Phonon Dispersion of Ice under Pressure

We report measurements of the phonon dispersion of ice $\mathrm{I}h$ under hydrostatic pressure up to 0.5 GPa, at 140 K, using inelastic neutron scattering. They reveal a pronounced softening of various low-energy modes, in particular, those of the transverse acoustic phonon branch in the [100] direction and polarization in the hexagonal plane. We demonstrate with the aid of a lattice dynamical model that these anomalous features in the phonon dispersion are at the origin of the negative thermal expansion (NTE) coefficient in ice below 60 K. Moreover, extrapolation to higher pressures shows that the mode frequencies responsible for the NTE approach zero at $\sim 2.5\mathrm{G}\mathrm{P}\mathrm{a}$, which explains the known pressure-induced amorphization (PIA) in ice. These results give the first clear experimental evidence that PIA in ice is due to a lattice instability, i.e., mechanical melting.

Figure 1

Neutron spectra of the lowest basal TA phonon in ice $\mathrm{I}h$ ($T=140\mathrm{K}$) at $P=0.05$ and $0.5\mathrm{G}\mathrm{P}\mathrm{a}$ (lower and upper curves, respectively) [measured at $(-1+q,2,0)$ with $q=1/2$, at $(-1+2q,2-q,0)$ with $q=1/3$ and at $(-1+2q,2-q,0)$ with $q=0.1$, respectively].

Figure 2

Top: Measured phonon dispersion of ice $\mathrm{I}h$ ($T=140\mathrm{K}$) at $P=0.05\mathrm{G}\mathrm{P}\mathrm{a}$ (circles) and $P=0.5\mathrm{G}\mathrm{P}\mathrm{a}$ (stars). The solid and dashed lines are fits using the lattice dynamical model discussed in the text. Bottom: Mode Grüneisen parameters ${\gamma }_s$ of the transverse and longitudinal acoustic branches (thick and thin lines, respectively) resulting from the evaluation of the model at $P=0.05$ and $0.05\mathrm{G}\mathrm{P}\mathrm{a}$.

Figure 3

Phonon density of states (PDOS) ($T=140\mathrm{K}$) at $P=0.05$ and $0.5\mathrm{G}\mathrm{P}\mathrm{a}$. Inset: linear thermal expansion coefficient $\alpha$ of ${\mathrm{D}}_2\mathrm{O}$ ice $\mathrm{I}h$ reconstructed (solid line) from the measured phonon dispersion in comparison with literature data (dashed lines).

Figure 4

Born's stability criteria as a function of pressure. $B_2$ and $B_3$ are almost simultaneously violated at $P_c\sim 2.5\mathrm{G}\mathrm{P}\mathrm{a}$.