# 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. 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, equal 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 in: Donnay, G. *Width of albite-twinning lamellae*, Am. Mineral., **25** (1940) 578-586, where the case D(**L**_{T}) ⊂ D(**L**_{ind}) was however overlooked.

### See also

Chapter 3.3 of *International Tables of Crystallography, Volume D*