TedJanssen at 13:12, 18 May 2009

2009-05-18T13:12:12Z

TedJanssen at 14:11, 16 May 2009

2009-05-16T14:11:34Z

New page

[[Aperiodic crystal]]

Cristal ap&eacute riodique (Fr.)

'''Definition'''

A ''periodic crystal'' is a structure with, ideally, sharp diffraction peaks on the positions
of a ''reciprocal lattice''. The structure then is invariant under the translations
of the ''direct lattice''. Periodicity here means lattice ''periodicity''. Any structure
without this property is ''aperiodic''. For example, an amorphous system is aperiodic.
An ''aperiodic crystal'' is a structure with sharp diffraction peaks, but without lattice periodicity.
Therefore, amorphous systems are not aperiodic crystals. The positions of the sharp diffraction peaks of an aperiodic crystal belong to a ''vector module'' of finite rank. This means that
the diffraction wave vectors are of the form
<math>\[ {\bf k}~=~\sum_{i=1}^n h_i{\bf a}_i^*,~~({\rm integer}~h_i). \]</math>
The basis vectors <math>{\bf a}_i^*</math> are supposed to be independent over the rational numbers, ''i.e''. when a linear combination of them with rational coefficients is zero, all coefficients are
zero. The minimum number of basis vectors is the ''rank'' of the vector module.
If the rank ''n'' is larger than the space dimension, the structure is not periodic, but aperiodic.

'''Applications'''

There are four classes of aperiodic structures, but these classes have an overlap:
''incommensurately modulated crystal phases'' , ''incommensurate composite structures'',
''quasicrystals'', and ''incommensurate magnetic structures''.

'''See also''': Incommensurately modulated crystal phases, incommensurate composite structures, quasicrystals, incommensurate magnetic structures.

@@ Line 2: / Line 2: @@
-Cristal ap&eacute riodique (Fr.)
+Cristal ap&eacute;riodique (Fr.)
 '''Definition'''
@@ Line 13: / Line 13: @@
 Therefore, amorphous systems are not aperiodic crystals. The positions of the sharp diffraction peaks of an aperiodic crystal belong to a ''vector module'' of finite rank. This means that
 the diffraction wave vectors are of the form
 <math>\[ {\bf k}~=~\sum_{i=1}^n h_i{\bf a}_i^*,~~({\rm integer}~h_i). \]</math>
 The basis vectors <math>{\bf a}_i^*</math> are supposed to be independent over the rational numbers, ''i.e''. when a linear combination of them with rational coefficients is zero, all coefficients are
 zero. The minimum number of basis vectors is the ''rank'' of the vector module.

User talk:TedJanssen - Revision history

TedJanssen at 13:12, 18 May 2009

TedJanssen at 14:11, 16 May 2009