Difference between revisions of "Atomic modulation function"
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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 | 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 | ||
| − | <math>r(n,j) = n~+~r_j+u_j | + | <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> | + | 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''. | + | <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^* | + | <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. | ||
[[Category: Fundamental crystallography]] | [[Category: Fundamental crystallography]] | ||
Revision as of 12:29, 12 May 2017
Fonction de modulation atomique (Fr). Funzione di modulazione atomica (It). 原子変調関数 (Ja).
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.