Difference between revisions of "Twin obliquity"

From Online Dictionary of Crystallography

Obliquité (Fr). Obliquità (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 = GT, 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