Brillouin neutron scattering in heavy water: Evidence for two-mode collective dynamics

A high resolution $\left(1.5\text{meV}\right)$ inelastic neutron scattering experiment was carried out, aiming at an accurate investigation of the high frequency and low momentum dynamic response in heavy water. The experimental data confirm the existence of a dispersionless mode, besides the ordinary longitudinal collective dynamics. A simplified model, based on the interaction of two vibrational branches, is proposed to interpret the observed features of the dynamic spectra. The validity extent of this scheme is proved by applying it to room temperature neutron and x-ray data, to temperature and pressure dependent x-ray data, and to room temperature neutron data of vibrational density of states. The overall successfull results provided by this model, in conjunction with the combined analysis of the x-ray and neutron data on collective dynamics, enable a deeper insight into the complex mechanisms of the water dynamics and provide a simple phenomenological explanation for the transition from ordinary to fast sound.

Figure 1

Summary of the excitation energies observed in water by inelastic neutron scattering experiments (triangles from Ref. 1, dots from Ref. 2), and by inelastic x-ray scattering experiments (circles from Ref. 5). The solid lines are the dispersion relations associated to the fast sound $(c_{\infty }=3200{\text{ms}}^{-1})$ and to the hydrodynamic sound $(c_0=1320{\text{ms}}^{-1})$, with the dashed lines representing the continuations. The dot-dashed line is the dispersionless mode discussed in Ref. 1. The curve with crosses on the right is the experimental density of states of light water measured in Ref. 8, and shown over the same energy scale. We note the correspondence between the peak in the density of states and the energy associated to the dispersionless mode.

Figure 2

Kinematic region $(Q,\hslash \omega )$ accessible to the present investigation ($80\text{−}\text{meV}$ final neutron energy and $1°$ scattering angle) and to the neutron experiment of Ref. 1 ($120\text{−}\text{meV}$ final neutron energy and $1°$ scattering angle). The dashed lines are the dispersion relations associated to the collective excitations in water, as described in Ref. 1.

Figure 3

Intensity of the vanadium standard versus energy transfer, measured at $Q=0.2$ and $0.4{\mathrm{\AA }}^{-1}$ (dots). The solid lines are Gaussian curves with $1.5\text{meV}$ FWHM describing the instrument resolution function [12].

Figure 4

Intensity from the water sample collected with $\left({\mathfrak{I}}^g\right)$ and without $\left(\mathfrak{I}\right)$ the honeycomb Cd grid (see text). The relative intensity difference at the $k$th energy point, $\Delta \mathfrak{I}=({\mathfrak{I}}_k-{\mathfrak{I}}_k^g)∕{\sum }_k{\mathfrak{I}}_k$, is shown versus the energy transfer at $Q=0.3{\mathrm{\AA }}^{-1}$. Dots: experimental data. Full line: curve calculated with the present MS simulation.

Figure 5

Dynamic structure factor $S_{\mathit{exp}}\left(Q\hslash \omega \right)$ of heavy water versus energy transfer and at the lowest wave vectors measured in the present neutron experiment. The experimental data (dots) and the best-fitting curves (lines), calculated according to the two-mode interaction model described in the text, are shown also on an expanded scale.

Figure 6

Same as Fig. 5, but at the largest wave vectors measured in the present neutron experiment. The bottom right panel shows, on an expanded scale, the experimental data (dots) in comparison with the resolution function (solid line) at $Q=0.4{\mathrm{\AA }}^{-1}$, to emphasize the effect of the instrument resolution on the side structures of the inelastic peak.

Figure 7

Wave-vector dependence of the strength integral $Z_1\left(Q\right)$ of the broad Lorentzian component $S_1^L(Q,\omega )$ of the neutron fit. The continuous line is the model function $f=f_0[1-\sin (Qd)∕(Qd\left)\right]$ fitting the $Z_1\left(Q\right)$ data (see text).

Figure 8

Wave-vector dependence of the strength integrals $Z_{-}\left(Q\right)$ and $Z_{+}\left(Q\right)$ associated to the $(-)$ and the $(+)$ vibrational modes, and described by $S_{-}(Q,\omega )$ and $S_{+}(Q,\omega )$ in the fit to the present neutron data (filled symbols). The results obtained in the neutron experiment of Ref. 1 are also shown by open symbols, for comparison purposes. The dashed lines are a guide to the eye.

Figure 9

Dispersion relations of the collective excitation in water. The continuous lines are the dispersion curves associated to $\hslash {\omega }_{-}\left(Q\right)$ and $\hslash {\omega }_{+}\left(Q\right)$ obtained by a global fit of the two-mode interaction model to the neutron data of the present and Ref. 1 experiments. The neutron data, that necessarily lie on the calculated curves, are identified by open circles. The x-ray data resulting from the experiment and the analysis described in Ref. 5 are also shown for comparison purposes (dots).

Figure 10

Dynamic structure factor $S_{\mathit{exp}}(Q,\hslash \omega )$ of light water at $T=277\mathrm{K}$ and normal pressure, versus energy transfer and at the wave vectors measured in the x-ray experiment reported in Ref. 6. The solid lines are the best-fitting curves resulting from the fitting of the x-ray data by means of the present interaction model, with all the nonlinear parameters already fixed by the companion fit to the neutron data (see text). The data and the calculated curves are shown also on an expanded scale.

Figure 11

Wave-vector dependence of the strength integrals $Z_{-}\left(Q\right)$ and $Z_{+}\left(Q\right)$ associated to the $(-)$ and $(+)$ vibrational modes, described by $S_{-}(Q,\omega )$ and $S_{+}(Q,\omega )$ in the fit to the x-ray data [6]. The solid lines are a guide to the eye.

Figure 12

Wave-vector dependence of the structure factors of hydrogen (filled and open circles) and oxygen (filled and open triangles), separately associated to the low-lying $(-)$ and the upper$(+)$ vibrational modes. The solid lines are analytic expressions for the hydrogen structure factors associated to the two modes.

Figure 13

Dynamic structure factor $S_{\mathit{exp}}(Q,\hslash \omega )$ of light water versus energy transfer, at constant density and at three temperature values. The x-ray experimental data (dots) are from Ref. 6. The solid lines are the best-fitting curves to the x-ray data obtained by applying the present interaction model.

Figure 14

Density of vibrational states in light water versus energy. The experimental neutron data (dots) from Ref. 8 are shown in comparison with the curve (solid line) calculated in the framework of the interaction model (see text). The two contributions to the calculated density of states, originating from the $\hslash {\omega }_{-}\left(Q\right)$ mode (short-dashed line) and the $\hslash {\omega }_{+}\left(Q\right)$ mode (long-dashed line) are also plotted to emphasize the different wave-vector dependence of the two contributions.

Figure 15

Temperature dependence of the fitting parameters $c_{\infty }$ (dots) and $\hslash {\omega }_0$ (triangles) obtained by fitting the present interaction model to the temperature-dependent x-ray data of Ref. 6.