Statistical Theory of Localized States of a Many-Impurity Crystal

A statistical theory of energy levels and position probability is derived for a one-dimensional, statistically dilute, many-impurity crystal. The equations are evaluated for a simple model. It is shown that two impurity bands are generated, one above and one below the host band, each merging without a gap onto the host band. If $U$ is greater than (less than) zero, the upper (lower) band is more dense. Each impurity-level electron is shown to be distributed in a wavelike manner along the impurities. The equations are derived for impurities differing both in their self-energy and in their nearest-neighbor coupling from the host monomers.