Difference between revisions of "Vector module"
From Online Dictionary of Crystallography
BrianMcMahon (talk | contribs) m |
BrianMcMahon (talk | contribs) (Added German translation (U. Mueller)) |
||
| (3 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| − | < | + | Synonyms: Z-module, Fourier module |
| + | |||
| + | |||
| + | <font color="blue">Module vectoriel</font> (''Fr''). <font color="red">Vektormodul</font> (''Ge''). <font color="black">Modulo vettoriale</font> (''It''). | ||
| − | |||
== Definition == | == Definition == | ||
| Line 15: | Line 17: | ||
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 ''m''. To indicate that this module | belong to a vector module of rank ''n'', larger than ''m''. To indicate that this module | ||
| − | exists in | + | exists in reciprocal space, it is sometimes called the ''Fourier module''. |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Latest revision as of 14:42, 20 November 2017
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 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. To indicate that this module exists in reciprocal space, it is sometimes called the Fourier module.