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

The density of states for a one-dimensional system of $r$ identical $\delta$-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 $\frac{r}{N}$ 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.