Twinning

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Oriented association and twinning

Crystals (also called individuals or components) or domains 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 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[math] \bar 3[/math]m crystal point group; {111} twin law; [math] \bar 3[/math]/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.

• twinning by merohedry
• twinning by pseudomerohedry
• twinning by reticular merohedry
• twinning by reticular pseudomerohedry
• twinning by metric merohedry
• twinning by reticular polyholohedry
• hybrid twins
• plesiotwins
• allotwins
• selective merohedry

Related articles

• twinning (effects of)
• twin index
• twin lattice
• twin law
• twin obliquity
• corresponding twins
• twinning (endemic conditions of)
• Twinning (effects of)

See also

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