Electronic Excess Energy in Modulated Positive-Background Model
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ELECTRONIC EXCESS ENERGY IN MODULATED POSITIVE-BACKGROUND MODEL H. YAHAUCHI Department of Engineering Materials, University of Windsor, Windsor, Ontario N9B 3P4, Canada ABSTRACT Electronic excess energy of a composition-modulated alloy system at absolute zero temperature is obtained using Hohenberg and Kohn's formula (for the ground-state enermy of an inhomogeneous electron gas) in a modulated positive-background model. Deoendence of the electronic excess energy on the modulation wavelength is studied. Two leading terms in the excess energy are examined to elucidate the limitation of this model. INTRODUCTION One may easily anticipate that free electrons in a compositionmodulated alloy are not homogeneously distributed, and question how significant is the contribution of such an inhomogeneous free electron gas to the excess (free) energy of such a system. A rigorous answer to this question would require an involved theoretical work. In the present work, it is shown that application of Hohenberg and Kohn's [11formula for the nround-state energy of an inhomogeneous electron gas to a modulated positive-background model [2-4] can provide a simple "first-step" method for answering the above question. A formula for the electronic excess energy obtained in the present modulated positive-background model will be compared with Cahn and Hilliard's [5-7] formula for the excess free energy of a nonuniform system, in order to find the corresponding contributions from the free
electron gas to the mixing- and gradient-energy. It should be noted that, in most alloys consisting of common metals, the electronic contribution does not count the whole magnitude of these energies. It may, however, be possible to assume that such electronic contributions are dominant in the mixing- and gradient-energy of alloys with a very low density of free electrons, since parallel works [8-11] showed that application of Hohonberg and Kohn's formula to a constant oositivebackground model for the metallic surface yielded theoretical values, for both electron work function and surface energy, which were in reasonable agreement with experimental values. MODEL In the present "modulated positive-background model", positive charges are assumed to be continuously distributed in accordance with the composition modulation wave. That is, for the sake of simplicity, discrete positive cores are smeared out to yield a continuous distribution of positive charges. This simple model would reasonably be valid for systems with a low free-electron density and with a long wavelength for the composition modulation. If the composition modulation, c(x), is given by a sinusoidal curve along the x-axis: c(x) = c0 + Acos(ks),
(1)
where co is the averane composition, A is the modulation amplitude and Mat. Res.
Soc. Symp. Proc. Vol. 21 (1984)
DElsevier
Science Publishing Co.,
Inc.
634
k is the wave number, the positive charge may be distributed as [3]: nx(X) = n0 [1+ ccos(kx)],
(2)
when the atomic volume difference between vA of an A atom and vB of a B atom defined by
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