# Mechanical and Thermal Properties for Uranium and U-6 Nb Alloy from First-Principles Theory
## 3 Results
### Lattice Dynamics  
In addition to the elasticity, lattice vibrations or phonons provide important thermal and structural information about the material. For example, soft phonons could signal structural phase transitions, while stiffer phonons usually imply greater elastic constants and  
\begin{table}
\begin{tabular}{l l l l l l} \hline
**Elastic** & \(\alpha^{\prime\prime}\)-U & \(\alpha^{\prime\prime}\)-U\({}_{14}\)Nb\({}_{2}\) & \(\alpha\)-U & \(\alpha\)-U- & \(\alpha\)-U\({}_{14}\)Nb\({}_{2}\) \\ \hline c\({}_{11}\) & 2.91 & 2.50 & 2.88 & 2.15 (2.10  
strength. The phonons also contain temperature-dependent information and parameters for thermodynamic modeling, such as the Debye temperature or the Gruneisen parameter.  
By relating the theoretical phonons with experimental observations, one can assess the accuracy and reliability of the model. If a material's calculated atomic density, crystal structure, elastic constants, and phonons all compare favorably with measurements, then the chemical bonding predicted by the model must be rather accurate as well. Recently, excellent phonons were calculated for a far more controversial material than \(\alpha\) uranium, namely \(\alpha\) plutonium [53], thus validating the electronic structure model assumed for plutonium [36].  
Our approach for modeling the \(\alpha\)-uranium phonon density of states is the same as that recently performed for \(\alpha\) plutonium [53]. One difference is that for plutonium, non-negligible spin and orbital magnetic moments are formed [54], while for \(\alpha\) uranium, the electrons do not spontaneously spin polarize. Nonetheless, effects of both spin-orbit interaction and orbital polarization are present and accounted for in our theory for uranium.  
In Figure 1, we show our DFT+OP phonon density of states together with experimental data [55] and least-square fits of force-constant models to experimental data [56]. In this figure, the notations are repeated from [55], and for more details we refer to that publication. The neutron-scattering data [55] show significant scatter, as do the force-constant models, but present in all data is the two-peak feature with one peak at 10 meV and the other around 12.5-14.5 meV. The DFT+OP model accurately reproduces this double-peak feature, while at the highest energies there is a shift between DFT+OP on one hand and the neutron data on the other, with the force-constant models in the middle.  
The discrepancy at the highest energies may be due to the lack of explicit temperature dependence of the phonons in the method, or that only harmonic lattice vibrations were considered in the model. One must also acknowledge that the experimental data may have rather large error bars, particularly at the highest energies (>12 meV) where the data points are more scattered; see Figure 1.  
The Debye temperature (\(\theta_{\text{D}}\)) can be expressed in terms of a Brillouin zone average of the phonon frequencies [57], and from the DFT+OP phonon DOS we determine \(\theta_{\text{D}}=240\) K. Experimentally, there is a wide range (183-259 K) of room-temperature Debye temperatures reported and listed in [55] and [58], with the 11 data-point average being \(\theta_{\text{D}}=217\) K. The Debye temperature is also known to be rather temperature dependent and decreases significantly with temperature [55].  
Figure 1: Measured (open squares and circles) [55] and force-constant models fitted to experiments (colored full or dashed lines) [56], and present DFT+OP (black line) phonon density of states for \(\alpha\) uranium at 300 K. The DFT+OP result was smoothened by a Gaussian function with a 0.2 eV width.  
We mention that another theoretical (DFT) study of \(\alpha\) uranium produced lattice dynamics in good agreement with neutron-scattering data as well [48]. In that study, the focus was on the low-temperature properties of \(\alpha\) uranium and specifically the distortions associated with the charge-density wave.