Twin lattice

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* = ∩_iH_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