# Modulated crystal structure

### From Online Dictionary of Crystallography

Structure cristalline modulée (*Fr*). Modulierte Kristallstruktur (*Ge*). Struttura cristallina modulata (*It*). 変調構造 (*Ja*). Estructura cristalina modulada (*Sp*).

## Definition

A *modulated crystal structure* is a density (or atom arrangement) that may be obtained from a density
(or atom arrangement) with space-group symmetry by a finite density change (or finite displacement
of each atom, respectively) that is (quasi)periodic. A function or a displacement field is periodic
if it is invariant under a lattice of translations. Then its Fourier transform consists of
δ-peaks on a reciprocal lattice that spans the space and is nowhere dense. A quasiperiodic
function has a Fourier transform consisting of δ-peaks on a vector module of finite rank. This means that the peaks may be indexed with integers using a finite number of basis vectors. If the modulation consists of deviations from the basic structure in the positions,
the modulation is *displacive* (see figure). When the probability distribution deviates from
that in the basic structure the modulation is *occupational*.

Model for a displacively modulated crystal structure. The basic structure is two-dimensional rectangular, with lattice constants *a* and *b*, the modulation wave vector is in the *b*-direction, the wavelength of the periodic modulation is λ such that λ/*b* is an irrational number.