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. Tetragonal space groups

H. Wondratscheke

1.4.1.4.6. Tetragonal space groups

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There are seven tetragonal crystal classes. The lattice may be P or I. The space groups of the three crystal classes 4, [\overline{4}] and 4/m have only one symmetry direction, [001]. The other four classes, 422, 4mm, [\overline{4}2m] and 4/m2/m2/m display three symmetry directions which are listed in the sequence [001], [100] and [[1\overline{1}0]].5

1.4.1.4.6.1. Tetragonal space groups with one symmetry direction

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In the space groups of the crystal class 4, rotation or screw rotation axes run in direction [001]; in the space groups of crystal class [\overline{4}] these are rotoinversion axes [\overline{4}]; and in crystal class 4/m both occur. The rotation 4 of the point group may be replaced by screw rotations 41, 42 or 43 in the space groups with a P lattice. If the lattice is I-centred, 4 and 42 or 41 and 43 occur simultaneously, together with [\overline{4}] rotoinversions.

In the space groups of crystal class 4/m with a P lattice, the rotations 4 can be replaced by the screw rotations 42 and the reflection m by the glide reflection n such that four space-group types with a P lattice exist: P4/m, [P4_2/m], P4/n and [P4_2/n]. Two more are based on an I lattice: I4/m and [I4_1/a]. In all these six space groups the short HM symbols and full HM symbols are the same.

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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