On the geometrical relationship between tilt and twist grain boundaries *

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Abstract
Within the framework of the coincident-site lattice (CSL) description of grain boundaries (GBs), the five macroscopic degrees of freedom (dof) are usually chosen to consist of a rotation axis and angle and the GB-plane normal. A different method is proposed for the characterization of these five independent parameters, by replacing the CSL misorientation by the GB-plane normals in the two halves of the bicrystal and by the twist component of the misorientation. The relevant expressions describing the transition from the one to the other are derived and illustrated. The new scheme has the advantage that (a) it readily allows one to visualize the unrelaxed atomic structure of the interface; (b) symmetrical GBs are easily identified; (c) the tilt and twist components of general GBs are readily apparent; and (d) the new scheme may be applied equally to GBs and to dissimilar-material (commensurate or incommensurate) interfaces. Within the new characterization scheme, the well-known fact that in cubic crystals with inversion symmetry, the symmetrical tilt GB on a given lattice plane may be considered as the most special of all twist boundaries, is easily recognized.