# Twinning

### 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 a 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 by a reflection [either plane (*reflection twin*) or center (*inversion twin*) of symmetry] or a rotation (*rotation twin*) form a *twin*.

**symmetry of a twin** - See *Eigensymmetry*

An element of symmetry crystallographically relating differently oriented crystals cannot belong to the individual. The element of symmetry that relates the individuals of a twin is called *twin element of symmetry* (or simply *twin element*) and the connected operation is a *twin operation*. The *Mallard's 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 centers*), 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 twin element is absent in the twin (e.g., *spinel twins*: *m**m* crystal point group; {111} twin law; /*m* twin point group).

### Classification of twins

Twins are classified following Friedel's *reticular* (i.e. lattice) *theory of twinning* (see: G. Friedel *Lecons de Cristallographie*, Nancy (1926) where reference to previous work of the author can be found; see also Friedel's law). 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 twin finally still depends on the matching of the crystal structures at the contact surface between the individuals.

### See also

- Chapter 1.3 of
*International Tables of Crystallography, Volume C* - Chapter 3.3 of
*International Tables of Crystallography, Volume D*