Electronic States of a Kronig-Penney Crystal with Random Atomic Positions

The density of states for a one-dimensional system of r identical δ-function atoms randomly distributed on N lattice sites (lattice spacing d) is derived in the limit as r and N approach infinity by a nonperturbational method. The case for which only one atom is allowed on a lattice site (F-D) and the case for which this restriction is dropped (B-E) are both treated. In the limit as r/N and d approach zero, keeping the average number of atoms per unit length fixed, a common limit for the density-of-states function is approached in the F-D and B-E cases. This limiting function is identical with the one found by Klauder using a Brueckner-like approximation.