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Difference between revisions of "Twin obliquity"

From Online Dictionary of Crystallography

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==References==
 
==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.
 
*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
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*Chapter 3.3 of International Tables of Crystallography, Volume D
 
  
 
==History==
 
==History==
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*Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.
 
*Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.
  
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==See also ==
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Chapter 1.3 of ''International Tables of Crystallography, Volume C''<br>
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Chapter 3.3 of ''International Tables of Crystallography, Volume D''<br>
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Revision as of 16:43, 26 April 2006

Obliquité de la macle (Fr). Obliquità del geminato (It)


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 = 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.


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.

See also

Chapter 1.3 of International Tables of Crystallography, Volume C
Chapter 3.3 of International Tables of Crystallography, Volume D