On the Periodicity of ⟨001⟩ Symmetrical Tilt Grain Boundaries

We report an application of the O-lattice theory to systematically analyse the structures of symmetrical tilt grain boundaries with the rotation axis of 〈001〉 and demonstrate a theoretical interpretation of the experimentally observed structures of a near Σ5 grain boundary in MgO in terms of the structural-units model and the periodicity of the O-points on the boundary. We further derive generalised decomposition formulae for the symmetrical tilt grain boundaries which are closely related to the distribution of irreducible rational numbers.