Atomic modulation function

Atomic Modulation Function

Fonction de modulation atomique (Fr.)

Definition

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]{\bf r}({\bf n},j)~=~{\bf n}+{\bf r}_j+{\bf u}_j({\bf n}+{\bf r}_j).[/math]

The occupation probability to find an atom of species A at the position [math]{\bf n}+{\bf r}_j[/math] is [math]p_A({\bf 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]{\bf u}({\bf n},j)[/math] and [math]p_A({\bf 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]{\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^*,~~(h_i~~{\rm integers},~n~{\rm finite}.)[/math]

Modulation functions may be continuous or discontinuous.