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, pp. 46-47
## Section 1.4.1.4.5. Orthorhombic space groups H. Wondratschek
^{e} |

To the orthorhombic crystal system belong the crystal classes 222, *mm*2 and 2/*m*2/*m*2/*m* with the Bravais types of lattices *P*, *C*, *A*, *F* and *I*. Four space groups with a *P* lattice belong to the crystal class 222, ten to *mm*2 and 16 to 2/*m*2/*m*2/*m*. Each of the basis vectors marks a symmetry direction; the lattice symbol is followed by characters representing the symmetry operations with respect to the symmetry directions along **a**, **b** and **c**.

We start with the full HM symbols. For a space group of crystal class 222 with a *P* lattice the HM symbol is thus `', where , , = 2 or . Conventionally one chooses a setting with the symbols *P*222, , and .

For the generation of the space groups of this crystal class only two non-translational generators are necessary, say and . However, it is not possible to indicate in the HM symbol whether the axes and intersect or not. This is decided by the third (screw) rotation : if , the axes and intersect, if , they do not. For this reason, is sometimes called an *indicator*. However, any two of the three rotations or screw rotations can be taken as the generators and the third one is then the indicator. Mathematically each element of a generating set is a generator independent of its possible redundancy.

In the space groups of crystal class *mm*2 the two reflections or glide reflections are the generators, the twofold rotation or screw rotation is generated by composition of the (glide) reflections. The position of the rotation axis relative to the intersection line of the two planes as well as its screw component are determined uniquely by the glide components of the reflections or glide reflections.

The rotation or screw rotation in the HM symbols of space groups of the crystal class *mm*2 could be omitted, and were omitted in older HM symbols. Nowadays they are included to make the orthorhombic HM symbols more homogeneous. Conventional symbols are, among others, *Pmm*2, , *Pba*2 and .

The 16 space groups with a *P* lattice in crystal class 2/*m*2/*m*2/*m* are similarly obtained by starting with the letter *P* and continuing with the point-group symbol, modified by the possible replacements for 2 and *a*, *b*, *c* or *n* for *m*. The conventional symbols are, among others, *P*2/*m*2/*m*2/*m*, , , or . The symbols and designate different space-group types, as is easily seen by looking at the screw rotations: has screw axes in the direction of **c** only, has screw axes in all three symmetry directions.

If the lattice is centred, the constituents in the same symmetry direction are not unique. In this case, according to the `simplest symmetry operation' rule, in general the simplest operation is chosen, *cf.* Section 1.5.4
.

*Examples*

In the HM symbol there are in addition screw rotations in the first two symmetry directions; additional glide reflections *b* occur in the first, and *n* in the second and third symmetry directions.

In *I*2/*b*2/*a*2/*m*, all rotations 2 are accompanied by screw rotations 2_{1}; *b* and *a* are accompanied by *c* and *m* is accompanied by *n*. The symmetry operations that are not listed in the full HM symbol can be derived by composition of the listed operations with a centring translation, *cf.* Section 1.4.2.4.

There are two exceptions to the `simplest symmetry operation' rule. If the *I* centring is added to the *P* space groups of the crystal class 222, one obtains two different space groups with an *I* lattice, each has 2 and 2_{1} operations in each of the symmetry directions. One space group is derived by adding the *I* centring to the space group *P*222, the other is obtained by adding the *I* centring to a space group . In the first case the twofold axes intersect, in the second they do not. According to the rules both should get the HM symbol *I*222, but only the space group generated from *P*222 is named *I*222, whereas the space group generated from is called . The second exception occurs among the cubic space groups and is due to similar reasons, *cf.* Section 1.4.1.4.8.

The *short HM symbols* for the space groups of the crystal classes 222 and *mm*2 are the same as the full HM symbols. In the short HM symbols for the space groups of the crystal class 2/*m*2/*m*2/*m* the symbols for the (screw) rotations are omitted, resulting in the short symbols *Pmmm*, *Pmma*, *Pmna*, *Pbam*, *Pnma*, *Cmcm* and *Ibam* for the space groups mentioned above.

These are HM symbols of space groups in conventional settings. It is less easy to find the conventional HM symbol and the space-group type from an unconventional short HM symbol. This may be seen from the following example:

*Question*: Given the short HM symbols *Pman*, *Pmbn* and *Pmcn*, what are the conventional descriptions of their space-group types, and are they identical or different?

*Answer*: A glance at the HM symbols shows that the second symbol does not describe any space-group type at all. The second symmetry direction is **b**; the glide plane is perpendicular to it and the glide component may be , or , but not .

In this case it is convenient to define the intersection of the three (glide) reflection planes as the site of the origin. Then all translation components of the generators are zero except the glide components.