Comparison of the Spherical Model and the Clapp-Moss Equations with the Ising Model

Three simple analytical approximations are elaborated for calculation of the short-range order parameters and their Fourier transform in disordered binary alloys with many-body atomic interactions of arbitrary order and effective radius of action. Within all the approximations the account of non-pair atomic interactions is reduced to the replacement of the pair mixing potential by the effective concentration-dependent potential. From the form of such an effective potential it is concluded that the temperature-concentration phase diagram of an alloy with non-pair atomic interactions is asymmetrical with respect to the equiatomic composition and that the effect on the structural properties of the alloy of such interactions decreases with concentration decrease. These conclusions are justified by simulation within the Monte Carlo method. On the basis of the comparison with the simulation data, an approximation which demonstrates high numerical accuracy of results in a wide temperature-concentration interval is determined and it is shown that its accuracy increases with the increase of the effective radius of atomic interactions. We prove the general invariance of the statistical-thermodynamic properties of binary alloys with respect to a number of transformations which results in symmetry of the phase diagrams of alloys with respect to equiatomic composition in a particular case of the presence of nothing but concentration-independent pair atomic interactions.

mentioning

confidence: 99%

Analytical description of the short-range order in alloys with many-body atomic interactions

Chepulskii¹

1998

“…Several authors have considered the spherical model (SM) (see, for instance, Ziman 1979) as an alternative to the KCM theory (Philhours andHall 1968, Hoffmann 1972). Hoffmann has pointed out an interesting circumstance in which the SM results for correlations are rather good and, in particular, far exceed KCM ones in accuracy.…”

mentioning

confidence: 99%

A simple and accurate theory of short-range order in alloys

Tokar¹,

Masanskii²,

Grishchenko³

1990

The authors propose a very simple method of calculating short-range order parameters alpha ij in disordered alloys using a series expansion in powers of a parameter gamma =exp(-1/ xi ) where xi is the dimensionless correlation length of the pair correlation function. In authors' approximation the sum rule alpha ii=1 is satisfied exactly, unlike in previous theories. In the zeroth order their approach leads to the spherical model results. The high accuracy of the theory developed is illustrated by comparing its results with those of Monte Carlo simulation and estimating the pair interactions from diffuse scattering data. In the latter problem pair-interaction potentials are explicitly expressed in terms of experimentally determined quantities.

“…-the well known formula previously obtained within the spherical model [4][5][6][7][8][9][10][11][12][13][14] and below named the spherical model approximation;…”

Section: Introductionmentioning

confidence: 99%

Anaytical methods for calculation of the short-range order in alloys: II. Numerical accuracy study

Chepulskii

Bugaev

1998

The numerical accuracy of the analytical approximations elaborated in part I and of a variety of other approximations advanced to date is examined, accepting the data on the short-range order parameters obtained by the Monte Carlo method as a standard. The approximations exhibiting the most promise for description of actual alloys and other lattice systems (for instance, interstitial alloys, semiconductors and magnetics) including those with a long-range character of atomic interactions are revealed. It is stated that, in the framework of the lattice gas model within a modified thermodynamic perturbation theory, the choice of the inverse effective number of atoms interacting with one fixed atom as a small parameter of the cumulant expansion along with taking account of the contributions from only linked irreducible diagrams ensures the fastest convergence of expansion. It is demonstrated that the use of the grand canonical ensemble yields the highest numerical accuracy in the statistical-thermodynamic description within the framework of all approximations elaborated in part I.