From Online Dictionary of Crystallography
Obliquité de la macle (Fr). Obliquità del geminato (It). 双晶傾斜 (Ja)
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) cosω = <uvw|hkl> = uh + vk + wl
where <| stands for a 1x3 row matrix and |> for a 3x1 column matrix.
It follows that
cosω = (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 thuas 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.
- Friedel, G. (1920) Contribution à l'étude géométrique des macles. Bull Soc fr Minér., 43 246-295
- Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.
- Donnay, J.D.H. and Donnay, G. (1959) Twinning, section 3.1.9 in International Tables for X-Ray Crystallography, Vol. III. Birmingham: Kynoch Press.
- 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