Coincidence-site lattices

The possibility that two arbitrary lattices, 1 and 2, have a coincidence-site lattice (CSL) in common is examined. Let T be the 3 × 3 matrix that maps a basis of lattice 1 onto a basis of lattice 2 and let ||T|| be the absolute value of its determinant. It may be assumed that ||T|| ≥ 1. There is a CSL if, and only if, T is rational. The main result is that the density ratio, Σ₂, of coincidence points to points of lattice 2 is equal to the least positive integer n such that nT and n||T||T^-1 are integral matrices. A basis for the CSL can be determined quickly if lattices 1 and 2 are related by a rotation.