Microscopic collective dynamics in liquid neon-deuterium mixtures: Inelastic neutron scattering and quantum simulations

In this paper a combined neutron scattering and quantum simulation study of the collective dynamics in liquid $\mathrm{Ne}\text{−}{\mathrm{D}}_2$ mixtures, at a temperature of $T=30\mathrm{K}$ and in the wave-vector transfer range 4 ${\mathrm{nm}}^{-1}<q<51{\mathrm{nm}}^{-1}$, is presented. Two ${\mathrm{D}}_2$ concentrations are investigated, one close to 25% molar and the other close to 50% molar, together with pure Ne. The dynamic structure factor for the centers of mass of the two molecular species is extracted from the neutron scattering data and subsequently compared with that obtained from three different quantum simulation methods, such as ring polymer molecular dynamics and two slightly different versions of the Feynman-Kleinert approach. A general agreement is found, even though some discrepancies both among simulations, and between simulations and experimental data, can be observed. In order to clarify the physical meaning of the present spectroscopic results, an analysis of the longitudinal current spectral maxima is carried out showing the peculiarities of the ${\mathrm{D}}_2$ center-of-mass dynamics in these mixtures. A comparison with the centroid molecular dynamics results obtained for the ${\mathrm{D}}_2$ center-of-mass self-dynamics in the same liquid mixtures is finally proposed.

Figure 1

Examples of the data reduction procedure explained in the main text and here applied to sample No. (4), measured with $E_i=40.0$ meV: panels (a) and (b) describe the removal of the empty cell signal (red line and error bars) from the sample-plus-cell raw spectrum (black line and error bars); panels (c) and (d) show the subtraction of the simulated multiple scattering signal (green line) from the processed spectrum (black line and error bars); and panels (e) and (f) compare the simulated incoherent spectrum $I_{\mathrm{inc}}(q,\omega )$ from ${\mathrm{D}}_2$ (red line) with the experimental one, $I_E(q,\omega )$ (blue line and error bars), containing both coherent and incoherent components. Panels (a), (c), and (e) and (b), (d), and (f) refer to $q$ values equal to 18.25 and 35.75 ${\mathrm{nm}}^{-1}$, respectively.

Figure 2

Experimental neutron spectra (fully processed) for sample No. (3) (pure Ne, $T=30.02\mathrm{K}$ as in Table 1) plotted for selected $q$ values. Curves have been vertically shifted for graphic reasons. Panel (a) contains data measured with an incoming neutron energy $E_i=40.0$ meV, from $q=15.75{\mathrm{nm}}^{-1}$ (bottom) to 58.25 ${\mathrm{nm}}^{-1}$ (top), while panel (b) shows those obtained with $E_i=7.1$ meV, from $q=19.0{\mathrm{nm}}^{-1}$ (bottom) to 26.0 ${\mathrm{nm}}^{-1}$ (top).

Figure 3

Experimental neutron spectra (fully processed) for sample No. (4) (${\mathrm{D}}_2/\mathrm{Ne}$ mixture, $x\left[{\mathrm{D}}_2\right]=23\%, T=30.01\mathrm{K}$ as in Table 1) plotted for selected $q$ values. Curves have been vertically shifted for graphic reasons. Panel (a) contains data measured with an incoming neutron energy $E_i=40.0$ meV, from $q=8.25{\mathrm{nm}}^{-1}$ (bottom) to 50.75 ${\mathrm{nm}}^{-1}$ (top), while panel (b) shows those obtained with $E_i=7.1$ meV, from $q=6.0{\mathrm{nm}}^{-1}$ (bottom) to 24.0 ${\mathrm{nm}}^{-1}$ (top).

Figure 4

Experimental neutron spectra (fully processed) for sample No. (5) (${\mathrm{D}}_2/\mathrm{Ne}$ mixture, $x\left[{\mathrm{D}}_2\right]=49\%, T=30.03\mathrm{K}$ as in Table 1) plotted for selected $q$ values. Curves have been vertically shifted for graphic reasons. Panel (a) contains data measured with an incoming neutron energy $E_i=40.0$ meV, from $q=8.25{\mathrm{nm}}^{-1}$ (bottom) to 53.25 ${\mathrm{nm}}^{-1}$ (top), while panel (b) shows those obtained with $E_i=7.1$ meV, from $q=6.0{\mathrm{nm}}^{-1}$ (bottom) to 24.0 ${\mathrm{nm}}^{-1}$ (top).

Figure 5

The $\omega$-integrated experimental spectra, $I_E\left(q\right)$, reported as solid symbols, together with the corresponding PIMC-simulated neutron structure factors, $I\left(q\right)$, multiplied for appropriate scale factors (full lines). Data associated to sample No. (3) are plotted in blue with triangles, those to sample No. (4) are plotted in red with circles, and those to sample No. (5) are plotted in green with squares. Dashed lines show that at high $q$ values the ${\mathrm{D}}_2$ incoherent component extends out of the experimentally accessible $\omega$ range (see the main text for details). In the inset the PIMC-simulated neutron structure factor, $I\left(q\right)$, for sample No. (5) is divided into its two components: coherent, $I_{\mathrm{coh}}\left(q\right)$, and incoherent, $I_{\mathrm{inc}}\left(q\right)$. In addition, the RPMD evaluation of $I_{\mathrm{coh}}\left(q\right)$ is also reported as magenta dots.

Figure 6

Coherent experimental spectra, $I_{\mathrm{coh}}^{\left(E\right)}(q,\omega )$, from sample No. (3), pure Ne, for $E_i=40.0$ meV are reported for selected $q$ values as black empty circles. Corresponding simulated data, $I_{\mathrm{coh}}(q,\omega )$ (see main text for details) are plotted as lines: blue for RPMD, green for FK-LPI, and red for FK-QCW.

Figure 7

Coherent experimental spectra, $I_{\mathrm{coh}}^{\left(E\right)}(q,\omega )$, from sample No. (3), pure Ne, for $E_i=7.1$ meV are reported for selected $q$ values as black empty circles. Corresponding simulated data, $I_{\mathrm{coh}}(q,\omega )$ (see main text for details) are plotted as lines: blue for RPMD, green for FK-LPI, and red for FK-QCW.

Figure 8

Coherent experimental spectra, $I_{\mathrm{coh}}^{\left(E\right)}(q,\omega )$, from sample No. (4), low ${\mathrm{D}}_2$-concentration mixture, for $E_i=40.0$ meV are reported for selected $q$ values as black empty circles. Corresponding simulated data, $I_{\mathrm{coh}}(q,\omega )$ (see main text for details) are plotted as lines: blue for RPMD, green for FK-LPI, and red for FK-QCW.

Figure 9

Coherent experimental spectra, $I_{\mathrm{coh}}^{\left(E\right)}(q,\omega )$, from sample No. (4), low ${\mathrm{D}}_2$-concentration mixture, for $E_i=7.1$ meV are reported for selected $q$ values as black empty circles. Corresponding simulated data, $I_{\mathrm{coh}}(q,\omega )$ (see main text for details) are plotted as lines: blue for RPMD, green for FK-LPI, and red for FK-QCW.

Figure 10

Coherent experimental spectra, $I_{\mathrm{coh}}^{\left(E\right)}(q,\omega )$, from sample No. (5), high ${\mathrm{D}}_2$-concentration mixture, for $E_i=40.0$ meV are reported for selected $q$ values as black empty circles. Corresponding simulated data, $I_{\mathrm{coh}}(q,\omega )$ (see main text for details) are plotted as lines: blue for RPMD, green for FK-LPI, and red for FK-QCW.

Figure 11

Coherent experimental spectra, $I_{\mathrm{coh}}^{\left(E\right)}(q,\omega )$, from sample No. (5), high ${\mathrm{D}}_2$-concentration mixture, for $E_i=7.1$ meV are reported for selected $q$ values as black empty circles. Corresponding simulated data, $I_{\mathrm{coh}}(q,\omega )$ (see main text for details) are plotted as lines: blue for RPMD, green for FK-LPI, and red for FK-QCW.

Figure 12

Generalized longitudinal current-current time correlation spectra, ${\tilde{g}}_L(q,\omega )$, derived from experimental neutron data with $E_i=40.0$ meV, are reported for selected $q$ values as follows: blue triangles for sample No. (3), red circles for sample No. (4), and green squares for sample No. (5). In the first case, related to pure Ne, ${\tilde{g}}_L(q,\omega )$ coincides with the standard longitudinal current-current time correlation spectrum, ${\tilde{c}}_L(q,\omega )$.

Figure 13

Dispersion curves ${\mathrm{\Omega }}_d\left(q\right)$ derived from the positions of the maxima of the longitudinal current-current time correlation spectra, ${\tilde{c}}_L(q,\omega )$, obtained according to Eq. (11) in the case of sample No. (3), i.e., pure Ne. Red line represents ${\mathrm{\Omega }}_d\left(q\right)$ for the FK-QCW simulated spectra, while blue and green lines stand for the dispersion curves from RPMD and FK-LPI, respectively. As for the experimental results, full squares (with error bars) are derived from neutron scattering data measured with $E_i=40.0$ meV, and empty circles (with error bars) from those measured with $E_i=7.1$ meV. The hydrodynamic low-$q$ behavior from speed of sound data in Ref. [16] is also represented as a black dotted straight line.

Figure 14

Pseudodispersion curves, ${\mathrm{\Omega }}_{\mathrm{pd}}\left(q\right)$, derived from the positions of the maxima of the generalized longitudinal current-current time correlation spectra, ${\tilde{g}}_L(q,\omega )$, obtained according to Eq. (13) in the case of sample No. (4) (full lines and full squares with error bars) and No. (5) (dashed lines and empty circles with error bars). Blue lines represent curves from the RPMD simulated spectra, green and red lines stand for those from FK-LPI and FK-QCW, respectively, while symbols with error bars stand for the corresponding experimental results measured with $E_i=40.0$ meV.

Figure 15

Position of the maxima of the lightweight, ${\tilde{c}}_L^{\left(l\right)}(q,\omega )$, and heavy, ${\tilde{c}}_L^{\left(h\right)}(q,\omega )$, longitudinal current-current time correlation spectra from RPMD simulations are reported as dashed lines and full lines, respectively. Both types of red lines are related to sample No. (5), both types of blue lines to sample No. (4); while the green full line refers to sample No. (3) and the black dashed line to pure liquid ${\mathrm{D}}_2$ [i.e., simulation No. (VIII) in Table 2]. The adimensional physical quantity on the abscissa is the reduced wave-vector transfer, $q{n}^{-1/3}$. The green dot-dashed line is the limiting ${\tilde{c}}_L^{\left(l\right)}(q,\omega )$ in case of an infinite ${\mathrm{D}}_2$ dilution (see main text for details), while the magenta dotted line represents the free recoil of the ${\mathrm{D}}_2$ center of mass evaluated for the density of sample No. (4). The hydrodynamic low-$q$ behavior from speed of sound data in Ref. [16] is also represented as a cyan dot-dashed straight line for sample No. (4), which almost coincides with the corresponding estimate for sample No. (5).

Figure 16

Spectra of the CMD velocity autocorrelation functions for Ne atoms (a) and ${\mathrm{D}}_2$ centers of mass (b). Blue and red full lines represent simulations for samples Nos. (4) and (5), respectively; while the black full line stands for the pure ${\mathrm{D}}_2$ case [i.e., No. (VIII) in Table 2]. As for the green lines, which are related sample No. (3), the full one is the pure Ne contribution, while the dotted one is the infinite-dilution spectrum of a single ${\mathrm{D}}_2$ impurity in neon. Vertical sticks mark the positions of the first maximum of the various $\hslash {\mathrm{\Omega }}_d$ curves reported in Fig. 15.