Anisotropy in cubic ${\mathrm{UO}}_2$ caused by electron-lattice interactions

Despite many years of research, the full complexity of the electron-lattice interactions in ${\mathrm{UO}}_2$ is not fully understood. We present x-ray inelastic scattering at low temperature showing that the interaction between electronic degrees of freedom and transverse acoustic phonons is strong only along the reciprocal space direction [100]. The anisotropy is reflected in the phonon-linewidth broadening, which persists also well above the Néel temperature. This intrinsic effect infers an anisotropy in the thermal conductivity, which has been observed, but which is formally forbidden in a cubic material. We have no model capable of connecting our experimental observations with the low thermal conductivity of ${\mathrm{UO}}_2$ below room temperature.

Figure 1

${\mathrm{UO}}_2$ single crystals used for experiments at low resolution $\mathrm{\Delta }\mathrm{E}=3$ meV (sample 1) and at high resolution $\mathrm{\Delta }\mathrm{E}=1.4$ meV (sample 2). The latter sample is a large [001]-oriented crystal, with a mosaic of ${0.05}^{\circ }$, used for earlier studies of the surface magnetism of ${\mathrm{UO}}_2$ [21].

Figure 2

Comparison of the inelastic TA[100] scans taken at room temperature at (6,0.4,0) for different samples. Sample 1 (left panel): High-quality sample with a good mosaic. The elastic line is nearly absent, and the phonon groups are well defined ($\mathrm{\Delta }\mathrm{E}=3.0$ meV). Sample 2 (right panel): High-resolution scans ($\mathrm{\Delta }\mathrm{E}=1.4$ meV) on the large-size and high-quality crystal.

Figure 3

Analysis of low-resolution IXS ($\mathrm{\Delta }\mathrm{E}=3.0$ meV) scans taken across the Brillouin zone $(6,k,0)$ at different temperatures for sample 1 (black diamonds, 300 K; blue open squares, 40 K; red circles, 5 K). The phonon groups are fit with a single-damped-harmonic-oscillator model deconvoluted with the experimental resolution (fit28 program, Ref. [19]). Top: TA[100] dispersion curve. Middle: Integrated phonon intensities normalized with the Bose factor. The line represent the inelastic structure factor (Str. Fac.) calculation. Bottom: TA(100) phonon linewidth deconvoluted with the experimental resolution.

Figure 4

Inelastic x-ray scattering scans taken at 20 K across the $(6,q,0)$ Brillouin zone. The transverse acoustic phonon TA[100] shows anticrossings that correspond to the two regions at $\mathrm{q}=0.45$ rlu with the acoustic magnon mode and at $\mathrm{q}=0.65$ rlu with the optic magnon mode. The color coding identifies branches on the dispersion curves of Fig. 5.

Figure 5

Dispersion curves measured at $\mathrm{T}=20$ K with inelastic x-ray scattering of transverse (solid colored symbols, TA) and longitudinal (open squares, LA) acoustic phonons in the [100] and [011] reciprocal lattice directions. The solid (dashed) lines represent the transverse (longitudinal) acoustic phonon branches calculated at room temperature. The dispersion of the magnetic excitations, determined by inelastic neutron scattering, is also indicated (gray circles) [12]. The $\mathrm{TA}({\mathrm{\Delta }}_5$) phonon branch splits into three branches, A, B, and C, as a result of the interactions with the acoustic magnon branch (branch A) and with the optic magnon branches (branches B and C). No interactions involving $\mathrm{TA}({\mathrm{\Sigma }}_4$) or ${\mathrm{TA}}^{\prime }({\mathrm{\Sigma }}_4$) are observed in the [011] direction. The color-coded rectangles refer to the energy linewidth broadening of the TA[100] branches in correspondence with the anticrossing regions.

Figure 6

Comparison of temperature dependence of TA phonons in the [100] and [011] directions. A finite phonon lifetime is clearly observed in TA[100] at 40 K (bottom panel), well above ${\mathrm{T}}_N=30.8$ K. Below ${\mathrm{T}}_N$ the TA[100] phonon splits in correspondence with the anticrossing region, as described in the text. The ${\mathrm{TA}}^{\prime }$[011] and TA[011] phonons have a linewidth close to the experimental resolution, as shown in the fit results presented in Tables 3 and 4, respectively.

Figure 7

Temperature dependence of TA[100] phonon linewidth (deconvoluted with the instrumental resolution). (a) shows the phonon broadening above ${\mathrm{T}}_N$, whereas (b) shows the evolution of linewidths of the different phonon branches across the two anticrossing regions (indicated by arrows), as described in the text.

Figure 8

Comparison of temperature dependence between the thermal conductivity difference $\mathrm{\Delta }\kappa =\kappa \left({\mathrm{ThO}}_2\right)-\kappa \left({\mathrm{UO}}_2\right)$ (solid blue diamonds) [13, 22], the FWHM of TA[100] averaged over values at the center of the zone (solid black squares), and the normalized quasielastic integrated intensities (open circles, shaded area). Lines are a guide for the eyes.