View source for Phase of a modulation

<font color="blue">Phase d'une modulation</font> (''Fr''). <font color="red">Modulationsphase</font> (''Ge''). <font color="black">Fase di une modulazione</font> (''It''). <font color="purple">変調の位相</font> (''Ja'').


== Definition ==

A [[modulated crystal structure|modulation ]] is a periodic or [[quasiperiodicity|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'''<sub>'''n'''j&alpha;</sub>, where &alpha;=''x,y,z'' in three dimensions.
Then for a modulation of finite rank the Fourier module ''M''<sup>*</sup> consists of the reciprocal vectors

<math> k~=~\sum_{i=1}^n h_i  a_i^*,</math>

and the displacement is given by

<math display="block">u_{n j\alpha} = \sum_{k\in M^*} \hat{u}_{j\alpha}( k)  \exp \Big( 2\pi  i \sum_m h_m  a_m^*.(n+r_j)+\varphi_{j\alpha}\Big), \quad(h_1,\dots,h_n\neq 0,\dots,0).</math>

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

<math> u_n~=~\hat{ u} ( k)\exp (2\pi i k. n+\varphi)+c.c.</math>

Here &phi; is the  ''phase of the modulation''. 
The embedded structure in superspace is

<math>( n+\hat{ u}( k) \exp (2\pi i  k.n+\varphi +r_I)+c.c,~r_I).</math>

r<sub>''I''</sub> is the internal coordinate, which changes the phase of the modulation. (In the literature
the internal coordinate ''r''<sub>''I''</sub> 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:

<math> k~=~\pi k_s~=~\sum_{i=1}^n h_i  (a_{Ei}^*,~ a_{Ii}^*).</math>

Then the embedding has components

<math>\Big( n_{j\alpha}+ r_{j\alpha}+\sum_{k\in M^*} \hat{u}_{j\alpha}( k)
         \exp [ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+  \varphi_{j\alpha}+\sum_m h_m a_{Im}^*.r_{I} ],\;\mathrm{r}_{I} \Big)</math>

Each plane wave for the modulation has a phase <math>\phi_{j\alpha}</math>  which is changed by changing the internal component   <math>r_I</math>, 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

<math>p( n)~=~\hat{p} ( k)\exp (2\pi i  k. n+\varphi)+ c.c.</math>

The embedding is the function

<math>p( n, r_I)~=~\hat{\rho}( k) \exp (2\pi i  k.n+\varphi +r_I)+c.c. </math>

In the general case

<math display="block">p( n)_{j} = \sum_{k\in M^*} \hat{p}_{j}( k)\exp \Big( 2\pi i \sum_m
  h_m  a_m^*.( n+ r_j)+\varphi_{j}\Big),\quad (h_1,\dots,h_n\neq 0,\dots,0)</math>

and the embedding is

<math>p( n, r_I)_j~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left[ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\varphi_{j}+\sum_m h_m a_{Im}^*.r_{I}\right].</math>

By a change of the internal coordinate ''r''<sub>''I''</sub> the phases <math>\phi_j</math> of the modulation functions change. 

== See also ==

*[[Incommensurate modulated structure]]

[[Category: Fundamental crystallography]]