X-ray inelastic scattering reveals phonon dispersion in high-pressure ice phases

Ice, the most fundamental hydrogen-bonded solid, reveals a remarkable structural complexity under pressure. Among the approximately 20 known ice phases, two high-pressure phases, ice VII and ice VIII (see Figure 1), emerge as noteworthy for several reasons. Firstly, their extraordinary stability under compression – these phases are able to endure pressures exceeding 50 GPa – makes them the two dominant forms in the phase diagram of water. Secondly, they demonstrate comparatively simple structures; and thirdly, ice VII is the fully hydrogen-disordered form of ice VIII. It is plausible that ice VII and VIII exist within the interiors of certain exoplanets. Given the fundamental role of water and ice, the dynamical properties of various forms of solid water appear to be of major academic and practical interest.

One way to investigate ice is by measuring its phonon dispersion. Phonons are collective excitations of molecular vibrations that govern a solid’s capacity to store heat, its thermal conductivity and, for most metallic systems, conventional superconductivity. After the crystal structure, phonons are therefore one of the most important properties characterising solids. However, there is very little experimental data on the full phonon dispersion of ice phases. Determining the dispersion curves of the ice VII and VIII phases has proven challenging until now, due to the difficulty of obtaining adequately sized single crystals with useful orientation. For ice VIII, single-crystal growth is unattainable as this phase is far from the melting line. Conversely, with ice VII, single crystals have to be grown at high temperatures in a high-pressure diamond anvil cell, which only yields microscopic crystals with arbitrary orientations.

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Fig. 1: Atomic structure of (a) ice VII and (b) ice VIII, with oxygen in red and hydrogen in white. In ice VII, the orientation of the molecules is disordered, resulting in four hydrogen sites around each oxygen, each occupied, on average, at 50%. Ice VIII is the ordered version of phase VII.

This study was undertaken to investigate phonon dispersion in ice VII and ice VIII, employing an innovative approach to address the aforementioned challenges. Recognising that all high-pressure phases can be decompressed to ambient pressure if the temperature remains below approximately 100 K, macroscopic samples of the ice phases were created under high pressure using a Paris-Edinburgh press. They were then returned to ambient pressure at 77 K, resulting in powder samples of approximately 100 mm³. Inelastic X-ray scattering (IXS) measurements of these samples were conducted at beamline ID28, providing spectra closely related to the phonon density of states (PhDOS, defined as phonon dispersion integrated over the first Brillouin zone of a crystal). These findings were combined with ab-initio calculations to determine the phonon dispersion curves of both phases.

Understanding the phonon dispersion of ice VIII is relatively straightforward as this phase exhibits hydrogen order, meaning the positions of hydrogen atoms remain fixed. In contrast, understanding the phonon dispersion of ice VII poses a greater challenge due to its disorder in the molecular orientations. Its recognised structure (space group Pn3m with oxygen at 0,0,0 and hydrogen at 0.42,0.42,0.42) is actually an average of instantaneous configurations, where each oxygen atom is surrounded by four hydrogen positions, each occupied on average only 50% of the time (see Figure 1). This proton-disordered system has been modelled theoretically by considering all distinct ordered atomic configurations compatible with a 2x2x2 ice VII supercell containing 16 water molecules, adhering to the Bernal-Fowler ice rules, totalling 52 configurations. Within this framework, phonon dispersion and spectra have been calculated as an average over these hypothetical ordered configurations, weighted by their statistical probability.

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Fig. 2: Phonon dispersion of ice VII, obtained by a combination of X-ray measurements and ab-initio calculations (coloured bands). Note the existence of gaps (shown by white arrows) in the [111] direction (upper panel) and between Γ and H (lower panel). The white points correspond to measurements on a single crystal. The white lines correspond to a classic Born – von Kármán model.

The data indicate that the dispersion of acoustic phonon branches in ice VII – those with the lowest energies (Figure 2) – exhibits typical characteristics of a regular body-centred cubic (bcc) solid. This implies that the dispersion is governed by the bcc symmetry of the oxygen sublattice, with the presence of hydrogen atoms (comprising two-thirds of all atoms) appearing to “average out”. However, numerical calculations reveal a surprising outcome: the prediction of gaps at certain locations in the Brillouin zone. These gaps would not be anticipated based solely on the crystallographic structure depicted in Figure 1. They arise due to the fact that the instantaneous local unit cells are larger than the average structure derived from diffraction. To validate this, previously unpublished data on a single crystal of ice VII obtained at ID28 in 1999 were retrieved. A detailed analysis of these constant-q scans across certain phonon branches appears to confirm the presence of these gaps (illustrated by white dots in Figure 2), despite the limited resolution and intensity of these data.

In summary, the significance of this study lies in the fact that the two phases – ice VII and ice VIII – are complementary counterparts, with ice VII representing the fully disordered form of ice VIII (Figure 1). This research has provided an opportunity to investigate the impact of orientational disorder on phonon dispersion within a relatively straightforward system like ice VII, which typically contains only two molecules per unit cell, on average. Additionally, this work corroborates recent findings from single-crystal neutron diffraction experiments conducted at the nearby Institut Laue-Langevin [2]. These findings reveal significant differences between the atomic distribution of hydrogen and the local structure of ice VII compared to the time and space-averaged standard model assumed in the Pn3m space group.

Principal publication and authors
Phonon dispersion and proton disorder of ice VII and VIII, G. Radtke (a), S. Klotz (a), M. Lazzeri (a), P. Loubeyre (b), M. Krisch (c), A. Bossak (c), Phys. Rev. Lett. 132, 056102 (2024); https://doi.org/10.1103/PhysRevLett.132.056102
(a) Sorbonne Université, IMPMC, Paris (France)
(b) CEA & Université Paris-Saclay, Bruyères le Châtel (France)
(c) ESRF

References
[1] V.F. Petrenko & R.W. Whitworth, Physics of Ice, Oxford University Press (Oxford, 1999).
[2] K. Yamashita et al., PNAS 119, e2208717119 (2022).