9.26: Twin obliquity
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 ) 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 ).