Figure 1

The dispersion of the mixed magnetovibrational-quadrupolar modes in the low-temperature phase of UO${}_2$ is shown along the principal crystallographic directions of the cubic lattice (Refs. 5 and6). Neutron groups have been collected in the constant-$\mathbf{Q}$ mode, at the reciprocal lattice points $(0,0,2-\xi )$, $(1-\xi ,1-\xi ,1)$, and $(1+\xi ,1+\xi ,1+\xi )$. Gray symbols indicate qualitatively smaller intensity than the black symbols. The broken lines and open squares indicate acoustic phonon branches measured at 270 K (Ref. 6). Excitations around 12 meV correspond to an optical quadrupolar branch. The experiments provide evidence of mixing between magnons and phonons, in the forms of large avoided crossings (ACs), for instance, at $\xi \sim$ 0.45 along the $\left[00\xi \right]$ direction. A portion of the UO${}_2$ $\left(1\overline{1}0\right)$ reciprocal lattice plane is shown in the upper panel. Filled and empty circles are points corresponding to crystal and magnetic structure Bragg reflections, respectively. Red (solid) arrows indicate neutron momentum transfer vectors; dark blue (dashed) arrows indicate quasi-momenta of collective excitations.Reuse & Permissions

Figure 2

Dipolar magnetic INS cross section along $\left[00\xi \right]$, calculated with the model described in Ref. 15 and discussed in this paper. The color scale represents intensity in arbitrary units. (a) Intensity map showing the results of RPA spin-wave calculations neglecting both superexchange and magnetoelastic quadrupole-quadrupole interactions. Bare acoustic phonon dispersion curves are shown by dashed white lines. (b) Intensity map obtained by taking into account quadrupole-quadrupole couplings. Dashed white lines show modes with sizable phonon component and cross section. Labels specify the character of the excitation branches: SA = spin acoustic, SO = spin optical, QA = quadrupolar acoustic, QO = quadrupolar optical, TA = transverse acoustic phonon, LA = longitudinal acoustic phonon. Vertical double arrows indicate avoided crossing positions. (c) Same calculation as in panel (b) with the ME coupling removed, i.e., by setting $g_A=g_B=0$ in Eq. (2). (d) Same calculation as in panel (b) with high-energy phonons removed. This is done by artificially inserting a smooth cutoff in ${\Theta }_{\alpha \sigma }\left(\mathbf{q}\right)$ in Eq. (3) for phonons with energy ${\epsilon }_{\sigma }\left(\mathbf{q}\right)>25$ meV. Panels (c) and (d) are commented at the end of Sec. V.Reuse & Permissions

Figure 3

Arrangement of the ordered magnetic moments (cones) and quadrupoles (ellipsoids) in the 3-$\mathbf{k}$ phase. The four sublattices are labeled by different colors. The figure also illustrates the splitting of the ground ${\Gamma }_5$ triplet below $T_N$ for two sublattices. The labels of nonvanishing off-diagonal matrix elements are referred to the local $z$ axis.Reuse & Permissions

Figure 4

Constant-$\mathbf{Q}$ energy scan recorded with IN22 at (a) $\mathbf{Q}=(1,1,1)-(0,0,0.3)$, and (b) $\mathbf{Q}=(2,2,0)+(0,0,0.3)$ ($\mathbf{P}\left|\right|\mathbf{Q}$; $T=2$ K). Red (filled) circles represent neutron counts in the spin-flip (SF) channel, open circles are non-spin-flip (NSF) counts. The SF data are fitted to the sum of four Gaussian lineshapes shown by dashed (blue) lines. The peak at 5 meV in the NSF channel at $\mathbf{Q}=(2,2,0.3)$ is a TA phonon (black dashed line). The corresponding portion of the calculated intensity map is shown as a guide to the energies of the excitations. The vertical dashed (white) line gives the $q$ value at which the energy scan has been performed.Reuse & Permissions

Figure 5

Dispersion along the $\left[00\xi \right]$ direction of the mixed magnon-phonon modes in UO${}_2$ at 2 K, near the anticrossing position at $\mathbf{q}=(0,0,0.45)$. The diagrams show detector counts normalized to monitor as a function of the energy transfer measured with IN14 at $\mathbf{Q}=(1,1,-1+\xi )$. A sloping background has been subtracted from the raw data. The inset shows calculated intensity maps in the energy-momentum transfer space [zoom of Fig. 2b]. For $\xi =$ 0.6, a small peak is visible at 6.5 meV (see Fig. 8).Reuse & Permissions

Figure 6

Longitudinal constant-$\mathbf{Q}$ energy scan at the anticrossing position $\mathbf{q}=(0,0,0.45)$. The experimental data have been measured with the IN14 spectrometer at $\mathbf{Q}=(0,0,-1.55)$, with the sample kept at $T=2$ K. A sloping background has been subtracted from the raw data. The two Gaussian curves (solid, red lines) have equal width.Reuse & Permissions

Figure 7

Constant-$\mathbf{Q}$ energy scans recorded with IN22 at (a)–(d) $\mathbf{Q}=(1,1,0.6)$ and $\mathbf{Q}=(2,2,0.4)$; (e)–(h) $\mathbf{Q}=(1,1,0.5)$ and $\mathbf{Q}=(2,2,0.5)$, corresponding to the quasiparticle wave vector $\mathbf{q}=(0,0,0.4)$ and $\mathbf{q}=(0,0,0.5)$, respectively. Experimental data are fitted to the sum of three Gaussian lineshapes, shown by dashed (blue) lines, and a constant background.Reuse & Permissions

Figure 8

Constant-$\mathbf{Q}$ energy scan recorded with IN22 at $\mathbf{Q}=(2,2,0)+(0,0,0.6)$ ($\mathbf{P}\left|\right|\mathbf{Q}$; $T=2$ K). (a) Neutron counts in the spin-flip (SF) channel; individual components of the spectrum are shown by dashed lines. (b) Non-spin-flip (NSF) neutron counts. The central peak at $\sim$9 meV (blue dashed line) is a transverse acoustic phonon, whereas the weaker peaks at $\sim$6.6 meV and $\sim$11.4 meV (red full lines) correspond to mixed phonon-quadrupolar acoustic modes.Reuse & Permissions

Figure 9

Constant-$\mathbf{Q}$ energy scan recorded with IN14 at $\mathbf{Q}=(0,0,1)$, corresponding to the $X$ point of the crystallographic cell. The experimental data are fitted to the sum of a Gaussian lineshape and a sloping background. The experimental error on the intensity is smaller than the size of the symbols.Reuse & Permissions

Figure 10

Left panel: excitations in the low-temperature phase of UO${}_2$ along $\Gamma X$. Circles are the same as in the left panel of Fig. 1, whereas blue diamonds are the excitations discussed in this paper. Right panel: dipolar magnetic INS cross section along $(1,1,1-\xi )$, calculated by the present model. Dashed white lines show modes with sizable phonon component and cross section. Note that, with $(1,1,1-\xi )$ being equivalent to $(0,0,\xi )$, this panel describes the same excitations as panel (b) in Fig. 2. However, the INS cross section is slightly different and both SO modes are visible.Reuse & Permissions