Superspace - Revision history

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

2017-11-20T09:07:32Z

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

BrianMcMahon: Style edits to align with printed edition

2017-05-22T12:54:52Z

Style edits to align with printed edition

BrianMcMahon at 11:49, 17 May 2017

2017-05-17T11:49:53Z

BrianMcMahon at 11:49, 17 May 2017

2017-05-17T11:49:02Z

BrianMcMahon: Style edits to align with printed edition

2017-05-17T11:46:22Z

Style edits to align with printed edition

MassimoNespolo: +cat, lang.

2014-12-04T05:48:52Z

+cat, lang.

GervaisChapuis at 12:48, 9 June 2014

2014-06-09T12:48:07Z

GervaisChapuis at 12:43, 9 June 2014

2014-06-09T12:43:41Z

BrianMcMahon at 15:25, 30 June 2010

2010-06-30T15:25:53Z

TedJanssen at 09:56, 19 May 2009

2009-05-19T09:56:08Z

← Older revision		Revision as of 09:07, 20 November 2017
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−	<~~Font~~ color="blue">Superespace, hyperespace </font>(''Fr''). <~~Font~~ color="black">Superspazio </font>(''It''); <~~Font~~ color="purple">超空間 </font>(''Ja'').	+	<font color="blue">Superespace, hyperespace</font> (''Fr''). <font color="red">Superraum</font> (''Ge''). <font color="black">Superspazio</font> (''It''). <font color="purple">超空間</font> (''Ja''). <font color="green">Superespacio</font> (''Sp'').
		+

	== Definition ==		== Definition ==

← Older revision		Revision as of 12:54, 22 May 2017
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−	<Font color="blue">Superespace, ~~Hyperespace~~ </font>(''Fr''). <Font color="black">Superspazio </font>(''It''); <Font color="purple">超空間 </font>(''Ja'').	+	<Font color="blue">Superespace, hyperespace </font>(''Fr''). <Font color="black">Superspazio </font>(''It''); <Font color="purple">超空間 </font>(''Ja'').

	== Definition ==		== Definition ==

@@ Line 39: / Line 39: @@
 &rho;<sub>s</sub>(r<sub>s</sub>) is lattice periodic. Its direct lattice &Sigma; is the dual of &Sigma;<sup>*</sup> and its basis
 vectors are ''a''<sub>is</sub> satisfying  ''a''<sub>si</sub>.''a''<sub>sj</sub><sup>*</sup> = &delta;<sub>ij</sub>. Therefore, the symmetry group
-of &rho;<sub>s</sub>(r<sub>s</sub>) is a space-group in the ''n''-dimensional superspace, where the dimension ''n'' is equal to the rank
+of &rho;<sub>s</sub>(r<sub>s</sub>) is a space group in the ''n''-dimensional superspace, where the dimension ''n'' is equal to the rank
 of the Fourier module. In the superspace the usual crystallographic notions and techniques may be applied. For point atoms, the function &rho;<sub>s</sub>(r<sub>s</sub>) is concentrated on surfaces
 of co-dimension equal to the dimension of the physical space (see figure).

@@ Line 43: / Line 43: @@
 of co-dimension equal to the dimension of the physical space (see figure).
-[[Image:IncEmbed.gif]]
+[[Image:IncEmbed.gif|frame|An example of the embedding of a one-dimensional modulated chain with two modulation wave vectors. It consists of an array of two-dimensional surfaces, with three-dimensional lattice periodicity. The intersection of the surfaces with the physical space (a line) gives the positions of the modulated structure.|right]]
 [[Category: Fundamental crystallography]]

@@ Line 1: / Line 1: @@
-<Font color="blue">Superespace, Hyperespace </font>(Fr.); <Font color="black">Superspazio </font>(It.); <Font color="purple">超空間 </font>(Ja.)
+<Font color="blue">Superespace, Hyperespace </font>(''Fr''). <Font color="black">Superspazio </font>(''It''); <Font color="purple">超空間 </font>(''Ja'').
 == Definition ==
@@ Line 8: / Line 8: @@
 == Applications ==
-Superspace is used to describe quasi-periodic structures (See [[aperiodic crystal]]). We denote the dimension of the physical space
+Superspace is used to describe quasi-periodic structures (see [[aperiodic crystal]]). We denote the dimension of the physical space
 by ''m''. Then a function &rho;('''r''') is quasi-periodic if its Fourier transform is given by
@@ Line 15: / Line 15: @@
 where the  ''Fourier module'' ''M''<sup>*</sup> is the set of vectors (Z-module) of the form
-<math> k~=~\sum_{i=1}^n h_i  a_i^* , ~~~(h_i~{\rm integers}),</math>
+<math> k~=~\sum_{i=1}^n h_i  a_i^*  ~~~(h_i~{\rm integers}),</math>
 for a basis  '''a'''<sub>i</sub><sup>*</sup> with the property that <math>\sum_i h_i  a_i^*=0</math> implies ''h''<sub>i</sub>=0 (all ''i'') if the indices
@@ Line 22: / Line 22: @@
 The vectors r<sub>s</sub> have two components:  '''r''' and  '''r'''<sub>''I''</sub>. The relation is written as
-<math>r_s~=~( r,~ r_I)</math>
+<math>r_s~=~( r,~ r_I).</math>
 In the superspace there is a reciprocal basis '''a'''<sub>si</sub><sup>*</sup> such that '''a'''<sub>i</sub><sup>*</sup> is the projection of '''a'''<sub>si</sub><sup>*</sup> on the subspace ''V''<sub>E</sub>.
@@ Line 41: / Line 41: @@
 of &rho;<sub>s</sub>(r<sub>s</sub>) is a space-group in the ''n''-dimensional superspace, where the dimension ''n'' is equal to the rank
 of the Fourier module. In the superspace the usual crystallographic notions and techniques may be applied. For point atoms, the function &rho;<sub>s</sub>(r<sub>s</sub>) is concentrated on surfaces
-of co-dimension equal to the dimension of the physical space (See Figure).
+of co-dimension equal to the dimension of the physical space (see figure).
 [[Image:IncEmbed.gif]]
-Figure Caption:  An example of the embedding of a one-dimensional modulated chain with two modulation
+An example of the embedding of a one-dimensional modulated chain with two modulation wave vectors. It consists of an array of two-dimensional surfaces, with three-dimensional lattice periodicity. The intersection of the surfaces with the physical space (a line) gives the positions of the modulated structure.
 [[Category: Fundamental crystallography]]

@@ Line 9: / Line 9: @@
 Superspace is used to describe quasi-periodic structures (See [[aperiodic crystal]]). We denote the dimension of the physical space
-by ''m''. Then a function &rho;('''r'''}) is quasi-periodic if its Fourier transform is given by
+by ''m''. Then a function &rho;('''r''') is quasi-periodic if its Fourier transform is given by
 <math>\rho( r)~=~\sum_{ k \in M^*}  \hat{\rho}( k \exp (2\pi i  k. r ),</math>

@@ Line 17: / Line 17: @@
 <math> k~=~\sum_{i=1}^n h_i  a_i^* , ~~~(h_i~{\rm integers}),</math>
-for a basis  '''a'''<sub>i</sub><sup>*</sup> with the property that <math>\sum_i h_i  a_i^*$=0</math> implies ''h''<sub>i</sub>=0 (all ''i'') if the indices
+for a basis  '''a'''<sub>i</sub><sup>*</sup> with the property that <math>\sum_i h_i  a_i^*=0</math> implies ''h''<sub>i</sub>=0 (all ''i'') if the indices
 ''h''<sub>i</sub> are integers. The number ''n'' of basis vectors is the  ''rank'' of the Fourier module.
@@ Line 26: / Line 26: @@
 In the superspace there is a reciprocal basis '''a'''<sub>si</sub><sup>*</sup> such that '''a'''<sub>i</sub><sup>*</sup> is the projection of '''a'''<sub>si</sub><sup>*</sup> on the subspace ''V''<sub>E</sub>.
 The reciprocal lattice &Sigma; in superspace then is projected
-on the Fourier module ''M''<su>*</sup>.
+on the Fourier module ''M''<sup>*</sup>.
 The function &rho;('''r''') then can be embedded as

@@ Line 6: / Line 6: @@
   '''Definition'''
-''Superspace'' is a Euclidean vector space with a preferred subspace ''V''<sub>E</sub> that has the dimension of
+''Superspace'' is a Euclidean vector space with a preferred (real) subspace  ''V''<sub>E</sub> that has the dimension of
 the physical space, usually three, but two for surfaces, and one for line structures.