A system of metrically invariant relations between the moduli squares of reciprocal-lattice vectors in one-, two- and three-dimensional space

Given the background of trial-and-error methods employed in recent automatic powder pattern indexing, an alternative route is suggested based on a generalization of the original Runge-de Wolff approach. For this purpose, a system of five metrically invariant relations between the squared moduli (Q values) of reciprocal-lattice vectors is developed that encompasses the earlier special relations. The five invariant relations correspond to a line, a zone, a bizone, a cone and a pencil configuration of reciprocal-lattice vectors. In particular, the zone configuration relates four vectors being arbitrarily distributed in a plane and as such allows one to identify among a set of measured Q values all quadruples that define reciprocal-lattice planes intersecting in space.