# Difference between revisions of "Twin operation"

### From Online Dictionary of Crystallography

m (fr, it) |
|||

Line 4: | Line 4: | ||

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*. | ||

+ | 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''. | ||

+ | |||

+ | ==References== | ||

+ | *Chapter 3.3 of International Tables of Crystallography, Volume D | ||

[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |

## Revision as of 15:55, 26 April 2006

Opération de maclage (*Fr*). Operazione di geminazione (*It*)

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

Let H_{i} be the oriented point group of the i-th individual of a twin. The intersection group of the oriented vector point groups H_{i} of the individuals is indicated as H* = ∩_{i}H_{i}. 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*.

## References

- Chapter 3.3 of International Tables of Crystallography, Volume D