# Mechanical and Thermal Properties for Uranium and U-6 Nb Alloy from First-Principles Theory
## 4 Discussion  
We have studied uranium metal and U-6 Nb alloy from the first-principles theory. For the uranium-niobium alloy, we focus on elastic moduli, and the essential information on its elasticity has not been reported until now. The elastic properties are important in terms of modeling, i.e., they can constrain interatomic potentials for molecular dynamics simulations, or they can explicitly be used in constitutive modeling. The elastic constants also explain the appreciated fact that the uranium-niobium alloys are more ductile than the pure metal because of a significant alloy-driven softening of the c\({}_{\rm ii}\) moduli. For both the metal and the alloy (\(\alpha\) or \(\alpha^{\prime}\) phase), we find that c\({}_{11}\), c\({}_{22}\), and c\({}_{33}\) are very different. In particular, c\({}_{22}\), that correlates with a change in the b lattice parameter, is significantly smaller than the other two. This crystal anisotropy shows up dramatically in the thermal-expansion behavior as well.  
Figure 3: The temperature dependence of the a (top panel), b (middle panel), and c (bottom panel) lattice parameters (Å) in \(\alpha\) uranium (solid circles). The dashed lines are guides to the eye only. The experimental behaviors [45] are shown as solid lines.  
The phonon density of states is calculated for \(\alpha\) uranium and compares well with neutron scattering. This fact, and that theoretical elastic moduli are in good agreement with measured data (particularly when compared at the same 0 K temperature), gives us confidence in the DFT+OP model used here for uranium and the uranium-niobium alloy. The Debye temperature obtained from the DFT+OP phonon DOS is consistent with that of the quasi-harmonic model, thus validating the latter as a sound approach for the thermal-expansion study.  
For \(\alpha\) uranium, we investigate the unusual thermal-expansion behavior. We show that the anisotropic and negative thermal expansion does not need to originate from exotic (magnetic) electronic structure. Instead, a delicate balance between zero-temperature energetic and temperature-dependent electronic and lattice-vibration contributions to the free energy fully explains the anomaly. In this approach, the electronic term is determined from DFT+OP electronic structure with a Fermi-Dirac broadening associated with the electronic density of states, and the phonon term is based on the Debye-Gruneisen quasi-harmonic scheme. The agreement between the observed anomalous lattice-constant anisotropy and that predicted by the model is satisfactory, at least below the highest temperatures.  
Conceptualization, P.S.; methodology, P.S. and L.H.Y.; software, P.S. and L.H.Y.; investigation, P.S. and L.H.Y.; writing--original draft preparation, P.S.; writing--review and editing, P.S., L.H.Y., A.L. and A.W. All authors have read and agreed to the published version of the manuscript. This research received no external funding. This work was performed under the auspices of the U.S. DOE by LLNL under Contract DE-AC52-07 NA27344. The authors declare no conflict of interest.