# Twinning

Maclage (Fr). Zwillingsbildung (Ge). Maclado (formación de macla) (Sp). двойникование (Ru). Geminazione (It)

# Oriented association and twinning

Crystals (also called individuals) belonging to the same phase form an oriented association if they can be brought to the same crystallographic orientation by translation, rotation or reflection. Individuals related by a translation form a parallel association; strictly speaking these individuals have the same orientation even without applying a translation. Individuals related either by a reflection (mirror plane or centre of symmetry) or a rotation form a twin.

• symmetry of a twin

An element of symmetry crystallographically relating differently oriented crystals cannot belong to the individual. The element of symmetry that relates the indivduals of a twin is called twinning element of symmetry and the connected operation is a twinning operation of symmetry. The Mallard law states that the twin element (i.e. the geometrical element relative to which the twining operation is defined) is restricted to a direct lattice element: lattice nodes (twin centres), lattice rows (twin axes) and lattice planes (twin planes).

In most twins the symmetry of a twin (twin point group) is that of the individual point group augmented by the symmetry of the twinning operation; however, a symmetry element that is oblique to the twinning element of symmetry is absent in the twin (e.g., spinel twins: m$\bar 3$m crystal point group; {111} twin law; $\bar 3$/m twin point group.

# Classification of twins

Twins are classified following Friedel's reticular (i.e. lattice) theory of twinning. This theory states that the presence, either in the lattice or a sublattice of a crystal, of (pseudo)symmetry elements is a necessary, even if not sufficient, condition for the formation of twins. In presence of the reticular necessary conditions, the formation of a twing finally still depends on the matching of the crystal structures at the contact surface between the individuals. The following categories of twins are described under the listed entries.

• twinning by metric merohedry

• twin obliquity

The twin obliquity is a measure of the distorsion of a (sub)lattice in twins by (reticular) pseudomerohedry.

• corresponding twins