International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 9.2, pp. 750-755
https://doi.org/10.1107/97809553602060000518

Chapter 9.2. Reduced bases

P. M. de Wolffa‡

a Laboratorium voor Technische Natuurkunde, Technische Hogeschool, Delft, The Netherlands

Footnotes

‡ Deceased.

1 Very often, the term `reduced cell' is used to indicate this normalized lattice description. To avoid confusion, we shall use `reduced basis', since it is actually a basis and some of the criteria are related precisely to the difference between unit cells and vector bases.
2 In a book on reduced cells and on retrieval of symmetry information from lattice parameters, Gruber (1978)[link] reformulated the main condition (i)[link] as a minimum condition on the sum [s = a + b + c]. He also examined the surface areas of primitive unit cells in a given lattice, which are easily shown to be proportional to the corresponding sums [s^{*} = a^{*} + b^{*} + c^{*}] for the reciprocal bases. He finds that if there are two or more non-congruent cells with minimum s (`Buerger cells'), these cells always have different values of [s^{*}]. Gruber (1989)[link] proposes a new criterion to replace the conditions (9.2.2.2a)[link][link][link]–(9.2.2.5f)[link], viz that, among the cells with the minimum s value, the one with the smallest value of [s^{*}] be chosen (which need not be the absolute minimum of [s^{*}] since that may occur for cells that are not Buerger cells). The analytic form of this criterion is identical to (9.2.2.2a)–(9.2.2.5e); only (9.2.2.5f)[link] is altered. For further details, see Chapter 9.3[link] .