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'') | ||
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== 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 | ||
− | + | <table border cellspacing=0 cellpadding=5 align=center> | |
− | <table border | ||
<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> | ||
+ | </tr> | ||
<tr> | <tr> | ||
<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> | ||
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<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> | ||
+ | <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> | ||
+ | <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> | ||
+ | </tr> | ||
</table> | </table> | ||
− | + | ||
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: | 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: |
Revision as of 11:48, 8 February 2012
Indice de macle (Fr). Indice di geminazione (It), 双晶指数 (Ja)
Contents
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 type | condition on hkl | condition on uvw | condition on X | n |
---|---|---|---|---|
P | none | none | X odd | n = X |
X even | n = X/2 | |||
C | h+k odd | none | none | n = X |
h+k even | u+v and w not both even | X odd | n = X | |
X even | n = X/2 | |||
u+v and w both even | X/2 odd | n = X/2 | ||
X/2 even | n = X/4 | |||
B | h+l odd | none | none | n = X |
h+l even | u+w and v not both even | X odd | n = X | |
X even | n = X/2 | |||
u+w and v both even | X/2 odd | n = X/2 | ||
X/2 even | n = X/4 | |||
A | k+l odd | none | none | n = X |
k+l even | v+w and u not both even | X odd | n = X | |
X even | n = X/2 | |||
v+w and u both even | X/2 odd | n = X/2 | ||
X/2 even | n = X/4 | |||
I | h+k+l odd | none | none | n = X |
h+k+l even | u, v and w not all odd | X odd | n = X | |
X even | n = X/2 | |||
u, v and w all odd | X/2 odd | n = X/2 | ||
X/2 even | n = X/4 | |||
F | none | u+v+w odd | none | n = X |
h, k, l not all odd | u+v+w even | X odd | n = X | |
X even | n = X/2 | |||
h, k, l all odd | u+v+w even | X/2 odd | n = X/2 | |
X/2 even | n = 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