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Difference between revisions of "Atomic modulation function"

From Online Dictionary of Crystallography

(Tidied translations and added German and Spanish (U. Mueller))
 
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[[Atomic Modulation Function]]
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<font color="blue">Fonction de modulation atomique</font> (''Fr''). <font color="red">Atomare Modulationsfunktion</font> (''Ge''). <font color="black">Funzione di modulazione atomica</font> (''It''). <font color="purple">原子変調関数</font> (''Ja''). <font color="green">Función de modulación atómica</font> (''Sp'').
  
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== Definition ==
  
<Font color="blue">Fonction de modulation atomique</font> (Fr.)
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A [[modulated crystal structure]] is a [[crystal pattern|structure]] that may be obtained from a crystalline system with [[space group]] symmetry, and therefore with [[lattice]] periodicity, by a regular displacement of atoms ([[displacive modulation]]) and/or change in the occupation probability of a site in the basic structure. The deviation from the positions in the basic structure are given by
  
'''Definition'''
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<math>r(n,j) = n~+~r_j+u_j (n+r_j).</math>
 
 
A modulated structure is a structure that may be obtained from a crystalline system
 
with space group symmetry, and therefore with lattice periodicity, by a regular
 
displacement of atoms (displacive modulation) and/or change in the occupation
 
probability of a site in the basic structure.  The deviation from the positions
 
in the basic structure are given by
 
 
 
<math>r(n,j) = n~+~r_j+u_j((n+r_j).</math>
 
  
 
The occupation probability to find an atom of species ''A'' at the position <math>n+r_j</math>
 
The occupation probability to find an atom of species ''A'' at the position <math>n+r_j</math>
is <math>p_A(n,j)</math>, where the sum over the species of the functions <math>p_A</math>
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is <math>p_A(n, j)</math>, where the sum over the species of the functions <math>p_A</math>
 
is one. Instead of a different species, one may have a vacancy. The functions
 
is one. Instead of a different species, one may have a vacancy. The functions
<math>u(n,j)</math> and <math>p_A(n,j)</math> are the  ''atomic modulation functions''.
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<math>u(n, j)</math> and <math>p_A(n, j)</math> are the  ''atomic modulation functions''.
 
For a crystal they should have Fourier modules of finite rank,  ''i.e''. the
 
For a crystal they should have Fourier modules of finite rank,  ''i.e''. the
 
functions have Fourier transforms with delta peaks on wave vectors  '''k''' of the form
 
functions have Fourier transforms with delta peaks on wave vectors  '''k''' of the form
  
  <math>k~=~\sum_{i=1}^n h_i a_i^*,~~(h_i~~{\rm integers},~n~{\rm finite}.)</math>
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<math>k~=~\sum_{i=1}^n h_i a_i^*~~(h_i~~{\rm integers},~n~{\rm finite}).</math>
  
 
Modulation functions may be continuous or discontinuous.
 
Modulation functions may be continuous or discontinuous.
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==See also==
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*[[Displacive modulation]]
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[[Category: Fundamental crystallography]]

Latest revision as of 18:07, 8 November 2017

Fonction de modulation atomique (Fr). Atomare Modulationsfunktion (Ge). Funzione di modulazione atomica (It). 原子変調関数 (Ja). Función de modulación atómica (Sp).

Definition

A modulated crystal structure is a structure that may be obtained from a crystalline system with space group symmetry, and therefore with lattice periodicity, by a regular displacement of atoms (displacive modulation) and/or change in the occupation probability of a site in the basic structure. The deviation from the positions in the basic structure are given by

[math]r(n,j) = n~+~r_j+u_j (n+r_j).[/math]

The occupation probability to find an atom of species A at the position [math]n+r_j[/math] is [math]p_A(n, j)[/math], where the sum over the species of the functions [math]p_A[/math] is one. Instead of a different species, one may have a vacancy. The functions [math]u(n, j)[/math] and [math]p_A(n, j)[/math] are the atomic modulation functions. For a crystal they should have Fourier modules of finite rank, i.e. the functions have Fourier transforms with delta peaks on wave vectors k of the form

[math]k~=~\sum_{i=1}^n h_i a_i^*~~(h_i~~{\rm integers},~n~{\rm finite}).[/math]

Modulation functions may be continuous or discontinuous.

See also