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Difference between revisions of "Vector module"

From Online Dictionary of Crystallography

 
(Added German translation (U. Mueller))
 
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[[Vector module]]
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Synonyms: Z-module, Fourier module
  
Module vectoriel (Fr.)
 
  
Synonymous: Z-module
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<font color="blue">Module vectoriel</font> (''Fr''). <font color="red">Vektormodul</font> (''Ge''). <font color="black">Modulo vettoriale</font> (''It'').
  
'''Definition'''
 
  
A  ''vector module'' is the set of vectors spanned by a number ''n'' of basis vectors
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== Definition ==
with integer coefficients. The basis vectors should be independent over the integers,
 
which means that any linear combination <math>\sum_i m_i {\bf a}_i</math> with ''m''<sub>i</sub> integers
 
is equal to zero if, and only if, all coefficients ''m''<sub>i</sub> 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'''
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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''<sub>i</sub> integers is equal to zero if, and only if, all coefficients ''m''<sub>i</sub> 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.
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== Comment ==
  
 
An ''n''-dimensional lattice in an ''n''-dimensional vector space is an example
 
An ''n''-dimensional lattice in an ''n''-dimensional vector space is an example
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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 ''n''.
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belong to a vector module of rank ''n'', larger than ''m''. To indicate that this module
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exists in reciprocal space, it is sometimes called the ''Fourier module''.
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[[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.