# Twinning

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# 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 centre (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 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 twin element 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 (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.

The following categories of twins are described under the listed entries.

# See also

• Chapter 1.3 of International Tables of Crystallography, Volume C
• Chapter 3.3 of International Tables of Crystallography, Volume D