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

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 47-48

Section 1.4.1.4.7. Trigonal, hexagonal and rhombohedral space groups

H. Wondratscheke

1.4.1.4.7. Trigonal, hexagonal and rhombohedral space groups

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Hexagonal and trigonal space groups are referred to a hexagonal coordinate system P with basis vector [{\bf c}\perp({\bf a},{\bf b})]. The basis vectors a and b span a hexagonal net and form an angle of 120°. The sequence of the representatives of the (up to three) symmetry directions is [001], [100] and [[1\overline{1}0]]. Usually, the seven trigonal space groups of the rhombohedral lattice system (or rhombohedral space groups for short) are described either with respect to a hexagonal coordinate system (triple hexagonal cell) or to a rhombohedral coordinate system (primitive rhombo­hedral cell).

1.4.1.4.7.1. Trigonal space groups

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Trigonal space groups are characterized by threefold rotation or screw rotation or rotoinversion axes in [001]. There may be in addition 2 and 21 axes in [100] or [[1\overline{1}0]], but only in one of these two directions. The same holds for reflections m or glide reflections c. The different possibilities are:

  • (1) There are only threefold axes 3 or 31 or 32 or [\overline{3}]. The short and the full HM symbols are P3, [P3_1], [P3_2], [P\overline{3}].

  • (2) There are in addition horizontal twofold axes. Their direction is either [100] or [[1\overline{1}0]]. The corresponding position of the HM symbol is marked by 2, the other (empty) position is marked by 1: P321, P312, [P3_121], [P3_112] etc. Note: P321 and P312 denote different space-group types.

  • (3) In addition to the threefold axes, there are reflection planes or glide planes with their representative normals in the horizontal directions [100] or [[1\overline{1}0]]. The corresponding position of the HM symbol is marked by m or c, the empty position is marked by 1: P3m1 or P31m etc.

  • (4) The main axis in [001] is [\overline{3}]. Because [\overline{3}] contains an inversion, the second or third position in the full HM symbol is marked by 2/m or 2/c, which leads to the HM symbols [P\,\overline{3}\,2/m\,1] or [P\,\overline{3}\,1\,2/m] etc. In the short HM symbol the `2/' is not kept: [P\overline{3}m1] or [P\overline{3}1m] etc.

1.4.1.4.7.2. Hexagonal space groups

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Hexagonal space groups have either one or three representative symmetry directions. The space groups of crystal classes 6, [\overline{6}] and 6/m have [001] as their single symmetry direction for the axis 6 or 6k for [k=1,\ldots, 5] or [\overline{6}], and for the plane m with its normal along [001]. The short and full HM symbols are the same. Examples are P6, [P6_4], [P\overline{6}] and [P6_3/m].

Space groups of crystal classes 622, 6mm, [\overline{6}2m] and 6/m2/m2/m have the representative symmetry directions [001], [100] and [[1\overline{1}0]]. As opposed to the trigonal HM symbols, in the hexagonal HM symbols no symmetry direction is `empty' and occupied by `1'.

In space groups of the crystal classes 622, 6mm and [\overline{6}2m] the short and full HM symbols are the same; in 6/m2/m2/m the short symbols are deprived of the parts `2/' of the full symbols. The full HM symbol [P\,6_3/m\,2/m\,2/c] is shortened to the short HM symbol [P6_3/mmc], the full HM symbol [P\,6_3/m\,2/c\,2/m] is shortened to [P6_3/mcm]. The two denote different space-group types.

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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