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Two neighbouring grains of the same phase with a lattice of hexagonal Bravais type are considered which have a three-dimensional lattice of symmetry translations in common, called the coincidence site lattice or CSL. The volume ratio of unit cells for the CSL and the original lattice is called the multiplicity Σ. The Σ-hex theorem gives Σ in terms of four integral parameters that describe the axis and angle of the rotation connecting the hexagonal lattices of the two neighbouring grains and in terms of their axial ratio c/a. Two types 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-hex theorem makes it possible to determine a lower and an upper bound for Σmin, the minimum value of the multiplicity of specific rotations for a given axial ratio. The lower bound can serve to determine systematically all specific rotations with c/a in a given interval and Σ not larger than some given value Σc. The bound is used to complete published tables. The upper bound is stronger than a similar bound given by Delavignette.
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