Lattice dynamics and molecular rotations in solid hydrogen deuteride: Inelastic neutron scattering study

In the present paper we report inelastic neutron scattering measurements on solid low-pressure hydrogen deuteride at three different temperatures (between 4.5 and 15.6 K) using the time-of-flight spectrometers BRISP at ILL (France) and TOSCA-II at ISIS, RAL (UK). The measured double-differential cross sections give access to the proton component of the HD self-inelastic structure factor. Processed BRISP data were employed to verify the applicability of the generalized Young and Koppel model to solid HD in our kinematic range and to obtain the mean-square displacement of the molecular centers of mass. In addition, a large broadening of the first two rotational peaks was observed. A reasonable result for the density of phonon states from TOSCA-II data has been obtained, although a rigorous extraction was not possible, due to the overlap among the various spectral components. The intensity loss in the extracted density of phonon states was interpreted as the effect the phonon-roton resonance in solid hydrogen deuteride. Finally the two Bose-corrected moments of the HD phonon spectrum, related to the molecular mean-square displacement and mean kinetic energy, were simulated through a path integral Monte Carlo code. The former quantity was compared to the mentioned experimental estimates.

Figure 1

Kinematic trajectories in the energy-momentum-transfer plane explored by the two neutron spectrometers employed. All the trajectories for the BRISP detectors are comprised between the two limiting curves (reported as full line), while the TOSCA trajectories are separately plotted for forwardscattering (dashed line) and backscattering (dotted line) detectors.

Figure 2

Examples of solid-HD fixed-angle spectra from BRISP (a) and TOSCA-II (b) plotted as histograms. In panel (a) sample-(1) spectra (as in Table ) are reported for angles of $1.01°$, $7.75°$, and $13.37°$ (bottom to top); while in panel (b) sample-(3) spectra (as in Table ) are reported for forwardscattering and backscattering detectors (bottom to top). In both panels curves are vertically shifted for graphic reasons.

Figure 3

Area of the rotational peaks in solid HD measured on BRISP as a function of the corresponding momentum-transfer value. Panel (a) shows $A_1$, the area of the elastic line (i.e., exhibiting final rotational quantum number $j^{\prime }=0$), while panel (b) reports $A_3$, the area of the rotational inelastic peak with $j^{\prime }=1$. Full squares stand for sample (1), while empty circles for sample (2), as explained in Table .

Figure 4

Single-scattering H part, $I_{1,\text{H}}\left[E,Q\left(E,{\theta }_F\right)\right]$, of the TOSCA-II neutron spectrum measured in forwardscattering at $T=10.6\text{K}$ (histogram), together with its simulated multiphonon component (dotted). The purely rotational transition $j=0\to j^{\prime }=1$ is visible around $E=11\text{meV}$, while two replicas of the density of phonon states show their maxima at about 5 and 16 meV, respectively.

Figure 5

Spectra of the $j=0\to j^{\prime }=2$ transition in solid HD and para-${\text{H}}_2$ obtained from different techniques: present HD neutron scattering (histogram), HD Raman scattering (Ref. 36) (dashed line), and para-${\text{H}}_2$ Raman scattering (Ref. 40) (squares plus solid line). All the three curves have been scaled and vertically shifted for graphic reasons, while the para-${\text{H}}_2$ one has also been horizontally shifted to account for the difference between the HD and the ${\text{H}}_2$ rotational constants.

Figure 6

Elastic (upper data) and inelastic (lower data) peak areas from solid HD at $T=4.53\text{K}$ [i.e., sample (1) in Table ] as a function of the momentum transfer. Solid dots with error bars are obtained from BRISP experimental data, while dashed line is the result of the generalized Young and Koppel model (see main text for details).

Figure 7

Radial pair distribution functions $r\left[g\left(r\right)-1\right]$ derived from PIMC simulations (empty symbols) compared to the same quantities (lines) obtained through a simple uncorrelated lattice calculation. Circles and full line stand for solid HD at $T=15.61\text{K}$, while lozenges and dashed line for the same sample at $T=10.6\text{K}$ and have been vertically shifted for graphic reasons.

Figure 8

One-phonon single-scattering H part, ${\tilde{I}}_{1,\text{H}}\left[E,Q\left(E,{\theta }_F\right)\right]$, of the TOSCA-II neutron spectrum measured in forwardscattering at $T=10.6\text{K}$ (empty circles with error/s), together with the same spectrum (full line) reconstructed from the density of phonon states extracted from the first part of the experimental data (full line in the inset). The purely rotational line centered at about 11 meV is not included in the reconstructed spectrum. The preliminary model of density of phonon states, used to evaluated the multiphonon component, is also reported in the inset as empty circles.

Figure 9

Phonon spectra in solid HD obtained from different techniques: present neutron scattering in connection with the no rotational transition (dotted line histogram) and with the $j=0\to j^{\prime }=1$ rotational transition (full line histogram), Raman scattering (Ref. 37) (full squares), and infrared absorption (Ref. 51) (empty circles). All the four curves have been scaled for graphic reasons.