# Atomic modulation function

### From Online Dictionary of Crystallography

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.