Inversion twins in orthorhombic 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. 403

Section 3.3.6.1. Inversion twins in orthorhombic 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

3.3.6.1. Inversion twins in orthorhombic crystals

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The (polar) 180° twin domains in a (potentially ferroelectric ) crystal of eigensymmetry $[{\cal H} = mm2]$ ( [m_xm_y2_z] ) and composite symmetry $[{\cal K} = 2/m\,2/m\,2/m]$ (e.g. in KTiOPO₄, NH₄LiSO₄, Li-formate monohydrate ) result from a group–subgroup relation of index [[i] = 2] with invariance of the symmetry framework (merohedral twins), but antiparallel orientation of the polar axes. The orientation relation between the two domain states is described by the coset $[k\times {\cal H}]$ of twin operations shown in Table 3.3.6.1, whereby the reflection in (001), [m_z] , is considered as the `representative' twin operation.

Table 3.3.6.1 | top | pdf |
Orthorhombic inversion twins: coset of alternative twin operations (twin law)

$[{\cal H}]$	$[k \times {\cal H} = m_z \times {\cal H} ]$
1	(normal to the polar axis [001])
	(normal to the polar axis)
	(normal to the polar axis)
	$[{\bar 1}]$ (inversion)

Hence, these twins can be regarded not only as reflection, but also as rotation or inversion twins. The composite symmetry, in black–white symmetry notation, is $[{\cal K} = {2_x'\over m_x}{2_y'\over m_y}{2_z\over m_z'}({\bar 1}{^\prime}),]$ whereby the primed symbols designate the (alternative) twin operations (cf. Section 3.3.5).

References

International Tables for Crystallography (2006). Vol. D. ch. 3.3, p. 403