International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 1.3, p. 10
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A twin consists of two or more single crystals of the same species but in different orientation, its twin components. They are intergrown in such a way that at least some of their lattice directions are parallel. The twin law describes the geometrical relation between the twin components. It specifies a symmetry operation, the twin operation, that brings one of the twin components into parallel orientation with the other. The corresponding symmetry element is called the twin element.
There are several kinds of twin laws:
In many cases, there does not exist a unique twin law, but a twin may be described equally well by more than one twin law. (a) If the crystal structure of the twin components contains an evenfold rotation or screw-rotation axis, an inversion twin cannot be distinguished from a reflection twin with twin plane perpendicular to that axis. (b) If the crystal structure contains a mirror or a glide plane, an inversion twin cannot be distinguished from a rotation twin with a twofold twin axis perpendicular to that plane. (c) If for a centrosymmetrical crystal structure the normal of a twin plane runs parallel to a lattice vector or a twin axis runs perpendicular to a net plane, the twin may be described equally well as a reflection twin or as a rotation twin.
The twin components are grown together in a surface called composition surface, twin interface or twin boundary. In most cases, the composition surfaces are low-energy surfaces with good structural fit. For a reflection twin, it is usually a plane parallel to the twin plane. The composition surface of a rotation twin may either be a plane parallel to the twin axis or be a non-planar surface with irregular shape.
If more than two components are twinned according to the same law, the twin is called a repeated twin or a multiple twin. If all the twin boundaries are parallel planes, it is a polysynthetic twin, otherwise it is called a cyclic twin. If the twin components are related to each other by more than one twin law, the shape and the mutual arrangement of the twin domains may be very irregular.
With respect to the formation process, one may distinguish between growth twins, transformation twins, and mechanical (deformation, glide) twins. Transformation twins result from phase transitions, e.g. of ferroelectric or ferromagnetic crystals. The corresponding twin domains are usually small and the number of such domains is high. Mechanical twinning is due to mechanical stress and may often be described in terms of shear of the crystal structure. This includes ferroelasticity.
Twins are observable by, for example, macroscopic or microscopic observation of re-entrant angles between crystal faces, by etching, by means of different extinction positions for the twin components between cross polarizers of a polarization microscope, by different rotation angles of the plane of polarization of a beam of plane-polarized light passing through the components of a twin showing optical activity, by a splitting of part of the X-ray diffraction spots (except for twins by merohedry), by means of domain contrast or boundary contrast in an X-ray topogram, or by investigation with a transmission electron microscope.
The phenomenon of twinning has frequently been described and discussed in the literature and it is impossible, therefore, to give a complete list of references. Further details may be learned, e.g. from a review article by Cahn (1954) or from appropriate textbooks. A comprehensive survey of X-ray topography of twinned crystals is given by Klapper (1987). The following papers are related to twinning by merohedry or pseudo-merohedry: Catti & Ferraris (1976), Grimmer (1984, 1989a, b), Grimmer & Warrington (1985), Donnay & Donnay (1974), Le Page, Donnay & Donnay (1984), Hahn (1981, 1984), Klapper, Hahn & Chung (1987), Flack (1987).
References
Cahn, R. W. (1954). Twinned crystals. Adv. Phys. 3, 363–445.Google ScholarCatti, M. & Ferraris, G. (1976). Twinning by merohedry and X-ray crystal structure determination. Acta Cryst. A32, 163–165.Google Scholar
Donnay, G. & Donnay, J. D. H. (1974). Classification of triperiodic twins. Can. Mineral. 12, 422–425.Google Scholar
Flack, H. D. (1987). The derivation of twin laws for (pseudo-) merohedry by coset decomposition. Acta Cryst. A43, 564–568.Google Scholar
Grimmer, H. (1984). The generating function for coincidence site lattices in the cubic system. Acta Cryst. A40, 108–112.Google Scholar
Grimmer, H. (1989a). Systematic determination of coincidence orientations for all hexagonal lattices with axial ratios c/a in a given interval. Acta Cryst. A45, 320–325.Google Scholar
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Grimmer, H. & Warrington, D. H. (1985). Coincidence orientations of grains in hexagonal materials. J. Phys. (Paris), 46, C4, 231–236.Google Scholar
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Hahn, Th. (1984). Twin domains and twin boundaries. Acta Cryst. A40, C-117.Google Scholar
Klapper, H. (1987). X-ray topography of twinned crystals. Prog. Cryst. Growth Charact. 14, 367–401.Google Scholar
Klapper, H., Hahn, Th. & Chung, S. J. (1987). Optical, pyroelectric and X-ray topographic studies of twin domains and twin boundaries in KLiSO4. Acta Cryst. B43, 147–159.Google Scholar
LePage, Y., Donnay, J. D. H. & Donnay, G. (1984). Printing sets of structure factors for coping with orientation ambiguities and possible twinning by merohedry. Acta Cryst. A40, 679–684.Google Scholar