Difference between revisions of "Twin index"
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= Twin index = | = Twin index = | ||
− | 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 ([[twinning (effects of)]]). The reciprocal ''n'' of the fraction 1/''n'' of (quasi)restored nodes is called ''twin index'' | + | A [[twinning|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)]]). 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. The twin index is then: | ||
+ | |||
+ | <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="1" cellspacing="2" cellpadding="2"> | ||
+ | <tr> | ||
+ | <th>Lattice type</th><th>condition on ''hkl''</th><th>condition on ''uvw''</th><th>condition on ''X''</th><th>''n''</th> | ||
+ | <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 rowspan="5" align="center">''F''</td><td>none</td><td>''u''+''v''+''w'' odd</td><td>none</td><td>''n'' = ''X''</td> | ||
+ | </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> | ||
+ | <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> | ||
+ | |||
+ | </table> | ||
+ | |||
+ | ==References== | ||
Chapter 3.3 of ''International Tables of Crystallography, Volume D''<br> | Chapter 3.3 of ''International Tables of Crystallography, Volume D''<br> | ||
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] |
Revision as of 15:01, 26 April 2006
Indice de macle (Fr). Indice di geminazione (It)
Twin index
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)). 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. The twin index is then:
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
</tr>
</tr>
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 |
References
Chapter 3.3 of International Tables of Crystallography, Volume D