Direction indices for crystal lattices

Abstract

Direction indices [uvw] of rational directions in crystal lattices are commonly restricted to integer numbers. This restriction is correct only when primitive unit cells are used. In the case of centred cells, however, direction indices may take fractional values too, because the first lattice node after the origin along a direction can have fractional coordinates in a centred basis. This evidence is very often overlooked and an undue simplification of direction indices to integer values is usually adopted. Although such a simplification does not affect the identification of the direction, it is potentially a source of confusion and mistakes in crystallographic calculations. A parallel is made with the incorrect restriction of Miller indices to relatively prime integers in centred cells.