Coincidence orientations of grains in rhombohedral materials

In experimental investigations and computer simulations of the structure and properties of grain boundaries, the results are usually discussed with reference to the special case of coincidence boundaries, where the two neighbouring grains have a three-dimensional lattice of symmetry translations in common. For historical reasons this lattice is called the coincidence site lattice or CSL. A systematic determination of CSL's for the case of grains with a lattice of rhombohedral Bravais type is presented. It is shown that a number of investigations of the structure of grain boundaries in alumina (α-Al₂O₃) have to be reinterpreted in the light of the present results. A central result is the Σ-rhomb theorem, which expresses the ratio Σ of unit-cell volumes of the CSL and the rhombohedral crystal lattice in terms of four integral parameters that describe the axis and angle of the rotation connecting the rhombohedral lattices of the two neighbouring grains and in terms of their axial ratio c/a. Two types of coincidence rotations, i.e. of rotations generating CSL's, may be distinguished, viz common rotations, which generate CSL's with the same Σ for every value of c/a, and specific rotations, which generate CSL's with a low value of Σ only for a few values of the axial ratio. The Σ-rhomb theorem makes it possible to determine systematically not only all common rotations with Σ up to a given maximum value Σ_c but also all specific rotations with Σ ≤ Σ_c and with c/a in any given interval about the experimental value of c/a for the material in question. It is shown that the multiplicities of the CSL's generated by a given rotation in a hexagonal and in a rhombohedral lattice with the same value of c/a differ by at most a factor 3.