Twin law

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= Twin law =
= Twin law =
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The '''twin law''' is the set of [[twin operation]]s mapping two individuals of a [[twin]]. It is obtained by [[coset]] decomposition of the point group of the [[twin lattice]] with respect to the point group of the individual. Each operation in the same coset is a possible twin operation that, from the lattice viepoint, is equivalent to any other operation in the same coset. Any of these can be taken as '''coset representative''' and indicated by the symbol of the twin element: <math> \bar 1</math>, [uvw] and (''hkl'') for the centre (''[[inversion twin]]''), direction of the rotation axis  (''[[rotation twin]]'') and [[Miller indices]] of the mirror plane (''[[reflection twin]]''), in the order. Except when one refers to a specific plane or direction, the symbols {''hkl''} or <''uvw''> have to be be used to indicate all the planes or directions which belong to the same [[coset]] and are therefore equivalent under the point group of the individual.
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The '''twin law''' is the set of [[twin operation]]s mapping two individuals of a [[twin]]. It is obtained by [[coset]] decomposition of the point group of the [[twin lattice]] with respect to the intersection group of the point groups of the individuals in their respective orientations. Each operation in the same coset is a possible twin operation that, from the lattice viepoint, is equivalent to any other operation in the same coset. Any of these can be taken as '''coset representative''' and indicated by the symbol of the twin element: <math> \bar 1</math>, [uvw] and (''hkl'') for the centre (''[[inversion twin]]''), direction of the rotation axis  (''[[rotation twin]]'') and [[Miller indices]] of the mirror plane (''[[reflection twin]]''), in the order. Except when one refers to a specific plane or direction, the symbols {''hkl''} or <''uvw''> have to be be used to indicate all the planes or directions which belong to the same [[coset]] and are therefore equivalent under the point group of the individual.
In case of [[TLQS twinning]] the equivalence of the operations in a coset is only approximate.
In case of [[TLQS twinning]] the equivalence of the operations in a coset is only approximate.

Revision as of 12:43, 17 March 2009

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Twin law

The twin law is the set of twin operations mapping two individuals of a twin. It is obtained by coset decomposition of the point group of the twin lattice with respect to the intersection group of the point groups of the individuals in their respective orientations. Each operation in the same coset is a possible twin operation that, from the lattice viepoint, is equivalent to any other operation in the same coset. Any of these can be taken as coset representative and indicated by the symbol of the twin element:  \bar 1, [uvw] and (hkl) for the centre (inversion twin), direction of the rotation axis (rotation twin) and Miller indices of the mirror plane (reflection twin), in the order. Except when one refers to a specific plane or direction, the symbols {hkl} or <uvw> have to be be used to indicate all the planes or directions which belong to the same coset and are therefore equivalent under the point group of the individual.

In case of TLQS twinning the equivalence of the operations in a coset is only approximate.

See also

Chapter 3.3 of International Tables of Crystallography, Volume D