# Atomic modulation function

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 $r(n,j) = n~+~r_j+u_j (n+r_j).$

The occupation probability to find an atom of species A at the position n + rj is pA(n,j), where the sum over the species of the functions pA is one. Instead of a different species, one may have a vacancy. The functions u(n,j) and pA(n,j) 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 $k~=~\sum_{i=1}^n h_i a_i^*~~(h_i~~{\rm integers},~n~{\rm finite}).$

Modulation functions may be continuous or discontinuous.