Periodic Impurities in a Periodic Lattice

Abstract

Because binary delta-function lattices of the Kronig-Penney type which have been previously studied are incapable of describing several important features of real crystals, the eigenvalue problem for a periodic linear chain $\cdots A\left(BB\cdots BB\right)A\left(BB\cdots BB\right)\cdots$ of arbitrary square-well, $A$ and $B$ atoms of an arbitrary concentration is taken up and solved. The methods used are generalizable to other binary or multi-nary chains. It is shown that a theorem of Saxon and Hutner and of Luttinger relating to the preservation of the common forbidden energies of pure $A$ and pure $B$ lattices in a mixed $A$, $B$ lattice is peculiar to their deltafunction representations of $A$ and $B$, and is without general validity.