Vectors and Planes

It may seem, after considering cubic systems, that any lattice plane (*hkl*) has a normal direction [*hkl*]. This is not always the case, as directions in a crystal are written in terms of the lattice vectors, which are not necessarily orthogonal, or of the same magnitude. A simple example is the case of in the (100) plane of a hexagonal system, where the direction [100] is actually at 120° (or 60° ) to the plane. The normal to the (100) plane in this case is [210]

The Weiss zone law states that:

If the direction [*UVW*] lies in the plane (*hkl*), then:

*hU* + *kV* + *lW* = 0

In a cubic system this is exactly analogous to taking the scalar product of the direction and the plane normal, so that if they are perpendicular, the angle between them, θ, is 90° , then cosθ = 0, and the direction lies in the plane. Indeed, in a **cubic** system, the scalar product can be used to determine the angle between a direction and a plane.

However, the Weiss zone law is more general, and can be shown to work for all crystal systems, to determine if a direction lies in a plane.

From the Weiss zone law the following rule can be derived:

The direction, [*UVW*], of the intersection of (*h*_{1}*k*_{1}*l*_{1}) and (*h*_{2}*k*_{2}*l*_{2}) is given by:

*U* = *k*_{1}*l*_{2} − *k*_{2}*l*_{1}

*V* = *l*_{1}*h*_{2} − *l*_{2}*h*_{1}

*W* = *h*_{1}*k*_{2} − *h*_{2}*k*_{1}

As it is derived from the Weiss zone law, this relation applies to all crystal systems, including those that are not orthogonal.