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

International Tables for Crystallography (2010). Vol. B, ch. 1.5, pp. 179-180   | 1 | 2 |

Section 1.5.4.2. Minimal domains

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

1.5.4.2. Minimal domains

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One can show that all irreps of [{\cal G}] can be built up from the irreps [\Gamma^{{\bf k}}] of [{\cal T}]. Moreover, to find all irreps of [{\cal G}] it is only necessary to consider one k vector from each orbit of k, cf. CDML, p. 31.

Definition. A simply connected part of the fundamental region which contains exactly one k vector of each orbit of k is called a minimal domain Φ.

The choice of the minimal domain is by no means unique. One of the difficulties in comparing the published data on irreps of space groups is due to the different minimal domains found in the literature.

The number of k vectors of each general k orbit in a fundamental region is always equal to the order of the point group [\overline{{\cal G}}] of [{\cal G}]; see Section 1.5.3.4[link]. Therefore, the volume of the minimal domain Φ in reciprocal space is [1/|\overline{{\cal G}}\, |] of the volume of the fundamental region. Now we can restrict the search for all irreps of [{\cal G}] to the k vectors within a minimal domain Φ.

In general, in representation theory of space groups the Brillouin zone is taken as the fundamental region and Φ is called a representation domain.4 Again, the volume of a representation domain in reciprocal space is [1/|\overline{{\cal G}}\,|] of the volume of the Brillouin zone. In addition, as the Brillouin zone contains for each k vector all k vectors of the star of k, by application of all symmetry operations [{\bi W} \in \overline{{\cal G}}] to Φ one obtains the Brillouin zone; cf. BC, p. 147. As the Brillouin zone may change its geometrical type depending on the lattice parameters, the type of the representation domain may also vary with varying lattice parameters; see the examples of Section 1.5.5[link].

The simplest crystal structures are the lattice-like structures that are built up of translationally equivalent points (centres of particles) only. For such a structure the point group [\overline{{\cal G}}] of the space group [{\cal G}] is equal to the point group [{\cal Q}] of its lattice L. Such point groups are called holohedral, the space group [{\cal G}] is called holosymmetric. There are seven holohedral point groups of three dimensions: [\overline{1}, 2/m, mmm, 4/mmm, \overline{3}m, 6/mmm] and [m\overline{3}m]. For the non-holosymmetric space groups [{\cal G}, \overline{{\cal G}} \,\lt \,{\cal Q}] holds. [The group [\cal Q] may have accidentally higher symmetry in addition, e.g. if the lattice of a monoclinic crystal structure is orthorhombic at a certain temperature, pressure and composition. In Section 8.2.6[link] of IT A it is shown how such accidental symmetry can be treated.]

In books on representation theory of space groups, holosymmetric space groups play a distinguished role. Their representation domains are called basic domains Ω. For holosymmetric space groups [\Omega = \Phi] holds. If [{\cal G}] is non-holosymmetric, i.e. [\overline{{\cal G}} \,\lt \,{\cal Q}] holds, Ω is defined by [{\cal Q}] and is smaller than the representation domain Φ by a factor which is equal to the index of [\overline{{\cal G}}] in [{\cal Q}]. In the literature these basic domains are considered to be of primary importance. In Miller & Love (1967)[link] only the irreps for the k vectors of the basic domains Ω are listed. Section 5.5 of BC and Davies & Cracknell (1976)[link] state that such a listing is not sufficient for the non-holosymmetric space groups because [\Omega \,\lt \,\Phi]. Section 5.5 of BC shows how to overcome this deficiency; Chapter 4 of CDML introduces new labels of k vectors for the parts of Φ not belonging to Ω.

The crystallographic analogue of the representation domain in direct space is the asymmetric unit, cf. IT A. According to its definition it is a simply connected smallest part of space from which by application of all symmetry operations of the space group the whole space is exactly filled. For each space-group type the asymmetric units of IT A belong to the same topological type independent of the lattice parameters. They are chosen as `simple' bodies by inspection rather than by applying clearly stated rules. Among the asymmetric units of the 73 symmorphic space-group types [{\cal G}_{0}] there are 31 parallelepipeds, 27 prisms (13 trigonal, 6 tetragonal and 8 pentagonal) for the non-cubic, and 15 pyramids (11 trigonal and 4 tetragonal) for the cubic [{\cal G}_{0}].

The asymmetric units of IT A – transferred to the groups [({\cal G})^{*}] of reciprocal space – are alternatives for the representation domains of the literature. They are formulated as closed bodies. Therefore, for inner points k, the asymmetric units of IT A fulfil the condition that each star of k is represented exactly once. For the surface, however, these conditions either have to be worked out or one gives up the condition of uniqueness and replaces exactly by at least in the definition of the minimal domain (see preceding footnote4). The examples of Section 1.5.5[link] show that the conditions for the boundary of the asymmetric unit and its special points, lines and planes are in many cases much easier to formulate than those for the representation domain. The asymmetric units used here are not always those of IT A. They may have been chosen differently in order to make them more convenient for our purpose or to adapt them more or less to the representation domains of CDML.

The k-vector coefficients. For each k vector one can derive a set of irreps of the space group [{\cal G}]. Different k vectors of a k orbit give rise to equivalent irreps. Thus, for the calculation of the irreps of the space groups it is essential to identify the orbits of k vectors in reciprocal space. This means finding the sets of all k vectors that are related by the operations of the reciprocal-space group [({\cal G})^{*}] according to equation (1.5.3.13)[link]. The classification of these k orbits can be done in analogy to that of the point orbits of the symmorphic space groups, as is apparent from the comparison of equations (1.5.3.14)[link] and (1.5.3.15)[link].

The classes of point orbits in direct space under a space group [{\cal G}] are well known and are listed in the space-group tables of IT A. They are labelled by Wyckoff letters. The stabilizer [{\cal S}_{\cal G}(X)] of a point X is called the site-symmetry group of X, and a Wyckoff position consists of all orbits for which the site-symmetry groups are conjugate subgroups of [{\cal G}]. Let [{\cal G}] be a symmorphic space group [{\cal G}_{0}]. Owing to the isomorphism between the reciprocal-space groups [({\cal G})^{*}] and the symmorphic space groups [{\cal G}_{0}], the complete list of special k vectors of [({\cal G})^{*}] is provided by the Wyckoff positions of [{\cal G}_{0}]. The groups [{\cal S}_{{\cal G}_{0}}(X)] and [\overline{{\cal G}}^{\,{\bf k}}] correspond to each other and the multiplicity of the Wyckoff position (divided by the number of centring vectors per unit cell for centred lattices) equals the number of arms of the star of k. Let the vectors t of L be referred to the conventional basis [({\bf a}_{i})^{T}] of the space-group tables of IT A, as defined in Chapters 2.1[link] and 9.1[link] of IT A. Then, for the construction of the irreps [\Gamma^{\bf k}] of [{\cal T}] the coefficients of the k vectors must be referred to the basis [({\bf a}_{j}^{*})] of reciprocal space dual to [({\bf a}_{i})^{T}] in direct space. These k-vector coefficients may be different from the conventional coordinates of [{\cal G}_{0}] listed in the Wyckoff positions of IT A.

Example

Let [{\cal G}] be a space group with an I-centred cubic lattice L, conventional basis [({\bf a}_{i})^{T}]. Then [{\bf L}^{*}] is an F-centred lattice. If referred to the conventional basis [({\bf a}_{j}^{*})] with [{\bf a}_{i}\cdot {\bf a}_{j}^{*} = 2\pi \delta_{ij}], the k vectors with coefficients 1 0 0, 0 1 0 and 0 0 1 do not belong to [{\bf L}^{*}] due to the `extinction laws' well known in X-ray crystallography. However, in the standard basis of [{\cal G}_{0}], isomorphic to [({\cal G})^{*}], the vectors 1 0 0, 0 1 0 and 0 0 1 point to the vertices of the face-centred cube and thus correspond to 2 0 0, 0 2 0 and 0 0 2 referred to the conventional basis [({\bf a}_{j}^{*})].

In the following, three bases and, therefore, three kinds of coefficients of k will be distinguished:

  • (1) Coefficients referred to the conventional basis [({\bf a}_{j}^{*})] in reciprocal space, dual to the conventional basis [({\bf a}_{i})^{T}] in direct space. The corresponding k-vector coefficients, [(k_{j})^{T}], will be called conventional coefficients.

  • (2) Coefficients of k referred to a primitive basis [({\bf a}_{pi}^{*})] in reciprocal space (which is dual to a primitive basis in direct space). The corresponding coefficients will be called primitive coefficients [(k_{pi})^{T}]. For a centred lattice the coefficients [(k_{pi})^{T}] are different from the conventional coefficients [(k_{i})^{T}]. In most of the physics literature related to space-group representations these primitive coefficients are used, e.g. by CDML.

  • (3) The coefficients of k referred to the conventional basis of [{\cal G}_{0}]. These coefficients will be called adjusted coefficients [(k_{ai})^{T}].

The relations between conventional and adjusted coefficients are listed for the different Bravais types of reciprocal lattices in Table 1.5.4.1[link], and those between adjusted and primitive coordinates in Table 1.5.4.2[link]. If adjusted coefficients are used, then IT A is as suitable for dealing with irreps as it is for handling space-group symmetry.

Table 1.5.4.1| top | pdf |
Conventional coefficients [(k_{i})^{T}] of k expressed by the adjusted coefficients [(k_{ai})] of IT A for the different Bravais types of lattices in direct space

Lattice types[k_{1}][k_{2}][k_{3}]
aP, mP, oP, tP, cP, rP [k_{a1}] [k_{a2}] [k_{a3}]
mA, oA [k_{a1}] [2k_{a2}] [2k_{a3}]
mC, oC [2k_{a1}] [2k_{a2}] [k_{a3}]
oF, cF, oI, cI [2k_{a1}] [2k_{a2}] [2k_{a3}]
tI [k_{a1} + k_{a2}] [-k_{a1} + k_{a2}] [2k_{a3}]
hP [k_{a1} - k_{a2}] [k_{a2}] [k_{a3}]
hR (hexagonal) [2k_{a1} - k_{a2}] [-k_{a1} + 2k_{a2}] [3k_{a3}]

Table 1.5.4.2| top | pdf |
Primitive coefficients [(k_{pi})^{T}] of k from CDML expressed by the adjusted coefficients [(k_{ai})] of IT A for the different Bravais types of lattices in direct space

Lattice types[k_{p1}][k_{p2}][k_{p3}]
aP, mP, oP, tP, cP, rP [k_{a1}] [k_{a2}] [k_{a3}]
mA, oA [k_{a1}] [k_{a2}\!-\!k_{a3}] [k_{a2}\!+\!k_{a3}]
mC, oC [k_{a1}\!-\!k_{a2}] [k_{a1}\!+\!k_{a2}] [k_{a3}]
oF, cF [k_{a2}\!+\!k_{a3}] [k_{a1}\!+\!k_{a3}] [k_{a1}\!+\!k_{a2}]
oI, cI [-k_{a1}\!+\!k_{a2}\!+\!k_{a3}] [k_{a1}\!-\!k_{a2}\!+\!k_{a3}] [k_{a1}\!+\!k_{a2}\!-\!k_{a3}]
tI [-k_{a1}\!+\!k_{a3}] [k_{a1}\!+\!k_{a3}] [k_{a2}\!-\!k_{a3}]
hP [k_{a1}\!-\!k_{a2}] [k_{a2}] [k_{a3}]
hR (hexagonal) [k_{a1}\!+\!k_{a3}] [-k_{a1}\!+\!k_{a2}\!+\!k_{a3}] [-k_{a2}\!+\!k_{a3}]

References

Davies, B. L. & Cracknell, A. P. (1976). Some comments on and addenda to the tables of irreducible representations of the classical space groups published by S. C. Miller and W. F. Love. Acta Cryst. A32, 901–903.
Miller, S. C. & Love, W. F. (1967). Tables of Irreducible Representations of Space Groups and Co-representations of Magnetic Space Groups. Boulder: Pruett Press.








































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