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Difference between revisions of "Twin lattice"

From Online Dictionary of Crystallography

(Twin lattice)
(Tidied translations and added German and Spanish (U. Mueller))
 
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<Font color="blue"> Réseau de la macle</Font>(''Fr'') <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 [[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 ([[twinning (effects of)]]. The lattice that is formed by the (quasi)restored nodes is the ''twin lattice''. It corresponds to the crystal lattice in [[twinning by merohedry]] and to a sublattice of the crystal (individual) in [[twinning 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.  
  
The symmetry of the twin lattice has to be compared with the intersection symmetry of the individuals in their respective orientation, which in general is a subgroup of the group of the individual. As a consequence, the symmetry of the twin lattice may be higher, equal or lower than the symmetry of the individual lattice.
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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).
  
==Related articles==
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[[Mallard's law]]
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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.
  
 
==See also==
 
==See also==
Chapter 3.3 of ''International Tables of Crystallography, Volume D''<br>
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*[[Mallard's law]]
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*Chapter 3.3 of ''International Tables for Crystallography, Volume D''
  
[[Category:Fundamental crystallography]]
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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