One-dimensional Chains with Random Spacing between Atoms

The calculation of the integrated density of states for positive electron energies of a one-dimensional chain of atoms whose spacing is given by a probability distribution function is reduced to the form of an integral equation. This equation is simplified considerably when the atoms are distributed at random. In this case, an explicit solution for the integrated density of states is found, which is rigorously valid for δ-function atomic potentials, and may be valid generally. It is shown that the solution gives good agreement with the machine results of Lax and Phillips (1958) for the δ-function case.