Chapter 8.2. Classifications of space groups, point groups and lattices

One of the main tasks of theoretical crystallography is to sort the infinite number of conceivable crystal patterns into a finite number of classes, where the members of each class have certain properties in common. In such a classification, each crystal pattern is assigned only to one class. The elements of a class are called equivalent, the classes being equivalence classes in the mathematical sense of the word. Sometimes the word `type' is used instead of `class'.

An important principle in the classification of crystals and crystal patterns is symmetry, in particular the space group of a crystal pattern. The different classifications of space groups discussed here are displayed in Fig. 8.2.1.1.

Figure 8.2.1.1| top | pdf | Classifications of space groups. In each box, the number of classes, e.g. 32, and the section in which the corresponding term is defined, e.g. 8.2.4, are stated.

Classification of crystals according to symmetry implies three steps. First, criteria for the symmetry classes have to be defined. The second step consists of the derivation and complete listing of the possible symmetry classes. The third step is the actual assignment of the existing crystals to these symmetry classes. In this chapter, only the first step is dealt with. The space-group tables of this volume are the result of the second step. The third step is beyond the scope of this volume.

The finest commonly used classification of three-dimensional space groups, i.e. the one resulting in the highest number of classes, is the classification into the 230 (crystallographic) space-group types.¹ The word `type' is preferred here to the word `class', since in crystallography `class' is already used in the sense of `crystal class', cf. Sections 8.2.3 and 8.2.4. The classification of space groups into space-group types reveals the common symmetry properties of all space groups belonging to one type. Such common properties of the space groups can be considered as `properties of the space-group types'.

The practising crystallographer usually assumes the 230 space-group types to be known and to be described in this volume by representative data such as figures and tables. To the experimentally determined space group of a particular crystal structure, e.g. of pyrite FeS₂, the corresponding space-group type No. 205 of International Tables is assigned. Two space groups, e.g. those of FeS₂ and CO₂, belong to the same space-group type if their symmetries correspond to the same entry in International Tables.

The rigorous definition of the classification of space groups into space-group types can be given in a more geometric or a more algebraic way. Here matrix algebra will be followed, by which primarily the classification into the 219 so-called affine space-group types is obtained.² For this classification, each space group is referred to a primitive basis and an origin. In this case, the matrices of the symmetry operations consist of integral coefficients and holds. Two space groups and are then represented by their matrix groups and . These two matrix groups are now compared.

Equation (8.2.2.1) is an equivalence relation for space groups. The corresponding classes are called affine space-group types. By this definition, one obtains 17 plane-group types for and 219 space-group types for , see Fig. 8.2.1.1. Listed in the space-group tables are 17 plane-group types for and 230 space-group types for . Obviously, the equivalence definition of the space-group tables differs somewhat from the one used above. In practical crystallography, one wants to distinguish between right- and left-handed screws and does not want to change from a right-handed to a left-handed coordinate system. In order to avoid such transformations, the matrix P of equation (8.2.2.1) is restricted by the additional condition . Using matrices with only, 11 space-group types of split into pairs, which are the so-called pairs of enantiomorphic space-group types. The Hermann–Mauguin and Schoenflies symbols (in parentheses) of the pairs of enantiomorphic space-group types are , , , , , , , , , and . In order to distinguish between the two definitions of space-group types, the first is called the classification into the 219 affine space-group types and the second the classification into the 230 crystallographic or positive affine or proper affine space-group types, see Fig. 8.2.1.1. Both classifications are useful.

In Section 8.1.6 , symmorphic space groups were defined. It can be shown (with either definition of space-group type) that all space groups of a space-group type are symmorphic if one of these space groups is symmorphic. Therefore, it is also possible to speak of types of symmorphic and non-symmorphic space groups. In , symmorphic space groups do not occur in enantiomorphic pairs. This does happen, however, in .

The so-called space-group symbols are really symbols of `crystallographic space-group types'. There are several different kinds of symbols (for details see Part 12 ). The numbers denoting the crystallographic space-group types and the Schoenflies symbols are unambiguous but contain little information. The Hermann–Mauguin symbols depend on the choice of the coordinate system but they are much more informative than the other notations.

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.

The widely used term `crystal class' corresponds to the `geometric crystal class' described in this section, and must be distinguished from the `arithmetic' crystal class, introduced in Section 8.2.3. Geometric crystal classes classify the space groups and their point groups, i.e. the symmetry groups of the external shape of macroscopic crystals. Classification by morphological symmetry was done long before space groups were known. In Section 8.1.6 , the reasons are stated why the two seemingly different classifications agree, namely that of space groups according to their matrix groups , and that of macroscopic crystals according to the `point groups' of their sets of face normals.

To define geometric crystal classes, we again compare the matrix parts of the space groups.

In contrast to the definition of arithmetic crystal classes, nonprimitive bases are admitted. To express this definition in matrix terms, we refer to equation (8.2.3.2) of the previous section.

In contrast to arithmetic crystal classes where P is a unimodular integral matrix, for geometric crystal classes only a real matrix P is required. Thus, the restriction is no longer necessary, det(P) may have any value except zero.

Clearly, space groups of the same arithmetic crystal class always obey condition (8.2.4.1). Thus, the geometric crystal classes form a classification not only of space groups and space-group types but also of arithmetic crystal classes. There are ten geometric crystal classes in and 32 geometric crystal classes in ; see Fig. 8.2.1.1. As is a matrix representation of the point group of a space group, the definition may be restated as follows:

This definition may also be used to classify point groups, via their matrix groups, into geometric crystal classes of point groups. Moreover, the geometric crystal classes provide a classification of the finite groups of integral matrices. Again, matrix groups of the same arithmetic crystal class always belong to the same geometric crystal class.

Enantiomorphism of geometric crystal classes may occur in dimensions greater than three, as it does for arithmetic crystal classes.

Every space group has a vector lattice L of translation vectors. The elements of the point group of are symmetry operations of L. The lattice L of , however, may have additional symmetry in comparison with .

The symmetry of a vector lattice L is its point group according to the following definition:

The inversion is always a symmetry operation of L, even if does not contain inversions. If, for instance, belongs to space-group type , its point group is 6mm but the point symmetry of L is . Thus, the point group of the lattice L is of higher order than the point group of .

Other symmetry operations of L may also have no counterpart in . Space groups of type , for instance, have inversions but no reflections across `vertical' mirror planes. The point symmetry of their lattices again is , i.e. in this case too there are more elements in the point group of L than in the point group of .

For purposes of classification, lattices L will now be considered independently of their space groups . Associated with each vector lattice L is a finite group of integral matrices which describes the point group of L with respect to some primitive basis of L. This matrix group is a member of an arithmetic crystal class; cf. Section 8.2.3. Thus, there are some arithmetic crystal classes with matrix groups of lattices, e.g. the arithmetic crystal class . Other arithmetic crystal classes, however, are not associated with lattices, like or 6mmP. One can distinguish these two cases with the following definition:

By this definition, each lattice is associated with a Bravais class. On the other hand, each matrix group of a Bravais class represents the point group of a lattice referred to an appropriate primitive basis. Closer inspection shows that there are five Bravais classes of and 14 of . With the use of Bravais classes, lattices may be classified using the following definition:

Thus, five Bravais types of lattices exist in , and 14 in . This classification can be transferred from vector lattices L to point lattices L. To each point lattice L a vector lattice L is uniquely assigned. Thus, one can define Bravais types of point lattices via the Bravais types of vector lattices by the definition:

Usually the Bravais types are called `the five' or `the 14 Bravais lattices' of or . It must be emphasized, however, that `Bravais lattices' are not individual lattices but types (or classes) of all lattices with certain common properties. Geometrically, these common properties are expressed by the `centring type' and the well known relations between the lattice parameters, provided the lattices are referred to conventional bases, cf. Chapters 2.1 and 9.1 . In these chapters a nomenclature of Bravais types is presented.

Another plausible classification of space groups and space-group types, as well as of arithmetic crystal classes, is based on the lattice of the space group. One is tempted to use the definition: `Two space groups are members of the same class if their lattices belong to the same Bravais type'. There is, however, a difficulty which will become apparent by an example.

It was shown in Section 8.2.5 with the two examples of space groups and that the lattice L of the space group may systematically have higher symmetry than the point group of . The lattice L, however, may also accidentally have higher symmetry than because the lattice parameters may have special metrical values.

This is of importance for the practising crystallographer, because quite often difficulties arise in the interpretation of X-ray powder diagrams if no single crystals are available. In some cases, changes of temperature or pressure may enable one to determine the true symmetry of the substance. The example shows, however, that the lattices of different space groups of the same space-group type may have different symmetries. The possibility of accidental lattice symmetry prevents the direct use of lattice types for a rigorous classification of space-group types.

Such a classification is possible, however, via the point group of the space group and its matrix groups. Referred to a primitive basis, the point group of is represented by a finite group of integral matrices which belongs to some arithmetic crystal class. This matrix group can be uniquely assigned to a Bravais class: It either belongs already to a Bravais class, e.g. for space groups Pmna and , or it may be uniquely connected to a Bravais class by the following two conditions:

This assignment leads to the definition:

By this definition, the space group mentioned above belongs to the Bravais flock of , despite the fact that the Bravais class of the lattice may be as a result of accidental symmetry.

Obviously, to each Bravais class a Bravais flock corresponds. Thus, there exist five Bravais flocks of plane groups and 14 Bravais flocks of space groups, see Fig. 8.2.1.1, and the Bravais flocks may be denoted by the symbols of the corresponding Bravais classes; cf. Section 8.2.5.

Though Bravais flocks themselves are of little practical importance, they are necessary for the definition of crystal families and lattice systems, as described in Sections 8.2.7 and 8.2.8.

Another classification of space groups, which is a classification of geometric crystal classes and Bravais flocks as well, is that into crystal families.

There are four crystal families in (oblique m, rectangular o, square t and hexagonal h) and six crystal families in [triclinic (anorthic) a, monoclinic m, orthorhombic o, tetragonal t, hexagonal h and cubic c]; see Fig. 8.2.1.1.

The classification into crystal families is a rather universal crystallographic concept as it applies to many crystallographic objects: space groups, space-group types, arithmetic and geometric crystal classes of space groups, point groups (morphology of crystals), lattices and Bravais types of lattices.

Remark: In most cases of and , the lattices of a given crystal family of lattices have the same point symmetry (for the symbols, see Table 2.1.2.1 ): rectangular op and oc in ; monoclinic mP and mS, orthorhombic oP, oS, oF and oI, tetragonal tP and tI, cubic cP, cF and cI in . Only to the hexagonal crystal family in do lattices with two different point symmetries belong: the hexagonal lattice type hP with point symmetry and the rhombohedral lattice type hR with point symmetry . In and higher dimensions, such cases are much more abundant.

Usually, the same type of coordinate system, the so-called `conventional coordinate system', is used for all space groups of a crystal family, for instance `hexagonal axes' for both hexagonal and rhombohedral lattices; cf. Chapters 2.1 , 2.2 and 9.1 . Other coordinate systems, however, may be used when convenient. To avoid confusion, the use of unconventional coordinate systems should be stated explicitly.

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.