Difference between revisions of "Vector module"
From Online Dictionary of Crystallography
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The Bragg peaks for the crystal fall on the positions of the reciprocal lattice. | 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 | More generally, the Bragg peaks of an ''m''-dimensional aperiodic crystal structure | ||
| − | belong to a vector module of rank ''n'', larger than '' | + | belong to a vector module of rank ''n'', larger than ''m''. |
Revision as of 19:05, 18 May 2009
Module vectoriel (Fr.)
Synonymous: Z-module
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 mi integers is equal to zero if, and only if, all coefficients mi 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.