Actions

Difference between revisions of "Twin index"

From Online Dictionary of Crystallography

(Twin index)
Line 4: Line 4:
 
= 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 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

References

Chapter 3.3 of International Tables of Crystallography, Volume D