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, and 4/m have only one symmetry direction, [001]. The other four classes, 422, 4mm, and 4/m2/m2/m display three symmetry directions which are listed in the sequence [001], [100] and .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 these are rotoinversion axes ; 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 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/n and . Two more are based on an I lattice: I4/m and . 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 , with k = 1, 2 or 3, and a, b, c, n or d, respectively. The constituent persists. Full HM symbols of space groups are, among others, , , and .

The full and short HM symbols agree for the space groups that belong to the crystal classes 422, 4mm and . 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 instead of .

#### 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 . However, in the tertiary symmetry direction , there occur reflections m and glide reflections g with a glide vector . 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 and Section 1.5.4 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.