# Twin lattice

### From Online Dictionary of Crystallography

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** = ∩_{i}*H*_{i} be the intersection group of the individuals in their respective orientations, *D*(*H**) the holohedral supergroup (proper or trivial) of *H**, *D*(**L**_{T}) the point group of the twin lattice and *D*(**L**_{ind}) the point group of the individual lattice. *D*(**L**_{T}) 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**_{ind}).

- When
*D*(**L**_{T}) =*D*(**L**_{ind}) and the two lattices have the same orientation, twinning is by merohedry (twin index = 1). When at least some of the symmetry elements of*D*(**L**_{T}) are differently oriented from the corresponding ones of*D*(**L**_{ind}), twinning is by reticular polyholohedry (twin index > 1, twin obliquity = 0) or reticular pseudopolyholohedry (twin index > 1, twin obliquity > 0). - When
*D*(**L**_{T}) ≠*D*(**L**_{ind}) twinning is by pseudomerohedry (twin index = 1, twin obliquity > 0), reticular merohedry (twin index > 1, twin obliquity = 0) or reticular pseudomerohedry (twin index > 1, twin obliquity > 0).

## 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*(**L**_{T}) ⊂ *D*(**L**_{ind}) was however overlooked.

## See also

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