# 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 (*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}}\]

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:

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

### History

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