Twin obliquity
From Online Dictionary of Crystallography
Obliquité de la macle (Fr). Zwillingsschiefe (Ge). Obliquità del geminato (It). 双晶傾斜 (Ja).
The concept of obliquity was introduced by Friedel in 1920 as a measure of the overlap of the lattices on the individuals forming a twin.
Let us indicate with [u ' v ' w '] the direction exactly perpendicular to a twin plane (hkl), and with (h ' k ' l ') the plane perpendicular to a twin axis [uvw]. [u ' v ' w '] is parallel to the reciprocal lattice vector [hkl]* and (h ' k ' l ') is parallel to the reciprocal lattice plane (uvw)*. The angle between [uvw] and [u ' v ' w '] or, which is the same, between (hkl) and (h ' k ' l '), is called the obliquity ω.
The vector in direct space [uvw] has length L(uvw); the reciprocal lattice vector [hkl]* has length L*(hkl). The obliquity ω is thus the angle between the vectors [uvw] and [hkl]*; the scalar product between these two vectors is
L(uvw) L*(hkl) cos ω = <uvw|hkl> = uh + vk + wl
where <| stands for a 1×3 row matrix and |> for a 3×1 column matrix.
It follows that
cos ω = (uh + vk + wl)/L(uvw)L*(hkl)
where L(uvw) = <uvw|G|uvw>1/2 and L*(hkl) = <hkl|G*|hkl>1/2, G and G* being the metric tensors in direct and reciprocal space, respectively.
Notice that G* = G−1 (and thus G = G*−1) and that the matrix representation of the metric tensor is symmetric and coincides thus with its transpose (G = GT, G* = G*T).
When the twin operation is of order higher than two, an imperfect overlap of lattice nodes may correspond to zero obliquity. For example, a pseudo-tetragonal crystal twinned by a fourfold rotation about the direction of pseudo-symmetry would produce a small deviation from the exact overlap of the lattice nodes, yet the obliquity is zero because the direction is perpendicular to a lattice plane. These cases are called zero-obliquity TLQS twinning and require a linear, instead of angular, measure of the deviation from the lattice overlap.
History
- Friedel, G. (1920). Bull. Soc. Fr. Minér. 43 246-295. Contribution à l'étude géométrique des macles.
- Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.
- Donnay, J. D. H. and Donnay, G. (1959). International Tables for X-ray Crystallography (1959), Vol. III, ch. 3.1.9. Birmingham: Kynoch Press.
See also
- Chapter 1.3 of International Tables for Crystallography, Volume C
- Chapter 3.3 of International Tables for Crystallography, Volume D
- Nespolo, M. and Ferraris, G. (2007). Acta Cryst. A63, 278-286. Overlooked problems in manifold twins: twin misfit in zero-obliquity TLQS twinning and twin index calculation. (Discusses how to deal with zero-obliquity TLQS twinning.)