Twin obliquity

Obliquité de la macle (Fr). 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 = G^T, 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 of Crystallography, Volume C
• Chapter 3.3 of International Tables of Crystallography, Volume D
• How to deal with zero-obliquity TLQS twinning: Acta Cryst. (2007). A63, 278-286