Difference between revisions of "Zone axis"
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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''). |
| − | < | ||
| − | 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]] | + | == 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 | ||
| + | |||
| + | <center> | ||
| + | ''uh'' + ''vk'' + ''wl'' = 0. | ||
| + | </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: | ||
| + | |||
| + | <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}}} = | ||
| + | {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: | ||
<center> | <center> | ||
| − | + | <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> | ||
</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 == | |
| + | *[[Miller indices]] | ||
| + | *[[Polar lattice]] | ||
| + | *[[Reciprocal lattice]] | ||
| + | *[[Zone]] | ||
| − | [[ | + | [[Category:Morphological crystallography]] |
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.