Difference between revisions of "Displacive modulation"
From Online Dictionary of Crystallography
BrianMcMahon (talk | contribs) m |
BrianMcMahon (talk | contribs) (Tidied translations and added German and Spanish (U. Mueller)) |
||
| (3 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| − | < | + | <font color="blue">Modulation displacive</font> (''Fr''). <font color="red">Displazive Modulation</font> (''Ge''). <font color="purple">変位型変調</font> (''Ja''). <font color="green">Modulación displaciva</font> (''Sp''). |
| + | |||
For a ''displacively modulated crystal phase'', the positions of the atoms are | 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 | displaced from those of a basis structure with space group symmetry (an ordinary | ||
| − | crystal). The displacements are given by the | + | [[crystal]]). The displacements are given by the [[atomic modulation function]] '''u'''<sub>j</sub>('''r'''), where ''j'' indicates the ''j''th atom in the [[unit cell]] of the basic structure. |
| − | where ''j'' indicates the ''j''th atom in the unit cell of the basic structure. | ||
<math> r( n,j)~=~ n+ r_j+ u_j( n+ r_j).</math> | <math> r( n,j)~=~ n+ r_j+ u_j( n+ r_j).</math> | ||
| Line 12: | Line 12: | ||
<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> | ||
| + | |||
| − | + | [[Category: Fundamental crystallography]] | |
Latest revision as of 13:51, 10 November 2017
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]