Twinning of rhombohedral crystals

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. 406

Section 3.3.6.5. Twinning of rhombohedral crystals

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

3.3.6.5. Twinning of rhombohedral crystals

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In some rhombohedral crystals such as corundum Al₂O₃ (Wallace & White, 1967), calcite CaCO₃ or FeBO₃ (calcite structure) (Kotrbova et al., 1985; Klapper, 1987), growth twinning with a `twofold twin rotation around the threefold symmetry axis [001]' (similar to the Dauphiné twins in low-temperature quartz described above) is common. Owing to the eigensymmetry $[{\bar 3}2/m]$ (order 12), the following 12 twin operations form the coset (twin law). They are described here in hexagonal axes:

(i) three rotations around the threefold axis : , , $[6^5 = 6^{-1}]$ ;
(ii) three twofold rotations around the axes , , $[[1{\bar 1}0]]$ ;
(iii) three reflections across the planes $[(10{\bar 1}0)]$ , $[(1{\bar 1}00)]$ , $[(01{\bar 1}0)]$ ;
(iv) three rotoinversions around the threefold axis : $[{\bar 6}{^1}]$ , $[{\bar 6}{^3} = m_z]$ and $[{\bar 6}{^5} = {\bar 6}{^{-1}}]$ .

Some of these twin elements are shown in Fig. 3.3.6.4. They include the particularly conspicuous twin reflection plane [m_z] perpendicular to the threefold axis [001]. The composite symmetry is $[{\cal K} = {6'\over m'} ({\bar 3}) {2\over m} {2'\over m'} \ \ ({\rm order} \ \ 24).]$

Figure 3.3.6.4 | top | pdf |

Twin intergrowth of `obverse' and `reverse' rhombohedra of rhombohedral FeBO₃. (a) `Obverse' rhombohedron with four of the 12 alternative twin elements. (b) `Reverse' rhombohedron (twin orientation). (c) Interpenetration of both rhombohedra, as observed in penetration twins of FeBO₃. (d) Idealized skeleton of the six components (exploded along [001] for better recognition) of the `obverse' orientation state shown in (a). The components are connected at the edges along the threefold and the twofold eigensymmetry axes. The shaded faces are $[\{10{\bar 1}0\}]$ and (0001) coinciding twin reflection and contact planes with the twin components of the `reverse' orientation state. Parts (a) to (c) courtesy of R. Diehl, Freiburg.

It is of interest that for FeBO₃ crystals this twin law always, without exception, forms penetration twins (Fig. 3.3.6.4), whereas for the isotypic calcite CaCO₃ only (0001) contact twins are found (Fig. 3.3.6.5). This aspect is discussed further in Section 3.3.8.6.

Figure 3.3.6.5 | top | pdf |

Contact growth twin of calcite with the same twin law as FeBO₃ in Fig. 3.3.6.4. Conspicuous twin element: twin reflection plane (0001), coinciding with the composition plane (0001).

References

Klapper, H. (1987). X-ray topography of twinned crystals. In Progress in crystal growth and characterization, Vol. 14, edited by P. Krishna. pp. 367–401. Oxford: Pergamon.Google Scholar

Kotrbova, M., Kadeckova, S., Novak, J., Bradler, J., Smirnov, G. V. & Shvydko, Yu. V. (1985). Growth and perfection of flux-grown FeBO₃ and ⁵⁷FeBO₃ crystals. J. Cryst. Growth, 71, 607–614.Google Scholar

Wallace, C. A. & White, E. A. D. (1967). The morphology and twinning of solution-grown corundum crystals. In Crystal growth, edited by H. S. Peiser (Supplement to Phys. Chem. Solids), pp. 431–435. Oxford: Pergamon.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, p. 406