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. 47

Section 1.4.1.4.6.2. Tetragonal space groups with three symmetry directions

H. Wondratscheke

1.4.1.4.6.2. Tetragonal space groups with three symmetry directions

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There are four crystal classes with three symmetry directions each. In the corresponding space-group symbols the constituents 2, 4 and m may be replaced by [2_1], [4_k] with k = 1, 2 or 3, and a, b, c, n or d, respectively. The constituent [\overline{4}] persists. Full HM symbols of space groups are, among others, [P4_22_12], [P4_2bc], [P\overline{4}2c] and [I\,4_1/a\,2/c\,2/d].

The full and short HM symbols agree for the space groups that belong to the crystal classes 422, 4mm and [\overline{4}2m]. Only for the space groups of 4/m2/m2/m have the short HM symbols lost their twofold rotations or screw rotations leading, e.g., to the symbol [I4_1/acd] instead of [I\,4_1/a\,2/c\,2/d].

Example

In P4mm, to the primary symmetry direction [001] belong the rotation 4 and its powers, to the secondary symmetry direction [100] belongs the reflection [m_{100}]. However, in the tertiary symmetry direction [[1\overline{1}0]], there occur reflections m and glide reflections g with a glide vector [\textstyle{{1 \over 2}}({\bf a}+{\bf b})]. Such glide reflections are not listed in the `symmetry operations' blocks of the space-group tables if they are composed of a representing general position and an integer translation, as happens here (cf. Section 1.4.2.4[link] and Section 1.5.4[link] for a detailed discussion of the additional symmetry operations generated by combinations with integer translations). Glide reflections may have complicated glide vectors. If these do not fit the labels a, b, c, n or d, they are frequently called g.








































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