Generalized Cluster Theory of the Coherent-Potential Approximations for Disordered Systems

The coherent-potential approximation (CPA) has been generalized to a cluster theory of CPA (${\alpha }_0$) where the effects due to all possible scatterings from clusters consisting of all atoms residing within the ${\alpha }_0$th-shell radius are taken into account in a self-consistent manner. The present theory is more general than most of the existing theories of CPA. The pair theory of CPA as proposed by Cyrot-Lackmann and Ducastelle is discussed in detail in the light of our theory. The limitations of their theory are discussed and it is shown that this pair theory is a special case of our theory. By considering an infinitely large cluster our theory can be made entirely self-consistent. The accuracy of our theory however depends on the size of the cluster (or the magnitude of ${\alpha }_0$) that we choose for practical computations. The inverse of the number of nearest neighbor, $z^{-1\mathrm{}}$, is argued to be a good expansion parameter. By applying the ideas of elementary perturbation theory to our $T$-matrix equations we have developed a diagrammatic-expansion scheme for our theory, which enables us to estimate the errors resulting from the choice of a finite-size cluster. It is shown that our diagrammatic scheme when applied to the single-site $T$-matrix equations gives a result which is equivalent to Yonezawa's cumulant method, but our method is more straightforward, easier to understand and has a strong theoretical foothold. This diagrammatic method is then extended to the case of multiple scattering. When applied to the cases of pair and triplet scatterings, our method produces results which are in agreement with those of Nickel and Krumhansl.