Aperiodic crystal

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<Font color="blue">Cristal ap&eacute;riodique</font> (''Fr''). <Font color="black">Cristallo aperiodico</font> (''It''). <Font color="purple">非周期性結晶</font> (''Ja'')
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<Font color="blue">Cristal ap&eacute;riodique</font> (''Fr''). <Font color="black">Cristallo aperiodico</font> (''It''). <Font color="purple">非周期性結晶</font> (''Ja'').
==Definition==
==Definition==
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The definition of aperiodic crystal was included in the definition of crystal proposed by the IUCr Commission on Aperiodic structures (1): by ''crystal'' we mean any solid having an essentially discrete diffraction diagram and ''aperiodic crystal'' we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent. As an extension, the latter term will also include those crystals in which three-dimensional periodicity is too weak to describe significant correlations in the atomic configuration, but which can be properly described by crystallographic methods developed for actual aperiodic crystals.
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The definition of aperiodic crystal was included in the definition of ''crystal'' proposed by the IUCr Commission on Aperiodic Structures (International Union of Crystallography, 1992): by ''crystal'' we mean any solid having an essentially discrete diffraction diagram and ''aperiodic crystal'' we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent. As an extension, the latter term will also include those crystals in which three-dimensional periodicity is too weak to describe significant correlations in the atomic configuration, but which can be properly described by crystallographic methods developed for actual aperiodic crystals.
For practical purposes, however, many scientists currently working in the field use a narrower definition of aperiodic crystal, namely
For practical purposes, however, many scientists currently working in the field use a narrower definition of aperiodic crystal, namely
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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
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
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<math>\textbf{k}=\sum_{i=1}^n h_i \textbf{a}_i^*, ( \textrm{integer}\ h_i).</math>  
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<math>\textbf{k}=\sum_{i=1}^n h_i \textbf{a}_i^* ( \textrm{integer}\ h_i).</math>  
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The basis vectors  <math>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.
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The basis vectors  <math>\textbf{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==
==Applications==
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* and [[incommensurate magnetic structure]]s
* and [[incommensurate magnetic structure]]s
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== References ==
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== Reference==
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(1) [http://dx.doi.org/10.1107/S0108767392008328''Acta Cryst.'' (1992), '''A48''', 928, '''Terms of reference''' of the IUCr commission on aperiodic crystals.]
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[https://doi.org/10.1107/S0108767392008328 International Union of Crystallography (1992). ''Acta Cryst.'' A'''48''', 928. ''Terms of reference of the IUCr Commission on Aperiodic Crystals''.]
[[Category:Fundamental crystallography]]
[[Category:Fundamental crystallography]]

Latest revision as of 12:07, 12 May 2017

Cristal apériodique (Fr). Cristallo aperiodico (It). 非周期性結晶 (Ja).

Definition

The definition of aperiodic crystal was included in the definition of crystal proposed by the IUCr Commission on Aperiodic Structures (International Union of Crystallography, 1992): by crystal we mean any solid having an essentially discrete diffraction diagram and aperiodic crystal we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent. As an extension, the latter term will also include those crystals in which three-dimensional periodicity is too weak to describe significant correlations in the atomic configuration, but which can be properly described by crystallographic methods developed for actual aperiodic crystals.

For practical purposes, however, many scientists currently working in the field use a narrower definition of aperiodic crystal, namely

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

\textbf{k}=\sum_{i=1}^n h_i \textbf{a}_i^* ( \textrm{integer}\ h_i).

The basis vectors \textbf{a}_i^* 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:

Reference

International Union of Crystallography (1992). Acta Cryst. A48, 928. Terms of reference of the IUCr Commission on Aperiodic Crystals.