On the determination of dense coincidence site lattice planes

The concept of coincidence site lattice (CSL) is used in descriptions of geometry of some intercrystalline boundaries. In face-centered cubic and body-centered cubic metals, atoms are located at the nodes of lattices, and results concerning lattice nodes are applicable to atomic sites. One of the criteria for special boundary configurations is that the boundary passes through a plane with a high density of coinciding atomic sites. Hence, there is an issue of identification of such planes. This paper describes a simple and reliable method for determining the planes with high densities of coinciding lattice nodes. The key elements of the procedure are the Hermite normal form of an integer matrix and Niggli reduction of the CSL basis. In its general form, the method is applicable to arbitrary three-dimensional lattices possessing a common three-dimensional sublattice. The densest and second-densest planes are determined for low-Σ CSLs of cubic lattices.