Difference between revisions of "Twin index"
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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''). |
− | = | + | == Definition == |
− | A [[ | + | 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'''. |
+ | |||
+ | 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: | ||
+ | |||
+ | <div align="center"> | ||
+ | ''n'' = ''X''/''f'', ''X'' = |''uh''+''vk''+''wl''| | ||
+ | </div> | ||
+ | |||
+ | 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> | ||
+ | <tr> | ||
+ | <th>Lattice type</th><th>Condition on ''hkl''</th><th>Condition on ''uvw''</th><th>Condition on ''X''</th><th>''n''</th> | ||
+ | </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> | ||
+ | </tr> | ||
+ | <tr><td>X even</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td rowspan="5" align="center">''C''</td><td>''h+k'' odd</td><td>none</td><td>none</td><td>''n'' = ''X''</td> | ||
+ | </tr> | ||
+ | <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> | ||
+ | </tr> | ||
+ | <tr><td>''X'' even</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | <tr><td rowspan="2">''u+v'' and ''w'' both even</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | <tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td rowspan="5" align="center">''B''</td><td>''h+l'' odd</td><td>none</td><td>none</td><td>''n'' = ''X''</td> | ||
+ | </tr> | ||
+ | <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> | ||
+ | </tr> | ||
+ | <tr><td>''X'' even</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | <tr><td rowspan="2">''u+w'' and ''v'' both even</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | <tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td rowspan="5" align="center">''A''</td><td>''k+l'' odd</td><td>none</td><td>none</td><td>''n'' = ''X''</td> | ||
+ | </tr> | ||
+ | <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> | ||
+ | </tr> | ||
+ | <tr><td>''X'' even</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | <tr><td rowspan="2">''v+w'' and ''u'' both even</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | <tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td rowspan="5" align="center">''I''</td><td>''h+k+l'' odd</td><td>none</td><td>none</td><td>''n'' = ''X''</td> | ||
+ | </tr> | ||
+ | <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> | ||
+ | </tr> | ||
+ | <tr><td>''X'' even</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | <tr><td rowspan="2">''u'', ''v'' and ''w'' all odd</td><td>''X''/2 odd</td><td>''n'' = ''X''/2</td> | ||
+ | </tr> | ||
+ | <tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <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> | ||
+ | <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><td>''X'' even</td><td>''n'' = ''X''/2</td> | ||
+ | </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> | ||
+ | </tr> | ||
+ | <tr><td>''X''/2 even</td><td>''n'' = ''X''/4</td> | ||
+ | </tr> | ||
+ | |||
+ | </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 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 | ||
+ | <div align="center"> | ||
+ | ''n'' = ''N''Ξ/ξ. | ||
+ | </div> | ||
+ | |||
+ | == 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'' | ||
+ | |||
+ | [[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).
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 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 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 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