Crystal dynamics and thermal properties of neptunium dioxide

We report an experimental and theoretical investigation of the lattice dynamics and thermal properties of the actinide dioxide ${\mathrm{NpO}}_2$. The energy-wave-vector dispersion relation for normal modes of vibration propagating along the $\left[001\right], \left[110\right]$, and $\left[111\right]$ high-symmetry lines in ${\mathrm{NpO}}_2$ at room temperature has been determined by measuring the coherent one-phonon scattering of x rays from an $\sim 1.2$-mg single-crystal specimen, the largest available single crystal for this compound. The results are compared against ab initio phonon dispersions computed within the first-principles density functional theory in the generalized gradient approximation plus Hubbard $U$ correlation (GGA+$U$) approach, taking into account third-order anharmonicity effects in the quasiharmonic approximation. Good agreement with the experiment is obtained for calculations with an on-site Coulomb parameter $U=4$ eV and Hund's exchange $J=0.6$ eV in line with previous electronic structure calculations. We further compute the thermal expansion, heat capacity, thermal conductivity, phonon linewidth, and thermal phonon softening, and compare with available experiments. The theoretical and measured heat capacities are in close agreement with another. About 27% of the calculated thermal conductivity is due to phonons with energy higher than 25 meV ($\sim 6\mathrm{THz}$), suggesting an important role of high-energy optical phonons in the heat transport. The simulated thermal expansion reproduces well the experimental data up to about 1000 K, indicating a failure of the quasiharmonic approximation above this limit.

Figure 1

Dispersion of longitudinal acoustic phonons propagating along the $\left[100\right]$ direction (top panel) and transverse acoustic phonons propagating along the $\left[011\right]$ direction (bottom panel). The data have been collected at the scattering vector $\mathit{Q}=\hslash ({\mathit{k}}_i-{\mathit{k}}_f)$ specified in each panel, ${\mathit{k}}_i$ and ${\mathit{k}}_f$ being the wave vector of incident and scattered photons, respectively.

Figure 2

Experimental IXS spectra collected for representative phonon branches of ${\mathrm{NpO}}_2$ at constant reciprocal space positions $\mathit{Q}$ indicated by the labels. Optic phonon branches TO1 and LO2 have a peak intensity weaker by 1 or 2 orders of magnitude than acoustic branches. Intensity data for each panel and corresponding inset are scaled by the same factor.

Figure 3

Phonon dispersion relations of ${\mathrm{NpO}}_2$ measured at room temperature along three high-symmetry directions by IXS (symbols) and calculated (solid lines) by first-principles density functional theory in the GGA+$U$ approach ($U=4$ eV, $J=0.6$ eV). $\xi$ is the phonon wave vector component in reciprocal lattice units (r.l.u.) of $2\pi /a$, where $a$ is the lattice parameter. Transverse (longitudinal) modes are indicated by black circles (blue squares). The labels of the branches refer to the irreducible representations of the wave vector little group. The computed total phonon density of states (pDOS) is shown in the right panel by the solid black line. Partial pDOS for Np and O atoms are indicated by dotted blue line and dashed red line, respectively. The Np atom contribution to the pDOS dominates below $\sim 25$ meV, whereas the higher-energy contributions are dominated by the light O atom vibrations. The pDOS simulated for ${\mathrm{UO}}_2$ is shown for comparison (thin dashed green line).

Figure 4

Constant-pressure heat capacity of ${\mathrm{NpO}}_2$ (black solid line) obtained as the sum of the calculated vibrational contribution (blue dashed line), the dilation contribution calculated by the Grüneisen relation (green dashed line), Ref. [43], and the Schottky contribution due to crystal field (CF) excitations (blue dot-dashed line), calculated on the basis of the CF level scheme derived from inelastic neutron scattering experiments [5, 44, 45]. Experimental data are taken from Ref. [46] (a, red solid circles), Ref. [12] (b, red open circles), and Ref. [47] (c, red dot line). The $\lambda$-type anomaly at 25 K is associated with the transition to the multipolar ordered phase of ${\mathrm{NpO}}_2$. Experimental data for ${\mathrm{ThO}}_2$ (d, blue open squares) are taken from Ref. [13].

Figure 5

Calculated (red solid line) and experimental thermal expansion of ${\mathrm{NpO}}_2$ (blue dashed line, from Ref. [43]). The inset shows the calculated (black solid line) bulk modulus of ${\mathrm{NpO}}_2$ compared with the experimental value provided by high-pressure X-ray diffraction measurements [49] (red circle) and the analytical estimation given in Ref. [50] (green dashed line).

Figure 6

Calculated temperature dependence of the thermal conductivity of the fcc ${\mathrm{NpO}}_2$ crystal using the full solution of the Boltzmann equation (black line) from first-principles anharmonic lattice dynamics calculations. Simulations based on the RTA approximation provide undistinguishable results in the shown temperature range. The experimental thermal conductivity curve given in Ref. [53] is shown by the red circles with connecting line.

Figure 7

Phonon linewidth distribution calculated along the high-symmetry lines for fcc ${\mathrm{NpO}}_2$ at 300 K (left panel) and accumulated phonon thermal conductivity $\tilde{\kappa }$ as a function of the phonon energy (right panel). Different colors in the linewidth distribution indicate different phonon branches, while the $q$-dependent linewidth is depicted by the width of the branch. The accumulated phonon thermal conductivity is shown (right panel) by the red line, while its derivative with respect to the energy is depicted by the blue line.

Figure 8

Variation of the half of the phonon linewidth for ${\mathrm{NpO}}_2$ with respect to the phonon frequency. Different phonon branches are indicated with different colors, while the average phonon linewidth is represented by a black line.

Figure 9

GGA+$U$ at 0 K (red line), GGA+$U$+SOC at 0 K (green dashed line), and GGA+$U$ at 300 K (blue dashed line) calculated phonon dispersions (left panel) and corresponding phonon density of states (right panel) of ${\mathrm{NpO}}_2$. Note that the spin-orbit coupling has a relatively small influence on the ab initio computed phonon dispersions, comparable to that of an increased temperature.

Figure 10

Phonon linewidth distribution calculated along the high-symmetry lines for fcc ${\mathrm{UO}}_2$ at 300 K (left panel) and accumulated phonon thermal conductivity $\tilde{\kappa }$ as a function of the phonon energy (right panel). Different colors in the linewidth distribution indicate different phonon branches, and the $q$-dependent linewidth is depicted by the width of the branch. The accumulated phonon thermal conductivity is shown (right panel) by the red line, while its derivative with respect to the energy is depicted by the blue line.