Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 177-178   | 1 | 2 |

Section Representations of the translation group [{\cal T}] and the reciprocal lattice

M. I. Aroyoa* and H. Wondratschekb

aDepartamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: Representations of the translation group [{\cal T}] and the reciprocal lattice

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

Let [{\cal G}] 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 [({\bi I}, {\bi t}_{bi})^{N_{i}} = ({\bi I}, N_{i}) = ({\bi I}, {\bi o}) \eqno(]to hold, where [{\bi t}_{bi} = (1, 0, 0)], (0, 1, 0) or (0, 0, 1) and [N_{i}] is a large integer for i = 1, 2 or 3, respectively. Then for any lattice translation (I, t), [ ({\bi I}, {\bi N}{\bi t}) = ({\bi I}, {\bi o}) \eqno(]holds, where Nt is the column [(N_{1}t_{1}, N_{2}t_{2}, N_{3}t_{3})]. If the (infinitely many) translations mapped in this way onto (I, o) form a normal subgroup [{\cal T}_{1}] of [{\cal G}], then the mapping described by ([link] is a homomorphism. There exists a factor group [{\cal G}' = {\cal G}/{\cal T}_{1}] of [{\cal G}] relative to [{\cal T}_{1}] with translation subgroup [{\cal T}' = {\cal T}/{\cal T}_{1}] 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 [{\cal T}] is Abelian, [{\cal T}'] is also Abelian. Replacing the space group [{\cal G}] by [{\cal G}'] means that the especially well developed theory of representations of finite groups can be applied, cf. Lomont (1959)[link], Jansen & Boon (1967)[link]. For convenience, the prime ′ will be omitted and the symbol [{\cal G}] will be used instead of [{\cal G}']; [{\cal T}'] will be denoted by [{\cal T}] in the following.

Because [{\cal T}] (formerly [{\cal T}']) is Abelian, its irreps [\Gamma ({\cal T})] are one-dimensional and consist of (complex) roots of unity. Owing to equations ([link] and ([link], the irreps [\Gamma^{q_{1} q_{2} q_{3}} [({\bi I}, {\bi t})]] of [{\cal T}] have the form [ \Gamma^{q_{1} q_{2} q_{3}} [({\bi I}, {\bi t})] = \exp \left[-2\pi i \left(q_{1} {t_{1} \over N_{1}} + q_{2} {t_{2} \over N_{2}} + q_{3} {t_{3} \over N_{3}}\right)\right], \eqno(]where t is the column [(t_{1}, t_{2}, t_{3})], [q_{j} = 0, 1, 2, \ldots,N_{j} - 1], [j = 1, 2, 3], and [t_{k}] and [q_{j}] are integers.

Given a primitive basis [{\bf a}_{1}, {\bf a}_{2}, {\bf a}_{3}] of L, mathematicians and crystallographers define the basis of the dual or reciprocal lattice [{\bf L}^{*}] by [{\bf a}_{i} \cdot {\bf a}_{j}^{*} = \delta_{ij}, \eqno(]where [{\bf a}\cdot {\bf a}^{*}] is the scalar product between the vectors and [\delta_{ij}] is the unit matrix (see e.g. Chapter 1.1, Section 1.1.3[link] ). Texts on the physics of solids redefine the basis [{\bf a}_{1}^{*}, {\bf a}_{2}^{*}, {\bf a}_{3}^{*}] of the reciprocal lattice [{\bf L}^{*}], lengthening each of the basis vectors [{\bf a}_{j}^{*}] by the factor [2\pi]. Therefore, in the physicist's convention the relation between the bases of direct and reciprocal lattice reads (cf. BC, p. 86): [{\bf a}_{i} \cdot {\bf a}_{j}^{*} = 2\pi \delta_{ij}. \eqno(]In the present chapter only the physicist's basis of the reciprocal lattice is employed, and hence the use of [{\bf a}_{j}^{*}] should not lead to misunderstandings. The set of all vectors K,2 [{\bf K} = k_{1}{\bf a}_{1}^{*} + k_{2}{\bf a}_{2}^{*} + k_{3}{\bf a}_{3}^{*}, \eqno(][k_{i}] integer, is called the lattice reciprocal to L or the reciprocal lattice [{\bf L}^{*}].3

If one adopts the notation of IT A, Part 5, the basis of direct space is denoted by a row [({\bf a}_{1}, {\bf a}_{2}, {\bf a}_{3})^{T}], where [(\;)^{T}] means transposed. For reciprocal space, the basis is described by a column [({\bf a}_{1}^{*}, {\bf a}_{2}^{*}, {\bf a}_{3}^{*})].

To each lattice generated from a basis [({\bf a}_{i})^{T}] a reciprocal lattice is generated from the basis [({\bf a}_{j}^{*})]. Both lattices, L and [{\bf L}^{*}], can be compared most easily by referring the direct lattice L to its conventional basis [({\bf a}_{i})^{T}] as defined in Chapters 2.1[link] and 9.1[link] of IT A. In this case, the lattice L may be primitive or centred. If [({\bf a}_{i})^{T}] forms a primitive basis of L, i.e. if L is primitive, then the basis [({\bf a}_{j}^{*})] forms a primitive basis of [{\bf L}^{*}]. If L is centred, i.e. [({\bf a}_{i})^{T}] is not a primitive basis of L, then there exists a centring matrix P, [ 0 \,\lt\, \det ({\bi P}) \,\lt\, 1], by which three linearly independent vectors of L with rational coefficients are generated from those with integer coefficients, cf. IT A, Table[link] .

Moreover, P can be chosen such that the set of vectors [ ({\bf p}_{1}, {\bf p}_{2}, {\bf p}_{3})^{T} = ({\bf a}_{1}, {\bf a}_{2}, {\bf a}_{3})^{T} {\bi P} \eqno(]forms a primitive basis of L. Then the basis vectors [({\bf p}_{1}^{*}, {\bf p}_{2}^{*}, {\bf p}_{3}^{*})] of the lattice reciprocal to the lattice generated by [({\bf p}_{1}, {\bf p}_{2}, {\bf p}_{3})^{T}] are determined by [ ({\bf p}_{1}^{*}, {\bf p}_{2}^{*}, {\bf p}_{3}^{*}) = {\bi P}^{-1} ({\bf a}_{1}^{*}, {\bf a}_{2}^{*}, {\bf a}_{3}^{*}) \eqno(]and form a primitive basis of [{\bf L}^{*}].

Because [ \det ({\bi P}^{-1})\, \gt \,1], not all vectors K of the form ([link] belong to [{\bf L}^{*}]. If [k_{1}, k_{2}, k_{3}] are the (integer) coefficients of these vectors K referred to [({\bf a}_{j}^{*})] and [k_{p1}{\bf p}_{1}^{*} + k_{p2}{\bf p}_{2}^{*} + k_{p3}{\bf p}_{3}^{*}] are the vectors of [{\bf L}^{*}], then [ {\bf K} = (k_{j})^{T} ({\bf a}_{j}^{*}) = (k_{j})^{T} {\bi P}({\bf p}_{i}^{*}) = (k_{pi})^{T} ({\bf p}_{i}^{*})] is a vector of [{\bf L}^{*}] if and only if the coefficients [ (k_{p1}, k_{p2}, k_{p3})^{T} = (k_{1}, k_{2}, k_{3})^{T} {\bi P} \eqno(]are integers. In other words, [(k_{1}, k_{2}, k_{3})^{T}] has to fulfil the equation [ (k_{1}, k_{2}, k_{3})^{T} = (k_{p1}, k_{p2}, k_{p3})^{T} {\bi P}^{-1}. \eqno(]

As is well known, the Bravais type of the reciprocal lattice [{\bf L}^{*}] 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 [({\bf a}_{i})^{T}], then [ {\bi W}^{-1}] is the matrix of the same symmetry operation of the reciprocal lattice but referred to the dual basis [({\bf a}_{i}^{*})]. 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 [{\bf L}^{*}] relative to the symmetry elements of [\overline{{\cal G}}] and due to the reference to the different bases [({\bf a}_{i})^{T}] and [({\bf a}_{i}^{*})]. For example, if L has the point symmetry (Hermann–Mauguin symbol) [\overline{3}m1], then the symbol for the point symmetry of [{\bf L}^{*}] is [\overline{3}1m] and vice versa.


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.

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