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

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

Section Cubic space groups

H. Wondratscheke Cubic space groups

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There are five cubic crystal classes combined with the three types of lattices P, F and I in which the cubic space groups are classified. The two symmetry directions [100] and [111] are the representative directions in the space groups of the crystal classes 23 and [2/m\,\overline{3}]. A third representative symmetry direction, [[1\overline{1}0]], is added for space groups of the crystal classes 432, [\overline{4}3m] and [4/m\,\overline{3}\,2/m].6

In the full HM symbol the symmetry is described as usual. Examples are [P2_13], [F\,2/d\,\overline{3}], [P4_332], [F\overline{4}3c], [P\,4_2/m\,\overline{3}\,2/n] and finally No. 230, [I\,4_1/a\,\overline{3}\,2/d]. The short HM symbols of the noncentrosymmetric space groups (those of crystal classes 23, 432 and [\overline{4}3m]) are the same as the full HM symbols. In the short HM symbols of centrosymmetric space groups of the crystal classes [2/m\,\overline{3}] and [4/m\,\overline{3}\,2/m] the rotations or screw rotations are omitted with the exception of the rotations 3 and rotoinversions [\overline{3}] which represent the symmetry in direction [111]. Thus, in the examples listed above, [Fd\overline{3}], [Pm\overline{3}n] and [Ia\overline{3}d] are the short HM symbols differing from the full HM symbols.

As in the orthorhombic space groups I222 and [I2_12_12_1], there is the pair I23 and [I2_13] in which the `simplest symmetry operation' rule is violated. In both space groups twofold rotations and screw rotations around a, b and c occur simultaneously. In I23 the rotation axes intersect, in [I2_13] they do not. The first space group can be generated by adding the I-centring to the space group P23, the second is obtained by adding the I-centring to the space group [P2_13].

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