axes, so is . (b) If is a non-rhombohedral trigonal or a cubic space group, then a rhombohedral subgroup is always referred to hexagonal axes. (c) A non-rhombohedral subgroup of a rhombohedral space group is referred to its conventional setting.

Remarks

 Rule (a) provides a clear definition, in particular for the axes of isomorphic subgroups. Rule (b) has been followed in the subgroup tables because the rhombohedral setting is rarely used in crystallography. Rule (c) implies that monoclinic subgroups of rhombohedral space groups are referred to the setting unique axis b'. There is a peculiarity caused by the two settings of the rhombohedral space groups. The rhomb­ohedral lattice appears to be centred in the hexagonal axes setting, whereas it is primitive in the rhombohedral axes setting. Therefore, there are trigonal subgroups of a rhombohedral space group which are listed in the block Loss of centring translations' for the hexagonal axes setting of but are listed in the block Enlarged unit cell' when is referred to rhombohedral axes. Although unnecessary and not done for other space groups with primitive lattices, the lineis listed for the rhombohedral axes setting.

#### Example 2.1.2.5.7

, No. 146. Maximal klassenglei­che subgroups of index 2 and 3. Comparison of the data for the settings hexagonal axes' and rhombohedral axes'. The data for the general position and the generators are omitted.

HEXAGONAL AXES

RHOMBOHEDRAL AXES

The sequence of the blocks has priority over the classification by increasing index. Therefore, in the setting hexagonal axes', the subgroups of index 3 precede the subgroup of index 2.

In the tables, the lattice relations are simpler for the setting hexagonal axes'.

The complete general position is listed for the maximal k-subgroups of index 3 in the setting hexagonal axes'; only the generator is listed for rhombohedral axes.