# Phase of a modulation

### From Online Dictionary of Crystallography

Phase d'une modulation (*Fr*). Fase di une modulazione (*It*). 変調の位相 (*Ja*).

## Definition

A modulation is a periodic or quasiperiodic, scalar or vector function. In the former case, the phase measures the progress along one periodic direction. The periodic or quasiperiodic function may be developed into plane waves.
The *phase(s) of the modulation* is (are) the phase(s) of elementary plane waves which describe the modulation.

## Details

A *displacive modulation* may be written as follows. For the *j*th atom in the unit cell **n** the displacement
has *m* components **u**_{njα}, where α=*x,y,z* in three dimensions.
Then for a modulation of finite rank the Fourier module *M*^{*} consists of the reciprocal vectors

and the displacement is given by

For the simplest case, with one modulation vector, one polarization direction and one atom per unit cell, this becomes

Here φ is the *phase of the modulation*.
The embedded structure in superspace is

r_{I} is the internal coordinate, which changes the phase of the modulation. (In the literature
the internal coordinate *r*_{I} is sometimes denoted by *t*.)

For the general case, a vector **k** from the Fourier module is the projection of a vector of the
reciprocal lattice in superspace, and this has an external and an internal component:

Then the embedding has components

Each plane wave for the modulation has a phase φ_{jα} which is changed by changing the internal component *r*_{I}, an (*n-m*)-dimensional vector in internal space.

For an occupation modulation, the situation is quite similar, but simpler because the occupation function is a scalar. For the simplest case one has

The embedding is the function

In the general case

and the embedding is

By a change of the internal coordinate *r*_{I} the phases φ_{j} of the modulation functions change.