Incommensurate magnetic structure - Revision history

BrianMcMahon: Added German and Spanish translations (U. Mueller)

2017-11-14T17:15:27Z

Added German and Spanish translations (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-15T14:09:17Z

Style edits to align with printed edition

MassimoNespolo: lang, links

2015-03-20T14:06:49Z

lang, links

BrianMcMahon at 14:05, 6 February 2012

2012-02-06T14:05:56Z

BrianMcMahon at 14:53, 30 June 2010

2010-06-30T14:53:29Z

TedJanssen at 18:22, 18 May 2009

2009-05-18T18:22:58Z

TedJanssen at 15:48, 18 May 2009

2009-05-18T15:48:13Z

TedJanssen at 14:06, 18 May 2009

2009-05-18T14:06:13Z

New page

[[Incommensurate magnetic structure]]

Structure magnétique incommensurable

'''Definition'''

An ''incommensurate magnetic structure'' is a structure in which the magnetic moments are
ordered, but without periodicity that is commensurate with that of the nuclear structure
of the crystal. In particular, the magnetic moments have a spin density with wave vectors
that have at least one irrational component with respect to the reciprocal lattice
of the atoms. Or, in the case of localized moments, the spin function '''S'''( '''n'''+'''r'''j)
(where the $j$th atom has position {\bf r}$_j$ in the unit cell) has Fourier components
with irrational indices with respect to the reciprocal lattice of the crystal.

'''Details'''

When the atoms of the basic structure are at positions '''n'''+'''r'''j, where '''r'''j
is the position of the ''j''th atom in the unit cell, then the spin function for
an incommensurate magnetic structure is

<math>{\bf S}({\bf n}+{\bf r}_j)~=~\sum_{\bf k} \hat{\bf S}({\bf k})_j \exp \left(2\pi i {\bf k}.({\bf n}+{\bf r}_j)\right),~~{\bf k}=\sum_{i=1}^n h_i {\bf a}_i^*~~(h_i ~~{\rm integer}).</math>

This spin structure is incommensurate if one component of the basis vectors '''a'''i*
is irrational. Incommensurate magnetic structures may be linear, but occur as quite
complicated, like fan structures etc. as well. Especially, in rare-earth compounds very
complicated magnetic phase diagrams have been found.

← Older revision		Revision as of 17:15, 14 November 2017
Line 1:		Line 1:
−	<~~Font~~ color="blue">Structure magnétique incommensurable</font> (''Fr''). <~~Font~~ color="black">Struttura magnetica incommensurabile</font> (''It''). <~~Font~~ color="purple">非整合磁気構造</font> (''Ja'').	+	<font color="blue">Structure magnétique incommensurable</font> (''Fr''). <font color="red">Inkommensurable Magnetstruktur</font> (''Ge''). <font color="black">Struttura magnetica incommensurabile</font> (''It''). <font color="purple">非整合磁気構造</font> (''Ja''). <font color="green">Estructura magnética inconmensurable structure</font> (''Sp'').

	== Definition ==		== Definition ==

@@ Line 1: / Line 1: @@
-<Font color="blue">Structure magn&eacute;tique incommensurable</font> (''Fr'').　<Font color="black">Struttura magnetica incommensurabile</font> (''It''). <Font color="purple">非整合磁気構造</font> (''Ja'').
+<Font color="blue">Structure magn&eacute;tique incommensurable</font> (''Fr''). <Font color="black">Struttura magnetica incommensurabile</font> (''It''). <Font color="purple">非整合磁気構造</font> (''Ja'').
 == Definition ==
@@ Line 17: / Line 17: @@
 an incommensurate magnetic structure is
-<math> S( n+ r_j)~=~\sum_ {k} \hat{S}( k)_j \exp \left(2\pi i  k.( n+ r_j)\right),~~ k=\sum_{i=1}^n h_i  a_i^*~~(h_i ~~{\rm integer}).</math>
+<center>
+<math>S( n+ r_j) = \sum_ {k} \hat{S}( k)_j \exp \left[2\pi i  k.( n+
 This spin structure is incommensurate if one component of the basis vectors  '''a'''<sub>i</sub><sup>*</sup>
 is irrational. Incommensurate magnetic structures may be linear, but occur as quite

@@ Line 1: / Line 1: @@
-<Font color="blue">Structure magn&eacute;tique incommensurable</font> (Fr.)
+<Font color="blue">Structure magn&eacute;tique incommensurable</font> (''Fr'').　<Font color="black">Struttura magnetica incommensurabile</font> (''It''). <Font color="purple">非整合磁気構造</font> (''Ja'').
 == Definition ==
-An  ''incommensurate magnetic structure'' is a structure in which the magnetic moments are
+An  ''incommensurate magnetic structure'' is a [[crystal pattern|structure]] in which the magnetic moments are
-ordered, but without  periodicity that is commensurate with that of the nuclear structure
+ordered, but without periodicity that is commensurate with that of the nuclear structure
 of the crystal. In particular, the magnetic moments have a spin density with wave vectors
 that have at least one irrational component with respect to the reciprocal lattice
 of the atoms. Or, in the case of localized moments, the spin function  '''S'''('''n'''+'''r'''<sub>''j''</sub>)
-(where the ''j''th atom has position '''r'''<sub>''j''</sub> in the unit cell) has Fourier components
+(where the ''j''th atom has position '''r'''<sub>''j''</sub> in the [[unit cell]]) has Fourier components
 with irrational indices with respect to the reciprocal lattice of the crystal.

@@ Line 7: / Line 7: @@
 of the crystal. In particular, the magnetic moments have a spin density with wave vectors
 that have at least one irrational component with respect to the reciprocal lattice
-of the atoms. Or, in the case of localized moments, the spin function  '''S'''('''n'''+'''r'''<sub>j</sub>)
+of the atoms. Or, in the case of localized moments, the spin function  '''S'''('''n'''+'''r'''<sub>''j''</sub>)
-(where the $j$th atom has position {\bf r}$_j$ in the unit cell) has Fourier components
+(where the ''j''th atom has position '''r'''<sub>''j''</sub> in the unit cell) has Fourier components
 with irrational indices with respect to the reciprocal lattice of the crystal.
@@ Line 17: / Line 17: @@
 an incommensurate magnetic structure is
- <math> S( n+ r_j)~=~\sum_ {k} \hat{S}( k)_j \exp \left(2\pi i  k.( n+ r_j)\right),~~ k=\sum_{i=1}^n h_i  a_i^*~~(h_i ~~{\rm integer}).</math>
+<math> S( n+ r_j)~=~\sum_ {k} \hat{S}( k)_j \exp \left(2\pi i  k.( n+ r_j)\right),~~ k=\sum_{i=1}^n h_i  a_i^*~~(h_i ~~{\rm integer}).</math>
 This spin structure is incommensurate if one component of the basis vectors  '''a'''<sub>i</sub><sup>*</sup>
@@ Line 23: / Line 23: @@
 complicated, like fan structures etc. as well. Especially, in rare-earth compounds very
 complicated magnetic phase diagrams have been found.

@@ Line 20: / Line 20: @@
   an incommensurate magnetic structure is
-  <math>{\bf S}({\bf n}+{\bf r}_j)~=~\sum_{\bf k} \hat{\bf S}({\bf k})_j \exp \left(2\pi i {\bf k}.({\bf n}+{\bf r}_j)\right),~~{\bf k}=\sum_{i=1}^n h_i {\bf a}_i^*~~(h_i ~~{\rm integer}).</math>
+  <math> S( n+ r_j)~=~\sum_ {k} \hat{ S}( k)_j \exp \left(2\pi i  k.( n+ r_j)\right),~~ k=\sum_{i=1}^n h_i  a_i^*~~(h_i ~~{\rm integer}).</math>
   This spin structure is incommensurate if one component of the basis vectors  '''a'''<sub>i</sub><sup>*</sup>