Irreducible representations of space groups and the reciprocal-space group

Aroyo, M. I.; Wondratschek, H.

doi:10.1107/97809553602060000553

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.5, p. 165 | 1 | 2 |

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

M. I. Aroyo^a ^* and H. Wondratschek^b

^a Faculty of Physics, University of Sofia, bulv. J. Boucher 5, 1164 Sofia, Bulgaria , and ^bInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: aroyo@phys.uni-sofia.bg

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), $[k_{i} = q_{i}/N_{i}]$ and $[{\bf k} = {\textstyle\sum_{k = 1}^{3}} k_{i} {\bf a}_{i}^{*}]$ , equation (1.5.3.4) 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) 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 $[{\bf k} = {\textstyle\sum_{i = 1}^{3}} k_{pi} {\bf p}_{i}^{*}]$ .

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). 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 $[\bar{{\cal G}}]$ . The following definitions are useful:

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

Definition. The set of all matrices $[{\bi W} \in \bar{{\cal G}}]$ 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 $[\bar{{\cal G}} ^{{\bf k}}]$ of k. The vector k is called general if $[\bar{{\cal G}} ^{{\bf k}} = \{{\bi I}\}]$ ; otherwise $[\bar{{\cal G}} ^{{\bf k}} \gt \{{\bi I}\}]$ and k is called special.

The little co-group $[\bar{{\cal G}} ^{{\bf k}}]$ is a subgroup of the point group $[\bar{{\cal G}}]$ . Consider the coset decomposition of $[\bar{{\cal G}}]$ relative to $[\bar{{\cal G}} ^{\bf k}]$ .

Definition. If $[ \{{\bi W}_{m}\}]$ is a set of coset representatives of $[\bar{{\cal G}}]$ relative to $[\bar{{\cal G}} ^{{\bf k}}]$ , 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 $[|\bar{{\cal G}} |]$ of the point group $[\bar{{\cal G}}]$ divided by the order $[|\bar{{\cal G}} ^{{\bf k}}|]$ of the symmetry group $[\bar{{\cal G}} ^{{\bf k}}]$ of k. If k is general, then there are $[|\bar{{\cal G}} |]$ vectors from the orbit of k in each fundamental region and $[|\bar{{\cal G}} |]$ arms of the star. If k is special with little co-group $[\bar{{\cal G}} ^{{\bf k}}]$ , 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 $[|\bar{{\cal G}}\ |/|\bar{{\cal G}} ^{\bf k}|]$ .

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

Equation (1.5.3.14) 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 $[\bar{{\cal G}} ^{{\bf k}}]$ 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, and the star of k corresponds to a set of representatives of X in $[{\cal G}_{0}]$ . (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 $[\bar{{\cal G}}]$ and the translation group of the reciprocal lattice $[{\bf L}^{*}]$ of $[{\cal G}]$ is called the reciprocal-space group of $[{\cal G}]$ .

The elements of $[{\cal G}^{*}]$ are the operations $[ ({\bi W}, {\bi K}) = ({\bi I}, {\bi K})({\bi W}, {\bi o})]$ with $[{\bi W} \in \bar{{\cal G}}]$ and $[{\bf K} \in {\bf L}^{*}]$ . 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. Following Wintgen (1941), 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 1.5.1.

References

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

International Tables for Crystallography (2006). Vol. B. ch. 1.5, p. 165