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Zone axis

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 (h1,k1,l1), (h2,k2,l2) are given by:

 

\[\frac{u}{\begin{vmatrix}
k_1 &l_1 \\ 
k_2 &l_2
\end{vmatrix}}=\frac{v}{\begin{vmatrix}
l_1 &h_1 \\ 
l_2 &h_2
\end{vmatrix}}=\frac{w}{\begin{vmatrix}
h_1 &k_1 \\ 
h_2 &k_2\\
\end{vmatrix}}\]

 

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

Three lattice planes have a common zone axis (are in zone) if their Miller indices (h1,k1,l1), (h2,k2,l2), (h3,k3,l3) satisfy the relation:


\begin{vmatrix} h_1 & k_1 & l_1\\
h_2 & k_2 & l_2\\
h_3 & k_3 & l_3\\
\end{vmatrix} = 0

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

History

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