Displacive modulation - Revision history

BrianMcMahon: Tidied translations and added German and Spanish (U. Mueller)

2017-11-10T13:51:10Z

Tidied translations and added German and Spanish (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-13T13:18:37Z

Style edits to align with printed edition

MassimoNespolo: lang, links

2015-03-20T13:56:28Z

lang, links

MassimoNespolo: cat

2015-02-26T07:55:52Z

cat

BrianMcMahon at 12:05, 6 February 2012

2012-02-06T12:05:05Z

TedJanssen at 18:19, 18 May 2009

2009-05-18T18:19:34Z

TedJanssen at 15:47, 18 May 2009

2009-05-18T15:47:05Z

TedJanssen at 13:58, 18 May 2009

2009-05-18T13:58:05Z

New page

[[Displacive Modulation]]

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'''),
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>

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>

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>

← Older revision		Revision as of 13:51, 10 November 2017
Line 1:		Line 1:
−	<~~Font~~ color="blue">Modulation displacive </font>(''Fr''). <~~Font~~ color="purple">変位型変調</font>(''Ja'').	+	<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'').

@@ 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>
-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
-<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]]

@@ Line 1: / Line 1: @@
-<Font color="blue">Modulation displacive </font>(Fr.)
+<Font color="blue">Modulation displacive </font>(''Fr''). <Font color="purple">変位型変調</font>(''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''  '''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.
 <math> r( n,j)~=~ n+ r_j+ u_j( n+ r_j).</math>

← Older revision		Revision as of 07:55, 26 February 2015
Line 16:		Line 16:

	<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]]

← Older revision		Revision as of 12:05, 6 February 2012
Line 1:		Line 1:
−	~~[[Displacive Modulation]]~~
−
−
	<Font color="blue">Modulation displacive </font>(Fr.)		<Font color="blue">Modulation displacive </font>(Fr.)
−
−	~~'''Definition'''~~

	For a ''displacively modulated crystal phase'', the positions of the atoms are		For a ''displacively modulated crystal phase'', the positions of the atoms are

@@ Line 11: / Line 11: @@
 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({\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>
+<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]]
 Modulation displacive (Fr.)
   '''Definition'''