A New Geometrical Method for Constructing Coincident Site Lattices for Cubic Crystals

–A simple independent method for geometrical construction of two-dimensional coincidence site lattices (CSL) based on vector representation of lattice sites is put forward. In this method, the construction of such CSLs is achieved by mutual rotation of a unique vector, termed a sigma generating vector, which has the magnitude of a non-prime integer and is present in an identically oriented superimposed zero-layer lattice of two identical cubic crystals. The angle of rotation of the bi-crystal is expressed in terms of some unique nonprime integers that describe the orientation of the sigma generating vector within the lattices of each constituting crystal. The method provides a better mathematical explanation of why the magnitude of the sigma generating vector referred to as Ʃ can be used to designate a periodic boundary in cubic crystals by a certain value of Ʃ . The cell parameters of the two-dimensional CSL thus formed can be described in terms of both the magnitude (termed as Σ) of the square root of the sigma generating vector as well as the lattice parameters of the constituting crystals. Addition of a third axis along the rotation axis of these two-dimensional CSLs yields related three dimensional CSLs. Keywords––coincident site lattices, grain boundaries, network of two-dimensional lattice unit in cubic crystals, sigma-