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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 42   | 1 | 2 |

## Section 1.3.2.6.1. Terminology

G. Bricognea

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#### 1.3.2.6.1. Terminology

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Let be the subset of 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 , and hence will be called the standard lattice in . Any other nonstandard' n-dimensional lattice Λ in is the image of this standard lattice by a general linear transformation.

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

Finally, we will denote by I the unit cube and by the subset