International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 178-179   | 1 | 2 |

Section 1.5.3.4. Irreducible representations of space groups and the reciprocal-space group

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:  wmpararm@lg.ehu.es

1.5.3.4. Irreducible representations of space groups and the reciprocal-space group

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Let [({\bf a}_{i})^{T}] be a conventional basis of the lattice L of the space group [{\cal G}]. From (1.5.3.6)[link], [k_{i} = q_{i}/N_{i}] and [{\bf k} = {\textstyle\sum_{k = 1}^{3}} k_{i} {\bf a}_{i}^{*}], equation (1.5.3.4)[link] can be written [ \Gamma^{q_{1} q_{2} q_{3}} [({\bi I}, {\bi t})] = \Gamma^{{\bf k}} [({\bi I},{\bi t})] = \exp [-i {\bf k} \cdot {\bf t}]. \eqno(1.5.3.12)]Equation (1.5.3.12)[link] has the same form if a primitive basis [({\bf p}_{i})^{T}] of L has been chosen. In this case, the vector k is given by [{\bf k} = {\textstyle\sum_{i = 1}^{3}} k_{pi} {\bf p}_{i}^{*}] with rational coefficients [k_{pi}].

Let a primitive basis [({\bf p}_{i})^{T}] be chosen for the lattice L. The set of all vectors k (known as wavevectors) forms a discontinuous array. Consider two wavevectors k and [{\bf k}' = {\bf k} + {\bf K}], where K is a vector of the reciprocal lattice [{\bf L}^{*}]. Obviously, k and [{\bf k}'] describe the same irrep of [{\cal T}]. Therefore, to determine all irreps of [{\cal T}] 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 [{\bf L}^{*}] fills the reciprocal space without gap or overlap. Such a region is called a fundamental region of [{\bf L}^{*}]. (The nomenclature in literature is not quite uniform. We follow here widely adopted definitions.)

The fundamental region of [{\bf L}^{*}] is not uniquely determined. Two types of fundamental regions are of interest in this chapter:

  • (1) The first Brillouin zone is that range of k space around o for which [|{\bf k}| \leq |{\bf K} - {\bf k}|] holds for any vector [{\bf K} \in {\bf L}^{*}] (Wigner–Seitz cell or domain of influence in k space). Visually, it is the region of endpoints of the k vectors that are nearer to the origin than to any other point of the reciprocal lattice. The Brillouin zone is a centrosymmetric body bounded by centrosymmetric planes which bisect perpendicularly the connecting lines from the origin to the neighbouring reciprocal-lattice points. The Brillouin zone is used in books and articles on irreps of space groups.

  • (2) The crystallographic unit cell in reciprocal space, for short unit cell, is the set of all k vectors with [0 \leq k_{i} \,\lt \,1]. It corresponds to the unit cell used in crystallography for the description of crystal structures in direct space.

Let k be some vector according to (1.5.3.12) and W be the matrices of [\overline{{\cal G}}]. The following definitions are useful:

Definition. The set of all vectors [{\bf k}'] fulfilling the condition[{\bf k}' = {\bf k}{\bi W} + {\bf K}, \quad {\bi W} \in \overline{{\cal G}}, \quad {\bf K} \in {\bf L}^{*} \eqno(1.5.3.13)]is called the orbit of k.

Definition. The set of all matrices [{\bi W} \in \overline{{\cal G}}] for which[ {\bf k} = {\bf k}{\bi W} + {\bf K},\quad {\bf K} \in {\bf L}^{*} \eqno(1.5.3.14)]forms a group which is called the little co-group [\overline{{\cal G}} ^{\,{\bf k}}] of k. The vector k is called general if [\overline{{\cal G}}^{\,{\bf k}} = \{{\bi I}\}]; otherwise [\overline{{\cal G}}^{\,{\bf k}} \,\gt\, \{{\bi I}\}] and k is called special.

In words: The k vector is called general if its little co-group [\overline{\cal G}^{\bf k}] is the identity of [\overline{\cal G}]. Otherwise, [\overline{\cal G}^{\bf k}] is a non-trivial subgroup of the point group [\overline{\cal G}] of [\cal G], [\overline{\cal G}\, >\, \overline{\cal G}^{\bf k}], and k is called a special vector of the reciprocal space.

Equation (1.5.3.14)[link] for k resembles the equation [ {\bi x} = {\bi W}{\bi x} + {\bi t}, \quad {\bf t} \in {\bf L} \eqno(1.5.3.15)]by which the fixed points of the symmetry operation [({\bi W}, {\bi t})] of a symmorphic space group [{\cal G}_{0}] are determined. Indeed, the orbits of k defined by (1.5.3.13) correspond to the point orbits of [{\cal G}_{0}], the little co-group [\overline{{\cal G}}^{\,{\bf k}}] of k corresponds to the site-symmetry group of that point X whose coordinates [(x_{i})] have the same values as the vector coefficients [(k_{i})^{T}] of k.

Consider the coset decomposition of [\overline{{\cal G}}] relative to [\overline{{\cal G}} ^{\bf k}].

Definition. If [ \{{\bi W}_{m}\}] is a set of coset representatives of [\overline{{\cal G}}] relative to [\overline{{\cal G}}^{\,{\bf k}}], then the set [\{{\bf k}{\bi W}_{m}\}] is called the star of k and the vectors [ {\bf k}{\bi W}_{m}] are called the arms of the star.

The number of arms of the star of k is equal to the order [|\overline{{\cal G}} |] of the point group [\overline{{\cal G}}] divided by the order [|\overline{{\cal G}}^{\,{\bf k}}|] of the symmetry group [\overline{{\cal G}}^{\,{\bf k}}] of k. If k is general, then there are [|\overline{{\cal G}} |] vectors from the orbit of k in each fundamental region and [|\overline{{\cal G}} |] arms of the star. If k is special with little co-group [\overline{{\cal G}}^{\,{\bf k}}], 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 [|\overline{{\cal G}}\ |/|\overline{{\cal G}} ^{\bf k}|].

Definition. The group of all elements [({\bi W}, {\bi w}) \in {\cal G}] for which [{\bi W} \in \overline{{\cal G}}^{\,{\bf k}}] is called the little group [{\cal L} ^{\bf k}] of k.

The analogue of the little group [{\cal L} ^{{\bf k}}] is rarely considered in crystallography.

All symmetry operations of [{\cal G}_{0}] may be obtained as combinations of an operation that leaves the origin fixed with a translation of L, i.e. are of the kind [ ({\bi W}, {\bi t}) = ({\bi I}, {\bi t})({\bi W}, {\bi o})]. We now define the analogous group for the k vectors. Whereas [{\cal G}_{0}] 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.

Definition. The group [({\cal G})^{*}] which is the semidirect product of the point group [\overline{{\cal G}}] and the translation group of the reciprocal lattice [{\bf L}^{*}] of [{\cal G}] is called the reciprocal-space group of [{\cal G}].

By this definition, the reciprocal-space group [({\cal G})^*] is isomorphic to a symmorphic space group [{\cal G}_0]. The elements of [{\cal G}_0] are the operations [({\bi W},{\bi t}_{\bi K})=({\bi I},{\bi t}_{\bi K})({\bi W},{\bi o})] with [{\bi W}\in \overline{\cal G}] and [{\bi t}_{\bi K}\in {\bf L}] with the coefficients of K. In order to emphasize that [({\cal G})^{*}] 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 [({\cal G})^{*}] it follows that space groups of the same type define the same type of reciprocal-space group [({\cal G})^{*}]. Moreover, as [({\cal G})^{*}] 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 [({\cal G})^{*}]; for arithmetic crystal class see Section 1.5.3.2[link]. Following Wintgen (1941)[link], the types of reciprocal-space groups [({\cal G})^{*}] are listed for the arithmetic crystal classes of space groups, i.e. for all space groups [{\cal G}], in Appendix A1.5.1[link].

References

Wintgen, G. (1941). Zur Darstellungstheorie der Raumgruppen. Math. Ann. 118, 195–215.








































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