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

International Tables for Crystallography (2006). Vol. A. ch. 9.3, p. 756

Section 9.3.1. Further kinds of reduced cells

B. Grubera

aDepartment of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, CZ-11800 Prague 1, Czech Republic

9.3.1. Further kinds of reduced cells

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In Section 9.2.2,[link] a `reduced basis' of a lattice is defined which permits a unique representation of this lattice. It was introduced into crystallography by Niggli (1928)[link] and incorporated into International Tables for X-ray Crystallography (1969)[link], Vol. I. Originating from algebra (Eisenstein, 1851[link]), a reduced basis is defined in a rather complicated manner [conditions ([link] ) to ( in Section 9.2.2[link] ] and lacks any geometrical meaning. A cell spanned by a reduced basis is called the Niggli cell.

However, unique primitive cells may be introduced also in other ways that – unlike the Niggli cell1 – have significant geometrical features based mainly on extremal principles (Gruber, 1989[link]). We shall describe some of them below.

If a (primitive) cell of the lattice L fulfils the condition [a + b + c = \min] on the set of all primitive cells of L, we call it a Buerger cell. This cell need not be unique with regard to its shape in the lattice. There exist lattices with 1, 2, 3, 4 and 5 (but not more) Buerger cells differing in shape. The uniqueness can be achieved by various additional conditions. In this way, we can arrive at the following four reduced cells:

  • (i) the Buerger cell with minimum surface;2

  • (ii) the Buerger cell with maximum surface;

  • (iii) the Buerger cell with minimum deviation;3

  • (iv) the Buerger cell with maximum deviation.

Equivalent definitions can be obtained by replacing the term `surface' in (i)[link] and (ii)[link] by the expression [\sin \alpha + \sin \beta + \sin \gamma] or [\sin \alpha \sin \beta \sin \gamma,] and by replacing the `deviation' in (iii) and (iv) by [|\cos \alpha | + |\cos \beta | + |\cos \gamma |] or [|\cos \alpha \cos \beta \cos \gamma |.] A Buerger cell can agree with more than one of the definitions [(\hbox{i}),\ (\hbox{ii}),\ (\hbox{iii}),\ (\hbox{iv}). \eqno(] For example, if a lattice has only one Buerger cell, then this cell agrees with all the definitions in ([link]. However, there exist also Buerger cells that are in agreement with none of them. Thus, the definitions ([link] do not imply a partition of Buerger cells into classes.

It appears that case (iv)[link] coincides with the Niggli cell. This is important because this cell can now be defined by a simple geometrical property instead of a complicated system of conditions.

Further reduced cells can be obtained by applying the definitions ([link] to the reciprocal lattice. Then, to a Buerger cell in the reciprocal lattice, there corresponds a primitive cell with absolute minimum surface4 in the direct lattice.

The reduced cells according to the definitions ([link] can be recognized by means of a table and found in the lattice by means of algorithms. Detailed mutual relationships between them have been ascertained.


First citationInternational Tables for X-ray Crystallography (1969). Vol. I, 3rd ed., edited by N. F. M. Henry & K. Lonsdale, pp. 530–535. Birmingham: Kynoch Press.Google Scholar
First citationEisenstein, 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 Scholar
First citationGruber, B. (1989). Reduced cells based on extremal principles. Acta Cryst. A45, 123–131.Google Scholar
First citationNiggli, P. (1928). Kristallographische und strukturtheoretische Grundbegriffe. Handbuch der Experimentalphysik, Vol. 7, Part 1. Leipzig: Akademische Verlagsgesellschaft.Google Scholar

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