International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 177-178
Section 1.5.3.3. Representations of the translation group and the reciprocal lattice^{a}Departamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and ^{b}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
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, assuming 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 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.
References
Jansen, L. & Boon, M. (1967). Theory of Finite Groups. Applications in Physics: Symmetry Groups of Quantum Mechanical Systems. Amsterdam: North-Holland.Lomont, J. S. (1959). Applications of Finite Groups. New York: Academic Press.