Difference between revisions of "Twin operation"
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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 | + | 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 "colour" and K is a thus a chromatic vector point group, known as ''twin point group''. | 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''. | ||
Revision as of 10:29, 26 February 2007
Opération de maclage (Fr). Operazione di geminazione (It)
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 twin point group.
See also
Chapter 3.3 of International Tables of Crystallography, Volume D