Difference between revisions of "Twin obliquity"
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| − | + | <font color="blue">Obliquité de la macle</font> (''Fr''). <font color="black">Obliquità del geminato</font> (''It''). <font color="purple">双晶傾斜</font> (''Ja''). | |
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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'''. | 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 |
<div align="center"> | <div align="center"> | ||
| − | L(''uvw'') L*(''hkl'') cosω = <''uvw''|''hkl''> = ''uh'' + ''vk'' + ''wl'' | + | ''L''(''uvw'') ''L''*(''hkl'') cos ω = <''uvw''|''hkl''> = ''uh'' + ''vk'' + ''wl'' |
</div> | </div> | ||
| − | where <| stands for a | + | where <| stands for a 1×3 row matrix and |> for a 3×1 column matrix. |
It follows that | It follows that | ||
<div align="center"> | <div align="center"> | ||
| − | cosω = (''uh'' + ''vk'' + ''wl'')/L(''uvw'')L*(''hkl'') | + | cos ω = (''uh'' + ''vk'' + ''wl'')/''L''(''uvw'')''L''*(''hkl'') |
</div> | </div> | ||
| − | where L(''uvw'') = <''uvw''|'''G'''|''uvw''><sup>1/2</sup> and L*(''hkl'') = <''hkl''|'''G'''*|''hkl''><sup>1/2</sup>, '''G''' and '''G'''* being the metric tensors in direct and reciprocal space, respectively. | + | where ''L''(''uvw'') = <''uvw''|'''G'''|''uvw''><sup>1/2</sup> and ''L''*(''hkl'') = <''hkl''|'''G'''*|''hkl''><sup>1/2</sup>, '''G''' and '''G'''* being the metric tensors in direct and reciprocal space, respectively. |
| − | Notice that '''G'''* = '''G'''<sup> | + | Notice that '''G'''* = '''G'''<sup>−1</sup> (and thus '''G''' = '''G'''*<sup>−1</sup>) and that the matrix representation of the metric tensor is symmetric and coincides thus with its transpose ('''G''' = '''G'''<sup>''T''</sup>, '''G'''* = '''G'''*<sup>''T''</sup>). |
| + | 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== | ==History== | ||
| − | *Friedel, G. (1920) | + | *Friedel, G. (1920). ''Bull. Soc. Fr. Minér''. '''43''' 246-295. ''Contribution à l'étude géométrique des macles''. |
| − | *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. |
| − | *Donnay, J.D.H. and Donnay, G. (1959) | + | *Donnay, J. D. H. and Donnay, G. (1959). ''International Tables for X-ray Crystallography'' (1959), Vol. III, ch. 3.1.9. Birmingham: Kynoch Press. |
==See also == | ==See also == | ||
| − | *Chapter 1.3 of ''International Tables | + | *Chapter 1.3 of ''International Tables for Crystallography, Volume C'' |
| − | *Chapter 3.3 of ''International Tables | + | *Chapter 3.3 of ''International Tables for Crystallography, Volume D'' |
| + | *Nespolo, M. and Ferraris, G. (2007). [https://doi.org/10.1107/S0108767307012135 ''Acta Cryst.'' A'''63''', 278-286]. ''Overlooked problems in manifold twins: twin misfit in zero-obliquity TLQS twinning and twin index calculation.'' (Discusses how to deal with zero-obliquity TLQS twinning.) | ||
[[Category:Twinning]] | [[Category:Twinning]] | ||
Latest revision as of 14:20, 20 November 2017
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) cos ω = <uvw|hkl> = uh + vk + wl
where <| stands for a 1×3 row matrix and |> for a 3×1 column matrix.
It follows that
cos ω = (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).
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). Bull. Soc. Fr. Minér. 43 246-295. Contribution à l'étude géométrique des macles.
- Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.
- Donnay, J. D. H. and Donnay, G. (1959). International Tables for X-ray Crystallography (1959), Vol. III, ch. 3.1.9. Birmingham: Kynoch Press.
See also
- Chapter 1.3 of International Tables for Crystallography, Volume C
- Chapter 3.3 of International Tables for Crystallography, Volume D
- Nespolo, M. and Ferraris, G. (2007). Acta Cryst. A63, 278-286. Overlooked problems in manifold twins: twin misfit in zero-obliquity TLQS twinning and twin index calculation. (Discusses how to deal with zero-obliquity TLQS twinning.)