# Twinning

## Other languages

Twin: Macles (Fr). Zwillinge (Ge). Maclas (Sp). двойники (Ru). Geminati (It)

Twinning: Maclage (Fr). Zwillingsbildung (Ge). Maclado (formación de macla) (Sp). двойникование (Ru).  ? (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).

• twin law

The twin law is indicated by the symbol of the twinning element of symmetry: $\bar 1$, [uvw] and (hkl) for the centre of symmetry, direction of the rotation axis and Miller indices of the mirror plane, in the order. Usually, instead of the single (hkl) plane, the symbol {hkl} is used to indicate all the planes equivalent for symmetry.

# Classification of twins

Twins are classified following Friedel reticular (i.e. lattice) theory of twinning which indicates the presence, either in the lattice or a sublattice of a crystal, of (pseudo)symmetry elements as necessary, even if not sufficient, condition for the formation of twins. In presence of the reticular necessary conditions, the formation of a twing finally depends on the matching of the crystal structures at the contact surface between the individuals.

• twinning by merohedry

The twin operation belongs to to the point group of the lattice but not to the point group of the crystal. Therefore, the point group of the crystal must be a subgroup of the point group of the lattice, i.e. the crystal shows only a part (merohedry) of the symmetry elements belonging to the its lattice which, instead, shows holohedry (complete symmetry). The twinning element of symmetry may (Class I of twins by merohedry) or may not belong to the Laue class of the crystal (Class II of twins by merohedry): consequences are discussed under solving the crystal strcuture of twins. - Examples - Class I: the mirror plane m acts as twinning operator in crystals with point group 2 (Laue group 2/m). Class II:

• twinning by pseudomerohedry

• twinning by reticular merohedry

• twinning by reticular pseudomerohedry

• overlap of lattices

• twin lattice

• twin index

• twin obliquity