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Difference between revisions of "Twin lattice"

From Online Dictionary of Crystallography

(Tidied translations and added German and Spanish (U. Mueller))
 
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<Font color="black"> Reticolo del geminato </Font>(''It'')
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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'').
  
= Twin lattice =
 
  
A [[twinning]] 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 ([[twinning (effects of)]]. The lattice that is formed by the (quasi)restored nodes is the ''twin lattice''. It corresponds to the crystal lattice in [[twinnning by merohedry]] and to a sublattice of the crystal (individual) in [[twinnning by reticular merohedry]].
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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''* = &cap;<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>).
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*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).
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*When ''D''('''L'''<sub>''T''</sub>) &ne; ''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==
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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>) &sub; ''D''('''L'''<sub>''ind''</sub>) was however overlooked.
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==See also==
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*[[Mallard's law]]
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*Chapter 3.3 of ''International Tables for Crystallography, Volume D''
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[[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* = ∩iHi be the intersection group of the individuals in their respective orientations, D(H*) the holohedral supergroup (proper or trivial) of H*, D(LT) the point group of the twin lattice and D(Lind) the point group of the individual lattice. D(LT) 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(Lind).


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(LT) ⊂ D(Lind) was however overlooked.

See also

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