Difference between revisions of "Twin obliquity"
From Online Dictionary of Crystallography
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<p><font color="blue"> Obliquité de la macle</font>(<i>Fr</i>). <font color="black"> Obliquità del geminato</font>(<i>It</i>)
<p><font color="blue"> Obliquité de la macle</font> (<i>Fr</i>). <font color="black"> Obliquità del geminato</font> (<i>It</i>)
Revision as of 14:32, 26 April 2006
Obliquité de la macle (Fr). Obliquità del geminato (It)
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).
- 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