Difference between revisions of "Displacive modulation"

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 u_j(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]

@@ Line 1: / Line 1: @@
-[[Displacive Modulation]]
+<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'').
-Modulation displacive (Fr.)
- '''Definition'''
 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'''<sub>j</sub>('''r'''),
+[[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>{\bf r}({\bf n},j)~=~{\bf n}+{\bf r}_j+{\bf u}_j({\bf n}+{\bf r}_j).</math>
+<math> r( n,j)~=~ n+ r_j+ u_j( n+ r_j).</math>
 The modulation function has a Fourier expansion
-<math>{\bf u}_j({\bf r})~=~\sum_{\bf k} \hat{\bf u}({\bf k}) \exp (2\pi i {\bf k}.{\bf r}),~{\rm with~}{\bf k}=\sum_{i=1}^n h_i {\bf 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 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>
-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({\bf n},j)_{\alpha}~=~ n_{\alpha}+ r_{j\alpha}+A_{j\alpha} \sin \left(2\pi i {\bf q}.({\bf n}+{\bf r}_j)+\phi_{j\alpha}\right),   (\alpha=x,y,z).</math>
+[[Category: Fundamental crystallography]]

Difference between revisions of "Displacive modulation"

From Online Dictionary of Crystallography

Latest revision as of 13:51, 10 November 2017