The Band Structure of Mixed Linear Lattices

Abstract

A reformulation of Saxon and Hutner's scattering matrix method for lattices is introduced, and a connection shown between the characteristic matrix for going across a cell in the lattice and proper Lorentz transformations. This provides a convenient geometrical model for analysing mixed lattices of A and B atoms. Luttinger's proof of a theorem on the persistence of forbiddenness, in the mixed lattice, of energies forbidden in the pure component lattices is generalized. It is shown also that there are energy levels in mixed lattices whose forbidden or allowed character is invariant to any re-arrangement of lattice atoms, and that there are physically distinguishable lattices having identical band structures. The major problem of finding the probability that a given energy is allowed or forbidden in a random mixed lattice is taken up. A method for finding the moments of the relevant random variable is provided, and the complete analysis for the Poisson limit of a few B atoms sprinkled in a long A lattice is given, with quantitative conclusions as to the persistence of allowedness in mixed lattices, the effect of strong forbiddenness in the pure B lattice on the band structure of the mixed lattice, and the like.