# Twin obliquity

### From Online Dictionary of Crystallography

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*) 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** = **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) 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.

## 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