Section 1.5.3. Basic concepts

Group theory is the proper tool for studying symmetry in science. The elements of the crystallographic groups are rigid motions (isometries) with regard to performing one after another. The set of all isometries that map an object onto itself always fulfils the group postulates and is called the symmetry or the symmetry group of that object; the isometry itself is called a symmetry operation. Symmetry groups of crystals are dealt with in this chapter. In addition, groups of matrices with regard to matrix multiplication (matrix groups) are considered frequently. Such groups will sometimes be called realizations or representations of abstract groups.

Many applications of group theory to physical problems are closely related to representation theory, cf. Rosen (1981) and references therein. In this section, matrix representations Γ of finite groups are considered. The concepts of homomorphism and matrix groups are of essential importance.

A group is a homomorphic image of a group if there exists a mapping of the elements of onto the elements of that preserves the multiplication relation (in general several elements of are mapped onto one element of ): if and , then holds for all elements of and (the image of the product is equal to the product of the images). In the special case of a one-to-one mapping, the homomorphism is called an isomorphism.

A matrix group is a group whose elements are non-singular square matrices. The law of combination is matrix multiplication and the group inverse is the inverse matrix. In the following we will be concerned with some basic properties of finite matrix groups relevant to representations.

Let and be two matrix groups whose matrices are of the same dimension. They are said to be equivalent if there exists a (non-singular) matrix S such that holds. Equivalence implies isomorphism but the inverse is not true: two matrix groups may be isomorphic without being equivalent. According to the theorem of Schur-Auerbach, every finite matrix group is equivalent to a unitary matrix group (by a unitary matrix group we understand a matrix group consisting entirely of unitary matrices).

A matrix group is reducible if it is equivalent to a matrix group in which every matrix M is of the form see e.g. Lomont (1959), p. 47. The group is completely reducible if it is equivalent to a matrix group in which for all matrices R the submatrices X are O matrices (consisting of zeros only). According to the theorem of Maschke, a finite matrix group is completely reducible if it is reducible. A matrix group is irreducible if it is not reducible.

A (matrix) representation of a group is a homomorphic mapping of onto a matrix group . In a representation Γ every element is associated with a matrix . The dimension of the matrices is called the dimension of the representation.

The above-mentioned theorems on finite matrix groups can be applied directly to representations: we can restrict the considerations to unitary representations only. Further, since every finite matrix group is either completely reducible into irreducible constituents or irreducible, it follows that the infinite set of all matrix representations of a group is known in principle once the irreducible representations are known. Naturally, the question of how to construct all nonequivalent irreducible representations of a finite group and how to classify them arises.

Linear representations are especially important for applications. In this chapter only linear representations of space groups will be considered. Realizations and representations are homomorphic images of abstract groups, but not all of them are linear. In particular, the action of space groups on point space is a nonlinear realization of the abstract space groups because isometries and thus symmetry operations of space groups are nonlinear operations. The same holds for their description by matrix-column pairs (W, w),¹ by the general position, or by augmented matrices, see IT A, Part 8 . Therefore, the isomorphic matrix representation of a space group, mostly used by crystallographers and listed in the space-group tables of IT A as the general position, is not linear.

In crystallography one deals with real crystals. In many cases the treatment of the crystal is much simpler, but nevertheless describes the crystal and its properties very well, if the real crystal is replaced by an `ideal crystal'. The real crystal is then considered to be a finite piece of an undisturbed, periodic, and thus infinitely extended arrangement of particles or their centres: ideal crystals are periodic objects in three-dimensional point space , also called direct space. Periodicity means that there are translations among the symmetry operations of ideal crystals. The symmetry group of an ideal crystal is called its space group .

Space groups are of special interest for our problem because:

Therefore, space groups are introduced here in a slightly more detailed manner than the other concepts. In doing this we follow the definitions and symbolism of IT A, Part 8 .

To each space group belongs an infinite set of translations, the translation lattice of . The lattice forms an infinite Abelian invariant subgroup of . For each translation its translation vector is defined. The set of all translation vectors is called the vector lattice L of . Because of the finite size of the atoms constituting the real crystal, the lengths of the translation vectors of the ideal crystal cannot be arbitrarily small; rather there is a lower limit for their length in the range of a few Å.

When referred to a coordinate system , consisting of an origin O and a basis , the elements , i.e. the symmetry operations of the space group , are described by matrix-column pairs (W, w) with matrix part W and column part w. The translations of are represented by pairs , where I is the unit matrix and is the column of coefficients of the translation vector . The basis can always be chosen such that all columns and no other columns of translations consist of integers. Such a basis is called a primitive basis. For each vector lattice L there exists an infinite number of primitive bases.

The space group can be decomposed into left cosets relative to : The coset representatives form the finite set , with , where o is the column consisting of zeros only. The factor group is isomorphic to the point group of (called in books on representation theory) describing the symmetry of the external shape of the macroscopic crystal and being represented by the matrices . If V can be chosen such that all , then is called a symmorphic space group . A symmorphic space group can be recognized easily from its conventional Hermann–Mauguin symbol which does not contain any screw or glide component. In terms of group theory, a symmorphic space group is the semidirect product of and , cf. BC, p. 44. In symmorphic space groups (and in no others) there are site-symmetry groups which are isomorphic to the point group of .

Space groups can be classified into 219 (affine) space-group types either by isomorphism or by affine equivalence; the 230 crystallographic space-group types are obtained by restricting the transformations available for affine equivalence to those with positive determinant, cf. IT A, Section 8.2.1. Many important properties of space groups are shared by all space groups of a type. In such a case one speaks of properties of the type. For example, if a space group is symmorphic, then all space groups of its type are symmorphic, so that one normally speaks of a symmorphic space-group type.

With the concept of symmorphic space groups one can also define the arithmetic crystal classes: Let be a symmorphic space group referred to a primitive basis and its set of coset representatives with for all columns. To all those space groups can be assigned for which a primitive basis can be found such that the matrix parts of their sets V are the same as those of , only the columns may differ. In this way, to a type of symmorphic space groups , other types of space groups are assigned, i.e. the space-group types are classified according to the symmorphic space-group types. These classes are called arithmetic crystal classes of space groups or of space-group types.

There are 73 arithmetic crystal classes corresponding to the 73 types of symmorphic space groups; between 1 and 16 space-group types belong to an arithmetic crystal class. A matrix-algebraic definition of arithmetic crystal classes and a proposal for their nomenclature can be found in IT A, Section 8.2.3 ; see also Section 8.3.4 and Table 8.3.4.1 .

For representation theory we follow the terminology of BC and CDML.

Let be referred to a primitive basis. For the following, the infinite set of translations, based on discrete cyclic groups of infinite order, will be replaced by a (very large) finite set in the usual way. One assumes the Born–von Karman boundary conditions to hold, where , (0, 1, 0) or (0, 0, 1) and is a large integer for i = 1, 2 or 3, respectively. Then for any lattice translation (I, t), holds, where Nt is the column . If the (infinitely many) translations mapped in this way onto (I, o) form a normal subgroup of , then the mapping described by (1.5.3.3) is a homomorphism. There exists a factor group of relative to with translation subgroup which is finite and is sometimes called the finite space group.

Only the irreducible representations (irreps) of these finite space groups will be considered. The definitions of space-group type, symmorphic space group etc. can be transferred to these groups. Because is Abelian, is also Abelian. Replacing the space group by means that the especially well developed theory of representations of finite groups can be applied, cf. Lomont (1959), Jansen & Boon (1967). For convenience, the prime ′ will be omitted and the symbol will be used instead of ; will be denoted by in the following.

Because (formerly ) is Abelian, its irreps are one-dimensional and consist of (complex) roots of unity. Owing to equations (1.5.3.2) and (1.5.3.3), the irreps of have the form where t is the column , , and and are integers.

Given a primitive basis of L, mathematicians and crystallographers define the basis of the dual or reciprocal lattice by where is the scalar product between the vectors and is the unit matrix (see e.g. Chapter 1.1, Section 1.1.3 ). Texts on the physics of solids redefine the basis of the reciprocal lattice , lengthening each of the basis vectors by the factor . Therefore, in the physicist's convention the relation between the bases of direct and reciprocal lattice reads (cf. BC, p. 86): In the present chapter only the physicist's basis of the reciprocal lattice is employed, and hence the use of should not lead to misunderstandings. The set of all vectors K,² integer, is called the lattice reciprocal to L or the reciprocal lattice .³

If one adopts the notation of IT A, Part 5, the basis of direct space is denoted by a row , where means transposed. For reciprocal space, the basis is described by a column .

To each lattice generated from a basis a reciprocal lattice is generated from the basis . Both lattices, L and , can be compared most easily by referring the direct lattice L to its conventional basis as defined in Chapters 2.1 and 9.1 of IT A. In this case, the lattice L may be primitive or centred. If forms a primitive basis of L, i.e. if L is primitive, then the basis forms a primitive basis of . If L is centred, i.e. is not a primitive basis of L, then there exists a centring matrix P, , by which three linearly independent vectors of L with rational coefficients are generated from those with integer coefficients, cf. IT A, Table 5.1.3.1 .

Moreover, P can be chosen such that the set of vectors forms a primitive basis of L. Then the basis vectors of the lattice reciprocal to the lattice generated by are determined by and form a primitive basis of .

Because , not all vectors K of the form (1.5.3.7) belong to . If are the (integer) coefficients of these vectors K referred to and are the vectors of , then is a vector of if and only if the coefficients are integers. In other words, has to fulfil the equation

As is well known, the Bravais type of the reciprocal lattice is not necessarily the same as that of its direct lattice L. If W is the matrix of a (point-) symmetry operation of the direct lattice, referred to its basis , then is the matrix of the same symmetry operation of the reciprocal lattice but referred to the dual basis . This does not affect the symmetry because in a (symmetry) group the inverse of each element in the group also belongs to the group. Therefore, the (point) symmetries of a lattice and its reciprocal lattice are always the same. However, there may be differences in the matrix descriptions due to the different orientations of L and relative to the symmetry elements of and due to the reference to the different bases and . For example, if L has the point symmetry (Hermann–Mauguin symbol) , then the symbol for the point symmetry of is and vice versa.

Let be a conventional basis of the lattice L of the space group . From (1.5.3.6), and , equation (1.5.3.4) can be written Equation (1.5.3.12) has the same form if a primitive basis of L has been chosen. In this case, the vector k is given by .

Let a primitive basis be chosen for the lattice L. The set of all vectors k (known as wavevectors) forms a discontinuous array. Consider two wavevectors k and , where K is a vector of the reciprocal lattice . Obviously, k and describe the same irrep of . Therefore, to determine all irreps of it is necessary to consider only the wavevectors of a small region of the reciprocal space, where the translation of this region by all vectors of fills the reciprocal space without gap or overlap. Such a region is called a fundamental region of . (The nomenclature in literature is not quite uniform. We follow here widely adopted definitions.)

The fundamental region of is not uniquely determined. Two types of fundamental regions are of interest in this chapter:

Let k be some vector according to (1.5.3.12) and W be the matrices of . The following definitions are useful:

The little co-group is a subgroup of the point group . Consider the coset decomposition of relative to .

The number of arms of the star of k is equal to the order of the point group divided by the order of the symmetry group of k. If k is general, then there are vectors from the orbit of k in each fundamental region and arms of the star. If k is special with little co-group , then the number of arms of the star of k and the number of k vectors in the fundamental region from the orbit of k is .

Equation (1.5.3.14) for k resembles the equation by which the fixed points of the symmetry operation of a symmorphic space group are determined. Indeed, the orbits of k defined by (1.5.3.13) correspond to the point orbits of , the little co-group of k corresponds to the site-symmetry group of that point X whose coordinates have the same values as the vector coefficients of k, and the star of k corresponds to a set of representatives of X in . (The analogue of the little group is rarely considered in crystallography.)

All symmetry operations of may be obtained as combinations of an operation that leaves the origin fixed with a translation of L, i.e. are of the kind . We now define the analogous group for the k vectors. Whereas is a realization of the corresponding abstract group in direct (point) space, the group to be defined will be a realization of it in reciprocal (vector) space.

The elements of are the operations with and . In order to emphasize that is a group acting on reciprocal space and not the inverse of a space group (whatever that may mean) we insert a hyphen `-' between `reciprocal' and `space'.

From the definition of it follows that space groups of the same type define the same type of reciprocal-space group . Moreover, as does not depend on the column parts of the space-group operations, all space groups of the same arithmetic crystal class determine the same type of ; for arithmetic crystal class see Section 1.5.3.2. Following Wintgen (1941), the types of reciprocal-space groups are listed for the arithmetic crystal classes of space groups, i.e. for all space groups , in Appendix 1.5.1.