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From Online Dictionary of Crystallography

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[[Superspace]]
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<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'').
  
  
<Font color="blue">Superespace, Hyperespace </font>(Fr.)
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== Definition ==
  
'''Definition'''
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''Superspace'' is a Euclidean vector space with a preferred (real) subspace ''V''<sub>E</sub> that has the dimension of
 
 
''Superspace'' is a Euclidean vector space with a preferred subspace ''V''<sub>E</sub> that has the dimension of
 
 
the physical space, usually three, but two for surfaces, and one for line structures.
 
the physical space, usually three, but two for surfaces, and one for line structures.
  
'''Applications'''
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== Applications ==
  
Superspace is used to describe quasi-periodic structures (See [[aperiodic crystal]]). We denote the dimension of the physical space
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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
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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>
 
<math>\rho( r)~=~\sum_{ k \in M^*}  \hat{\rho}( k \exp (2\pi i  k. r ),</math>
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where the  ''Fourier module'' ''M''<sup>*</sup> is the set of vectors (Z-module) of the form
 
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>
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<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
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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.
 
''h''<sub>i</sub> are integers. The number ''n'' of basis vectors is the  ''rank'' of the Fourier module.
 
   
 
   
 
The vectors r<sub>s</sub> have two components:  '''r''' and  '''r'''<sub>''I''</sub>. The relation is written as
 
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>
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<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>.
 
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
 
The reciprocal lattice &Sigma; in superspace then is projected
on the Fourier module ''M''<su>*</sup>.
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on the Fourier module ''M''<sup>*</sup>.
 
The function &rho;('''r''') then can be embedded as
 
The function &rho;('''r''') then can be embedded as
 
   
 
   
<math> \rho_s(r_s)~=~\sum_{k_s\in\Sigma^*} \hat{\rho}(\pi k_s={\bf k}) \exp (2\pi i k_s.r_s).</math>
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<math> \rho_s(r_s)~=~\sum_{k_s\in\Sigma^*} \hat{\rho}(\pi k_s= k) \exp (2\pi i k_s.r_s).</math>
 
   
 
   
 
The function &rho;('''r''') is the restriction of &rho;<sub>s</sub>(r<sub>s</sub>) to the subspace  
 
The function &rho;('''r''') is the restriction of &rho;<sub>s</sub>(r<sub>s</sub>) to the subspace  
 
   
 
   
<math> \rho({\bf r})~=~\rho_s({\bf r},0).</math>
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<math> \rho( r)~=~\rho_s( r,0).</math>
  
 
   
 
   
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&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
 
&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  
 
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
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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 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).
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of co-dimension equal to the dimension of the physical space (see figure).
 
   
 
   
[[Image:IncEmbed.gif]]
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[[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]]
  
Figure Caption: An example of the embedding of a one-dimensional modulated chain with two modulation
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[[Category: Fundamental crystallography]]
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.
 

Latest revision as of 09:07, 20 November 2017

Superespace, hyperespace (Fr). Superraum (Ge). Superspazio (It). 超空間 (Ja). Superespacio (Sp).


Definition

Superspace is a Euclidean vector space with a preferred (real) subspace VE that has the dimension of the physical space, usually three, but two for surfaces, and one for line structures.

Applications

Superspace is used to describe quasi-periodic structures (see aperiodic crystal). We denote the dimension of the physical space by m. Then a function ρ(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]

where the Fourier module M* 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]

for a basis ai* with the property that [math]\sum_i h_i a_i^*=0[/math] implies hi=0 (all i) if the indices hi are integers. The number n of basis vectors is the rank of the Fourier module.

The vectors rs have two components: r and rI. The relation is written as

[math]r_s~=~( r,~ r_I).[/math]

In the superspace there is a reciprocal basis asi* such that ai* is the projection of asi* on the subspace VE. The reciprocal lattice Σ in superspace then is projected on the Fourier module M*. The function ρ(r) then can be embedded as

[math] \rho_s(r_s)~=~\sum_{k_s\in\Sigma^*} \hat{\rho}(\pi k_s= k) \exp (2\pi i k_s.r_s).[/math]

The function ρ(r) is the restriction of ρs(rs) to the subspace

[math] \rho( r)~=~\rho_s( r,0).[/math]


Because the Fourier components in superspace belong to reciprocal vectors of Σ*, the function ρs(rs) is lattice periodic. Its direct lattice Σ is the dual of Σ* and its basis vectors are ais satisfying asi.asj* = δij. Therefore, the symmetry group of ρs(rs) 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 ρs(rs) is concentrated on surfaces of co-dimension equal to the dimension of the physical space (see figure).

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.