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