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[[Incommensurate Composite Crystal]]
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<font color="blue">Cristal composite incommensurable</font> (''Fr''). <font color="red">Inkommensurabler Kompositkristall</font> (''Ge''). <font color="black">Cristallo composito incommensurabile</font> (''It''). <font color="purple">非整合複合結晶</font> (''Ja''). <font color="green">Estructura inconmensurable compuesta crystal</font> (''Sp'').
  
 
Cristal composite (Fr.)
 
 
'''Definition'''
 
  
 
An ''incommensurate composite crystal'' is a compound with two or more (''N'') subsystems that are
 
An ''incommensurate composite crystal'' is a compound with two or more (''N'') subsystems that are
 
themselves modulated structures, with basis structures that are mutually incommensurate.
 
themselves modulated structures, with basis structures that are mutually incommensurate.
Each subsystem (numbered by &nu;) has a reciprocal lattice for its basic structure with three basis vectors
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Each subsystem (numbered by &nu;) has a reciprocal lattice for its basic structure with three basis vectors
'''a'''<sub>i</sub><sup>*&nu;</sup>. There is a basis of the module of diffraction spots that has at most
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<math>a_i^{*\nu}</math>. There is a basis of the [[vector module]] of diffraction spots that has at most
3''N'' basis vectors  '''A'''<sub>j</sub><sup>*</sup> such that
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3''N'' basis vectors  <math>A_j^*</math> such that
  
<math>{\bf a}_i^{*\nu}~=~\sum_{j=1}^n Z_{ij}^{\nu} {\bf A}_j^* ~~~(i=1,2,3),</math>
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<math> a_i^{*\nu}~=~\sum_{j=1}^n Z_{ij}^{\nu} A_j^* ~~~(i=1,2,3),</math>
  
where Z<sub>ij</sub><sup>&nu;</sup> are integer coefficients.
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where <math>Z_{ij}^{\nu}</math> are integer coefficients.
 
If ''n'' is larger than the dimension of space (three), the composite crystal is an aperiodic crystal.
 
If ''n'' is larger than the dimension of space (three), the composite crystal is an aperiodic crystal.
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''n'' is the rank of the vector module.
  
'''Applications'''
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== Applications ==
  
 
Examples are intergrowth crystals and adsorbed monolayers. To the former belongs
 
Examples are intergrowth crystals and adsorbed monolayers. To the former belongs
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by AsF<sub>6</sub> octahedra. Another example is nonadecane in the channels of a urea
 
by AsF<sub>6</sub> octahedra. Another example is nonadecane in the channels of a urea
 
host crystal.
 
host crystal.
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[[Category:Fundamental crystallography]]

Latest revision as of 17:15, 14 November 2017

Cristal composite incommensurable (Fr). Inkommensurabler Kompositkristall (Ge). Cristallo composito incommensurabile (It). 非整合複合結晶 (Ja). Estructura inconmensurable compuesta crystal (Sp).


An incommensurate composite crystal is a compound with two or more (N) subsystems that are themselves modulated structures, with basis structures that are mutually incommensurate. Each subsystem (numbered by ν) has a reciprocal lattice for its basic structure with three basis vectors [math]a_i^{*\nu}[/math]. There is a basis of the vector module of diffraction spots that has at most 3N basis vectors [math]A_j^*[/math] such that

[math] a_i^{*\nu}~=~\sum_{j=1}^n Z_{ij}^{\nu} A_j^* ~~~(i=1,2,3),[/math]

where [math]Z_{ij}^{\nu}[/math] are integer coefficients. If n is larger than the dimension of space (three), the composite crystal is an aperiodic crystal. n is the rank of the vector module.

Applications

Examples are intergrowth crystals and adsorbed monolayers. To the former belongs Hg3-δAsF6 with two systems of Hg chains inside the host lattice formed by AsF6 octahedra. Another example is nonadecane in the channels of a urea host crystal.