Collective dynamics of liquid deuterium: Neutron scattering and approximate quantum simulation methods

We present an experimental and simulation study of the collective dynamics of liquid ${\mathrm{D}}_2$ in the range of exchanged wave vectors $3\le Q\le 14{\mathrm{nm}}^{-1}$. Neutron Brillouin scattering results for the center-of-mass dynamic structure factor of this moderately quantum fluid are compared, on an absolute scale, with those obtained by three different quantum simulation methods such as ring polymer molecular dynamics and two versions of the Feynman-Kleinert approach. The experimental data can be well described by dynamical models typically used for liquids. Some discrepancies show up both among simulations, and between simulations and experimental data. Such discrepancies are found to mainly concern the damping of the propagating sound modes.

Figure 1

Effect of the correction for multiple scattering. The raw intensity before the correction (blue circles) is compared with the corrected spectrum (red dots).

Figure 2

[(a)–(c)] Fully corrected single scattering intensity from the ${\mathrm{D}}_2$ liquid sample (black circles with error bars) at the $Q$ values specified in each panel. The fit of Eq. (21) to the neutron data (full-red curve) was based on a GH modeling [59] of $S_{\mathrm{K}}(Q,\omega )$ with the explicit imposition of the exact value of $M_{\mathrm{K}}^{\left(2\right)}$, which ensures, via Eq. (13), the conformance to the quantum sum rule dictated by Eq. (9). The quantum fit result for $S(Q,\omega$) is displayed as a cyan-dashed curve. Its elastic (black-dotted curve) and inelastic (black-dashed curve) contributions, adding up to the cyan curve, are also shown. The dominant component, as far as the central part of the neutron spectrum is concerned, is the contribution of the single-molecule term (magenta dot-dashed curve) we calculated as explained in the text, and normalized according to the fitted $C$ of Eq. (21). Finally, the green-thin curve accounts for a $\sim 0.5\%$ presence of ${\mathrm{H}}_2$ impurities allowing to better reproduce the experimental data around the almost coinciding energies (15 meV) pertaining to both the $1\to 2$ rotational transition of ${\mathrm{D}}_2$ and the $0\to 1$ one of ${\mathrm{H}}_2$. Details can be better appreciated in panel (d), which displays a zoom of panel (c).

Figure 3

[(a)–(c)] Experimental dynamic structure factor (black circles with error bars) obtained from Eq. (25). This quantity is in absolute units, although still affected by the experimental energy resolution. The red full line is the component due to collective modes resulting from the fit to $I^{\left(1\right)}(Q,E)$, i.e., apart from the factor $C{k}_{1}u\left(Q\right)/k_0$, corresponds to the cyan dashed curve of Fig. 2. Comparison is also made with the three different simulations of the quantum $S(Q,E)$ broadened by the instrumental resolution function: RPMD (blue-dashed line), FK-QCW (green-dotted line), and FK-LPI (dot-dashed magenta line). To better distinguish the different results for the central part of the spectrum we show in panel (d) a zoom of panel (c).

Figure 4

Static structure factor of ${\mathrm{D}}_2$ at $T=20.7\mathrm{K}, n=25.40{\mathrm{nm}}^{-3}$, obtained from simulations with $N=256$ particles and Trotter number $P=32$. Red-full stars: $S\left(Q\right)=F(Q,0)$ from RPMD; blue-full circles: $S_{\mathrm{K}}\left(Q\right)=F_{\mathrm{K}}(Q,0)$ from RPMD; black-solid line: $S\left(Q\right)$ from PIMC.

Figure 5

[(a)–(c)] Resolution free double differential cross section of liquid ${\mathrm{D}}_2$ at 20.7 K at three typical $Q$ values, as obtained from the fits to $I^{\left(1\right)}(Q,\omega )$ explained in the text. The displayed quantity is in absolute units. Note the very different amplitudes and widths of the spectra with increasing $Q$. (d) Zoom of the three spectra shown in panels [(a)–(c)] in order to better appreciate how the Brillouin peak (clearly visible at 5 ${\mathrm{nm}}^{-1}$, red curve) evolves as $Q$ grows, and tends to merge with the $0\to 1$ rotational line. Also visible are the low-intensity $1\to 0$ and $1\to 2$ lines at –7.4 meV and 14.8 meV, respectively.

Figure 6

Resolution free dynamic structure factor obtained from the fit to the neutron data (red-solid line) compared with three different evaluations of the quantum $S(Q,E)$: RPMD (blue-dashed line), FK-QCW (green -dotted line), and FK-LPI (dot-dashed magenta line). Note that FK-QCW performs better than RPMD at $Q=9{\mathrm{nm}}^{-1}$.

Figure 7

Kubo intermediate scattering function derived from the fit to the experimental data at $Q=12{\mathrm{nm}}^{-1}$ (red-solid line) compared with the corresponding RPMD (blue-dashed line), FK-QCW (green-dotted line), and FK-LPI (dot-dashed magenta line) estimates.

Figure 8

(a) RPMD estimate of the Kubo intermediate scattering function $F_{\mathrm{K}}(Q,t)$ of liquid ${\mathrm{D}}_2$ at $Q=4.11{\mathrm{nm}}^{-1}$ (black circles) and VE fit result (red-solid curve). Note the change of the spectral variable with respect to previous figures. The multi-exponential fit components are also shown separately, and specified in the legend. (b) Corresponding spectrum $S_{\mathrm{K}}(Q,\omega )/S_{\mathrm{K}}\left(Q\right)$, VE fit result, and fit components, following the legend of panel (a). Panels (c) and (d) show the time correlation and its spectrum, respectively, on a semilogarithmic scale in order to better appreciate the quality of the fit (red-solid curve).

Figure 9

Same as Fig. 8 at $Q=13.01{\mathrm{nm}}^{-1}$.

Figure 10

Same as Figs. 8 and 8 but in the case of a GH fit to the RPMD data.

Figure 11

Same as Fig. 9 but referring to the FK-QCW simulations. Note that the cutoff in time is shorter than in Fig. 9, due to some noise in these results that was absent in the RPMD data. Consequently, also the fit quality, though more than acceptable, is inferior to that of the RPMD case, as it is particularly evident from panels (c) and (d).

Figure 12

(a) Experimental (red-full circles), RPMD (blue-empty circles), and FK-QCW (green-empty stars) dispersion curve of the longitudinal modes. The hydrodynamic linear behavior is also shown as a dashed-black line. (b) The RPMD and FK-QCW values for the undamped frequency ${\mathrm{\Omega }}_{\mathrm{s}}\left(Q\right)$ are reported with the same symbols used in panel (a). The damping $z_{\mathrm{s}}\left(Q\right)$ of the sound modes turns out to nearly coincide in both the FK-QCW (green-empty diamonds) and RPMD (blue empty squares) cases. Full-red symbols are used for the corresponding experimental quantities.

Figure 13

Same as Fig. 2 but using the ideal gas lineshape in the calculation of $J_{\mathrm{s}}(Q,\omega )$ of Eq. (6) (magenta-chain curve).

Figure 14

$Q$ dependence of the amplitudes and dampings of the real modes considered in a GH modeling of the neutron data (labelled as ${\mathrm{R}}_0$, full red circles) and in a VE one of the simulated RPMD spectra, with ${\mathrm{R}}_1$ represented by the blue empty squares, and ${\mathrm{R}}_2$ by orange empty diamonds. In panel (a) the sum of the ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ VE contributions to the central RPMD spectrum is also compared with the effective one obtained from experiment.