Resonant scattering by impurities in metals and the Anderson model

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Abstract
The Anderson Hamiltonian, written in a representation where the extra orbital is not orthogonal to the conduction states, is used to derive a general theory of the electronic structure of dilute alloys. The theory describes both simple impurities in the over-complete or Wolff limit, and transition or rare-earth impurities where the scattering of the conduction electrons has a resonance. The extra-orbital of Anderson is shown to be identical to a bound state extracted from higher bands by the impurity potential, and overlapping the conduction band in energy. The resonant scattering of conduction electrons is described by a pseudopotential, which is singular in energy, in analogy to the theory of band structures of pure transition elements. The position and width of the resonance, as well as a direct scattering potential introduced by the non-orthogonality, are given in terms of Anderson's parameters. The resonance is narrowed by the non-orthogonality and disappears in the over-complete limit.