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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 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[math] \bar 3[/math]m crystal point group; {111} twin law; [math] \bar 3[/math]/m twin point group.

  • twin law

The twin law is indicated by the symbol of the twinning element of symmetry: [math] \bar 1[/math], [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 structure of twins. - Examples - Class I: in crystals with point group 2 (Laue group 2/m) the mirror plane m acts as twinning operator . Class II: in crystals with point group 4 (Laue group 4/m) a mirror plane m parallel to the foufold axis 4 acts as twinning operator.

  • twinning by pseudomerohedry

A lattice is said to be pseudosymmetric if at least a lattice row or/and a lattice plane approximately correspond to elements of symmetry; these elements can act as twinning operators (e.g., a monoclinic lattice with its oblique angle close to 90° is pseudo-orthorhombic and thus shows two pseudo twofold axes and two pseudo mirror planes).

  • twinning by reticular merohedry

In the presence of a sublattice displaying symmetry other than that of the crystal lattice, a symmetry element belonging to the sublattice point group but not to the crystal point group can act as twinning operator. If lattice and sublattice have the same point group but (some of) their elements of symmetry are differently oriented twins by polyholohedry can form.

  • twinning by reticular pseudomerohedry

Substituting lattice with sublattice, the definition of twinning by reticular pseudomerohedry corresponds to that given for the twinning by pseudomerohedry.

  • overlap of lattices

By effect of a twinning operation, both the direct and reciprocal lattice of the individuals forming a twin are overlapped. Overlapping (restoration) of nodes belonging to different individuals can be: (i) exact and total (twinning by merohedry); (ii) exact but partial (i.e. only a fraction of the nodes of an individual lattice is restored; twinning by reticular merohedry); (iii) total but approximate (twinning by pseudomerohedry), approximate and partial (twinning by reticular pseudomerohedry).

  • twin index

The reciprocal n of the fraction 1/n of (quasi)restored nodes is called twin index

  • twin lattice

The lattice that is formed by the (quasi)restored nodes is the twin lattice. It corresponds to the crystal lattice in twins by (pseudo)merohedry and to a sublattice of the crystal (individual) in twins by reticular (pseudo)merohedry.

  • twin obliquity

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

  • twinning by metric merohedry

  • corresponding twins

Other categories of twins

Endemic conditions for twinning

See also

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