Alloys Coherent Potential Approximation

We have used the basis set of the Linear-Muffin-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 the Vosko-Wilk-Nusair parametrization for the exchange-correlation energy density and potential. In conjunction with this we have treated the alloying effects for random and partially ordered phases with a multisublattice generalization of the coherent potential approximation (CPA). 

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... 

In the realm of theory also, greater demands will be made. As such studies (37—39) as those of Cu—Ni (Fig. 13) and Ag—Pd (Fig. 14) have shown, the d levels of the two species in transition metal alloys tend to maintain their atomic identities, at least when the levels in the pure components are sufficiently well separated in energy. However, neither calculation nor experiment has been done with refinement sufficient for quantitative testing of a theory, such as the coherent potential approximation, designed to describe the d band behavior. In pure metals and intermetallic compounds, band calculations can be compared directly with experiment if transition probabilities and relaxation effects are understood. With care they can be used also in evaluation of the effective interelectronic terms which enter equations such as (18a). Unfortunately, one cannot, by definition, produce a set of selfconsistent band calculation results for a matrix of specific valence electron snpmdl.. . configurations thus, direct estimates for I of Eq. (18a) or F of Eq. (18b) cannot be made. However, band calculations for a set of systems can indicate whether or not it is reasonable to factor level shifts into volume and electron count terms, in the manner of Eqs. (18a) and (23). When this cannot be done, one must revert to a more general expression for a level shift, such as Eq. (1). 

Figure 4. Schematic representation of the coherent potential approximation for a substu-tionally disordered alloy. The vertical strip in (h) denotes the effective self-energy, which is a complex quantity, to be determined by the compatibility requirement between die local and average description.

The first method for solving the MST problem in angular momentum representation was made by Korringa [43] and Kohn and Rostocker [44] separately. The method came to be called the KKR method for electronic structure calculations and used the Green s function technique from Chapter 3 to solve the electronic structure problem. The separation into potential- and structure dependent parts made the method conceptually clean and also speeded up calculations, since the structural dependent part could be calculated once and for all for each structure. Furthermore, the Green s function technique made the method very suitable for the treatment of disordered alloys, since the Coherent Potential Approximation [45] could easily be implemented. 

Furthermore, a chapter which discusses different ways to perform calculations for disordered systems is presented, including a description of the Coherent Potential Approximation and the Screened Generalized Perturbation Method. A comparison between the Exact Muffin-Tin Orbital method and the Projector Augmented-Wave method in the case of systems exhibiting both compositional and magnetic disordered is included as well as a case study of the MoRu alloy, where the theoretical and experimental discrepancies are discussed. 

Contents H.Matsuda Atoms as Constituents of Matter. - T. Tsuneto System of Protons and Electrons. - T. Tsuneto Helium. - T. Tsuneto Superfluid Helium 3. - T.Matsubara Metals. - T.Matsubara Non-metals. - T.Matsubara Localized Electron Approximation. - T.Murao Magnetism. - T.Murao Magnetic Properties of Dilute Alloys - the Kondo Effect. - H.Matsuda Random Systems. - F. Yonezawa Coherent Potential Approximation (CPA).- References.- Subject Index. 

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