International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 40-41   | 1 | 2 |

Section 1.3.2.6.1. Terminology

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.6.1. Terminology

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Let [{\bb Z}^{n}] be the subset of [{\bb R}^{n}] consisting of those points with (signed) integer coordinates; it is an n-dimensional lattice, i.e. a free Abelian group on n generators. A particularly simple set of n generators is given by the standard basis of [{\bb R}^{n}], and hence [{\bb Z}^{n}] will be called the standard lattice in [{\bb R}^{n}]. Any other `non-standard' n-dimensional lattice Λ in [{\bb R}^{n}] is the image of this standard lattice by a general linear transformation.

If we identify any two points in [{\bb R}^{n}] whose coordinates are congruent modulo [{\bb Z}^{n}], i.e. differ by a vector in [{\bb Z}^{n}], we obtain the standard n-torus [{\bb R}^{n}/{\bb Z}^{n}]. The latter may be viewed as [({\bb R}/{\bb Z})^{n}], i.e. as the Cartesian product of n circles. The same identification may be carried out modulo a non-standard lattice Λ, yielding a non-standard n-torus [{\bb R}^{n}/\Lambda]. The correspondence to crystallographic terminology is that `standard' coordinates over the standard 3-torus [{\bb R}^{3}/{\bb Z}^{3}] are called `fractional' coordinates over the unit cell; while Cartesian coordinates, e.g. in ångströms, constitute a set of non-standard coordinates.

Finally, we will denote by I the unit cube [[0, 1]^{n}] and by [C_{\varepsilon}] the subset [C_{\varepsilon} = \{{\bf x} \in {\bb R}^{n}\|x_{j}| \;\lt\; \varepsilon \hbox{ for all } j = 1, \ldots, n\}.]








































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