# Displacive modulation

### From Online Dictionary of Crystallography

##### Revision as of 13:51, 10 November 2017 by BrianMcMahon (talk | contribs) (Tidied translations and added German and Spanish (U. Mueller))

Modulation displacive (*Fr*). Displazive Modulation (*Ge*). 変位型変調 (*Ja*). Modulación displaciva (*Sp*).

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 *j*th atom in the unit cell of the basic structure.

[math] r( n,j)~=~ n+ r_j+ u_j( n+ r_j).[/math]

The modulation function has a Fourier expansion

[math] u_j( r)~=~\sum_ k \hat{ u}( k) \exp (2\pi i k. r),~with~ k=\sum_{i=1}^n h_i 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(n,j)_{\alpha}~=~ n_{\alpha}+ r_{j\alpha}+A_{j\alpha} \sin [\left(2\pi i q. n+ r_j)+\phi_{j\alpha}\right)], (\alpha=x,y,z).[/math]