Difference between revisions of "Twin operation"
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| − | <font color="blue">Opération de maclage</font> (''Fr'') | + | <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* = ∩<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*. | + | 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''* = ∩<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 | + | 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 == | ||
| − | + | *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