Twins of twins

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, p. 412

Section 3.3.6.14. Twins of twins

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

3.3.6.14. Twins of twins

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This term is due to Henke (2003) and refers to the simultaneous occurrence (superposition) of two or more different twin types (twin laws) in one and the same crystal. In twins of twins, one `generation' of twin domains is superimposed upon the other, each with its own twin law. This may occur as a result of:

(1) two successive phase transitions, each with its own twinning scheme, or
(2) one phase transition with loss of two kinds of symmetry elements, or
(3) a phase transition superimposed on an existing growth twin.

Typical examples are:

(i) the cubic–tetragonal ( $[m{\bar 3}m \Longleftrightarrow 4mm]$ ) phase transition of BaTiO₃, described above. Here, 90° domains (due to the loss of the diagonal mirror planes) are superimposed by 180° domains (due to the loss of the inversion centres);
(ii) a similar case (tetragonal–monoclinic) is provided by the `type case' of Henke (2003), (NO)₂VCl₆;
(iii) ammonium lithium sulfate exhibits pseudo-hexagonal growth-sector twins upon which lamellae of ferroelectric 180° domains are superimposed.

In this context, the term complete twin should be noted. It was coined by Curien & Donnay (1959) for the symmetry description of a crystal containing several merohedral twin laws. Their preferred example was quartz, but there are many relevant cases:

(i) The complete merohedral `twins of twins' of quartz, i.e. the superposition of the Dauphiné, Brazil and Leydolt twins , can be formulated as follows: $[\eqalign{\hbox{Dauphin\'e twin law: } 321 &\Rightarrow 6'(3)22'\cr\hbox{Brazil twin law: } 321 &\Rightarrow {\bar 3}{^\prime}(3)2/m'1({\bar 1}{^\prime})\cr\hbox{Leydolt twin law: } 321 &\Rightarrow {\bar 6}{^\prime}(3)2m' = 3/m'2m'.}]$ Combination = `complete twin': $[6'(3)/m'\,2/m'\,2'/m'({\bar 1}{^\prime})]$ ; this symmetry corresponds to the hexagonal holohedral point group $[6/m\,2/m\,2/m]$ , cf. Example 3.3.6.3.
(ii) Another example is provided by KLiSO₄ (crystal class 6), extensively investigated by Klapper et al. (1987): $[\eqalign{\hbox{Inversion twins: }6 &\Rightarrow 6/m'({\bar 1}{^\prime})\cr\hbox{Reflection twins: }6 &\Rightarrow 6m'm'\cr\hbox{Rotation twins: }6 &\Rightarrow 62'2'.}]$ Combination = `complete twin': $[ 6/m'2'/m'2'/m'({\bar 1}{^\prime})]$ ; this symmetry is isomorphic to the complete-twin symmetry of quartz, given above, and to the hexagonal holohedral point group $[6/m\,2/m\,2/m]$ .

References

Curien, H. & Donnay, J. D. H. (1959). The symmetry of the complete twin. Am. Mineral. 44, 1067–1071.Google Scholar

Henke, H. (2003). Crystal structures, order–disorder transition and twinning of the Jahn–Teller system (NO)₂VCl₆. Z. Kristallogr. 218, 617–625.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

International Tables for Crystallography (2006). Vol. D. ch. 3.3, p. 412