Hydrogen self-dynamics in liquid ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixtures studied through inelastic neutron scattering

We have measured the dynamic structure factor of liquid para-hydrogen mixed with normal deuterium $(T=20 \mathrm{K}$) at two different concentration levels using incoherent inelastic neutron scattering. This choice has been made since the presence of ${\mathrm{D}}_2$ modifies the self-dynamics of ${\mathrm{H}}_2$ in a highly nontrivial way, acting both on its pseudophononic and its diffusive parts in a tunable way. After an accurate data reduction, recorded neutron spectra were studied through the modified Young and Koppel model and the ${\mathrm{H}}_2$ center-of-mass self-dynamics structure factor was finally extracted for the two mixtures. Some physical quantities (i.e., self-diffusion coefficient and mean kinetic energy) were determined and compared with accurate quantum calculations, which, in addition, also provided estimates of the velocity autocorrelation function for the ${\mathrm{H}}_2$ centers of mass. These estimates, in conjunction with the Gaussian approximation, were used to simulate the ${\mathrm{H}}_2$ center-of-mass self-dynamics structure factor in the same range as the experimental one. The agreement between measured and calculated spectra was globally good, but some discrepancies proved the unquestionable breakdown of the Gaussian approximation in these semiquantum systems at a level comparable to that already observed in pure liquid para-hydrogen.

Figure 1

Main panel: experimental raw spectral data as in Table 1, corrected for the kinematic factor, in the angular range $21.{83}^{\circ }<\theta <25.{92}^{\circ }$ (average value: $\theta =23.{43}^{\circ }$). The empty can spectrum is plotted as a dashed black line and the pure liquid ${\mathrm{D}}_2$ one as a dotted red line, while the dotted-dashed green line and the full blue line represent the two ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixtures at $c\left[{\mathrm{H}}_2\right]=24\%$ and $50\%$, respectively. Inset: same data as in the main panel, but zoomed to show the experimental statistical accuracy.

Figure 2

Experimental estimates of the self-part of the para-hydrogen center-of-mass dynamic structure factor $S_{s,\text{c.m.}}(Q,E)$, in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture sample No. (4) (i.e., $c\left[{\mathrm{H}}_2\right]=24\%$, see Table 1) shown for $Q$ values ranging from 1.4 ${\AA }^{-1}$ (black line, the most peaked and leftmost) to 4.6 ${\AA }^{-1}$ (pink line, the least peaked and rightmost).

Figure 3

Experimental estimates of the self-part of the para-hydrogen center-of-mass dynamic structure factor $S_{s,\text{c.m.}}(Q,E)$, in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture sample No. (5) (i.e., $c\left[{\mathrm{H}}_2\right]=50\%$, see Table 1) shown for $Q$ values ranging from 1.4 ${\AA }^{-1}$ (black line, the most peaked and leftmost) to 4.6 ${\AA }^{-1}$ (pink line, the least peaked and rightmost).

Figure 4

Simulations of the velocity autocorrelation function spectrum $f\left(\omega \right)$ for ${\mathrm{H}}_2$ and ${\mathrm{D}}_2$ centers of mass calculated in the thermodynamic conditions Nos. (vi), (viii), (x), and (xiii) (see Table 2 for details). ${\mathrm{H}}_2$ spectra are plotted for Nos. (vi) (dotted blue line), (viii) (dashed red line), and (x) (full black line), i.e., for $c\left[{\mathrm{H}}_2\right]=50.0\%, c\left[{\mathrm{H}}_2\right]=32.8\%$, and $c\left[{\mathrm{H}}_2\right]=25.0\%$, respectively. A single ${\mathrm{D}}_2$ spectrum is reported for No. (xiii) (dashed-dotted green line), which corresponds to the pure deuterium simulated sample.

Figure 5

Three-dimensional plot of the Kubo-transformed velocity autocorrelation functions ${〈\vec{v}\left(0\right)·\vec{v}\left(t\right)〉}^{\left(k\right)}$ for the ${\mathrm{H}}_2$ centers of mass, simulated in the thermodynamic conditions Nos. (o)–(xii) (see Table 2 for details). For graphic reasons, data have been normalized making use of their $t=0$ values $〈{\vec{v}}^2〉$.

Figure 6

GA calculations of the self-part of the para-hydrogen center-of-mass dynamic structure factor $S_{s,\text{c.m.}}^{\left(\text{GA}\right)}(Q,E)$ in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture sample No. (x) (i.e., $c\left[{\mathrm{H}}_2\right]=25.0\%$, see Table 2) shown for $Q$ values ranging from 1.4 ${\AA }^{-1}$ (black line, the most peaked and leftmost) to 4.6 ${\AA }^{-1}$ (pink line, the least peaked and rightmost).

Figure 7

GA calculations of the self-part of the para-hydrogen center-of-mass dynamic structure factor $S_{s,\text{c.m.}}^{\left(\text{GA}\right)}(Q,E)$ in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture sample No. (vi) (i.e., $c\left[{\mathrm{H}}_2\right]=50.0\%$, see Table 2) shown for $Q$ values ranging from 1.4 ${\AA }^{-1}$ (black line, the most peaked and leftmost) to 4.6 ${\AA }^{-1}$ (pink line, the least peaked and rightmost).

Figure 8

Main panel: GA calculations of the self-part of the para-hydrogen center-of-mass dynamic structure factor $S_{s,\text{c.m.}}^{\left(\text{GA}\right)}(Q,E)$ in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture sample No. (vi) (i.e., $c\left[{\mathrm{H}}_2\right]=50.0\%$, see Table 2) shown for $Q$ values ranging from 2.6 to 3.4 ${\AA }^{-1}$ (full black line). The purely diffusional components are also reported (dashed red line) together with the single-pseudophonon terms (dotted blue line). All plots do not include the instrumental resolutions and have been vertically shifted (0.04 ${\mathrm{meV}}^{-1}$) for graphic reasons. Inset: separation of the corresponding simulated $f\left(\omega \right)$ (green full line) into diffusional (dashed red line) plus solidlike (dotted blue line) terms used for the calculations plotted in the main figure.

Figure 9

Experimental estimates of full width at half maximum of the para-hydrogen center-of-mass dynamic structure factor $S_{s,\text{c.m.}}(Q,E)$ in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture samples Nos. (4) (full black squares with error bars) and (5) (empty red circles with error bars). Only the lowest momentum transfer values (i.e., 1.4 ${\AA }^{-1}\le Q\le 2.4{\AA }^{-1}$) have been considered for such an analysis. Full black and dashed red lines represent data best fits obtained assuming a jump-diffusion model for samples Nos. (4) and (5), respectively. Dotted black and dotted-dashed red lines stand for analogous data best fits, but using a continuous diffusion model.

Figure 10

Experimental estimates of the West-scaling response function $F_e(y,Q)$ in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture sample No. (5) (empty black circles), and their best multi-Gaussian fits (full red lines). Only the highest momentum transfer values (i.e., 3.6 ${\AA }^{-1}\le Q\le 4.6{\AA }^{-1}$) have been considered for such an analysis. Spectra, derived from data plotted in Fig. 3, have been vertically shifted (0.1 Å) for graphic reasons.

Figure 11

Non-Gaussian components of the self-part of the para-hydrogen center-of-mass dynamic structure factor ${\delta }_{n\text{GA}}{S}_{s,\text{c.m.}}(Q,E)$, in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture sample No. (4) (i.e. $c\left[{\mathrm{H}}_2\right]=24\%$, see Table 1) shown for some selected $Q$ values ranging from 1.4 ${\AA }^{-1}$ (black full line, bottom) to 4.2 ${\AA }^{-1}$ (full orange line, top). Spectra have been vertically shifted (0.002 ${\mathrm{meV}}^{-1}$) for graphic reasons.

Figure 12

Non-Gaussian components of the self-part of the para-hydrogen center-of-mass dynamic structure factor ${\delta }_{n\text{GA}}{S}_{s,\text{c.m.}}(Q,E)$, in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture sample No. (5) (i.e., $c\left[{\mathrm{H}}_2\right]=50\%$, see Table 1) shown for some selected $Q$ values ranging from 1.4 ${\AA }^{-1}$ (black full line, bottom) to 4.2 ${\AA }^{-1}$ (full orange line, top). Spectra have been vertically shifted (0.002 ${\mathrm{meV}}^{-1}$) for graphic reasons.

Figure 13

Main panel: non-Gaussian components of the self-part of the para-hydrogen center-of-mass dynamic structure factor ${\delta }_{n\text{GA}}{S}_{s,\text{c.m.}}(Q=2.2{\AA }^{-1},E)$, in the ${\mathrm{H}}_2-{\mathrm{D}}_2$ mixture samples No. (4) (thin blue histogram) and (5) (thick red histogram), together with the corresponding Fourier transforms of $exp[-{\sigma }^2\left(0\right)t^2/\left(2{\hslash }^2\right)-Q^2{\gamma }_1\left(t\right)]Q^4{\gamma }_1^2\left(t\right)/2$, plotted as thin black dashes and thick green dashes, respectively. The latter two (simulated) spectra have been scaled for graphic reasons. Inset: modulus (symbols plus line) of the first non-Gaussian coefficient ${\alpha }_2\left(t\right)$ for samples Nos. (4) (full blue circles) and (5) (empty red circles). These two full lines are only a guide for the eye, while the dotted black line represents $|{\alpha }_2\left(t\right)|$ for pure para-hydrogen elaborated from data in Ref. [7].