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Difference between revisions of "Twin operation"

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<font color="blue">Opération de maclage</font> (''Fr'')<Font color="black">Operazione di geminazione</Font> (''It''); <Font color="purple">双晶操作</Font> (''Ja'').
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<font color="blue">Opération de maclage</font> (''Fr''). <font color="red">Zwillingsoperation</font> (''Ge''). <font color="black">Operazione di geminazione</font> (''It''). <font color="purple">双晶操作</font> (''Ja''). <font color="green">Operación de macla</font> (''Sp'').
  
  
 
The operation (action) of an element of symmetry that generates a ''[[twin]]''.
 
The operation (action) of an element of symmetry that generates a ''[[twin]]''.
  
Let H<sub>i</sub> be the oriented [[point group]] of the i-th individual of a [[twin]]. The intersection group of the oriented vector point groups H<sub>i</sub> of the individuals is indicated as H* = &cap;<sub>i</sub>H<sub>i</sub>. The symmetry of a twin is identified in [[vector space]] by a point group K which is a supergroup of H*. The [[coset]] decomposition of K with respect to H* gives the possible [[twin law]]s, each coset representing a [[twin law]], and each operation in a coset representing a twin operation; the operations in a coset are equivalent under the operations of H*.
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Let ''H<sub>i</sub>'' be the oriented [[point group]] of the ''i''-th individual of a [[twin]]. The intersection group of the oriented vector point groups ''H<sub>i</sub>'' of the individuals is indicated as ''H''* = &cap;<sub>''i''</sub>''H<sub>i</sub>''. The symmetry of a twin is identified in [[vector space]] by a point group ''K'' which is a supergroup of ''H''*. The [[coset]] decomposition of ''K'' with respect to ''H''* gives the possible [[twin law]]s, each coset representing a [[twin law]], and each operation in a coset representing a twin operation; the operations in a coset are equivalent under the operations of ''H''*.
  
Operations in H describe the vector point symmetry of the individuals, whereas those in the cosets obtained by decomposing K in terms of H* connect the different individuals. To underline their different nature, the twin operations are often associated with a "colour" and K is a thus a chromatic vector point group, known as ''twin point group''.
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Operations in ''H'' describe the vector point symmetry of the individuals, whereas those in the cosets obtained by decomposing ''K'' in terms of ''H''* connect the different individuals. To underline their different nature, the twin operations are often associated with a 'colour' and ''K'' is a thus a chromatic vector point group, known as a ''twin point group''.
  
 
== See also ==
 
== See also ==
  
<br> Chapter 3.3 of ''International Tables of Crystallography, Volume D''<br>
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*Chapter 3.3 of ''International Tables for Crystallography, Volume D''
  
 
[[Category:Twinning]]
 
[[Category:Twinning]]

Latest revision as of 14:21, 20 November 2017

Opération de maclage (Fr). Zwillingsoperation (Ge). Operazione di geminazione (It). 双晶操作 (Ja). Operación de macla (Sp).


The operation (action) of an element of symmetry that generates a twin.

Let Hi be the oriented point group of the i-th individual of a twin. The intersection group of the oriented vector point groups Hi of the individuals is indicated as H* = ∩iHi. The symmetry of a twin is identified in vector space by a point group K which is a supergroup of H*. The coset decomposition of K with respect to H* gives the possible twin laws, each coset representing a twin law, and each operation in a coset representing a twin operation; the operations in a coset are equivalent under the operations of H*.

Operations in H describe the vector point symmetry of the individuals, whereas those in the cosets obtained by decomposing K in terms of H* connect the different individuals. To underline their different nature, the twin operations are often associated with a 'colour' and K is a thus a chromatic vector point group, known as a twin point group.

See also

  • Chapter 3.3 of International Tables for Crystallography, Volume D