Transformation twinning

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 414-415

Section 3.3.7.2. Transformation twinning

Th. Hahn^a ^* and H. Klapper^b

^a Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

3.3.7.2. Transformation twinning

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A solid-to-solid (polymorphic) phase transition is – as a rule – accompanied by a symmetry change. For displacive and order–disorder transitions , the symmetries of the `parent phase ' (prototype phase) $[{\cal G}]$ and of the `daughter phase' (deformed phase) $[{\cal H}]$ exhibit frequently, but not always, a group–subgroup relation. During the transition to the low-symmetry phase the crystal usually splits into different domains. Three cases of transformation-twin domains are distinguished:

(i) The symmetry operations suppressed during the transition belong to the point group $[{\cal G}]$ of the high-symmetry (prototype) phase, whereas the lattice, except for a small affine deformation, is unchanged (translationengleiche subgroup). In this case, the structures of the domains have different orientations and/or different handedness, both of which are related by the suppressed symmetry elements. Thus, the transition induces twins with the suppressed symmetry elements acting as twin elements (twin law). The number of orientation states is equal to the index $[[i] = \left\vert {\cal G} \right\vert/ \left\vert {\cal H} \right\vert]$ of the group–subgroup relation, i.e. to the number of cosets of $[{\cal G}]$ with respect to $[{\cal H}]$ , including $[{\cal H}]$ itself; cf. Section 3.3.4.1. If, for example, a threefold symmetry axis is suppressed, three domain states related by approximate 120° rotations will occur (for the problems of pseudo-n-fold twin axes, see Section 3.3.2.3.2). A further well known example is the α–β phase transformation of quartz at 846 K. On cooling from the hexagonal $[\beta]$ phase (point group 622) to the trigonal $[\alpha]$ phase (point group 32), the twofold rotation , contained in the sixfold axis of $[\beta]$ -quartz , is suppressed, and so are the other five rotations of the coset [cf. Example 3.3.6.3.1]. Consequently, two domain states appear (Dauphiné twins). These twins are usually described with the twofold axis along [001] as twin element.
(ii) If a lattice translation is suppressed without change of the point-group symmetry (klassengleiche subgroup), i.e. due to loss of cell centring or to doubling (tripling etc.) of a lattice parameter, translation domains (antiphase domains ) are formed (cf. Wondratschek & Jeitschko, 1976). The suppressed translation appears as the fault vector of the translation boundary (antiphase boundary) between the domains. Recently, translation domains were called `translation twins' (T-twins, Wadhawan, 1997, 2000), cf. Section 3.3.2.4, Note (7).
(iii) The two cases can occur together, i.e. point-group symmetry and translation symmetry are both reduced in one phase transition (general subgroup). Here caution in the counting of the number of domain states is advisable since now orientation states and translation states occur together.

Well known examples of ferroelastic transformation twins are K₂SO₄ (Example 3.3.6.7) and various perovskites (Example 3.3.6.13). Characteristic for non-merohedral (ferroelastic) transformation twins are their planar twin boundaries and the many parallel (lamellar) twin domains of nearly equal size. In contrast, the twin boundaries of merohedral (non-ferroelastic ) transformation twins, e.g. Dauphiné twins of quartz, often are curved, irregular and non-parallel.

Transformation twins are closely related to the topic of domain structures, which is extensively treated in Chapter 3.4 of this volume.

A generalization of the concept of transformation twins includes twinning due to structural relationships in a family of related compounds (`structural twins'). Here the parent phase is formed by the high-symmetry `basic structure' (`aristotype') from which the `deformed structures' and their twin laws, occurring in other compounds, can be derived by subgroup considerations similar to those for actual transformation twins. Well known families are ABX₃ (perovskites ) and A₂BX₄ (Na₂SO₄- and K₂SO₄-type compounds). In Example (3) of Section 3.3.9.2.4, growth twins among MeX₂ dichalcogenides are described in detail.

References

Wadhawan, V. K. (1997). A tensor classification of twinning in crystals. Acta Cryst. A53, 546–555.Google Scholar

Wadhawan, V. K. (2000). Introduction to ferroic materials, ch. 7. Amsterdam: Gordon and Breach.Google Scholar

Wondratschek, H. & Jeitschko, W. (1976). Twin domains and antiphase domains. Acta Cryst. A32, 664–666.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 414-415