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Difference between revisions of "Quasiperiodicity"

From Online Dictionary of Crystallography

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on positions
 
on positions
  
  <math>{\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^*,~~({\rm integers ~}h_i)  </math>
+
  <math> k~=~\sum_{i=1}^n h_i a_i^*,~~({\rm integers ~}h_i)  </math>
  
 
for basis vectors  '''a'''<sub>i</sub><sup>*</sup> in a space of dimension ''m''. If the basis vectors
 
for basis vectors  '''a'''<sub>i</sub><sup>*</sup> in a space of dimension ''m''. If the basis vectors
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A quasiperiodic function may be expressed in a convergent trigonometric series.
 
A quasiperiodic function may be expressed in a convergent trigonometric series.
  
   <math>f({\bf r})~=~\sum_{{\bf k}} A({\bf k}) \cos \left( 2\pi {\bf k}.{\bf r}+\phi ({\bf k}) \right). </math>
+
   <math>f( r)~=~\sum_k A(k) \cos \left( 2\pi k. r+\phi ( k) \right). </math>
  
 
It is a special case of an almost periodic function. An  ''almost periodic function''
 
It is a special case of an almost periodic function. An  ''almost periodic function''
 
is a function ''f''('''r''') such that for every small number &epsilon; there is
 
is a function ''f''('''r''') such that for every small number &epsilon; there is
 
a translation  '''a''' such that the difference between the function and the function shifted over
 
a translation  '''a''' such that the difference between the function and the function shifted over
'''a''' is smaller than the chosen quantity:
+
'''a''' is smaller than the chosen quantity:
  
  | <math>f({\bf r}+{\bf a})-f({\bf r}) |~<~ \epsilon~~{\rm for ~all~{\bf r}} .</math>  
+
  | <math>f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .</math>  
  
 
A quasiperiodic function is always an almost periodic function, but the converse
 
A quasiperiodic function is always an almost periodic function, but the converse

Revision as of 18:28, 18 May 2009

Quasiperiodicity


Quasi-periodicité (Fr.)

Definition

A function is called quasiperiodic if its Fourier transform consists of δ-peaks on positions

[math] k~=~\sum_{i=1}^n h_i  a_i^*,~~({\rm integers ~}h_i)  [/math]

for basis vectors ai* in a space of dimension m. If the basis vectors form a basis for the space (n equal to the space dimension, and linearly independent basis vectors over the real numbers) then the function is lattice periodic. If n is larger than the space dimension, then the function is aperiodic.

Comment

Sometimes the definition includes that the function is not lattice periodic.

A quasiperiodic function may be expressed in a convergent trigonometric series.

 [math]f( r)~=~\sum_k A(k) \cos \left( 2\pi  k. r+\phi ( k) \right). [/math]

It is a special case of an almost periodic function. An almost periodic function is a function f(r) such that for every small number ε there is a translation a such that the difference between the function and the function shifted over a is smaller than the chosen quantity:

| [math]f(r+ a)-f( r) |~\lt ~ \epsilon~~{\rm for ~all~ r} .[/math] 

A quasiperiodic function is always an almost periodic function, but the converse is not true.

The theory of almost-periodic functions goes back to the work by H. Bohr.