Difference between revisions of "Zone axis"
From Online Dictionary of Crystallography
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| − | <Font color="blue">Axe de zone</Font> (''Fr'') | + | <Font color="blue">Axe de zone</Font> (''Fr''). <Font color="red">Zonenachse</Font> (''Ge''). <Font color="green">Eje de zona</Font> (''Sp''). <Font color="brown">Ось зоны</Font> (''Ru''). <Font color="black"> Asse di zona </Font>(''It''). <Font color="purple">晶帯軸</Font> (''Ja''). |
== 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 | + | 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> | ||
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\begin{vmatrix} h_1 & k_1\\ | \begin{vmatrix} h_1 & k_1\\ | ||
h_2 & k_2\\ \end{vmatrix}} } | h_2 & k_2\\ \end{vmatrix}} } | ||
| − | </math> | + | .</math> |
</center> | </center> | ||
| − | Conversely, any crystal face can be determined if one knows two zone axes parallel to it. | + | 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> | ||
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== See also == | == See also == | ||
*[[Miller indices]] | *[[Miller indices]] | ||
| − | *[[ | + | *[[Polar lattice]] |
| − | *[[ | + | *[[Reciprocal lattice]] |
| − | *[[ | + | *[[Zone]] |
[[Category:Morphological crystallography]] | [[Category:Morphological crystallography]] | ||
Revision as of 18:34, 17 May 2017
Axe de zone (Fr). Zonenachse (Ge). Eje de zona (Sp). Ось зоны (Ru). Asse di zona (It). 晶帯軸 (Ja).
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.