Phase of a modulation - Revision history

BrianMcMahon: Added German translation (U. Mueller)

2017-11-17T09:42:20Z

Added German translation (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-16T12:41:36Z

Style edits to align with printed edition

MassimoNespolo: lang, links

2015-03-20T13:50:19Z

lang, links

MassimoNespolo: /* See also */ cat + 1 broken link fixed and 1 broken link removed

2015-02-26T07:57:10Z

‎See also: cat + 1 broken link fixed and 1 broken link removed

BrianMcMahon at 17:11, 7 February 2012

2012-02-07T17:11:33Z

BrianMcMahon at 17:00, 7 February 2012

2012-02-07T17:00:31Z

TedJanssen at 15:22, 19 May 2009

2009-05-19T15:22:15Z

TedJanssen at 18:59, 18 May 2009

2009-05-18T18:59:22Z

TedJanssen at 18:58, 18 May 2009

2009-05-18T18:58:29Z

TedJanssen at 18:56, 18 May 2009

2009-05-18T18:56:49Z

← Older revision		Revision as of 09:42, 17 November 2017
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−	<font color="blue">Phase d'une modulation</font> (''Fr''). <font color="black">Fase di une modulazione</font> (''It''). <font color="purple">変調の位相</font> (''Ja'').	+	<font color="blue">Phase d'une modulation</font> (''Fr''). <font color="red">Modulationsphase</font> (''Ge''). <font color="black">Fase di une modulazione</font> (''It''). <font color="purple">変調の位相</font> (''Ja'').
		+

	== Definition ==		== Definition ==

@@ Line 16: / Line 16: @@
 and the displacement is given by
-<math>u_{ n j\alpha}~=~\sum_{ k\in M^*} \hat{u}_{j\alpha}( k)\exp \left( 2\pi i \sum_m h_m  a_m^*.(n+r_j)+\phi_{j\alpha}\right),~~~(h_1,\dots,h_n\neq 0,\dots,0).</math>
+<math display="block">u_{n j\alpha} = \sum_{k\in M^*} \hat{u}_{j\alpha}( k)  \exp \Big( 2\pi  i \sum_m h_m  a_m^*.(n+r_j)+\varphi_{j\alpha}\Big), \quad(h_1,\dots,h_n\neq 0,\dots,0).</math>
-For the simplest case with one modulation vector, one polarization direction and one atom per unit cell  this becomes
+For the simplest case, with one modulation vector, one polarization direction and one atom per unit cell, this becomes
-<math> u_n~=~\hat{ u} ( k)\exp (2\pi i k. n+\phi)+c.c..</math>
+<math> u_n~=~\hat{ u} ( k)\exp (2\pi i k. n+\varphi)+c.c.</math>
 Here &phi; is the  ''phase of the modulation''.
 The embedded structure in superspace is
-<math>( n+\hat{ u}( k) \exp (2\pi i  k.n+\phi +r_I)+c.c,~r_I).</math>
+<math>( n+\hat{ u}( k) \exp (2\pi i  k.n+\varphi +r_I)+c.c,~r_I).</math>
 r<sub>''I''</sub> is the internal coordinate, which changes the phase of the modulation. (In the literature
@@ Line 36: / Line 36: @@
 Then the embedding has components
 <math>
- \left(  n_{j\alpha}+ r_{j\alpha}+\sum_{ k\in M^*} \hat{u}_{j\alpha}( k)\exp \left[ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j\alpha}+\sum_m h_m a_{Im}^*.r_{I}\right],~~{\rm r}_{I}\right)</math>
+<math>\Big( n_{j\alpha}+ r_{j\alpha}+\sum_{k\in M^*} \hat{u}_{j\alpha}( k)
 Each plane wave for the modulation has a phase <math>\phi_{j\alpha}</math>  which is changed by changing the internal component   <math>r_I</math>, an (''n-m'')-dimensional vector in internal space.
@@ Line 44: / Line 45: @@
 For the simplest case one has
-<math>p( n)~=~\hat{p} ( k)\exp (2\pi i  k. n+\phi)+ c.c.</math>
+<math>p( n)~=~\hat{p} ( k)\exp (2\pi i  k. n+\varphi)+ c.c.</math>
 The embedding is the function
-<math>p( n, r_I)~=~\hat{\rho}( k) \exp (2\pi i  k.n+\phi +r_I)+c.c. </math>
+<math>p( n, r_I)~=~\hat{\rho}( k) \exp (2\pi i  k.n+\varphi +r_I)+c.c. </math>
 In the general case
-<math>p( n)_{j}~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left( 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j}\right),~~~(h_1,\dots,h_n\neq 0,\dots,0).</math>
+<math display="block">p( n)_{j} = \sum_{k\in M^*} \hat{p}_{j}( k)\exp \Big( 2\pi i \sum_m
 and the embedding is
-<math>p( n, r_I)_j~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left[ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j}+\sum_m h_m a_{Im}^*.r_{I}\right].</math>
+<math>p( n, r_I)_j~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left[ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\varphi_{j}+\sum_m h_m a_{Im}^*.r_{I}\right].</math>
 By a change of the internal coordinate ''r''<sub>''I''</sub> the phases <math>\phi_j</math> of the modulation functions change.
@@ Line 62: / Line 64: @@
 == See also ==
-[[incommensurate modulated structure]]
+*[[Incommensurate modulated structure]]
 [[Category: Fundamental crystallography]]

@@ Line 1: / Line 1: @@
-<font color="blue">Phase de la modulation</font> (Fr).
+<font color="blue">Phase d'une modulation</font> (''Fr''). <font color="black">Fase di une modulazione</font> (''It''). <font color="purple">変調の位相</font> (''Ja'').
 == Definition ==
-A modulation (see [[modulated crystal structure]]) is a periodic or quasiperiodic, scalar or vector function. In the former case, the phase measures
+A [[modulated crystal structure|modulation ]] is a periodic or [[quasiperiodicity|quasiperiodic]], scalar or vector function. In the former case, the phase measures the progress along one periodic direction. The periodic or quasiperiodic function may be developed into plane waves.
 The  ''phase(s) of the modulation'' is (are) the phase(s) of elementary plane waves which describe the modulation.

@@ Line 64: / Line 64: @@
 == See also ==
-[[modulation function]]
+[[incommensurate modulated structure]]
-[[incommensurate modulated crystal structure]].
+[[Category: Fundamental crystallography]]

@@ Line 1: / Line 1: @@
-<font color="blue">Phase de la modulation</font> (Fr.)
+<font color="blue">Phase de la modulation</font> (Fr).
 == Definition ==
@@ Line 8: / Line 8: @@
 The  ''phase(s) of the modulation'' is (are) the phase(s) of elementary plane waves which describe the modulation.
- '''Details'''
+== Details ==
 A  ''displacive modulation'' may be written as follows. For the ''j''th atom in the unit cell '''n''' the displacement
@@ Line 22: / Line 22: @@
 For the simplest case with one modulation vector, one polarization direction and one atom per unit cell  this becomes
- <math> u_n~=~\hat{ u} ( k)\exp (2\pi i k. n+\phi)+c.c..</math>
+<math> u_n~=~\hat{ u} ( k)\exp (2\pi i k. n+\phi)+c.c..</math>
 Here &phi; is the  ''phase of the modulation''.
 The embedded structure in superspace is
- <math>( n+\hat{ u}( k) \exp (2\pi i  k.n+\phi +r_I)+c.c,~r_I).</math>
+<math>( n+\hat{ u}( k) \exp (2\pi i  k.n+\phi +r_I)+c.c,~r_I).</math>
 r<sub>''I''</sub> is the internal coordinate, which changes the phase of the modulation. (In the literature
@@ Line 35: / Line 35: @@
 reciprocal lattice in  [[superspace]], and this has an external and an internal component:
- <math> k~=~\pi k_s~=~\sum_{i=1}^n h_i  (a_{Ei}^*,~ a_{Ii}^*).</math>
+<math> k~=~\pi k_s~=~\sum_{i=1}^n h_i  (a_{Ei}^*,~ a_{Ii}^*).</math>
 Then the embedding has components
@@ Line 46: / Line 46: @@
 For the simplest case one has
- <math>p( n)~=~\hat{p} ( k)\exp (2\pi i  k. n+\phi)+ c.c.</math>
+<math>p( n)~=~\hat{p} ( k)\exp (2\pi i  k. n+\phi)+ c.c.</math>
 The embedding is the function
@@ Line 54: / Line 54: @@
 In the general case
- <math>p( n)_{j}~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left( 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j}\right),~~~(h_1,\dots,h_n\neq 0,\dots,0).</math>
+<math>p( n)_{j}~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left( 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j}\right),~~~(h_1,\dots,h_n\neq 0,\dots,0).</math>
 and the embedding is
- <math>p( n, r_I)_j~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left[ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j}+\sum_m h_m a_{Im}^*.r_{I}\right].</math>
+<math>p( n, r_I)_j~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left[ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j}+\sum_m h_m a_{Im}^*.r_{I}\right].</math>
 By a change of the internal coordinate ''r''<sub>''I''</sub> the phases <math>\phi_j</math> of the modulation functions change.
- '''See also''':  [[modulation function]], [[incommensurate modulated crystal structure]].
+== See also ==

← Older revision		Revision as of 18:59, 18 May 2009
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	By a change of the internal coordinate ''r''<sub>''I''</sub> the phases φ<sub>j</sub> of the modulation functions change.		By a change of the internal coordinate ''r''<sub>''I''</sub> the phases φ<sub>j</sub> of the modulation functions change.

−	'''See also''': [[~~Modulation~~ function]], [[incommensurate modulated crystal structure]].	+	'''See also''': [[modulation function]], [[incommensurate modulated crystal structure]].

@@ Line 60: / Line 60: @@
 and the embedding is
-  <math>p( n, r_I)_{j}~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left[ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j}+\sum_m h_m a_{Im}^*.r}_{I}\right].</math>
+  <math>p( n, r_I)_j~=~\sum_{ k\in M^*} \hat{p}_{j}( k)\exp \left[ 2\pi i \sum_m h_m  a_m^*.( n+ r_j)+\phi_{j}+\sum_m h_m a_{Im}^*.r_{I}\right].</math>
 By a change of the internal coordinate ''r''<sub>''I''</sub> the phases &phi;<sub>j</sub> of the modulation functions change.
   '''See also''':  [[Modulation function]], [[incommensurate modulated crystal structure]].