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Difference between revisions of "Twinning by metric merohedry"

From Online Dictionary of Crystallography

 
(Tidied translations and added German and Spanish (U. Mueller))
 
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<p><font color="blue"> Mériédrie métrique</font>(<i>Fr</i>). <font color="black"> Meroedria metrica</font>(<i>It</i>)
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<font color="blue">Maclage par mériédrie métrique</font> (''Fr''). <font color="red">Verzwillingung durch metrische Meroedrie</font> (''Ge''). <font color="black">Geminazione per meroedria metrica</font> (''It''). <font color="green">Macla por meroedría métrica</font> (''Sp'').
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Twinning by metric merohedry is a special case of [[twinning by merohedry]] which occurs when:
 
Twinning by metric merohedry is a special case of [[twinning by merohedry]] which occurs when:
* the lattice of the individual has accidentally a specialized metric which corresponds to a higher holohedry
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* the lattice of the individual has accidentally a specialized metric which corresponds to a higher holohedry,
* the twin operation belongs to this higher holohedry only
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* the twin operation belongs to this higher holohedry only.
  
If H is the individual point group, D(H) the corresponding holohedry and D(L) the point group of the lattice, twinning by metric merohedry corresponds to D(L) &sup; D(H) &supe; H
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If ''H'' is the individual point group, ''D''(''H'') the corresponding holohedry and ''D''(''L'') the point group of the lattice, twinning by metric merohedry corresponds to ''D''(''L'') &sup; ''D''(''H'') &supe; ''H''.
  
 
Twinning by metric merohedry can be seen as the degeneration of [[twinning by reticular merohedry]] to [[twin index]] 1, or of [[twinning by pseudomerohedry]] to [[twin obliquity]] zero.
 
Twinning by metric merohedry can be seen as the degeneration of [[twinning by reticular merohedry]] to [[twin index]] 1, or of [[twinning by pseudomerohedry]] to [[twin obliquity]] zero.
  
 
==Example==
 
==Example==
A monoclinic crystal of point group H = 2 with angle &beta; = 90º has a orthorhombic lattice. It may undergo two types of twinning by merohedry:
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A monoclinic crystal of point group ''H'' = 2 with angle &beta; = 90º has an orthorhombic lattice. It may undergo two types of twinning by merohedry:
* if the twin operation belongs to the monoclinic holohedry D(H) = 2/''m'', twinning is the classical twinning by merohedry, also termed [[twinning by syngonic merohedry]];
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* if the twin operation belongs to the monoclinic holohedry ''D''(''H'') = 2/''m'', twinning is the classical twinning by merohedry, also termed ''twinning by syngonic merohedry'';
* if the twin operation belongs to the orthorhombic holohedry D(L) = ''mmm'', twinning is by metric merohedry.
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* if the twin operation belongs to the orthorhombic holohedry ''D''(''L'') = ''mmm'', twinning is by metric merohedry.
  
 
==Historical note==
 
==Historical note==
Friedel (1904, p. 143; 1926, p. 56-57) called metric merohedry '''mériédrie d’ordre supérieur''' (higher order merohedry)  but stated that it was either unlikely or equivalent to a pseudo-merohedry of low obliquity. Nowadays several examples of true metric merohedry (within experimental uncertainty) are known.
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Friedel (1904, p. 143; 1926, pp. 56-57) called metric merohedry '''mériédrie d’ordre supérieur''' (higher order merohedry)  but stated that it was either unlikely or equivalent to a pseudo-merohedry of low obliquity. Nowadays several examples of true metric merohedry (within experimental uncertainty) are known.
  
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== References ==
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*Friedel, G. (1904). ''Étude sur les groupements cristallins.'' Extrait du ''Bulletin de la Société de l'Industrie minérale'', Quatrième série, Tomes III et IV. Saint-Étienne, Société de l'imprimerie Théolier J. Thomas et C., 485 pp.
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*Friedel, G. (1926). ''Leçons de Cristallographie.'' Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.
  
[[Category: Fundamental crystallography]]
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[[Category:Twinning]]

Latest revision as of 14:30, 20 November 2017

Maclage par mériédrie métrique (Fr). Verzwillingung durch metrische Meroedrie (Ge). Geminazione per meroedria metrica (It). Macla por meroedría métrica (Sp).


Twinning by metric merohedry is a special case of twinning by merohedry which occurs when:

  • the lattice of the individual has accidentally a specialized metric which corresponds to a higher holohedry,
  • the twin operation belongs to this higher holohedry only.

If H is the individual point group, D(H) the corresponding holohedry and D(L) the point group of the lattice, twinning by metric merohedry corresponds to D(L) ⊃ D(H) ⊇ H.

Twinning by metric merohedry can be seen as the degeneration of twinning by reticular merohedry to twin index 1, or of twinning by pseudomerohedry to twin obliquity zero.

Example

A monoclinic crystal of point group H = 2 with angle β = 90º has an orthorhombic lattice. It may undergo two types of twinning by merohedry:

  • if the twin operation belongs to the monoclinic holohedry D(H) = 2/m, twinning is the classical twinning by merohedry, also termed twinning by syngonic merohedry;
  • if the twin operation belongs to the orthorhombic holohedry D(L) = mmm, twinning is by metric merohedry.

Historical note

Friedel (1904, p. 143; 1926, pp. 56-57) called metric merohedry mériédrie d’ordre supérieur (higher order merohedry) but stated that it was either unlikely or equivalent to a pseudo-merohedry of low obliquity. Nowadays several examples of true metric merohedry (within experimental uncertainty) are known.

References

  • Friedel, G. (1904). Étude sur les groupements cristallins. Extrait du Bulletin de la Société de l'Industrie minérale, Quatrième série, Tomes III et IV. Saint-Étienne, Société de l'imprimerie Théolier J. Thomas et C., 485 pp.
  • Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.