Abstract
We study the accuracy and convergence of the real space cluster expansion (RSCE) for the total energies of the Pd-rich PdX (X=Ru, Rh) alloys, which are used to study the phase stability and phase equilibria of the Pd-rich PdX alloys. In the present RSCE, the X atoms of minor element are treated as impurities in Pd. Then-body interaction energies (IEs) among X impurities in Pd, being used in the expansion of the total energies of the Pd-rich PdX alloys, are determined uniquely and successively from the low body to high body, by the full-potential Korringa-Kohn-Rostoker (FPKKR) Green's function method (FPKKR) for the perfect and impurity systems (Pd-host and Xnin Pd,n=1~4), combined with the generalized gradient approximation in the density functional theory. In the previous paper, we showed that the RSCE, in which the perturbed potentials due to the insertion of Xnimpurities in Pd were redetermined self-consistently up to the first-nearest neighboring (nn) host atoms around Xnimpurities, reproduce fairly well (the error of ~ 0.2mRy per atom) the FPKKR-band-calculation result of the ordered Pd3Rh alloy in L12structure, but a little wrongly (the error of ~ 0.7mRy per atom) for the ordered Pd3Ru alloy in L12structure. In the present paper, we show that this small RSCE error for the Pd3Ru alloy is corrected very well (from ~ 0.7mRy to ~ 0.1mRy per atom) by enlarging the self-consistent region for the perturbed potentials up to the 2nd-nn host atoms around Run impurities in Pd. We also clarify the correction for each value of the n-body (n=1~ 4) IEs.
Highlights
We recently succeeded in quantitatively reproducing the X-concentration dependence of the observed solvus temperatures of the Pd-rich PdX (X=Ru, Rh) alloys in fcc structure [10], by the free energy calculations based on the cluster variation method (CVM) [11,12,13]
We obtained the nice agreement for the Pd3Rh alloy, while slightly wrong result for the Pd3Ru alloy. We considered that this difference between the real space cluster expansion (RSCE) results for X=Ru and for X=Rh might be due to the difference of the delocalization of the perturbed potentials in the Xn impurity systems, depending on the delocalization of 4d orbitals of X impurities
In the previous study we found that the present RSCE reproduces fairly well the band calculation result for the ordered Pd3Rh alloy in L12 structure, while a little wrongly for the Pd3Ru alloy in L12 structure [14]
Summary
We recently succeeded in quantitatively reproducing the X-concentration dependence of the observed solvus temperatures of the Pd-rich PdX (X=Ru, Rh) alloys in fcc structure [10], by the free energy calculations based on the cluster variation method (CVM) [11,12,13]. We obtained the nice agreement (the error of ~ 0.2mRy per atom) for the Pd3Rh alloy, while slightly wrong result (the error of ~ 0.7mRy per atom) for the Pd3Ru alloy We considered that this difference between the RSCE results for X=Ru and for X=Rh might be due to the difference of the delocalization of the perturbed potentials in the Xn impurity systems, depending on the delocalization of 4d orbitals of X impurities. It is noted that in these calculations, the perturbed potentials due to the insertion of X impurities were self-consistently redetermined up to the first-nn host atoms around Xn impurities in Pd, as in the usual KKR calculations for the impurity systems [9]
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