Alloys rigid band model

Figure 3 Compositional dependence of the stacking fault energy calculated from the rigid-band model (solid line) compared with the more accurate results from the LKKR-CPA calculation (dashed line) for the Al-Cu alloy system.

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. 

The discovery by ultraviolet photoelectron spectroscopy that the rigid-band model is not even applicable to the bulk of alloys such as Ni-Cu and Pd-Ag. The d-electrons of copper or silver in such alloys experience a potential that is predominantly the same as in the metals before alloying. For instance, upon alloying the d-bands of copper and nickel remain discernible in the alloy and no common d-band is formed, as is supposed by the rigid-band theory. 

Fig. 33. (0 Lattice parameters a (upper curve) and c (lower curve), (ii) Curie points, (iii) saturation magnetic moments at RT (solid circles) and LN2 (open circles), and (iv) anisotropy fields N (symbols as in iii) versus Co concentration in YFeio, Co,V2 alloys. The occurrence of maxima in 7 c(x) and A/,(x) plots can be explained, as for the Slater-Pauling curve, in terms of the rigid band model in which holes are present in both 3 subbands in the Fe-rich samples. (Jurczyk and Chistyakov 1989.)...

This model allows one to estimate the applicability of the phenomenological rigid band approach. This approach claims that the shape of DOS for carbides (nitrides) coincides with that of their ternary alloys, and explains the concentration-dependent changes in the alloy properties by changes in the position of the Fermi energy, which is, in turn, determined by the VEC in the system. Thus, the rigid band model can be used for qualitative interpretation of the properties of cubic alloys containing... 

Next, we focus on the general trend of electronic structure of transition metals. It is known that the electronic structure of transition metal alloys can be described by means of the so-called rigid band model as a first approximation, which states that the electronic band structure is unchanged upon substitutional alloying and the electronic structure is simply described... 

Fig. 7.13 Schematic representation of the ionic rigid-band and metallic common-band models of bonding in binary AB systems. The quantities and give the free-atom energy levels, whereas the positions of fA and B in the metallic bond reflect the small shift which takes place on alloy formation in order to maintain local charge neutrality.

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 EH MO method is not in complete agreement with the rigid band picture of Ag-Pd alloys. It predicts that 0.3-0.4 electron per Ag atom is transferred from Ag to Pd when Ag atoms are added to random lattice positions of the 15-atom model. The number of d holes on bulk and surface Pd atoms in the cluster are shown as a function of composition in Fig. 14. Here bulk Pd atoms have more holes than do surface Pd atoms. The number of Pd d holes decreases with added Ag but does not equal zero at 60% Ag. 

To identify the contributions of the various ingredients entering our model we consider the Pdo,8 80,2 alloy in more detail. Figs.8 a-d show the pressure-composition isotherms corresponding to values for D, a, K and rigid-band shift different from those used for the isotherms in Fig. 7a. 

There are several phenomenological models of the electronic structure of such solid solutions based on the rigid band approximation. Such models use either experimental data, e.g., the models of Lesnaya (1981), Zhurakovsky and Nemtchenko (1989), or describe ternary system spectra as a superposition of the valence bands of the starting binary alloys. 

In order to comprehensively show the chemical dissociation process of CO on metal surfaces, electronic structure calculations have been performed for simple models. We have chosen two methods for the present analyses. The first method is the Discrete Variational Xa (DV-Xa) method, which is the first-principles molecular orbital calculation using Slater s Xa fimctional for the electron many body term [21]. This method is applied for the electronic structural analyses of CO adsorption on metal surfaces. The second method is the Full-Potential Linear Muffin-Tin Orbital (FP-LMTO) method, which is the first-principles band structure calculation method [22]. The FP-LMTO implementation code of LmtART [23, 24] is used for the calculations of the density of states (DOS) of non-magnetic fee iron phase. We discuss the electronic structure of transition metal alloys from the rigid band analyses using this DOS. The local density approximation (LDA) parameterized by Vosko et al. [25] is used for the present FP-LMTO calculations. The tetrahedron... 

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