Zone axis

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


{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}} }

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

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