Single site approximation

The CPA has proved to be an enormously successful tool in the study of alloys, and has been implemented within various frameworks, such as the TB, linear muffin-tin orbital and Korringa-Kohn-Rostoker (Kumar et al 1992, Turek et al 1996), and is still considered to be the most satisfactory single-site approximation. Efforts to do better than the single-site CPA have focused on multi-site (or cluster) CPA s (see, e.g., Gonis et al 1984, Turek et al 1996), in which a central site and its set of nearest neighbours are embedded in an effective medium. Still, for present purposes, the single-site version of the CPA suffices, and we derive the necessary equations here, within the framework of the TB model. 

In the single-site approximation, scattering from different sites are taken to be independent, so (6.16) becomes... 

To this point, the formalism has been quite general, and from here we could proceed to derive any one of several single-site approximations (such as the ATA, for example). However, we wish to focus on the desired approach, the CPA. To do so, we recall that our aim is to produce a (translationally invariant) effective Hamiltonian He, which reflects the properties of the exact Hamiltonian H (6.2) as closely as possible. With that in mind, we notice that the closer the choice of unperturbed Hamiltonian Ho (6.4) is to He, then the smaller are the effects of the perturbation term in (6.7), and hence in (6.10). Clearly, then, the optimal choice for H0 is He. Thus, we have... 

Interstitial collagenases catalyze the degradation of types I, II, and III collagen. Cleavage occurs at a single site, approximately of the distance from the carboxy-terminal end. The f and degradation products are unstable and are further broken down by either the same collagenases or other proteases. The initial bond cleaved is either a Gly-Leu or a Gly-Ile. 

In order to solve the multiple-scattering problem for the alloy, one replaces the original random alloy by an ordered lattice of effective scatterers, or the so-called effective medium. The effective medium is described by a site U (but not a sort i) dependent (complex) coherent potential. Therefore, the eflFective medium has the symmetry of the underlying crystal lattice. In the single site approximation the properties of these effective atoms have to be determined self-consistently by the condition that the scattering of electrons of real atoms embedded in the effective medium vanish on the average. For a disordered binary alloy this idea is schematically illustrated in Fig. 2. 

The first term from the right hand side of Eq. (13) assures the proper normalization of the one-electron states for the optimized overlapping potential. In fact, within the single site approximation for the impurity Green function, Eq. (12) gives the exact number of states at the Fermi leveT . ... 

Since the impurity problem in both the CPA and the Poisson equations is treated within the single site approximation, the Coulomb system of a particular alloy component may contain a non-zero net charge. The effect of the charge misfit on the spherical potential is taken into account using the screened impurity modef-, i.e. an additional shift of... 

To treat the effects of statistical correlations on alloy properties, one invariably needs to go beyond the single-site approximation [3]. A number of non-self-consistent cluster theories were introduced for this purpose that allowed the treatment of a finite cluster of atoms embedded in a medium determined in some way - usually through the self-consistent condition of the CPA [3]. These methodologies, however, do not yield an improved self-energy and are not of further interest to us in this review. 

B. Velicky, S. Kirkpatrick, and E. H. Ehrenreich, Single-site approximations in the electronic theory of simple binary alloys, Phys. Rev. 175, 747-66... 

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