Twin obliquity

Obliquité de la macle (Fr). Obliquità del geminato (It)

The concept of obliquity was introduced by Friedel in 1920 (Bull Soc fr Minér., 43 246-295) 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) 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 thus with its transpose (G = G^T, G* = G*^T).

References

• 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