Combining density functional theory and Monte Carlo neutron transport calculations to study the phonon density of states of $\mathrm{U}{\mathrm{O}}_2$ up to 1675 K by inelastic neutron scattering

Significant advances in the use of atomistic simulation techniques, such as ab initio density functional theory and the molecular dynamics method, made it possible to predictively calculate properties of materials. In parallel, low-energy neutron scattering instruments and data analysis tools available in different institutes become mature for providing high-quality data for experimental validation purposes. Despite such experimental and theoretical improvements, the accurate modeling of experimental neutron-weighted multiphonon spectra for $\mathrm{U}{\mathrm{O}}_2$ over a broad temperature range still remains an issue. Combining prior phonon density of states (PDOS) from density functional theory and Monte Carlo inelastic neutron scattering calculations in a Bayesian fitting procedure is a valuable alternative approach to assess the partial PDOS of uranium and oxygen in $\mathrm{U}{\mathrm{O}}_2$ from 294 to 1675 K for improving the comparison to experiment and exploring the first-principles calculation hypothesis.

Figure 1

Ab initio phonon density of states of $\mathrm{U}{\mathrm{O}}_2$ at $T$ = 0 K [6] and $T$ = 294 K [8]. The dot-dashed lines locate the main vibration modes of $\mathrm{U}{\mathrm{O}}_2$ calculated at T = 294 K. The two first peaks at 10 and 20 meV correspond to the translational (TA) and longitudinal (LA) acoustic phonon modes of the uranium atoms. The structures at around 30 meV, 55 meV, and 70 meV are dominated by the optical phonon modes (TO1, LO1, TO2, LO2) of the oxygen atoms.

Figure 2

Simplified top view of the IN4 and IN6 time-of-flight spectrometers of the Institute Laue-Langevin.

Figure 3

Examples of background contributions at room temperature in the case of the neutron-weighted multiphonon spectrum of $\mathrm{U}{\mathrm{O}}_2$ deduced from the IN6 data measured with a quartz tube in a thin niobium sample-holder tube (a) and a thick niobium sample-holder tube (b).

Figure 4

Experimental neutron-weighted multiphonon spectrum ${\rho }_{\exp }^T$ of $\mathrm{U}{\mathrm{O}}_2$ deduced from the IN6 data. The dot-dashed lines show the energy positions of the vibration modes of $\mathrm{U}{\mathrm{O}}_2$ calculated at T = 294 K (Fig. 1).

Figure 5

Comparison of the neutron-weighted multiphonon spectrum ${\rho }_{\exp }^T$ and ${\rho }_{\mathrm{th}}^T$ deduced from the IN6 data and the tripoli4 simulations, respectively. The calculations were performed at T = 294 K by using the PDOS shown in Fig. 1. The theoretical curves in plots (a) and (b) were obtained within the single neutron scattering and one-phonon approximations, respectively. The theoretical curve in the bottom plot (c) accounts for both the multiple neutron scattering effects and the multiphonon contribution.

Figure 6

Prior and posterior tripoli4 simulations compared to the neutron-weighted multiphonon spectra deduced from the IN6 data.

Figure 7

Experimental symmetric form of the dynamic structure factor $S_{\exp }^T$ without the contribution of the quasielastic neutron scattering peak [Eq. (9)] measured at $\theta ={90}^{\circ }$ with the IN6 spectrometer. The solid red lines correspond to the posterior tripoli4 simulations obtained after the optimization of the partial phonon density of states ${\rho }_X^T$ of uranium and oxygen in $\mathrm{U}{\mathrm{O}}_2$.

Figure 8

Temperature-dependent partial phonon density of states ${\rho }_X^T$ of uranium and oxygen in $\mathrm{U}{\mathrm{O}}_2$ deduced from the IN6 data. The corresponding values are reported in the Appendix (Tables 1 and 2).

Figure 9

Comparison of the neutron-weighted multiphonon spectra ${\rho }_{\exp }^T$ and ${\rho }_{\mathrm{th}}^T$ deduced from the IN4 data and the tripoli4 simulations, respectively. The calculations were performed at 300, 600, and 900 K by using the partial PDOS obtained from the IN6 data (Fig. 8).

Figure 10

The red filled circles (connected by the red “eye-guide” curves) represent the B factors (in ${\AA }^2$) calculated with Eq. (11) by using the partial PDOS obtained from the IN6 data (Fig. 8). They are compared with experimental values reported in Refs. [28, 33, 34, 35, 36, 37].