Coherent-potential and averaget-matrix approximations for disordered muffin-tin alloys. I. Formalism

The average density of states $〈\rho \mathrm{}\left(E\right)\mathrm{}〉$ and the component charge density associated with an $A\mathrm{}\left(B\right)\mathrm{}$ atom in the alloy, $〈{\rho }_{A\mathrm{}\left(B\right)\mathrm{}}\mathrm{}\left(E\right)\mathrm{}〉$, are discussed for the disordered alloy $A_x{B}_{1-x}$ within the framework of the muffintin Hamiltonian. A new version of the average $t$-matrix (ATA) is developed. The structure in the spectral density function, $〈\rho (\overrightarrow{k},E)〉$, in the coherent-potential approximation (or the new ATA) is seen to result from not only the Bloch-type states in the medium of coherent-potential effective atoms $t_{\mathrm{CP}}$ (or the average $t$-atoms $〈t〉$) but also from non-Bloch-type impurity levels arising when a single $A$ or $B$ atom is embedded in an otherwise perfect effective medium. The proposed ATA equations would allow a simple yet reliable treatment of many aspects of the electronic spectrum of disordered transition and noble-metal alloys.