7.4: 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:
\[uh + vk + wl = 0 \nonumber \]
This is the so-called Weiss law.
The indices of the zone axis defined by two lattice planes ( h 1 , k 1 , l 1 ), ( h 2 , k 2 , l 2 ) are given by:
\[\frac{u}{\begin{vmatrix}
k_1 &l_1 \\
k_2 &l_2
\end{vmatrix}}=\frac{v}{\begin{vmatrix}
l_1 &h_1 \\
l_2 &h_2
\end{vmatrix}}=\frac{w}{\begin{vmatrix}
h_1 &k_1 \\
h_2 &k_2\\
\end{vmatrix}} \nonumber \]
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 ( h 1 , k 1 , l 1 ), ( h 2 , k 2 , l 2 ), ( h 3 , k 3 , l 3 ) satisfy the relation:
\[\begin{vmatrix} h_1 & k_1 & l_1\\h_2 & k_2 & l_2\\h_3 & k_3 & l_3\\\end{vmatrix}=0 \nonumber \]