# 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** = **G**^{T}, **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