International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 1.5, p. 191
Section 1.5.6. Conclusions^{a}Departamento de Fisíca de la Materia Condensada, Facultad de Cienca y Technología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain , and ^{b}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
International Tables for Crystallography Volume A can serve as a basis for the classification of irreps of space groups by using the concept of reciprocal-space groups. The main features of the crystallographic classification scheme are as follows.
Data on the independent parameter ranges are essential to make sure that exactly one k vector per orbit is represented in the representation domain or in the asymmetric unit. Such data are often much easier to calculate for the asymmetric unit of the unit cell than for the representation domain of the Brillouin zone, in particular if a uni-arm description has been chosen, cf. Section 1.5.5. Such data can not be found in the cited tables of irreps.
The uni-arm description unmasks those k vectors which lie on the boundary of the Brillouin zone but belong to a Wintgen position which also contains inner k vectors, see the example of the lines Λ and F in and . Such k vectors can not give rise to little-group representations obtained from projective representations of the little co-group .
The consideration of the basic domain in relation to the representation domain is unnecessary. It may even be misleading because special k-vector subspaces of frequently belong to more general types of k vectors in . Space groups with non-holohedral point groups can be referred to their reciprocal-space groups directly without reference to the types of irreps of the corresponding holosymmetric space group.
In principle both approaches are equivalent: the traditional one by Brillouin zone, basic domain and representation domain and the crystallographic one by unit cell and asymmetric unit. Moreover, it is not difficult to relate one approach to the other, see Figs. and Tables 1.5.5.1 to 1.5.5.7. The conclusions show that the crystallographic approach for the description of irreps of space groups has several advantages as compared with the traditional approach. Owing to these advantages, CDML have already accepted the crystallographic approach for triclinic and monoclinic space groups. However, the advantages are not restricted to such low symmetries. In particular, the simple boundary conditions and shapes of the asymmetric units result in simple equations for the boundaries and shapes of volume elements and facilitate numerical calculations, integrations etc. If there are special reasons to prefer k vectors inside or on the boundary of the Brillouin zone to those outside, then the advantages and disadvantages of both approaches have to be compared in order to find the optimal way to solve the problem.
The crystallographic approach may be realized in three different ways: