Phase of a modulation
From Online Dictionary of Crystallography
Phase d'une modulation (Fr). Modulationsphase (Ge). 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 jth atom in the unit cell n the displacement has m components unjα, where α=x,y,z in three dimensions. Then for a modulation of finite rank the Fourier module M* consists of the reciprocal vectors
[math] k~=~\sum_{i=1}^n h_i a_i^*,[/math]
and the displacement is given by
[math]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 φ 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]
rI is the internal coordinate, which changes the phase of the modulation. (In the literature the internal coordinate rI 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]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 rI the phases [math]\phi_j[/math] of the modulation functions change.