InternationalSpace-group symmetryTables for Crystallography Volume A Edited by M. I. Aroyo © International Union of Crystallography 2015 |
International Tables for Crystallography (2015). Vol. A, ch. 1.4, p. 47
## Section 1.4.1.4.6. Tetragonal space groups H. Wondratschek
^{e} |

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, 4*mm*, and 4/*m*2/*m*2/*m* display three symmetry directions which are listed in the sequence [001], [100] and .^{5}

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 4_{1}, 4_{2} or 4_{3} in the space groups with a *P* lattice. If the lattice is *I*-centred, 4 and 4_{2} or 4_{1} and 4_{3} 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 4_{2} and the reflection *m* by the glide reflection *n* such that four space-group types with a *P* lattice exist: *P*4/*m*, , *P*4/*n* and . Two more are based on an *I* lattice: *I*4/*m* and . In all these six space groups the short HM symbols and full HM symbols are the same.

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, 4*mm* and . Only for the space groups of 4/*m*2/*m*2/*m* have the short HM symbols lost their twofold rotations or screw rotations leading, *e.g.*, to the symbol instead of .

*Example*

In *P*4*mm*, 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*.