# Difference between revisions of "Twin lattice"

### From Online Dictionary of Crystallography

BrianMcMahon (talk | contribs) (Tidied translations and added German and Spanish (U. Mueller)) |
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− | + | <font color="blue">Réseau de la macle</font> (''Fr''). <font color="red">Zwillingsgitter</font> (''Ge''). <font color="black">Reticolo del geminato</font> (''It''). <font color="purple">双晶格子</font> (''Ja''). <font color="green">Red de la macla</font> (''Sp''). | |

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− | The lattice that is formed by the (quasi)restored nodes is the ''twin lattice''. | + | A [[twin operation]] overlaps both the direct and reciprocal lattices of the individuals that form a twin; consequently, the nodes of the individual lattices are [[twinning (effects of)|overlapped (restored)]] to some extent. The (sub)lattice that is formed by the (quasi)restored nodes is the ''twin lattice''. In case of non-zero [[twin obliquity]] the twin lattice suffers a slight deviation at the composition surface. |

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+ | Let ''H''* = ∩<sub>''i''</sub>''H''<sub>''i''</sub> be the intersection group of the individuals in their respective orientations, ''D''(''H''*) the holohedral supergroup (proper or trivial) of ''H''*, ''D''('''L'''<sub>''T''</sub>) the point group of the twin lattice and ''D''('''L'''<sub>''ind''</sub>) the point group of the individual lattice. ''D''('''L'''<sub>''T''</sub>) either coincides with ''D''(''H''*) (case of zero [[twin obliquity]]) or is a proper supergroup of it (case of non-zero [[twin obliquity]]): it can be higher than, equal to or lower than ''D''('''L'''<sub>''ind''</sub>). | ||

+ | *When ''D''('''L'''<sub>''T''</sub>) = ''D''('''L'''<sub>''ind''</sub>) and the two lattices have the same orientation, twinning is by [[twinning by merohedry|merohedry]] ([[twin index]] = 1). When at least some of the symmetry elements of ''D''('''L'''<sub>''T''</sub>) are differently oriented from the corresponding ones of ''D''('''L'''<sub>''ind''</sub>), twinning is by [[twinning by reticular polyholohedry|reticular polyholohedry]] ([[twin index]] > 1, [[twin obliquity]] = 0) or reticular pseudopolyholohedry ([[twin index]] > 1, [[twin obliquity]] > 0). | ||

+ | *When ''D''('''L'''<sub>''T''</sub>) ≠ ''D''('''L'''<sub>''ind''</sub>) twinning is by [[twinning by pseudomerohedry|pseudomerohedry]] ([[twin index]] = 1, [[twin obliquity]] > 0), [[twinning by reticular merohedry|reticular merohedry]] ([[twin index]] > 1, [[twin obliquity]] = 0) or [[twinning by reticular pseudomerohedry|reticular pseudomerohedry]] ([[twin index]] > 1, [[twin obliquity]] > 0). | ||

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+ | ==History== | ||

+ | The definition of twin lattice was given by G. Donnay [(1940). ''Am. Mineral.'' '''25''', 578-586. ''Width of albite-twinning lamellae''] where the case ''D''('''L'''<sub>''T''</sub>) ⊂ ''D''('''L'''<sub>''ind''</sub>) was however overlooked. | ||

+ | |||

+ | ==See also== | ||

+ | *[[Mallard's law]] | ||

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

+ | |||

+ | [[Category:Twinning]] |

## Latest revision as of 14:13, 20 November 2017

Réseau de la macle (*Fr*). Zwillingsgitter (*Ge*). Reticolo del geminato (*It*). 双晶格子 (*Ja*). Red de la macla (*Sp*).

A twin operation overlaps both the direct and reciprocal lattices of the individuals that form a twin; consequently, the nodes of the individual lattices are overlapped (restored) to some extent. The (sub)lattice that is formed by the (quasi)restored nodes is the *twin lattice*. In case of non-zero twin obliquity the twin lattice suffers a slight deviation at the composition surface.

Let *H** = ∩_{i}*H*_{i} be the intersection group of the individuals in their respective orientations, *D*(*H**) the holohedral supergroup (proper or trivial) of *H**, *D*(**L**_{T}) the point group of the twin lattice and *D*(**L**_{ind}) the point group of the individual lattice. *D*(**L**_{T}) either coincides with *D*(*H**) (case of zero twin obliquity) or is a proper supergroup of it (case of non-zero twin obliquity): it can be higher than, equal to or lower than *D*(**L**_{ind}).

- When
*D*(**L**_{T}) =*D*(**L**_{ind}) and the two lattices have the same orientation, twinning is by merohedry (twin index = 1). When at least some of the symmetry elements of*D*(**L**_{T}) are differently oriented from the corresponding ones of*D*(**L**_{ind}), twinning is by reticular polyholohedry (twin index > 1, twin obliquity = 0) or reticular pseudopolyholohedry (twin index > 1, twin obliquity > 0). - When
*D*(**L**_{T}) ≠*D*(**L**_{ind}) twinning is by pseudomerohedry (twin index = 1, twin obliquity > 0), reticular merohedry (twin index > 1, twin obliquity = 0) or reticular pseudomerohedry (twin index > 1, twin obliquity > 0).

## History

The definition of twin lattice was given by G. Donnay [(1940). *Am. Mineral.* **25**, 578-586. *Width of albite-twinning lamellae*] where the case *D*(**L**_{T}) ⊂ *D*(**L**_{ind}) was however overlooked.

## See also

- Mallard's law
- Chapter 3.3 of
*International Tables for Crystallography, Volume D*