# Mechanical and Thermal Properties for Uranium and U-6 Nb Alloy from First-Principles Theory
## 3 Results  
### Elastic Constants  
Elastic constants reflect important material properties related to the strength and character of the interatomic bonding. They directly connect to macroscopic measurables such as ductility, brittleness, and resistance to shear and pressure. Furthermore, elasticity is an essential ingredient in constitutive modeling and involved in the shape memory effect that exists in the U-Nb materials. Elastic moduli can also be used to constrain interatomic potentials that are developed and employed in molecular dynamics simulations.  
Here, we focus on the uranium-niobium alloy with about 6 wt.% niobium (\(\sim\)14 at.% Nb), often denoted U-6 Nb (approximated here by an ordered U\({}_{14}\)Nb\({}_{2}\) compound). Because we are interested in the influence of niobium on the elastic properties, we explore unalloyed uranium as well, and include its ground-state orthorhombic \(\alpha\) phase in the study as a reference in addition to the monoclinic \(\alpha\)\" (U-6 Nb) phase.  
The calculated elastic constants for uranium and U\({}_{14}\)Nb\({}_{2}\) in the \(\alpha\)\" and \(\alpha\) phases are listed in Table 1. We are not aware of either calculations or measurements for the uranium-niobium alloy, but for \(\alpha\) uranium the data are consistent with others' modeling [40; 48; 49; 50; 51] and rather close to the experimental data [52], particularly when extrapolated to zero temperature. The good comparison between experiment and modeling for \(\alpha\) uranium gives us confidence that the theoretical framework is also sound and appropriate for the uranium-niobium alloy.  
The obtained elastic moduli are not dramatically different between U\({}_{14}\)Nb\({}_{2}\) and uranium in either the \(\alpha\) or \(\alpha^{\prime\prime}\) phase, and that is likely because only 12.5% of the uranium atoms are replaced by niobium atoms. However, we find a general trend of substantial elastic softening in both the \(\alpha\) and \(\alpha^{\prime\prime}\) phase. All c\({}_{\rm{ii}}\) moduli are significantly smaller for the alloy containing niobium. The bulk modulus, however, is relatively uniform between uranium and the uranium-niobium alloy in both phases. The shear modulus (G\({}_{\rm{V}}\)), on the other hand, is notably smaller for the alloy. Consequently, the Pugh ratio B\({}_{\rm{V}}\)/G\({}_{\rm{V}}\) increases by an appreciable 30-40% for the alloy, suggesting a great improvement in ductility over the pure element.  
We also performed reference calculations with 1/16 niobium (U\({}_{15}\)Nb) in the \(\alpha\) phase only, and in this case the ductility (in terms of the Pugh ratio) increased by about 17% relative to elemental uranium. There is a proportionality between niobium content and ductility, at least for smaller amounts of niobium. The reason for the better ductility is directly traced to the softening of the c\({}_{\rm{ii}}\) single-crystal elastic constants that we compute for the alloy (Table 1).  
Another interesting observation in Table 1 is that c\({}_{11}\), c\({}_{22}\), and c\({}_{33}\) are quite different for the studied phases. These three moduli correspond to the energy cost to independently strain along the a, b, and c axis, respectively. Consequently, there is a great deal of anisotropy in the crystal both in the \(\alpha\) and \(\alpha^{\prime\prime}\) phases for uranium and the uranium-niobium alloy. Particularly, c\({}_{22}\) stands out and is smaller than the other two. This is correlated with the anisotropic behavior of the b lattice constant with temperature, and we shall return to this in Section 3.3.  
Finally, the present U\({}_{14}\)Nb\({}_{2}\) elastic constants in the \(\alpha^{\prime\prime}\) and \(\alpha\) phases have practical value because they can help constrain interatomic potentials or guide constitutive modeling for the shape memory U-6 Nb alloy.