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Difference between revisions of "Twin index"

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<Font color="black"> Indice di geminazione </Font> (''It'')
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<font color="blue">Indice de macle</font> (''Fr''). <font color="red">Zwillingsindex</font> (''Ge''). <font color="black">Indice di geminazione</font> (''It''). <font color="purple">双晶指数</font> (''Ja''). <font color="green">Índice de macla</font> (''Sp'').
  
  
= Twin index =
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== Definition ==
  
A [[twinning]] operation overlaps both the direct and reciprocal lattices of the individuals that form a twin; consequently, the nodes of the individual lattices are overlapped (restored) to some extent ([[overlap of lattices]]). The reciprocal ''n'' of the fraction 1/''n'' of (quasi)restored nodes is called ''twin index''
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A [[twin operation]] overlaps both the direct and reciprocal lattices of the individuals that form a twin; consequently, the nodes of the individual lattices are overlapped (restored) to some extent ([[twinning (effects of)|twinning]]). The reciprocal ''n'' of the fraction 1/''n'' of (quasi)-restored nodes is called the '''twin index'''.
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Let (''hkl'') be the twin plane and [''uvw''] the lattice direction (quasi)-normal to it. Alternatively, let [''uvw''] be the twin axis and (''hkl'') the lattice plane (quasi)-normal to it. For ''twofold operations'' (180º rotations or reflections) the twin index is:
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<div align="center">
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''n'' = ''X''/''f'', ''X'' = |''uh''+''vk''+''wl''|
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</div>
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where ''f'' depends on the [[direct lattice|lattice type]] and on the parities of ''X'', ''h'', ''k'', ''l'', ''u'', ''v'' and ''w'', as in the following table.
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<table border cellspacing=0 cellpadding=5 align=center>
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<tr>
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<th>Lattice type</th><th>Condition on ''hkl''</th><th>Condition on ''uvw''</th><th>Condition on ''X''</th><th>''n''</th>
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</tr>
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<tr>
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<td rowspan="2" align="center">''P''</td><td rowspan="2">none</td><td rowspan="2">none</td><td>X odd</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td>X even</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr>
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<td rowspan="5" align="center">''C''</td><td>''h+k'' odd</td><td>none</td><td>none</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td rowspan="4">''h+k'' even</td><td rowspan="2">''u+v'' and ''w'' not both even</td><td>''X'' odd</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td>''X'' even</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td rowspan="2">''u+v'' and ''w'' both even</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
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</tr>
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<tr>
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<td rowspan="5" align="center">''B''</td><td>''h+l'' odd</td><td>none</td><td>none</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td rowspan="4">''h+l'' even</td><td rowspan="2">''u+w'' and ''v'' not both even</td><td>''X'' odd</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td>''X'' even</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td rowspan="2">''u+w'' and ''v'' both even</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
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</tr>
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<tr>
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<td rowspan="5" align="center">''A''</td><td>''k+l'' odd</td><td>none</td><td>none</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td rowspan="4">''k+l'' even</td><td rowspan="2">''v+w'' and ''u'' not both even</td><td>''X'' odd</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td>''X'' even</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td rowspan="2">''v+w'' and ''u'' both even</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
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</tr>
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<tr>
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<td rowspan="5" align="center">''I''</td><td>''h+k+l'' odd</td><td>none</td><td>none</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td rowspan="4">''h+k+l'' even</td><td rowspan="2">''u'', ''v'' and ''w'' not all odd</td><td>''X'' odd</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td>''X'' even</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td rowspan="2">''u'', ''v'' and ''w'' all odd</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
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</tr>
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<tr>
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<td ''Italic text''rowspan="5" align="center">''F''</td><td>none</td><td>''u''+''v''+''w'' odd</td><td>none</td><td>''n'' = ''X''</td>
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</tr>
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<tr>
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<td rowspan="2">''h'', ''k'', ''l'' not all odd</td><td rowspan="2">u+v+w even</td><td>''X'' odd</td><td>''n'' = ''X''</td>
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</tr>
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<tr><td>''X'' even</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr>
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<td rowspan="2">''h'', ''k'', ''l'' all odd</td><td rowspan="2">u+v+w even</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td>
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</tr>
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<tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
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</tr>
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</table>
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When the twin operation is a rotation of higher degree about [''uvw''], in general the rotational symmetry of the two-dimensional mesh in the (''hkl'') plane no longer coincides with that of the twin operation. The degree of restoration of lattice nodes must now take into account the two-dimensional coincidence index &Xi; for a plane of the family (''hkl''), which defines a super mesh in the [[twin lattice]]. Moreover, such a super mesh may exist in &xi; planes out of ''N'', depending on where is located the intersection of the [''uvw''] twin axis with the plane. The twin index ''n'' is finally given by
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<div align="center">
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''n'' = ''N''&Xi;/&xi;.
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</div>
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== References ==
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*Chapter 3.1.9 of ''International Tables for X-ray Crystallography'' (1959)
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==History==
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*Friedel, G. (1904). ''Étude sur les groupements cristallins. Extrait du Bullettin de la Société de l'Industrie minérale, Quatrième série'', Tomes III et IV. Saint-Étienne, Société de l'imprimerie Thèolier J. Thomas et C., 485 pp.
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*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 for Crystallography, Volume C''
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*Chapter 3.3 of ''International Tables for Crystallography, Volume D''
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[[Category:Twinning]]

Latest revision as of 14:11, 20 November 2017

Indice de macle (Fr). Zwillingsindex (Ge). Indice di geminazione (It). 双晶指数 (Ja). Índice de macla (Sp).


Definition

A twin operation overlaps both the direct and reciprocal lattices of the individuals that form a twin; consequently, the nodes of the individual lattices are overlapped (restored) to some extent (twinning). The reciprocal n of the fraction 1/n of (quasi)-restored nodes is called the twin index.

Let (hkl) be the twin plane and [uvw] the lattice direction (quasi)-normal to it. Alternatively, let [uvw] be the twin axis and (hkl) the lattice plane (quasi)-normal to it. For twofold operations (180º rotations or reflections) the twin index is:

n = X/f, X = |uh+vk+wl|

where f depends on the lattice type and on the parities of X, h, k, l, u, v and w, as in the following table.

Lattice typeCondition on hklCondition on uvwCondition on Xn
PnonenoneX oddn = X
X evenn = X/2
Ch+k oddnonenonen = X
h+k evenu+v and w not both evenX oddn = X
X evenn = X/2
u+v and w both evenX/2 oddn = X/2
X/2 evenn = X/4
Bh+l oddnonenonen = X
h+l evenu+w and v not both evenX oddn = X
X evenn = X/2
u+w and v both evenX/2 oddn = X/2
X/2 evenn = X/4
Ak+l oddnonenonen = X
k+l evenv+w and u not both evenX oddn = X
X evenn = X/2
v+w and u both evenX/2 oddn = X/2
X/2 evenn = X/4
Ih+k+l oddnonenonen = X
h+k+l evenu, v and w not all oddX oddn = X
X evenn = X/2
u, v and w all oddX/2 oddn = X/2
X/2 evenn = X/4
Fnoneu+v+w oddnonen = X
h, k, l not all oddu+v+w evenX oddn = X
X evenn = X/2
h, k, l all oddu+v+w evenX/2 oddn = X/2
X/2 evenn = X/4


When the twin operation is a rotation of higher degree about [uvw], in general the rotational symmetry of the two-dimensional mesh in the (hkl) plane no longer coincides with that of the twin operation. The degree of restoration of lattice nodes must now take into account the two-dimensional coincidence index Ξ for a plane of the family (hkl), which defines a super mesh in the twin lattice. Moreover, such a super mesh may exist in ξ planes out of N, depending on where is located the intersection of the [uvw] twin axis with the plane. The twin index n is finally given by

n = NΞ/ξ.

References

  • Chapter 3.1.9 of International Tables for X-ray Crystallography (1959)


History

  • Friedel, G. (1904). Étude sur les groupements cristallins. Extrait du Bullettin de la Société de l'Industrie minérale, Quatrième série, Tomes III et IV. Saint-Étienne, Société de l'imprimerie Thèolier J. Thomas et C., 485 pp.
  • Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.

See also

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