2.12 Space Groups
We showed in Section 1.4 that repetition of an object (e.g. an atom or group of atoms) in a crystal is carried out by operations of rotation, reflection (or inversion) and translation. We have also described in this chapter the consistent combinations of rotations and inversions that can occur in crystals, and in Section 1.8 we described the possible translations that can occur in crystals to produce the 14 Bravais lattices, or space lattices.
However, we have so far not made any attempt to combine the operations of rotation (and inversion) with translation, except briefly in Section 1.5, where it was shown that only one‐, two‐, three‐, four‐, and sixfold rotation axes are compatible with translational symmetry. A full description of the symmetry of a crystal involves a description of the way in which all of the symmetry elements are distributed in space. This is called the space group. There are 230 different crystallographic space groups8; each one gives the fullest description of the symmetry elements present in a crystal possessing that group.
The rotation axes and rotoinversion axes possible in crystals were discussed in Section 2.1 and the possible translations in Section 1.8. An enumeration of the way these axes can be consistently combined with the translation is therefore an enumeration of the possible space groups. Space groups are most important in the solution of crystal structures. They are also very useful when establishing symmetry hierarchies in phase transitions in materials such as perovskites [10–12].
When an attempt is made to combine the operations of rotation and translation, the possibility arises naturally of what is called a screw axis. This involves repetition by rotation about an axis, together with translation parallel to that axis. Similarly, a repetition by reflection in a mirror plane may be combined with a translational component parallel to that plane to produce a glide plane. These will now each be considered in more detail.
2.12.1 Screw Axes
If, for example, a twofold rotational axis occurs in a crystal, then this means some structural unit or motif is arranged about this direction so that it is repeated by a rotation of 180° about the axis. The repetition shown in Figure 2.22a corresponds to a pure rotational diad axis. However, the rotation of 180° could be coupled with a translation of one‐half of the lattice repeat distance, t, in the direction of the axis to give the screw diad axis shown in Figure 2.22b, denoted by the symbol 21. The translation t/2 will be of a length of the order of the lattice parameters of the crystal and hence of the order of a few Ångstrom units (1 Å = 10−10 m), and so quite undetected by the naked eye. Macroscopically, the crystal containing a 21 axis would then show diad symmetry about that axis in the symmetry of its external faces or of its physical properties.
Figure 2.22 (a) A twofold rotation axis. (b) A 21 screw axis. (c) A glide plane normal to the surface of the paper
Although the presence of a screw axis, say a screw diad as in Figure 2.22b, indicates the presence of identical atomic motifs arranged about it so that they are related by a rotation plus translation, the screw axis must not be thought of as translating the translation vectors of the lattice. A screw axis and a pure rotation axis of the same order n repeat a translation in the same way. It follows that the rotational components of screw axes can only be 2π/1, 2π/2, 2π/3, 2π/4 and 2π/6, and that an n‐fold screw and an n‐fold pure rotation axis must have similar locations with respect to a similar set of translations. The angles between screws and between screws and rotations must therefore be the same as the permissible combinations listed in Table 1.2.
The various kinds of screw axis possible are shown in Table 2.3. An nN‐fold screw axis involves repetition by rotation through 360°/n with a translation of tN/n, where t is a lattice repeat vector parallel to the axis. There are five types of screw hexad; for example, 61 involves rotation through 60° and translation of t/6, while 65 involves rotation through 60° and translation of 5t/6. By drawing a diagram showing the repetition of objects by these axes, it is easily seen that 61 is a screw of opposite hand to 65, 41 to 43 and 62 to 64. A diagram to show this for 31 and 32 is given in Figure 2.23. In this figure, A and A′ are lattice points. The operation of 31 is straightforward, as shown in Figure 2.23a. When repeating an object according to 32, note that the object at height
Table 2.3 Screw axes in crystals
| Name | Symbol | Graphical symbol | Right‐handed screw translation along the axis in units of the lattice parameter |
| Screw diad | 21 |
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| Screw triads | 31 |
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