# Vector module

### From Online Dictionary of Crystallography

##### Revision as of 14:42, 20 November 2017 by BrianMcMahon (talk | contribs) (Added German translation (U. Mueller))

Synonyms: Z-module, Fourier module

Module vectoriel (*Fr*). Vektormodul (*Ge*). Modulo vettoriale (*It*).

## Definition

A *vector module* is the set of vectors spanned by a number *n* of basis vectors with integer coefficients. The basis vectors should be independent over the integers, which means that any linear combination [math]\sum_i m_i a_i[/math] with *m*_{i} integers is equal to zero if, and only if, all coefficients *m*_{i} are zero. The term Z-module is sometimes used to underline the condition that the coefficients are integers. The number of basis vectors is the *rank* of the vector module.

## Comment

An *n*-dimensional lattice in an *n*-dimensional vector space is an example
of a vector module, with rank *n*. In reciprocal space, the reciprocal lattice
corresponding to a crystallographic structure is a special case of a vector module.
The Bragg peaks for the crystal fall on the positions of the reciprocal lattice.
More generally, the Bragg peaks of an *m*-dimensional aperiodic crystal structure
belong to a vector module of rank *n*, larger than *m*. To indicate that this module
exists in reciprocal space, it is sometimes called the *Fourier module*.