Coherent potential approximation

In this section, we consider how to model a bulk (i.e., infinite) substitution-ally disordered binary alloy (DBA), in light of its intrinsic randomness. The fact that the DBA lacks periodicity means that the key tool of Bloch s theorem is inapplicable, so specialized methods (Ehrenreich and Schwartz 1976, Faulkner 1982, Yonezawa 1982, Turek et al 1996) must be used. 

One early and simple concept is the rigid-band model (Friedel 1958), wherein a fixed DOS is taken to represent an entire class of alloys (such as those composed of 3d transition metals). Individual alloys are distinguished solely by assigning to each a Fermi level, determined by the concentration of valence electrons. Unfortunately, this model is too much of an oversimplification, because, for example, the DOS is chosen empirically, and may not be clearly related to that for any of the constituent metals. 

In the virtual-crystal approximation (VCA) (Nordheim 1931), the site energy of an alloy atom is taken to be 

A better method is the average t-matrix approximation (ATA) (Korringa 1958), in which the alloy is characterized by an effective medium, which is determined by a non-Hermitean (or effective ) Hamiltonian with complex-energy eigenvalues. The corresponding self-energy is calculated (non-self- 

The improvement came in the form of the coherent-potential approximation (CPA) (Soven 1967, Taylor 1967, Velicky et al 1968), which remedied the lack of self-consistency exhibited by the ATA. The crux of this approach is that each lattice site has associated with it a complex self-consistent potential, called a coherent potential (CP). The CP gives rise to an effective medium with the important property that removing that part of the medium belonging to a particular site, and replacing it by the true potential, produces, on average, no further scattering. Because the CPA is used for our discussion of chemisorption on DBA s, its mathematical formulation is given below. 

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We have used the multisublattice generalization of the coherent potential approximation (CPA) in conjunction with the Linear-MufRn-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed for the local spin density approximation (LSDA) the Vosko-Wilk-Nusair parameterization". 

Figure 2.Virtual crystal approximation calculations (solid line) compared with coherent potential approximation calculations for Fe-Co (longdashed line), Fe-Ni (dot-dashed line) and Fe-Cu (dashed line). The fcc-bcc energy difference is shown as a function of the atomic number.

It would be interesting to compare these results with those of a CPA (coherent potential approximation) calculation. For a real test of both methods a system should be found which shows a more pronounced change when going from one phase to the other, than found for the system presented in this paper. 

Application of the Coherent Potential Approximation (CPA) alloy theory in connec-... 

Ffom a theoretical point of view, stacking fault energies in metals have been reliably calculated from first-principles with different electronic structure methods [4, 5, 6]. For random alloys, the Layer Korringa Kohn Rostoker method in combination with the coherent potential approximation [7] (LKKR-CPA), was shown to be reliable in the prediction of SFE in fcc-based solid solution [8, 9]. 

P.A.Korzhavhi, A.V.Ruban, I.A.Abrikosov and H.L.Skriver, Madelung energy for random metallic alloys in the coherent potential approximation , Phys. Rev. B51 5773 (1995) ... 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

The most essential progress from the point of view of application of this theory in catalysis and chemisorption has actually been achieved by the very first papers (48-50), where the so-called coherent potential approximation (CPA) was developed and applied. By means of this, photoemission data were explained in a quite satisfying way and the catalytic research got full theoretical support for some of the ideas introduced in catalysis earlier on only semiempirical grounds (5) namely, individual components are distinguishable for molecules from the gas phase and the alloy atoms preserve very much of their metallic individuality also in alloys—something that was impossible according to the RBT and the early electronic theory of catalysis. 

The coherent potential approximation for a disordered alloy (7,2) provides a satisfactory framework for describing the effect of alloying within two extremes on the one hand, the rigid-band approximation, which supposes that band shapes do not alter upon alloying, and on the other hand, the minimum polarity model, which supposes the electron distribution of the elements forming an alloy to be similar to that in free atoms. 

The coherent potential approximation (1, 2) is a consistent theoretical frame, which unifies the different alloy models. In order to account for changes in the electronic nature of the atoms, the coherent potential approximation for a disordered alloy appears at present to be the best. It has been applied to single- and two-band systems (130a 130c). 

Gyorffy, B.L. (1972). Coherent-potential approximation for a nonoverlapping muffin-tin potential model of random substitutional alloys, Phys. Rev. B 5, 2382-2384. 

Nesbet, R.K. (1992). Full-potential revision of coherent-potential-approximation alloy theory, Phys. Rev. B 45, 13234-13238. 

We summarize below the main theoretical approaches to the binary alloy, i.e. those based on the coherent-potential approximation (CPA).156... 

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