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Difference between revisions of "Twin index"

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<Font color="blue"> Indice de macle </Font> (''Fr''). <Font color="black"> Indice di geminazione </Font> (''It''), <Font color="purple"> 双晶指数 </Font>(''Ja'')
 
<Font color="blue"> Indice de macle </Font> (''Fr''). <Font color="black"> Indice di geminazione </Font> (''It''), <Font color="purple"> 双晶指数 </Font>(''Ja'')
 
  
 
== Definition ==
 
== Definition ==
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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
 
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>
<table border="1" cellspacing="2" cellpadding="2">
 
 
<tr>
 
<tr>
 
<th>Lattice type</th><th>condition on ''hkl''</th><th>condition on ''uvw''</th><th>condition on ''X''</th><th>''n''</th>
 
<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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<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>
 
<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><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
 
<tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
 
</tr>
 
</tr>
 
  
 
<tr>
 
<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>
 
<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>
 
</tr>
 
</tr>
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<tr>
 
<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>
 
<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>
 
</tr>
 
</tr>
 
<tr><td>''X'' even</td><td>''n'' = ''X''/2</td>
 
<tr><td>''X'' even</td><td>''n'' = ''X''/2</td>
 
</tr>
 
</tr>
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<tr>
 
<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>
 
<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>
 
</tr>
 
</tr>
 
<tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
 
<tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td>
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</tr>
  
 
</table>
 
</table>
</center>
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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 does no longer coincide 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:
 
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 does no longer coincide 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:

Revision as of 11:48, 8 February 2012

Indice de macle (Fr). Indice di geminazione (It), 双晶指数 (Ja)

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 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 does no longer coincide 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 in 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 e 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 of Crystallography, Volume C
Chapter 3.3 of International Tables of Crystallography, Volume D