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

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

Section Trigonal space groups

H. Wondratscheke 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.

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