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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, p. 48

Section 1.4.1.4.7.3. Rhombohedral space groups

H. Wondratscheke

1.4.1.4.7.3. Rhombohedral space groups

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The rhombohedral lattice may be understood as an R-centred hexagonal lattice and then referred to the hexagonal basis. It has two kinds of symmetry directions, which coincide with the primary and secondary symmetry directions of the hexagonal lattice (owing to the R centrings, no symmetry operation along the tertiary symmetry direction of the hexagonal lattice is compatible with the rhombohedral lattice). On the other hand, the rhombohedral lattice may be referred to a (primitive) rhombohedral coordinate system with the lattice parameters a = b = c and α = β = γ. The HM symbol of a rhombohedral space group starts with R, its representative symmetry directions are [[001]_{\rm hex}] or [[111]_{\rm rhomb}] and [[100]_{\rm hex}] or [[1\overline{1}0]_{\rm rhomb}]. In this section the rhombohedral primitive cell is used. The rotations 3 and the rotoinversions [\overline{3}] are accompanied by screw rotations 31 and 32. Rotations 2 about horizontal axes always alternate with 21 screw rotations and reflections m are accompanied by different glide reflections g with unconventional glide components. The additional operations mentioned are not listed in the full HM symbols.

The seven rhombohedral space groups belong to the five crystal classes 3, [\overline{3}], 32, 3m and [\overline{3}2/m]. In R3 and [R\overline{3}] only the first of the symmetry directions is occupied and listed in the full and short HM symbols. In the space groups of the other crystal classes the second symmetry direction [[1\overline{1}0]] is occupied by `2' or `m' or `c' or `2/m' or `2/c', leading to the full HM symbols R32, R3m, R3c, [R\overline{3}2/m] and [R\overline{3}2/c]. In the short HM symbols the `2/' parts of the last two symbols are skipped: [R\overline{3}m] and [R\overline{3}c].








































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