Crystal directions, planes and the zone axis

Directions

Consider a crystal lattice with unit cell edges a, b and c.  A crystal direction [uvw] is parallel to the direction joining the origin of the crystal lattice with the point with coordinates (ua, vb, wc)

Planes

A plane with Miller indices (hkl) passes through the three points (a/h,0,0), (0, b/k,0) and (0,0, c/l) on the edges of the unit cell. The set of parallel lattice planes passes through all similar points in the lattice. The plane d-spacing is the perpendicular distance from the origin to the closest plane and also the perpendicular distance between successive planes. In materials with cubic symmetry the crystal direction [uvw] and the normal to the plane (uvw) are parallel.

Zone Axis

The common direction shared by two crystal planes when they intersect is called the zone axis. A zone axis [uvw] is always perpendicular to the plane normal (hkl) that comprises the zone, and this relation is stated in the zone law: .

Symmetrically related planes and directions

The family of symmetrically related directions [uvw] is shown by the notation <uvw> and the family of symmetrically related planes (hkl) is shown by the notation {hkl}.