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, equal or lower than D(Lind).
- When D(LT) = D(Lind) 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(LT) are differently oriented from the corresponding ones of D(Lind), twinning is by reticular polyholohedry (twin index > 1, twin obliquity = 0) or reticular pseudopolyholohedry (twin index > 1, twin obliquity > 0).
- When D(LT) ≠ D(Lind) 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).
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(LT) ⊂ D(Lind) was however overlooked.
Chapter 3.3 of International Tables of Crystallography, Volume D