Section 8.2.8. Crystal systems and lattice systems⁴

At least three different classifications of space groups, crystallographic point groups and lattice types have been called `crystal systems' in crystallographic literature. Only one of them classifies space groups, crystallographic point groups and lattice types. It has been introduced in the preceding section under the name `crystal families'. The two remaining classifications are called here `crystal systems' and `lattice systems', and are considered in this section. Crystal systems classify space groups and crystallographic point groups but not lattice types. Lattice systems classify space groups and lattice types but not crystallographic point groups.

The `crystal-class systems' or `crystal systems' are used in these Tables. In and , the crystal systems provide the same classification as the crystal families, with the exception of the hexagonal crystal family in . Here, the hexagonal family is subdivided into the trigonal and the hexagonal crystal system. Each of these crystal systems consists of complete geometric crystal classes of space groups. The space groups of the five trigonal crystal classes 3, , 32, 3m and belong to either the hexagonal or the rhombohedral Bravais flock, and both Bravais flocks are represented in each of these crystal classes. The space groups of the seven hexagonal crystal classes 6, , , 622, 6mm, and , however, belong only to the hexagonal Bravais flock.

These observations will be used to define crystal systems by the concept of intersection. A geometric crystal class and a Bravais flock of space groups are said to intersect if there is at least one space group common to both. Accordingly, the rhombohedral Bravais flock intersects all trigonal crystal classes but none of the hexagonal crystal classes. The hexagonal Bravais flock, on the other hand, intersects all trigonal and hexagonal crystal classes, see Table 8.2.8.1.

Table 8.2.8.1| top | pdf | Distribution of trigonal and hexagonal space groups into crystal systems and lattice systems The hexagonal lattice system is also the hexagonal Bravais flock, the rhombohedral lattice system is the rhombohedral Bravais flock. Crystal system Crystal class Hexagonal lattice system Rhombohedral lattice system Hexagonal Bravais flock Rhombohedral Bravais flock Hexagonal , , ,     , , ,     6mm P6mm, P6cc, ,     622 P622, , ,     ,         6 P6, , , , ,   Trigonal , , ,   3m P3m1, P31m, P3c1, P31c R3m, R3c   32 P312, P321, , , , R32     3 P3, , R3

Using the concept of intersection, one obtains the definition:

There are four crystal systems in and seven in . The classification into crystal systems applies to space groups, space-group types, arithmetic crystal classes and geometric crystal classes, see Fig. 8.2.1.1. Moreover, via their geometric crystal classes, the crystallographic point groups are classified by `crystal systems of point groups'. Historically, point groups were the first to be classified by crystal systems. Bravais flocks of space groups and Bravais types of lattices are not classified, as members of both can occur in more than one crystal system. For example, P3 and belong to the same hexagonal Bravais flock but to different crystal systems, P3 to the trigonal, to the hexagonal crystal system. Thus, a crystal system of space groups does not necessarily contain complete Bravais flocks (it does so, however, in and in all crystal systems of , except for the trigonal and hexagonal systems).

The use of crystal systems has some practical advantages.

Whereas crystal systems classify geometric crystal classes and point groups, lattice systems classify Bravais flocks and Bravais types of lattices. Lattice systems may be defined in two ways. The first definition is analogous to that of crystal systems and uses once again the concept of intersection, introduced above.

There are four lattice systems in and seven lattice systems in . In and , the classification into lattice systems is the same as that into crystal families and crystal systems except for the hexagonal crystal family of . The space groups of the hexagonal Bravais flock (lattice letter P) belong to the twelve geometric crystal classes from 3 to , whereas the space groups of the rhombohedral Bravais flock (lattice letter R) only belong to the five geometric crystal classes 3, , 32, 3m and . Thus, these two Bravais flocks form the hexagonal and the rhombohedral lattice systems with 45 and 7 types of space groups, respectively.

The lattice systems provide a classification of space groups, see Fig. 8.2.1.1. Geometric crystal classes are not classified, as they can occur in more than one lattice system. For example, space groups and R3, both of crystal class 3, belong to the hexagonal and rhombohedral lattice systems, respectively.

The above definition of lattice systems corresponds closely to the definition of crystal systems. There exists, however, another definition of lattice systems which emphasizes the geometric aspect more. For this it should be remembered that each Bravais flock is related to the point symmetry of a lattice type via its Bravais class; cf. Sections 8.2.5 and 8.2.6. It can be shown that Bravais flocks intersect the same set of crystal classes if their Bravais classes belong to the same (holohedral) geometric crystal class. Therefore, one can use the definition:

According to this second definition, it is sufficient to compare only the Bravais classes instead of all space groups of different Bravais flocks. The comparison of Bravais classes can be replaced by the comparison of their holohedries; cf. Section 8.2.5. This gives rise to a special advantage of lattice systems, the possibility of classifying lattices and lattice types. (Such a classification is not possible using crystal systems.) All those lattices belong to the same lattice system of lattices for which the lattice point groups belong to the same holohedry. As lattices of the same lattice type always belong to the same holohedry, lattice systems also classify lattice types.

The adherence of a space group of the hexagonal crystal family to the trigonal or hexagonal crystal system and the rhombohedral or hexagonal lattice system is easily recognized by means of its Hermann–Mauguin symbol. The Hermann–Mauguin symbols of the trigonal crystal system display a `3' or `', those of the hexagonal crystal system a `6' or `'. On the other hand, the rhombohedral lattice system displays lattice letter `R' and the hexagonal one `P' in the Hermann–Mauguin symbols of their space groups.

It should be mentioned that the lattice system of the lattice of a space group may be different from the lattice system of the space group itself. This always happens if the lattice symmetry is accidentally higher than is required by the space group, e.g. for a monoclinic space group with an orthorhombic lattice, i.e. , or a tetragonal space group with cubic metrics, i.e. . These accidental lattice symmetries are special cases of metrical pseudo-symmetries. Owing to the anisotropy of the thermal expansion or the contraction under pressure, for special values of temperature and pressure singular lattice parameters may represent higher lattice symmetries than correspond to the symmetry of the crystal structure. The same may happen, and be much more pronounced, in continuous series of solid solutions owing to the change of cell dimensions with composition. Note that this phenomenon does not represent a new phase and a phase transition is not involved. Therefore, accidental lattice symmetries cannot be the basis for a classification in practice, e.g. for crystal structures or phase transitions. In contrast, structural pseudo-symmetries of crystals often lead to (displacive) phase transitions resulting in a new phase with higher structural and lattice symmetry.

In spite of its name, the classification of space groups into `lattice systems of space groups' does not depend on the accidental symmetry of the translation lattice of a space group.