Coherent-Potential Model of Substitutional Disordered Alloys

We introduce a model of a substitutional alloy based on the concept of an effective or coherent potential which, when placed on every site of the alloy lattice, will simulate the electronic properties of the actual alloy. The coherent potential is necessarily a complex, energy-dependent quantity. We evaluate the model for the simple case of a one-dimensional alloy of $\delta$-function potentials. In order to provide a basis for comparison, as well as to see if a simpler scheme will suffice, we also calculate the spectrum of the same alloy using the average $t$-matrix approximation introduced by Beeby. On the basis of these results, we conclude that the average $t$-matrix approximation is not adequate for the description of an actual transition-metal alloy, while the coherent-potential picture will provide a more reasonable facsimile of the density of states in such an alloy.