# Difference between revisions of "Incommensurate composite crystal"

### From Online Dictionary of Crystallography

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themselves modulated structures, with basis structures that are mutually incommensurate. | 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 | 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 | |

− | 3''N'' basis vectors | + | 3''N'' 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> | <math> a_i^{*\nu}~=~\sum_{j=1}^n Z_{ij}^{\nu} A_j^* ~~~(i=1,2,3),</math> | ||

− | where | + | 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. | ||

+ | ''n'' is the rank of the vector module. | ||

'''Applications''' | '''Applications''' |

## Revision as of 06:20, 19 May 2009

Incommensurate Composite Crystal

Cristal composite (Fr.)

Definition

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
3*N* 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
Hg_{3-δ}AsF_{6} with two systems of Hg chains inside the host lattice formed
by AsF_{6} octahedra. Another example is nonadecane in the channels of a urea
host crystal.