Difference between revisions of "Displacive modulation"
From Online Dictionary of Crystallography
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<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> | <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 | + | 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 |
| − | 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> | + | <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> |
[[Category: Fundamental crystallography]] | [[Category: Fundamental crystallography]] | ||
Revision as of 13:18, 13 May 2017
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.
[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]