Difference between revisions of "Twin obliquity"
From Online Dictionary of Crystallography
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| − | The concept of obliquity was introduced by Friedel in 1920 as a measure of the overlap of the [[Direct lattice|lattice]]s on the individuals forming a [[twin]]. | + | 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 [[Direct lattice|lattice]]s 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 ('' | + | 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 | 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 | ||
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==References== | ==References== | ||
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*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. | *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 1.3 of International Tables of Crystallography, Volume C | ||
Revision as of 13:22, 26 April 2006
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