Twinning by metric merohedry

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Maclage par mériédrie métrique(Fr); Geminazione per meroedria metrica(It)

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 a 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, 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.