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Difference between revisions of "Zone axis"

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<Font color="blue">Axe de zone</Font> (''Fr''). <Font color="red">Zonenachse</Font> (''Ge'').  
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<font color="blue">Axe de zone</font> (''Fr''). <font color="red">Zonenachse</font> (''Ge''). <font color="black">Asse di zona</font> (''It''). <font color="purple">晶帯軸</font> (''Ja''). <font color="brown">Ось зоны</font> (''Ru''). <font color="green">Eje de zona</font> (''Sp'').
<Font color="green">Eje de zona</Font> (''Sp''). <Font color="purple">Ось зоны</Font> (''Ru'').
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== Definition ==
 
== Definition ==
  
A zone axis is a lattice row parallel to the intersection of two (or more) families of lattices planes. It is denoted by [''u'' ''v'' ''w'']. A zone axis [''u'' ''v'' ''w''] is parallel to a family of lattice planes of [[Miller indices]] (''hkl'') if:
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A zone axis is a lattice row parallel to the intersection of two (or more) families of lattices planes. It is denoted by [''u'' ''v'' ''w'']. A zone axis [''u'' ''v'' ''w''] is parallel to a family of lattice planes of [[Miller indices]] (''hkl'') if
  
 
<center>
 
<center>
''uh'' + ''vk'' + ''wl'' = 0
+
''uh'' + ''vk'' + ''wl'' = 0.
 
</center>
 
</center>
 +
 +
This is the so-called Weiss law.
  
 
The indices of the zone axis defined by two lattice planes (<math> h_1, k_1, l_1 </math>), (<math> h_2, k_2, l_2</math>) are given by:
 
The indices of the zone axis defined by two lattice planes (<math> h_1, k_1, l_1 </math>), (<math> h_2, k_2, l_2</math>) are given by:
  
[[image:Zoneaxis-1.png|center]]
+
<center>
 +
<math>
 +
{u\over {
 +
\begin{vmatrix} k_1 & l_1\\
 +
k_2 & l_2\\ \end{vmatrix}}} =
 +
{v\over {
 +
\begin{vmatrix} l_1 & h_1\\
 +
l_2 & h_2\\ \end{vmatrix}}} =
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{w\over {
 +
\begin{vmatrix} h_1 & k_1\\
 +
h_2 & k_2\\ \end{vmatrix}} }
 +
.</math>
 +
</center>
 +
 
 +
Conversely, any crystal face can be determined if one knows two zone axes parallel to it. This is the zone law, or ''Zonenverbandgesetz''.
  
 
Three lattice planes have a common zone axis (''are in zone'') if their Miller indices (<math> h_1, k_1, l_1 </math>), (<math> h_2, k_2, l_2</math>), (<math> h_3, k_3, l_3</math>) satisfy the relation:
 
Three lattice planes have a common zone axis (''are in zone'') if their Miller indices (<math> h_1, k_1, l_1 </math>), (<math> h_2, k_2, l_2</math>), (<math> h_3, k_3, l_3</math>) satisfy the relation:
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h_2 & k_2 & l_2\\
 
h_2 & k_2 & l_2\\
 
h_3 & k_3 & l_3\\
 
h_3 & k_3 & l_3\\
\end{vmatrix} = 0</math>
+
\end{vmatrix} = 0.</math>
 
</center>
 
</center>
  
 +
== History ==
 +
 +
The notion of zone axis and the zone law were introduced by the German crystallographer Christian Samuel Weiss in 1804.
  
 
== See also ==
 
== See also ==
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*[[Miller indices]]
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*[[Polar lattice]]
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*[[Reciprocal lattice]]
 +
*[[Zone]]
  
[[Miller indices]]<br>
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[[Category:Morphological crystallography]]
[[reciprocal lattice]]
 
 
 
----
 
 
 
[[Category:Fundamental crystallography]]<br>
 

Latest revision as of 14:58, 20 November 2017

Axe de zone (Fr). Zonenachse (Ge). Asse di zona (It). 晶帯軸 (Ja). Ось зоны (Ru). Eje de zona (Sp).


Definition

A zone axis is a lattice row parallel to the intersection of two (or more) families of lattices planes. It is denoted by [u v w]. A zone axis [u v w] is parallel to a family of lattice planes of Miller indices (hkl) if

uh + vk + wl = 0.

This is the so-called Weiss law.

The indices of the zone axis defined by two lattice planes ([math] h_1, k_1, l_1 [/math]), ([math] h_2, k_2, l_2[/math]) are given by:

[math] {u\over { \begin{vmatrix} k_1 & l_1\\ k_2 & l_2\\ \end{vmatrix}}} = {v\over { \begin{vmatrix} l_1 & h_1\\ l_2 & h_2\\ \end{vmatrix}}} = {w\over { \begin{vmatrix} h_1 & k_1\\ h_2 & k_2\\ \end{vmatrix}} } .[/math]

Conversely, any crystal face can be determined if one knows two zone axes parallel to it. This is the zone law, or Zonenverbandgesetz.

Three lattice planes have a common zone axis (are in zone) if their Miller indices ([math] h_1, k_1, l_1 [/math]), ([math] h_2, k_2, l_2[/math]), ([math] h_3, k_3, l_3[/math]) satisfy the relation:

[math] \begin{vmatrix} h_1 & k_1 & l_1\\ h_2 & k_2 & l_2\\ h_3 & k_3 & l_3\\ \end{vmatrix} = 0.[/math]

History

The notion of zone axis and the zone law were introduced by the German crystallographer Christian Samuel Weiss in 1804.

See also