We provide a concurrent measurement of the hydrogen and oxygen nuclear kinetic energies in the water molecule across melting at 270 K in the solid phase and 276 K in the liquid phase. Experimental values are obtained by analyzing the neutron Compton profiles of each atomic species in a deep inelastic neutron scattering experiment. The concurrent measurement of the atom kinetic energy of both hydrogen and oxygen allows the estimate of the total kinetic energy per molecule due to the motion of nuclei, specifically 35.3 ± 0.8 and 34.8 ± 0.8 kJ/mol for the solid and liquid phases, respectively. Such a small difference supports results from ab initio simulations and phenomenological models from the literature on the mechanism of competing quantum effects across the phase change. Despite the experimental uncertainties, the results are consistent with the trend from state-of-the-art computer simulations, whereby the atom and molecule kinetic energies in the liquid phase would be slightly lower than in the solid phase. Moreover, the small change of nuclear kinetic energy across melting can be used to simplify the calculation of neutron-related environmental dose in complex locations, such as high altitude or polar neutron radiation research stations where liquid water and ice are both present: for neutron energies between hundreds of meV and tens of keV, the total scattering cross section per molecule in the two phases can be considered the same, with the macroscopic cross section only depending upon the density changes of water near the melting point.
I. INTRODUCTION
Quantum effects in water and water-related systems affect the properties of their condensed phases at the microscopic to macroscopic scales in isotope-specific ways. For this reason, a number of investigations have concentrated on the quantum effects on the lightest atom, hydrogen, in a number of materials. Over the past two decades, the topic has greatly benefited from joint approaches (see, e.g., Refs. 1–3) based on Deep Inelastic Neutron Scattering and non-classical computer simulations, such as semi-empirical models, or computer simulations based on Density Functional Theory or Path-Integral Molecular Dynamics (PIMD). These experimental and theoretical techniques can assess the kinetic energies of nuclei in molecular systems, which can be considered a thermometer of the magnitude of nuclear quantum effects, which have deep consequences, e.g., on biochemical reactions4,5 and imply changes in nominal isotope concentrations depending on physical and chemical variables.6,7
Among many molecular systems, water represents a particularly fascinating example. Beyond the importance of water for biology and life, deviations from the classical (non-quantum) dynamics of hydrogen and oxygen impact the modeling of neutron transport simulations for radiation protection, dose calculation, and the development of new neutron moderators at state-of-the-art neutron sources.8–11 Environmental radiation levels are affected by secondary neutrons, created by cosmic rays interacting with the atmosphere, whose energy spectrum is modified by moderation processes at ground levels.12,13 In such cases, the dose calculation related to fast and thermal neutrons in complex environments depends upon soil moisture and snow cover at mountain altitudes14 or at high-altitude Antarctic plateaus.15 In this context, the detailed knowledge of nuclear quantum effects in water just above and below the melting point can make radiation dose calculations more accurate in complex environments where ice and liquid water may coexist.
Despite the amount of published literature studying hydrogen dynamics, values for the kinetic energy [Nuclear Kinetic Energy (NKE)] of oxygen in water, either experimental or theoretical, are much less available. Ramirez and Herrero16 reported the value of the oxygen NKE from Path Integral Molecular Dynamics (PIMD), simulations of water in the gas, liquid, and solid phases as 48.5, 52.3, and 52.8 meV, respectively. Lin et al.,17 through an anisotropic quasiharmonic phonon calculation of open path integral Car-Parrinello molecular dynamics data for ice Ih at 269 K, estimated an oxygen kinetic energy of 56.4 meV. Ceriotti and Manoloupolos, performing imaginary-time PIMD with a carefully designed generalized Langevin equation, reported values of 54.6 meV and 58.1 for oxygen in H2O and D2O at 300 K, respectively,18 while Pinilla et al.,7 using PIMD approaches, obtained values of the oxygen NKE at the triple point of 52.7 and 52.4 meV for ice and liquid water, respectively. The first experimental values for the oxygen NKE in heavy water across melting19 provided values in the liquid and solid phases that were different by less than 1 meV (about 1.5% of the total), also supported by PIMD simulations. The relatively small difference (1 meV), despite the drastic changes in vibrational energies (about 12 meV shift for librations and 14 meV for stretching, in opposite directions20) and structure across the melting, was explained by a cancellation mechanism described as competing quantum effects. The mechanism of competing quantum effects was qualitatively explained in analogy to a two-level quantum system with an environment-dependent off-diagonal coupling, playing the role of hydrogen bonding. A phase transition, provoking a small change in the coupling, shifts the eigenvalues of the system by the same amount but in opposite directions.19 Upon melting, librations undergo a red-shift, while stretching increases in frequency. In a harmonic picture where the NKE can be evaluated through the average vibrational frequency weighted by the vibrational density of states, these shifts approximately cancel out.
The small difference in the oxygen NKE, , across melting was discussed in Ref. 16 using PIMD and suggested a slightly lower value in the liquid phase than in the solid, which was reflected also on the hydrogen kinetic energy, , and on the total kinetic energy per molecule, K. Similarly, using a semi-empirical method combining discrete molecular vibrations, librations, and translations in water, Finkelstein and Moreh21 showed an approximate continuity in the kinetic energy of both hydrogen and oxygen over a wide temperature range encompassing the liquid and solid forms. Pinilla et al.7 calculated both hydrogen and oxygen NKEs, reporting very similar values across melting, again with that in the solid phase being slightly higher than in the liquid for both nuclei. Larger differences in the oxygen NKE are expected and measured upon H-D substitution due to the reduced mass effect, as reported by Vos et al.22 for Ice Ih and its deuterated counterpart at a fixed temperature of 118 K using electron scattering at high momentum transfer.
Here, we provide the concurrent experimental determination of the NKEs of hydrogen and oxygen in liquid water and ice across melting using DINS. Combining the values from each nucleus in the molecule, we also provide the total nuclear kinetic energy per molecule, which can be directly compared with PIMD simulations from the literature.
II. EXPERIMENTAL
DINS experiments were performed at the VESUVIO spectrometer28,29 at the ISIS Neutron and Muon Source (UK).30 Two containers were used: a copper container, already used in Ref. 19, for the solid sample, and a TiZr container for the liquid sample. The containers were placed within the instrument’s closed circuit refrigerator and equipped with heaters for temperature control. The flat-faced containers were positioned with the large face perpendicular to the incident neutron beam, allowing the use of both front-scattering and back-scattering detectors on the instrument. In order to measure the energy transfer, the neutron energy of scattered neutrons was fixed using the 4.9 eV neutron capture resonance in 197Au foils at room temperature, while the neutron energy before the scattering event was calculated using the time-of-flight (t.o.f.) technique. DINS spectra for forward-scattering and back-scattering detectors were obtained using the foil-cycling31 and double-difference32 techniques, respectively. Examples of spectra from banks of detectors in the t.o.f. domain are reported in Fig. 1. In particular, the top panel reports back-scattering spectra featuring the signal from oxygen (left peak) partially isolated from the signal due to the container (right peak). On the other hand, the bottom panel, corresponding to front-scattering detectors, features a large signal due to neutron scattering from hydrogen (left peak) and an additional signal (right composite peak) corresponding to the overlap of NCPs from oxygen and the elements composing the container. The intensity and position of the peak from the container play a crucial role in the analysis of the oxygen NCP. Copper, used for the solid sample, has a bound scattering cross section of 8.03 b, while titanium and zirconium have bound scattering cross sections of 4.35 and 6.36 b, respectively.33 The higher scattering power of the copper container compared to the TiZr one can be easily appreciated in its higher relative intensity compared to the signal from water (hydrogen in forward scattering and oxygen in backward scattering, i.e., top and bottom panels in Fig. 1). However, while less intense, the composite peak of the TiZr container has an increased overlap with the oxygen NCP compared to copper, with titanium having a lower atomic mass than copper.
Raw time-of-flight spectra for ice in a copper container at 270 K (red crosses) and liquid water in a titanium–zirconium container at 276 K (green circles). The top panel corresponds to the sum of all backward scattering detectors, while the bottom panel corresponds to the sum of signals from forward-scattering detectors from 142 to 170 (scattering angles between 35° and 60°).
Raw time-of-flight spectra for ice in a copper container at 270 K (red crosses) and liquid water in a titanium–zirconium container at 276 K (green circles). The top panel corresponds to the sum of all backward scattering detectors, while the bottom panel corresponds to the sum of signals from forward-scattering detectors from 142 to 170 (scattering angles between 35° and 60°).
DINS spectra were analyzed separately for the forward and backward scattering detectors in order to isolate the hydrogen and oxygen NCPs, respectively. In both cases, the data were corrected for the multiple scattering contribution using a Monte Carlo approach,34 and the forward scattering detectors were corrected for the sample-dependent gamma background.31 The entire data reduction process followed the procedure described in Ref. 35.
III. RESULTS AND DISCUSSION
The corrected oxygen (top) and hydrogen (bottom) NCPs in ice at 270 K (left) and in liquid water at 276 K (right). Experimental data are reported as circles with error bars, and the best fit as red solid lines.
The corrected oxygen (top) and hydrogen (bottom) NCPs in ice at 270 K (left) and in liquid water at 276 K (right). Experimental data are reported as circles with error bars, and the best fit as red solid lines.
Values of the total nuclear kinetic energy of hydrogen and oxygen in H2O and deuterium and oxygen in D2O for liquid water and ice at temperatures near the melting point. The value of the total kinetic energy per molecule, K, is also reported.
H2O . | Phase . | T (K) . | (meV) . | (meV) . | K (kJ/mol) . | Reference . |
---|---|---|---|---|---|---|
DINS | Ice Ih | 270 | 157 ± 2 | 52 ± 5 | 35.3 ± 0.8 | a |
INS | Ice Ih | 271 | 153.7 ± 2 | 20 | ||
DINS | Ice Ih | 271 | 157 ± 2 | ⋯ | 39 | |
PIMD/NN | Ice | 273 | 155.5 | 52.7 | 35.1 | 40 |
PIMD/NN | Liquid | 273 | 154.7 | 52.5 | 34.9 | 40 |
DINS | Liquid | 276 | 155 ± 2 | 51 ± 5 | 34.8 ± 0.8 | a |
INS | Liquid | 276 | 152 ± 2 | ⋯ | 41 | |
SE | Liquid | 276 | 154 | ⋯ | 21 | |
DINS | Liquid | 300 | 146 ± 3 | ⋯ | 39 |
H2O . | Phase . | T (K) . | (meV) . | (meV) . | K (kJ/mol) . | Reference . |
---|---|---|---|---|---|---|
DINS | Ice Ih | 270 | 157 ± 2 | 52 ± 5 | 35.3 ± 0.8 | a |
INS | Ice Ih | 271 | 153.7 ± 2 | 20 | ||
DINS | Ice Ih | 271 | 157 ± 2 | ⋯ | 39 | |
PIMD/NN | Ice | 273 | 155.5 | 52.7 | 35.1 | 40 |
PIMD/NN | Liquid | 273 | 154.7 | 52.5 | 34.9 | 40 |
DINS | Liquid | 276 | 155 ± 2 | 51 ± 5 | 34.8 ± 0.8 | a |
INS | Liquid | 276 | 152 ± 2 | ⋯ | 41 | |
SE | Liquid | 276 | 154 | ⋯ | 21 | |
DINS | Liquid | 300 | 146 ± 3 | ⋯ | 39 |
D2O . | Phase . | T (K) . | (meV) . | (meV) . | K (kJ/mol) . | Reference . |
---|---|---|---|---|---|---|
DINS | Ice Ih | 274 | 108 ± 2 | 60 ± 4 | 26.6 ± 0.8 | 19 |
PIMD | Ice Ih | 274 | 108.3 | 55.7 | 26.3 | 19 |
DINS | Liquid | 280 | 112 ± 2 | 61 ± 3 | 27.5 ± 0.8 | 19 |
PIMD | Liquid | 280 | 108.7 | 55.6 | 26.3 | 19 |
D2O . | Phase . | T (K) . | (meV) . | (meV) . | K (kJ/mol) . | Reference . |
---|---|---|---|---|---|---|
DINS | Ice Ih | 274 | 108 ± 2 | 60 ± 4 | 26.6 ± 0.8 | 19 |
PIMD | Ice Ih | 274 | 108.3 | 55.7 | 26.3 | 19 |
DINS | Liquid | 280 | 112 ± 2 | 61 ± 3 | 27.5 ± 0.8 | 19 |
PIMD | Liquid | 280 | 108.7 | 55.6 | 26.3 | 19 |
Present work.
From the experimental point of view, the concurrent measurement of hydrogen and oxygen in a DINS experiment is a complex task because of the large difference in their scattering cross sections, by a factor of almost 20. In fact, the double-differential scattering cross section in a DINS experiment is weighted, in the incoherent approximation, by the total bound scattering cross sections that, for hydrogen and oxygen, are 82.03 and 4.232 b, respectively.33 The high scattering cross section of hydrogen requires only small amounts of samples in order to obtain a good statistics in a DINS experiment. On the other hand, larger quantities of samples are requested to measure the oxygen NCP. However, larger samples also imply larger contributions from multiple scatterings, which increase the complexity of its analysis. For this reason, the concurrent measurement of D and O in heavy water, where the difference in scattering cross sections is less than 2, was already achieved a decade ago.19 By comparing the top and bottom panels of Fig. 2, one can appreciate how, for a given quantity of sample, there is a marked difference in the quality of the experimental data, with the error bars in the hydrogen NCP much smaller than those in the oxygen one.
Experimental oxygen NKE in light water across the melting point in the solid (red circle) and liquid (green cross). The results from PIMD simulations in Ref. 7 for ice and liquid water at 273 K are reported as filled red and empty green squares, respectively. The prediction from the semi-empirical (SE) harmonic model is reported as a solid black line, and the Maxwell–Boltzmann classical prediction is reported as a dashed black line.
Experimental oxygen NKE in light water across the melting point in the solid (red circle) and liquid (green cross). The results from PIMD simulations in Ref. 7 for ice and liquid water at 273 K are reported as filled red and empty green squares, respectively. The prediction from the semi-empirical (SE) harmonic model is reported as a solid black line, and the Maxwell–Boltzmann classical prediction is reported as a dashed black line.
Experimental total kinetic energy in light water across the melting point in the solid (red circle) and liquid (green cross). The results from PIMD simulations in Ref. 40 for ice and liquid water at 273 K are reported as filled and empty red triangles, respectively. Finally, the temperature dependencies of K for ice and liquid water from Ref. 16 are reported as filled red and empty green diamonds.
Experimental total kinetic energy in light water across the melting point in the solid (red circle) and liquid (green cross). The results from PIMD simulations in Ref. 40 for ice and liquid water at 273 K are reported as filled and empty red triangles, respectively. Finally, the temperature dependencies of K for ice and liquid water from Ref. 16 are reported as filled red and empty green diamonds.
A similar small discrepancy in the total kinetic energy in the case of solid H2O can be appreciated in Fig. 5, where the DINS data from ice at 270 K are compared with the results from PIMD simulations from Ref. 44. In the same reference, isotope nuclear quantum effects were discussed by calculating the value of K for heavy ice as well, which is reported in Fig. 5 and compared with the corresponding experimental value of K from Ref. 19 (using the values of and reported in Table I). As opposed to the slight difference in the case of light ice, in the case of heavy ice, the experimental data are in good agreement, within the experimental uncertainties, with the results from Ref. 16.
The total kinetic energy per molecule obtained from DINS experiments for light ice (this work) and heavy ice (Ref. 19) are reported as red and blue circles with error bars. They are compared with the results from PIMD simulations from Ref. 44 for light ice (red diamonds) and heavy ice (blue diamonds).
The total kinetic energy per molecule obtained from DINS experiments for light ice (this work) and heavy ice (Ref. 19) are reported as red and blue circles with error bars. They are compared with the results from PIMD simulations from Ref. 44 for light ice (red diamonds) and heavy ice (blue diamonds).
Experimental total cross section per water molecule (red markers and error bars) compared with the short collision time prediction (green line).
Experimental total cross section per water molecule (red markers and error bars) compared with the short collision time prediction (green line).
IV. CONCLUSION
We have provided a concurrent experimental determination of the oxygen and hydrogen nuclear kinetic energies in water across melting in the ice and liquid water phases at 270 and 276 K, respectively. The experimental results support the idea of competing quantum effects, which leave the average kinetic energy of each nucleus in the water molecule approximately unchanged. Qualitatively, our experimental results are consistent with the slight decrease of hydrogen, oxygen, and total kinetic energies going from the solid to the liquid phase, as reported by state-of-the-art computer simulations.
The concurrent determination of both oxygen and hydrogen average nuclear kinetic energies allows us to provide a consistent approximation to the total scattering cross section of liquid water and ice between hundreds of meV and a few keV, using the short collision time approximation. Such simplification can be used to improve estimates of the fraction of ground albedo neutrons in environmental models and the resulting relevance of neutron attenuation during particle transport mechanisms55 in complex settings where liquid water and ice coexist across saturated atmospheric layers and mixed water/snow areas, such as high-altitude mountains56 and polar54,57 research stations.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support of the ISIS@MACH ITALIA Research Infrastructure, the hub of ISIS Neutron and Muon Source (UK), (MUR official registry No. U. 0008642.28-05-2020—April 16, 2020). The financial support from the Consiglio Nazionale delle Ricerche within the CNR-STFC Grant Agreement (No. 2021-2027) concerning collaboration in scientific research at the ISIS (UK) of STFC is gratefully acknowledged. The STFC Rutherford Appleton Laboratory is acknowledged for its access to neutron beam facilities.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Giovanni Romanelli: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Carla Andreani: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Alessio Bocedi: Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Roberto Senesi: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.