Actions

Difference between revisions of "Twin index"

From Online Dictionary of Crystallography

(Definition)
Line 4: Line 4:
 
== Definition ==
 
== Definition ==
  
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''
+
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 ''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:
+
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 ''binary operations'' (180º rotations or reflections) the twin index is:
  
 
<div align="center">
 
<div align="center">
Line 86: Line 86:
 
</table>
 
</table>
 
</center>
 
</center>
 +
 
== References ==
 
== References ==
  

Revision as of 09:42, 9 April 2007

Indice de macle (Fr). Indice di geminazione (It)


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 (effects of)|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 binary 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

</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

</center>

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