International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2015). Vol. A, ch. 1.4, pp. 46-47

Section 1.4.1.4.5. Orthorhombic space groups

H. Wondratscheke

1.4.1.4.5. Orthorhombic space groups

| top | pdf |

To the orthorhombic crystal system belong the crystal classes 222, mm2 and 2/m2/m2/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 mm2 and 16 to 2/m2/m2/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 `[PR_1R_2R_3]', where [R_1], [R_2], [R_3] = 2 or [2_1]. Conventionally one chooses a setting with the symbols P222, [P222_1], [P2_12_12] and [P2_12_12_1].

For the generation of the space groups of this crystal class only two non-translational generators are necessary, say [R_1] and [R_2]. However, it is not possible to indicate in the HM symbol whether the axes [R_1] and [R_2] intersect or not. This is decided by the third (screw) rotation [R_3]: if [R_3=R_1R_2=2], the axes [R_1] and [R_2] intersect, if [R_3=2_1], they do not. For this reason, [R_3] 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 mm2 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 mm2 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, Pmm2, [Pmc2_1], Pba2 and [Pca2_1].

The 16 space groups with a P lattice in crystal class 2/m2/m2/m are similarly obtained by starting with the letter P and continuing with the point-group symbol, modified by the possible replacements [2_1] for 2 and a, b, c or n for m. The conventional symbols are, among others, P2/m2/m2/m, [P\,2_1/m\,2/m\,2/a], [P\,2/m\,2/n\,2_1/a], [P\,2_1/b\,2_1/a\,2/m] or [P\,2_1/n\,2_1/m\,2_1/a]. The symbols [P\,2/m\,2/n\,2_1/a] and [P\,2_1/n\,2_1/m\,2_1/a] designate different space-group types, as is easily seen by looking at the screw rotations: [P\,2/m\,2/n\,2_1/a] has screw axes in the direction of c only, [P\,2_1/n\,2_1/m\,2_1/a] 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[link] .

Examples

In the HM symbol [C\,2/m\,2/c\,2_1/m] there are in addition [2_1] 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 I2/b2/a2/m, all rotations 2 are accompanied by screw rotations 21; 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[link].

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 21 operations in each of the symmetry directions. One space group is derived by adding the I centring to the space group P222, the other is obtained by adding the I centring to a space group [P2_12_12_1]. In the first case the twofold axes intersect, in the second they do not. According to the rules both should get the HM symbol I222, but only the space group generated from P222 is named I222, whereas the space group generated from [P2_12_12_1] is called [I2_12_12_1]. The second exception occurs among the cubic space groups and is due to similar reasons, cf. Section 1.4.1.4.8[link].

The short HM symbols for the space groups of the crystal classes 222 and mm2 are the same as the full HM symbols. In the short HM symbols for the space groups of the crystal class 2/m2/m2/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 [\textstyle{{1 \over 2}}{\bf a}], [\textstyle{{1 \over 2}}{\bf c}] or [\textstyle{{1 \over 2}}({\bf c}+{\bf a})], but not [\textstyle{{1 \over 2}}{\bf b}].

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.

  • (1) Pman. If one names the three (glide) reflections according to the directions of their normals by [m_{100}], [a_{010}] and [n_{001}], then [a_{010}n_{001}=2_{100}], [m_{100}a_{010}=2_{001}], while the composition [n_{001}m_{100}] results in a 21 screw rotation along [010].

    Clearly, the unconventional full HM symbol is [P\,2/m\,2_1/a\,2/n]. The procedure for obtaining from this symbol the conventional HM symbol [P\,2/m\,2/n\,2_1/a] (or short symbol Pmna) with the origin at the inversion centre is described in Chapter 1.5[link] .

  • (2) Pmcn. Using a nomenclature similar to that of (1), one obtains 21 screw axes along [100], [010] and [001] by the compositons [c_{010}n_{001}], [n_{001}m_{100}] and [m_{100}c_{010}], respectively. Thus the unconventional full HM symbol is [P\,2_1/m\,2_1/c\,2_1/n]. Again, the procedure of Chapter 1.5[link] results in the full HM symbol [P\,2_1/n\,2_1/m\,2_1/a] or the short symbol Pnma. The full HM symbols show that the two space-group types are different.








































to end of page
to top of page