Section 8.2.3. Arithmetic crystal classes

As space groups not only of the same type but also of different types have symmetry properties in common, coarser classifications can be devised which are classifications of both space-group types and individual space groups. The following classifications are of this kind. Again each space group is referred to a primitive basis and an origin.

Algebraically, this definition may be expressed as follows. Equation (8.2.2.1) of Section 8.2.2 relating space groups of the same type may be written more explicitly as follows: the matrix part of which is Space groups of different types belong to the same arithmetic crystal class if equation (8.2.3.2), but not equation (8.2.2.1) or equation (8.2.3.1), is fulfilled, e.g. space groups of types P2 and . This gives rise to the following definition:

By definition, both space groups and space-group types may be classified into arithmetic crystal classes. It is apparent from equation (8.2.3.2) that `arithmetic equivalence' refers only to the matrix parts and not to the column parts of the symmetry operations. Among the space-group types of each arithmetic crystal class there is exactly one for which the column parts vanish for a suitable choice of the origin. This is the symmorphic space-group type, cf. Sections 8.1.6 and 8.2.2. The nomenclature for arithmetic crystal classes makes use of this relation: The lattice letter and the point-group part of the Hermann–Mauguin symbol for the symmorphic space-group type are interchanged to designate the arithmetic crystal class, cf. de Wolff et al. (1985). This symbolism enables one to recognize easily the arithmetic crystal class to which a space group belongs: One replaces in the Hermann–Mauguin symbol of the space group all screw rotations and glide reflections by the corresponding rotations and reflections and interchanges then the lattice letter and the point-group part.

Remark: In order to belong to the same arithmetic crystal class, space groups must belong to the same geometric crystal class, cf. Section 8.2.4 and to the same Bravais flock; cf. Section 8.2.6. These two conditions, however, are only necessary but not sufficient.

There are 13 arithmetic crystal classes of plane groups in and 73 arithmetic crystal classes of space groups in , see Fig. 8.2.1.1. Arithmetic crystal classes are rarely used in practical crystallography, even though they play some role in structural crystallography because the `permissible origins' (see Giacovazzo, 2002) are the same for all space groups of one arithmetic crystal class. The classification of space-group types into arithmetic crystal classes, however, is of great algebraic consequence. In fact, the arithmetic crystal classes are the basis for the further classifications of space groups.

In , enantiomorphic pairs of space groups always belong to the same arithmetic crystal class. Enantiomorphism of arithmetic crystal classes can be defined analogously to enantiomorphism of space groups. It does not occur in and , but appears in spaces of higher dimensions, e.g. in ; cf. Brown et al. (1978).

In addition to space groups, equation (8.2.3.2) also classifies the set of all finite integral-matrix groups. Thus, one can speak of arithmetic crystal classes of finite integral-matrix groups. It is remarkable, however, that this classification of the matrix groups does not imply a classification of the corresponding point groups. Although every finite integral-matrix group represents the point group of some space group, referred to a primitive coordinate basis, there are no arithmetic crystal classes of point groups. For example, space-group types P2 and C2 both have point groups of the same type, 2, but referred to primitive bases their matrix groups are not arithmetically equivalent, i.e. there is no integral matrix P with , such that equation (8.2.3.2) holds.

The arithmetic crystal classes of finite integral-matrix groups are the basis for the classification of lattices into Bravais types of lattices: see Section 8.2.5. Even though the consideration of finite integral-matrix groups in connection with space groups is not common in practical crystallography, these matrix groups play a very important role in the classifications discussed in subsequent sections. Finite integral-matrix groups have the advantage of being particularly suitable for computer calculations.