Correlation and relativistic effects in U metal and U-Zr alloy: Validation ofab initioapproaches

Abstract

Ab initio calculations have been performed on all solid phases of U metal and U-Zr alloy, the basis of a promising metallic fuel for fast nuclear reactors. Based on generalized gradient approximation, both density functional theory (DFT) in its standard form and the so-called DFT plus Hubbard $U$ (DFT+$U$) modification are evaluated. The evolution of calculated energetics, volume, magnetic moments, electronic structure, and $f$-orbital occupation as functions of the effective Hubbard $U$ parameter, ${U}_{\mathrm{eff}}$, is carefully examined at ${U}_{\mathrm{eff}}$ from 0 to 4 eV. DFT is found to overestimate energetics, underestimate volume, downward shift some $f$ bands near Fermi level and overestimate $f$-orbital occupation against existing experimental and/or computational data. The error is \ensuremath{\sim}0.07 eV/atom in terms of enthalpy, which affects phase stability modeling for \ensuremath{\delta}(U,Zr) and \ensuremath{\gamma}(U,Zr). DFT+$U$ at ${U}_{\mathrm{eff}}=1\ensuremath{-}1.5$ eV offers clear improvement on these calculated properties (\ensuremath{\sim}0.05 eV/atom in terms of enthalpy) and in general still neither promotes ordered magnetic moments nor opens unphysical band gaps, which occur at higher ${U}_{\mathrm{eff}}$ values. The empirical ${U}_{\mathrm{eff}}$ values of 1--1.5 eV are close to but smaller than the theoretical estimations of 1.9--2.3 eV that we obtain from the linear response approach. ${U}_{\mathrm{eff}}$ is found to vary only slightly (\ensuremath{\le}0.24 eV) between different phases and at different compositions of U and U-Zr; thus, a single ${U}_{\mathrm{eff}}=1.24$ eV, which is the statistical optimal from energetic fitting, is suggested for both U and U-Zr. Besides correlation, the relativistic effect of spin-orbit coupling (SOC) is also systematically explored. SOC is found to lower energy, increase volume, and split the 5$f$ shell above Fermi level and reduce $f$-orbital occupation. The effect predominates in the unoccupied states and is very small on all these calculated ground state properties (\ensuremath{\sim}0.02 eV/atom in terms of enthalpy).