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

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

Section 1.5.4.1. Fundamental regions

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

1.5.4.1. Fundamental regions

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Different types of regions of reciprocal space may be chosen as fundamental regions, see Section 1.5.3.4[link]. The most frequently used type is the first Brillouin zone, which is the Wigner–Seitz cell (or Voronoi region, Dirichlet domain, domain of influence; cf. IT A, Chapter 9.1[link] ) of the reciprocal lattice. The Brillouin zone has the property of including the star of each k vector that belongs to it. Such a choice has three advantages:

  • (1) the Brillouin zone manifests the point symmetry of the reciprocal lattice [{\bf L}^{*}] of [{\cal G}];

  • (2) only k vectors of the boundary of the Brillouin zone may have little-group representations which are obtained from projective representations of the little co-group [\overline{{\cal G}}^{\,{\bf k}}], see e.g. BC, p. 156;

  • (3) for physical reasons, the Brillouin zone may be the most convenient fundamental region.

Of these advantages only the third may be essential. For the classification of irreps the minimal domains, see Section 1.5.4.2[link], are much more important than the fundamental regions. The minimal domain does not display the point-group symmetry anyway and the distinguished k vectors always belong to its boundary however the minimal domain may be chosen.

The serious disadvantage of the Brillouin zone is its often complicated shape which, moreover, depends on the lattice parameters of [{\bf L}^{*}], cf. Section 1.5.5.3[link]. The body that represents the Brillouin zone belongs to one of the five Fedorov polyhedra (more or less distorted versions of the cubic forms cube, rhombdodecahedron or cuboctahedron, of the hexagonal prism, or of the tetragonal elongated rhombdodecahedron). A more detailed description is that by the 24 symmetrische Sorten (Delaunay sorts) of Delaunay (1933a[link],b[link]), Figs. 11 and 12. According to this classification, the Brillouin zone may display three types of polyhedra of cubic, one type of hexagonal, two of rhombohedral, three of tetragonal, six of orthorhombic, six of monoclinic, and three types of triclinic symmetry.

For low symmetries the shape of the Brillouin zone is so variable that BC, p. 90 ff. chose a primitive unit cell of [{\bf L}^{*}] for the fundamental regions of triclinic and monoclinic crystals. This cell also reflects the point symmetry of [{\bf L}^{*}], it has six faces only, and although its shape varies with the lattice parameters all cells are affinely equivalent. For space groups of higher symmetry, BC and most other authors prefer the Brillouin zone.

Considering [{\bf L}^{*}] as a lattice, one can refer it to its conventional crystallographic lattice basis. Referred to this basis, the unit cell of [{\bf L}^{*}] is always an alternative to the Brillouin zone. With the exception of the hexagonal lattice, the unit cell of [{\bf L}^{*}] reflects the point symmetry, it has only six faces and its shape is always affinely equivalent for varying lattice parameters. For a space group [{\cal G}] with a primitive lattice, the above-defined conventional unit cell of [{\bf L}^{*}] is also primitive. If [{\cal G}] has a centred lattice, then [{\bf L}^{*}] also belongs to a type of centred lattice and the conventional cell of [{\bf L}^{*}] [not to be confused with the cell spanned by the basis [({\bf a}_{j}^{*})] dual to the basis [({\bf a}_{i})^{T}]] is larger than necessary. However, this is not disturbing because in this context the fundamental region is an auxiliary construction only for the definition of the minimal domain; see Section 1.5.4.2[link].

References

Delaunay, B. (1933a). Neue Darstellung der geometrischen Kristallographie. Z. Kristallogr. 84, 109–149.
Delaunay, B. (1933b). Berichtigung zur Arbeit “Neue Darstellung der geometrischen Kristallographie”. Z. Kristallogr. 85, 332.








































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