Conclusions

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, pp. 172-176 | 1 | 2 |

Section 1.5.5.4. Conclusions

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

1.5.5.4. Conclusions

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As has been shown, IT A can serve as a basis for the classification of irreps of space groups by using the concept of reciprocal-space groups:

(a) The asymmetric units of IT A are minimal domains of k space which are in many cases simpler than the representation domains of the Brillouin zones. However, the asymmetric units of IT A are not designed particularly for this use, cf. Section 1.5.4.2. Therefore, it should be checked whether they are the optimal choice for this purpose. Otherwise, other asymmetric units could easily be introduced.
(b) All k-vector stars giving rise to the same type of irreps belong to the same Wintgen position. In the tables they are collected in one box and are designated by the same Wintgen letter.
(c) The Wyckoff positions of IT A, interpreted as Wintgen positions, provide a complete list of the special k vectors in the Brillouin zone; the site symmetry of IT A is the little co-group $[\bar{{\cal G}} ^{{\bf k}}]$ of k; the multiplicity per primitive unit cell is the number of arms of the star of k.
(d) The Wintgen positions with 0, 1, 2 or 3 variable parameters correspond to special k-vector points, k-vector lines, k-vector planes or to the set of all general k vectors, respectively.
(e) The complete set of types of irreps is obtained by considering the irreps of one k vector per Wintgen position in the uni-arm description or one star of k per Wintgen position otherwise. A complete set of inequivalent irreps of $[{\cal G}]$ is obtained from these irreps by varying the parameters within the asymmetric unit or the representation domain of $[{\cal G}^{*}]$ .
(f) For listing each irrep exactly once, the calculation of the parameter range of k is often much simpler in the asymmetric unit of the unit cell than in the representation domain of the Brillouin zone.
(g) The consideration of the basic domain Ω in relation to the representation domain Φ is unnecessary. It may even be misleading, because special k-vector subspaces of Ω frequently belong to more general types of k vectors in Φ. Space groups $[{\cal G}]$ with non-holohedral point groups can be referred to their reciprocal-space groups $[{\cal G}^{*}]$ directly without reference to the types of irreps of the corresponding holosymmetric space group. If Ω is used, and if the representation domain Φ is larger than Ω, then in most cases the irreps of Φ can be obtained from those of Ω by extending the parameter ranges of k.
(h) The classification by Wintgen letters facilitates the derivation of the correlation tables for the irreps of a group–subgroup chain. The necessary splitting rules for Wyckoff (and thus Wintgen) positions are well known.

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 of IT A. Moreover, it is not difficult to relate one approach to the other, see the figures and Tables 1.5.5.1 to 1.5.5.4. The conclusions show that the crystallographic approach for the description of irreps of space groups has several advantages as compared to 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 again in order to find the optimal method for the solution of the problem.

The crystallographic approach may be realized in three different ways:

(1) In the uni-arm description one lists each k-vector star exactly once by indicating the parameter field of the representing k vector. Advantages are the transparency of the presentation and the relatively small effort required to derive the list. A disadvantage may be that there are protruding flagpoles or wings. Points of these lines or planes are no longer neighbours of inner points (an inner point has a full three-dimensional sphere of neighbours which belong to the asymmetric unit).
(2) In the compact description one lists each k vector exactly once such that each point of the asymmetric unit is either an inner point itself or has inner points as neighbours. Such a description may not be uni-arm for some Wintgen positions, and the determination of the parameter ranges may become less straightforward. Under this approach, all points fulfil the conditions for the asymmetric units of IT A, which are always closed. The boundary conditions of IT A have to be modified: in reality the boundary is not closed everywhere; there are frequently open parts (see Section 1.5.5.3).
(3) In the non-unique description one gives up the condition that each k vector is listed exactly once. The uni-arm and the compact descriptions are combined but the equivalence relations (∼) are stated explicitly for those k vectors which occur in more than one entry. Such tables are most informative and not too complicated for practical applications.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.5, pp. 172-176