Pseudo n-fold twin rotations (twin axes) with

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. 396-397

Section 3.3.2.3.2. Pseudo n-fold twin rotations (twin axes) with $[n\ge 3]$

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

3.3.2.3.2. Pseudo n-fold twin rotations (twin axes) with $[n\ge 3]$

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There is a long-lasting controversy in the literature, e.g. Hartman (1956, 1960), Buerger (1960b), Curien (1960), about the acceptance of three-, four- and sixfold rotation axes as twin elements, for the following reason:

Twin operations of order two (reflection, twofold rotation, inversion) are `exact', i.e. in a component pair they transform the orientation state of one component exactly into that of the other. There occur, in addition, many cases of multiple twins, which can be described by three-, four- and sixfold twin axes . These axes, however, are pseudo axes because their rotation angles are close to but not exactly equal to 120, 90 or 60°, due to metrical deviations (no matter how small) from a higher-symmetry lattice. A well known example is the triple twin (German: Drilling) of orthorhombic aragonite, where the rotation angle $[\gamma=]$ $[2\arctan b/a = 116.2^\circ]$ (which transforms the orientation state of one component exactly into that of the other) deviates significantly from the 120° angle of a proper threefold rotation (Fig. 3.3.2.4). Another case of n = 3 with a very small metrical deviation is provided by ammonium lithium sulfate (γ = 119.6°).

Figure 3.3.2.4 | top | pdf |

(a) Triple growth twin of orthorhombic aragonite, CaCO₃, with pseudo-threefold twin axis. The gap angle is 11.4°. The exact description of the twin aggregate by means of two symmetrically equivalent twin mirror planes (110) and ( $[{\bar 1}10]$ ) is indicated. In actual crystals, the gap is usually closed as shown in (b).

All these (pseudo) n-fold rotation twins, however, can also be described by (exact) binary twin elements , viz by a cyclic sequence of twin mirror planes or twofold twin axes . This is also illustrated and explained in Fig. 3.3.2.4. This possibility of describing cyclic twins by `exact' binary twin operations is the reason why Hartman (1956, 1960) and Curien (1960) do not consider `non-exact' three-, four- and sixfold rotations as proper twin operations.

The crystals forming twins with pseudo n-fold rotation axes always exhibit metrical pseudosymmetries. In the case of transformation twins and domain structures, the metrical pseudosymmetries of the low-symmetry (deformed) phase $[\cal{H}]$ result from the true structural symmetry $[{\cal G}]$ of the parent phase (cf. Section 3.3.7.2). This aspect caused several authors [e.g. Friedel, 1926, p. 435; Donnay (cf. Hurst et al., 1956); Buerger, 1960b] to accept these pseudo axes for the treatment of twinning. The present authors also recommend including three-, four- and sixfold rotations as permissible twin operations. The consequences for the definition of the twin law will be discussed in Section 3.3.4 and in Section 3.4.3 . For a further extension of this concept to fivefold and tenfold multiple growth twins, see Note (6) below and Example 3.3.6.8.

References

Buerger, M. J. (1960b). Introductory remarks. Twinning with special regard to coherence. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 3 and 5–7. Madrid: CSIC.Google Scholar

Curien, H. (1960). Sur les axes de macle d'ordre supérieur à deux. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 9–11. Madrid: CSIC.Google Scholar

Friedel, G. (1926). Lecons de cristallographie, ch. 15. Nancy, Paris, Strasbourg: Berger-Levrault. [Reprinted (1964). Paris: Blanchard].Google Scholar

Hartman, P. (1956). On the morphology of growth twins. Z. Kristallogr. 107, 225–237.Google Scholar

Hartman, P. (1960). Epitaxial aspects of the atacamite twin. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 15–18. Madrid: CSIC.Google Scholar

Hurst, V. J., Donnay, J. D. H. & Donnay, G. (1956). Staurolite twinning. Mineral. Mag. 31, 145–163.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 396-397