Recurrence properties of O-lattices and the classification of grain boundaries

A recurrence relation is shown to exist between O-lattices of rotation-related grain boundaries (GBs) when a suitable parametrization of the rotation angle is introduced. This relation allows the basis vectors of any O-lattice to be calculated by a simple vector addition if the basis vectors of any two orientations are known. Its main usefulness, however, lies in the fact that it induces a partition of the angular space into disjoint sets, which groups grain boundaries into a finite number of equivalence classes, each represented by a special singular boundary (normal form). This shows that the O-lattice theory contains within it a much sought after general classification scheme for interfaces independent of the crystal system and therefore completely general.