Displacive modulation

Displacive Modulation

Modulation displacive (Fr.)

Definition

For a displacively modulated crystal phase, the positions of the atoms are displaced from those of a basis structure with space group symmetry (an ordinary crystal). The displacements are given by the atomic modulation function u_j(r), where j indicates the jth atom in the unit cell of the basic structure.

[math]{\bf r}({\bf n},j)~=~{\bf n}+{\bf r}_j+{\bf u}_j({\bf n}+{\bf r}_j).[/math]

The modulation function has a Fourier expansion

[math]{\bf u}_j({\bf r})~=~\sum_{\bf k} \hat{\bf u}({\bf k}) \exp (2\pi i {\bf k}.{\bf r}),~{\rm with~}{\bf k}=\sum_{i=1}^n h_i {\bf a}_i^*,[/math]

with finite value of n. If n=1, the modulated structure is one-dimensionally modulated. A special case of a one-dimensionally modulated structure is

[math]r({\bf n},j)_{\alpha}~=~ n_{\alpha}+ r_{j\alpha}+A_{j\alpha} \sin \left(2\pi i {\bf q}.({\bf n}+{\bf r}_j)+\phi_{j\alpha}\right), (\alpha=x,y,z).[/math]