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Chemistry LibreTexts

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 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).