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Difference between revisions of "Twin lattice"

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<Font color="blue"> Réseau de la macle</Font> (''Fr''). <Font color="black"> Reticolo del geminato </Font>(''It''). <Font color="purple"> 双晶格子 </Font>(''Ja'')
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<Font color="blue"> Réseau de la macle</Font> (''Fr''). <Font color="black"> Reticolo del geminato </Font>(''It''). <Font color="purple"> 双晶格子 </Font>(''Ja'').
  
 
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.  
 
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.  
  
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, equal or lower than D('''L'''<sub>ind</sub>).
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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>).
*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>) = ''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>) &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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*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==
 
[[Mallard's law]]
 
  
 
==History==
 
==History==
The definition of twin lattice was given in: Donnay, G. ''Width of albite-twinning lamellae'', Am. Mineral., '''25''' (1940) 578-586, where the case D('''L'''<sub>T</sub>) &sub; D('''L'''<sub>ind</sub>) was however overlooked.
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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:Twinning]]
 
[[Category:Twinning]]

Revision as of 13:59, 17 May 2017

Réseau de la macle (Fr). Reticolo del geminato (It). 双晶格子 (Ja).

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