Twin obliquity - Revision history

MassimoNespolo: German

2025-05-09T08:10:29Z

German

BrianMcMahon: Tidied translations.

2017-11-20T14:20:35Z

Tidied translations.

BrianMcMahon: Style edits to align with printed edition

2017-05-17T14:24:22Z

Style edits to align with printed edition

BrianMcMahon: Style edits to align with printed edition

2017-05-17T14:08:55Z

Style edits to align with printed edition

MassimoNespolo: zero-obliquity TLQS

2015-06-07T01:51:12Z

zero-obliquity TLQS

MassimoNespolo at 17:32, 15 April 2007

2007-04-15T17:32:20Z

MassimoNespolo at 17:27, 15 April 2007

2007-04-15T17:27:19Z

MassimoNespolo at 09:42, 15 May 2006

2006-05-15T09:42:12Z

AndreAuthier at 16:43, 26 April 2006

2006-04-26T16:43:02Z

MassimoNespolo: ==History==

2006-04-26T15:07:09Z

==History==

← Older revision		Revision as of 08:10, 9 May 2025
Line 1:		Line 1:
−	<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'').	+	<font color="blue">Obliquité de la macle</font> (''Fr''). <font color="red">Zwillingsschiefe</font> (''Ge''). <font color="black">Obliquità del geminato</font> (''It''). <font color="purple">双晶傾斜</font> (''Ja'').

← Older revision		Revision as of 14:20, 20 November 2017
Line 1:		Line 1:
−	<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'').	+	<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'').

@@ Line 33: / Line 33: @@
 ==See also ==
-*Chapter 1.3 of ''International Tables of Crystallography, Volume C''
+*Chapter 1.3 of ''International Tables for Crystallography, Volume C''
-*Chapter 3.3 of ''International Tables of Crystallography, Volume D''
+*Chapter 3.3 of ''International Tables for Crystallography, Volume D''
-*How to deal with zero-obliquity TLQS twinning: [http://dx.doi.org/10.1107/S0108767307012135 Acta Cryst. (2007). A63, 278-286]
+*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]]

@@ Line 1: / Line 1: @@
-<p><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'')
+<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'').
@@ Line 7: / Line 6: @@
 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 &omega;'''.
-The vector in direct space [''uvw''] has length L(''uvw''); the [[reciprocal lattice]] vector [''hkl'']* has length L*(''hkl''). The obliquity &omega; 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 &omega; is thus the angle between the vectors [''uvw''] and [''hkl'']*; the scalar product between these two vectors is
 <div align="center">
-L(''uvw'') L*(''hkl'') cos&omega; = <''uvw''|''hkl''> = ''uh'' + ''vk'' + ''wl''
+''L''(''uvw'') ''L''*(''hkl'') cos &omega; = <''uvw''|''hkl''> = ''uh'' + ''vk'' + ''wl''
 </div>
-where <| stands for a 1x3 row matrix and |> for a 3x1 column matrix.
+where <| stands for a 1&times;3 row matrix and |> for a 3&times;1 column matrix.
 It follows that
 <div align="center">
-cos&omega; = (''uh'' + ''vk'' + ''wl'')/L(''uvw'')L*(''hkl'')
+cos &omega; = (''uh'' + ''vk'' + ''wl'')/''L''(''uvw'')''L''*(''hkl'')
 </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>-1</sup> (and thus '''G''' = '''G'''*<sup>-1</sup>) and that the matrix representation of the metric tensor is symmetric and coincides thuas with its transpose ('''G''' = '''G'''<sup>T</sup>, '''G'''* = '''G'''*<sup>T</sup>).
+Notice that '''G'''* = '''G'''<sup>&minus;1</sup> (and thus '''G''' = '''G'''*<sup>&minus;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==
-*Friedel, G. (1920) Contribution à l'étude géométrique des macles. ''Bull Soc fr Minér''., '''43''' 246-295
+*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) 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). ''International Tables for X-ray Crystallography'' (1959), Vol. III, ch. 3.1.9. Birmingham: Kynoch Press.
 ==See also ==

@@ Line 23: / Line 23: @@
 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>-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>).
+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 thuas with its transpose ('''G''' = '''G'''<sup>T</sup>, '''G'''* = '''G'''*<sup>T</sup>).
 ==History==
@@ Line 33: / Line 34: @@
 ==See also ==
-*Chapter 1.3 of ''International Tables of Crystallography, Volume C''<br>
+*Chapter 1.3 of ''International Tables of Crystallography, Volume C''
-*Chapter 3.3 of ''International Tables of Crystallography, Volume D''<br>
+*Chapter 3.3 of ''International Tables of Crystallography, Volume D''
 [[Category:Twinning]]

← Older revision		Revision as of 17:32, 15 April 2007
Line 1:		Line 1:
−	<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> (''Fr''). <font color="black"> Obliquità del geminato</font> (''It''). <font color="purple"> 双晶傾斜</font> (''Ja'')
	</p>		</p>

@@ Line 27: / Line 27: @@
 ==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
 ==History==
@@ Line 34: / Line 33: @@
 *Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.
 [[Category:Fundamental crystallography]]

@@ Line 3: / Line 3: @@
-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]].
+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]].
 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 &omega;'''.
@@ Line 29: / Line 29: @@
 *Chapter 1.3 of International Tables of Crystallography, Volume C
 *Chapter 3.3 of International Tables of Crystallography, Volume D
 [[Category:Fundamental crystallography]]