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, p. 750
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Unit cells are usually chosen according to the conventions mentioned in Chapter 9.1 so one might think that there is no need for another standard choice. This is not true, however; conventions based on symmetry do not always permit unambiguous choice of the unit cell, and unconventional descriptions of a lattice do occur. They are often chosen for good reasons or they may arise from experimental limitations such as may occur, for example, in high-pressure work. So there is a need for normalized descriptions of crystal lattices.
Accordingly, the reduced basis1 (Eisenstein, 1851; Niggli, 1928), which is a primitive basis unique (apart from orientation) for any given lattice, is at present widely used as a means of classifying and identifying crystalline materials. A comprehensive survey of the principles, the techniques and the scope of such applications is given by Mighell (1976). The present contribution merely aims at an exposition of the basic concepts and a brief account of some applications.
The main criterion for the reduced basis is a metric one: choice of the shortest three non-coplanar lattice vectors as basis vectors. Therefore, the resulting bases are, in general, widely different from any symmetry-controlled basis, cf. Chapter 9.1 .
References
Eisenstein, G. (1851). Tabelle der reducierten positiven quadratischen Formen nebst den Resultaten neuer Forschungen über diese Formen, insbesondere Berücksichtigung auf ihre tabellarische Berechung. J. Math. (Crelle), 41, 141–190.Google ScholarMighell, A. D. (1976). The reduced cell: its use in the identification of crystalline materials. J. Appl. Cryst. 9, 491–498.Google Scholar
Niggli, P. (1928). Kristallographische und strukturtheoretische Grundbegriffe. Handbuch der Experimentalphysik, Vol. 7, Part 1. Leipzig: Akademische Verlagsgesellschaft.Google Scholar