Displacive modulation

From Online Dictionary of Crystallography

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 uj(r), where j indicates the jth 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]