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 [{\cal H}] of a rhombohedral space group [{\cal G}] is listed in the same setting as [{\cal G}]: if [{\cal G}] is referred to hexagonal axes, so is [{\cal H}]; if [{\cal G}] is referred to rhombohedral axes, so is [{\cal H}].

  • (b) If [{\cal G}] is a non-rhombohedral trigonal or a cubic space group, then a rhombohedral subgroup [{\cal H} \,\lt\, {\cal G}] is always referred to hexagonal axes.

  • (c) A non-rhombohedral subgroup [{\cal H}] of a rhombohedral space group [{\cal G}] 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 [{\cal G}] which are listed in the block `Loss of centring translations' for the hexagonal axes setting of [{\cal G}] but are listed in the block `Enlarged unit cell' when [{\cal G}] is referred to rhombohedral axes. Although unnecessary and not done for other space groups with primitive lattices, the line[\bullet \,\, {\rm {\bf Loss\,\, of\,\, centring\,\, translations}} \quad\quad {\rm none}]is listed for the rhombohedral axes setting.

Example 2.1.2.5.7

[{\cal G}=R3], 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[\matrix{&\bullet\ {\bf Loss\ of\ centring\ translations}\hfill &\cr [3]\hfill&P3_2\ (145)\hfill & 0,1/3, 0\hfill\cr [3]\hfill& P3_1\ (144)\hfill & 1/3,1/3,0\hfill\cr[3]&P3\ (143)\hfill&\cr&\bullet\ {\bf Enlarged\ unit\ cell}\hfill&\cr [2]\hfill&{\bf a}'=-{\bf b}, {\bf b}'={\bf a}+{\bf b}, {\bf c}'=2{\bf c}\hfill&\cr&R3\ (146)\quad\quad\quad\quad\quad\quad -{\bf b}, {\bf a}+{\bf b}, 2{\bf c}\hfill\cr\ldots\hfill\cr}]

RHOMBOHEDRAL AXES[\matrix{&\bullet\ {\bf Loss\ of\ centring\ translations}\hfill &{\rm none}\cr &\bullet\ {\bf Enlarged\ unit\ cell}\hfill&\cr [2]\hfill&{\bf a}'={\bf a}+{\bf c}, {\bf b}'={\bf a}+{\bf b}, {\bf c}'={\bf b} + {\bf c}\hfill&\cr&R3\ (146)\phantom{_1}\quad\quad\quad {\bf a}+{\bf c}, {\bf a}+{\bf b}, {\bf b}+{\bf c}\hfill\cr[3]\hfill&{\bf a}'={\bf a}-{\bf b}, {\bf b}'={\bf b}-{\bf c}, {\bf c}'={\bf a}+{\bf b} + {\bf c}\hfill\cr&P3_2\ (145)\quad\quad\quad{\bf a}-{\bf b}, {\bf b}-{\bf c}, {\bf a}+{\bf b} + {\bf c}\hfill&0,1/3,-1/3\hfill\cr&P3_1\ (144)\quad\quad\quad{\bf a}-{\bf b}, {\bf b}-{\bf c}, {\bf a}+{\bf b} + {\bf c}\hfill&1/3,0,-1/3\hfill\cr&P3\ (143)\phantom{_1}\quad\quad\quad{\bf a}-{\bf b}, {\bf b}-{\bf c}, {\bf a}+{\bf b} + {\bf c}\hfill&\cr}]

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.








































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