# Displacive modulation

Modulation displacive (Fr). 変位型変調(Ja).

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.

$r( n,j)~=~ n+ r_j+ u_j( n+ r_j).$

The modulation function has a Fourier expansion

$u_j( r)~=~\sum_ k \hat{ u}( k) \exp (2\pi i k. r),~with~ k=\sum_{i=1}^n h_i a_i^*,$

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

$r(n,j)_{\alpha}~=~ n_{\alpha}+ r_{j\alpha}+A_{j\alpha} \sin \left(2\pi i q. n+ r_j)+\phi_{j\alpha}\right), (\alpha=x,y,z).$