# Mechanical and Thermal Properties for Uranium and U-6 Nb Alloy from First-Principles Theory
## 3 Results
### Thermal Expansion  
The thermal expansion of \(\alpha\) uranium is strongly anisotropic [45; 46]. Above room temperature, two (a and c) of the three lattice constants display a normal (and quite similar) positive expansion with temperature, while b anomalously shrinks or contracts with a negative thermal expansion. The volumetric thermal expansion, however, remains positive above room temperature. Negative thermal expansion is unusual in general, and among the actinides the \(\delta\)-phase of plutonium is probably best known for this exceptional behavior. For \(\delta\) plutonium, its origin has been proposed to be thermally excited longitudinal spin fluctuations [36]. For \(\alpha\) uranium, the formation of magnetic moments has been suggested [59] as an explanation for the anisotropic behavior. To our knowledge, a fundamental and detailed understanding of this phenomenon in \(\alpha\) uranium has not yet been provided. Regardless of cause, the expansion behavior of uranium is important since it relates to the thermal dependency of the mechanical properties and the shape memory effect in the uranium-niobium alloy. Because magnetic moments do not survive (collapse) in \(\alpha\) uranium in our first-principles model, any negative thermal expansion needs to have a different explanation than magnetism for the model to be correct. To explore a more straightforward origin, namely, a combination of conventional electronic and lattice-vibration thermal effects on the crystal structure (axial ratios), we study the free energy as a function of the b/a axial ratio and temperature. We simplify the model by keeping the c/a axial ratio constant at 1.755 and the internal parameter y at 0.1009. This is a reasonable simplification because the y parameter is rather insensitive to small changes in the lattice constants (not shown) and the a and c lattice constant expansions are nearly identical [45; 46] not too far above room temperature. Quasi-harmonic theory in the framework of the Debye-Gruneisen model is employed for the lattice-vibrational contribution to the free energy, and the electronic contribution is computed within the DFT+OP approach. The calculated quasi-harmonic Debye temperature is identical (240 K) to that obtained independently from our DFT+OP phonon calculation, and that reassures us of the appropriateness of the Debye-Gruneisen model for the vibrational contribution. In Figure 2, we show free energies, relative to the ground-state \(\alpha\)-uranium phase (b/a = 2.045, c/a = 1.755, and y = 0.1009), for structures with different b/a axial ratios as functions of temperature. The top and middle panel display the lattice and electronic free-energy contributions, respectively. We note that increasing temperature favors a decrease in the b/a axial ratio for both the lattice and electronic contributions to the free energy. These quantities are similar in magnitude and equally important for the net result. The electronic term at T = 0 K reflects the structural energetics at zero temperature and shows a weak energy dependence on the b/a axial ratio (\(\sim\)0.25 mRy/atom between b/a = 2.045 and b/a = 1.995). This weak energy dependence is essential because otherwise the b/a ratio would be insensitive to an increase in temperature (and it obviously is not). The bottom panel of Figure 2 shows the total free energy (F\({}_{\text{total}}\) = F\({}_{\text{electronic}}\) + F\({}_{\text{lattice}}\)) and one finds that the lowest energies are predicted to occur for different b/a ratios at different temperatures. First, we have b/a = 2.045 at zero temperature, and then we see transitions to (a) 2.035, (b) 2.025, (c) 2.015, (d) 2.005, and finally (e) 1.995 values of b/a with increasing temperatures. These transitions occur only because of the delicate balance between the structural energetics at zero temperature and the temperature-dependent lattice and electronic free-energy terms. The net effect of these contributions is a decrease in the b/a axial ratio with temperature for \(\alpha\) uranium.