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.

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, EKO, 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, EKH, 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.

The DINS technique,1,23 also referred to as Neutron Compton Scattering (NCS), allows mass-selective spectroscopy experiments with neutrons, whereby the signals from isotopes with different masses are treated independently owing to the Impulse Approximation (IA).24–26 Within the IA, the neutron scattering intensity is centered at the atomic recoil energy, whereas the dynamic structure factor S(Q, ω) can be expressed as the linear superposition of a Neutron Compton Profile (NCP), J(y), from each isotope j in a sample
(1)
where ℏω and Q are the neutron energy and wavevector transfer during the scattering process, Ij is a weighting factor considering the stoichiometry and bound scattering cross section of isotope j, Mj is the mass of the struck atom and,
(2)
is the isotope-specific West scaling variable.27 The dependence of the West scaling variable upon isotope mass allows the separation of the neutron scattering intensity into mass-separated isotope-specific peaks. The NCP is related to the nuclear momentum distribution, a probability density function measuring the momentum spread of a particle due to its spatial localization in a potential. The second moment of the nuclear momentum distribution is proportional to the nuclear kinetic energy of an atom which, for a light-weight quantum particle in a confining potential, is substantially higher than the classical prediction from a Maxwell–Boltzmann theory of a gas of non-interacting particles, namely 3kBT/2, where kB is the Boltzmann constant and T is the thermodynamic temperature of the system.

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.

FIG. 1.

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°).

FIG. 1.

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°).

Close modal

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.

Figure 2 shows the oxygen (top) and hydrogen (bottom) NCPs in their West-scaling domains from ice at 270 K (left) and liquid water at 276 K. In particular, we report the experimental Compton profiles, F̄(y,Q), corresponding to the sum over all backward scattering (oxygen) and forward scattering (hydrogen) detectors. Moreover, as the hydrogen NCP is symmetrized as a final step of the data correction,35 only the data for yH ≥ 0 Å−1 are reported. The best fit using a Gauss–Hermite model,37 including the convolution, for each detector, with the experimental resolution,36 is also reported for both nuclei as a red solid line. For oxygen (both liquid and ice) and for hydrogen in the liquid form, the Gauss–Hermite expansion was truncated at the leading Gaussian term, while for hydrogen in ice, the first Hermite polynomial (n = 4) was also used, confirming the higher anisotropy of the hydrogen momentum distribution in the solid phase than in the liquid one.38 In the case of the oxygen NCP, we report in the figure the yO range where the data were analyzed. The fitted values of the NCP width, expressed as the average NKE of hydrogen and oxygen, are reported in Table I together with other values reported in the literature. It is worth noting that the assessment of nuclear and total kinetic energies is approached by different authors in several ways and, consequently, using different units of measurement (meV, kJ/mol, and kcal/mol). In order to facilitate the comparison of our results with the theoretical values already published, we have adopted in each comparison the units used in the reference theoretical work. A conversion rule to change from one unit to another is as follows:
(3)
FIG. 2.

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.

FIG. 2.

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.

Close modal
TABLE I.

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.

H2OPhaseT (K)EKH (meV)EKO (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  
H2OPhaseT (K)EKH (meV)EKO (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  
D2OPhaseT (K)EKD (meV)EKO (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  
D2OPhaseT (K)EKD (meV)EKO (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  
a

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.

The experimental values for the oxygen NKE are in excellent agreement with those previously reported in Ref. 7 (see Fig. 3) and with those from Ref. 40 (see Table I), although the 0.2 meV difference predicted by the PIMD/NN simulations is too subtle to be observed in the experimental data. Our data also confirm the predictions from semi-empirical models,21 which imply a mild dependence of the kinetic energy on the phase change. The semi-empirical model is based on the assumption that within the harmonic approximation, the NKE can be predicted as half of the total energy related to a set of harmonic oscillators representing internal vibrations (discrete in nature) and external and lattice modes which, for water clusters linked by hydrogen bonding, correspond to hindered translations and librational modes. In this framework, and by assuming a discrete vibrational density of states, one has3,21,42,43
(4)
where the summation runs over the discrete vibrations, both internal (stretching and bending) and external (translations and librations), each with a frequency ωα. The parameters fα describe the energy uptake of oxygen from each harmonic oscillation. As hydrogen has the lowest mass in the molecule, it takes most of the energy from the internal vibrations, and the same is true for librational modes, as oxygen is near the molecule’s center of mass or axes of rotation. Of the parameters fα, the one corresponding to translations is the largest, as it is defined by the ratio of the oxygen mass to the mass of the molecule, ft = MO/(2MH + MO), and it needs to be considered three times for each spatial dimension. The second consideration has to do with the separation in energy of the different modes: internal modes are found at several hundreds of meV, libration modes in the range 70–80 meV, and translation modes at about 20–30 meV (see, e.g., Refs. 20, 21, and 42). In the temperature range around the melting point of water, the internal modes only occupy the vibrational ground state, so they are just contributing with a zero-point energy (ZPE) constant term, while the contribution from low-lying translational modes can be approximated as the one from free translations (knowing that coth(x) → 1/x as x → 0). Librational modes are in an intermediate regime because of the partial occupation of the first excited state. However, considering the overall contribution of librations to the oxygen NKE (with fα ≃ 5%) being about 3 meV out of about 50 meV, one can approximate the previous equation as
(5)
In this context, it was noted21 how the temperature dependence of oxygen NKE was affected very little by the phase change, although it implied a change in the definition of both fα and ωα. This is true because the competing quantum effects effectively cancel out the ZPE contribution in Eq. (5). The approximately linear dependence of EKO on T can be appreciated in Fig. 3 by comparison with the classical Maxwell–Boltzmann prediction: the semi-empirical model provides a NKE increased by about 15 meV compared to the classical limit, with a slope slightly decreased by a factor ft.
FIG. 3.

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.

FIG. 3.

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.

Close modal
The concurrent measurement of both the hydrogen and oxygen NKEs allowed the estimation of the total kinetic energy per molecule, K, which is sometimes reported in the literature.16,44 Figure 4 shows the experimental values from DINS as compared with the results from PIMD simulations in Ref. 40, obtained by combining the provided values of EKH and EKO, namely
(6)
for ice and liquid water at 273 K. When looking at the total kinetic energy, the small difference between the ice (larger) and liquid (smaller) values becomes more apparent, as both hydrogen and oxygen NKEs decrease upon melting according to the PIMD simulations, both from Refs. 40 and 16. The experimental results in Fig. 4 seem to suggest the same trend, although the difference is comparable to the experimental uncertainties. While the agreement with the results from Ref. 40 extends on the absolute scale, the values from Ref. 16 are lower by about 1 kJ/mol (about 3%).
FIG. 4.

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  (K=2EKH+EKO) 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.

FIG. 4.

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  (K=2EKH+EKO) 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.

Close modal

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 EKD and EKO 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.

FIG. 5.

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).

FIG. 5.

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).

Close modal
Finally, the values of the NKE for hydrogen and oxygen can be used to approximate the energy-dependent total scattering cross sections at hot-to-epithermal neutron energies. This can be performed in the formalism of the Short-Collision Time (SCT)45 approximation, whereby the neutron scattering from free nuclei is corrected by an effective temperature related to the NKE (3kBTeff/2 = ⟨EK⟩). The scattering cross section for each element becomes
(7)
with E the incident neutron energy and Aj the atomic mass number in units of the neutron mass, σb,j its bound scattering cross section.33 We note here the difference between the bound scattering cross section, a constant parameter not dependent upon the neutron energy and related to the neutron scattering length, and the total scattering cross section, σ(E), the result of the integration of the dynamic structure factor (or the double-differential scattering cross section) over all scattering angles and final energies of the neutron, which is a function of the incident neutron energy. The total scattering cross section of the molecule is—under the same assumptions of Eq. (1),
(8)
where Nj represents the stoichiometry of the molecule, and σf,j=σb,jAj2/(Aj+1)2 is the constant free-scattering cross-section parameter. The right-hand side of Eq. (8) only holds for energies above a few eV. The ability of Eq. (8) to reproduce the experimental data can be appreciated in Fig. 6, where the SCT approximation is exemplified in the case of ice at 270 K, using the DINS parameters obtained in this study to evaluate the effective temperatures affecting hydrogen and oxygen. The experimental data, collected on VESUVIO concurrently with the DINS measurement and making use of the capabilities for neutron transmission measurements,46 can be approximated by the SCT already at a few hundreds of meV, where one could expect more structured contributions from stretching modes in the multi-phonon-expansion approximation to the total scattering cross section.47,48 A similar agreement in the same energy range was also found for hydrogenated alcohols.49 As the NKEs for both oxygen and hydrogen across melting are found to be the same within the experimental error bars, one can safely assume that the total scattering cross section per molecule of liquid water and ice across melting can be considered equal, with the macroscopic cross section (i.e., the microscopic cross section multiplied by the number density of scattering centers) only affected by the temperature- and phase-dependent variations in the bulk density. It is also worth noting that the hydrogen NKE of pure light water across melting, presented here, is more similar in value to that of seawater at high pressure than to that of room-temperature water at ambient pressure.50 These findings are relevant in the context of neutron transport modeling of the thermal scattering law in applications ranging from environmental field-scale soil moisture analysis for the quantification of biomass and snow51,52 to investigations of neutron albedo in space weather ground-level and altitude observations,53,54 where water hydrogen in soil, air, liquid water, and snow determines the amount of ground albedo neutrons in the sensitive energy ranges.
FIG. 6.

Experimental total cross section per water molecule (red markers and error bars) compared with the short collision time prediction (green line).

FIG. 6.

Experimental total cross section per water molecule (red markers and error bars) compared with the short collision time prediction (green line).

Close modal

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.

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.

The authors have no conflicts to disclose.

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).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
C.
Andreani
,
D.
Colognesi
,
J.
Mayers
,
G. F.
Reiter
, and
R.
Senesi
, “
Measurement of momentum distribution of lightatoms and molecules in condensed matter systems using inelastic neutron scattering
,”
Adv. Phys.
54
,
377
469
(
2005
).
2.
M.
Ceriotti
,
W.
Fang
,
P. G.
Kusalik
,
R. H.
McKenzie
,
A.
Michaelides
,
M. A.
Morales
, and
T. E.
Markland
, “
Nuclear quantum effects in water and aqueous systems: Experiment, theory, and current challenges
,”
Chem. Rev.
116
,
7529
7550
(
2016
).
3.
R.
Moreh
,
W. C.
Sellyey
,
D.
Sutton
, and
R.
Vodhanel
, “
Widths of the 6.92 and 7.12 MeV levels in 16O and the influence of the effective temperature
,”
Phys. Rev. C
31
,
2314
2316
(
1985
).
4.
R. K.
Allemann
and
N. S.
Scrutton
,
Quantum Tunnelling in Enzyme-Catalysed Reactions
(
Royal Society of Chemistry
,
2009
).
5.
G. R.
Fleming
,
G. D.
Scholes
, and
Y.-C.
Cheng
, “
Quantum effects in biology
,”
Procedia Chem.
3
,
38
57
(
2011
), 22nd Solvay Conference on Chemistry.
6.
T. E.
Markland
and
B.
Berne
, “
Unraveling quantum mechanical effects in water using isotopic fractionation
,”
Proc. Natl. Acad. Sci. U. S. A.
109
,
7988
7991
(
2012
).
7.
C.
Pinilla
,
M.
Blanchard
,
E.
Balan
,
G.
Ferlat
,
R.
Vuilleumier
, and
F.
Mauri
, “
Equilibrium fractionation of H and O isotopes in water from path integral molecular dynamics
,”
Geochim. Cosmochim. Acta
135
,
203
216
(
2014
).
8.
A.
Viñales
,
J.
Dawidowski
, and
J.
Márquez Damián
, “
Neutron total cross-section model for liquids and its application to light water
,”
Ann. Nucl. Energy
38
,
1687
1692
(
2011
).
9.
M.
Martellucci
,
G.
Romanelli
,
S.
Valeri
,
D.
Cottone
,
C.
Andreani
, and
R.
Senesi
, “
The neutron cross section of barite-enriched concrete for radioprotection shielding in the range 1 meV–1 keV
,”
Eur. Phys. J. Plus
136
,
259
(
2021
).
10.
G.
Noguere
,
J. P.
Scotta
,
S.
Xu
,
E.
Farhi
,
J.
Ollivier
,
Y.
Calzavarra
,
S.
Rols
,
M.
Koza
, and
J. I.
Marquez Damian
, “
Temperature-dependent dynamic structure factors for liquid water inferred from inelastic neutron scattering measurements
,”
J. Chem. Phys.
155
,
024502
(
2021
).
11.
J. I.
Marquez Damian
,
J.
Dawidowski
,
R. J.
Granada
,
F.
Cantargi
,
G.
Romanelli
,
C.
Helman
,
M.
Krzystyniak
,
G.
Skoro
,
D.
Roubtsov
, and
D.
Roubtsov
, “
Experimental validation of the temperature behavior of the ENDF/B-VIII.0 thermal scattering kernel for light water
,”
EPJ Web Conf.
239
,
14001
(
2020
).
12.
L.
Dorman
,
Cosmic Rays in the Earth’s Atmosphere and Underground
(
Springer Science & Business Media
,
2013
), Vol.
303
.
13.
M.
Eisenbud
and
T. F.
Gesell
,
Environmental Radioactivity from Natural, Industrial and Military Sources: From Natural, Industrial and Military Sources
(
Elsevier
,
1997
).
14.
T.
Brall
,
V.
Mares
,
R.
Bütikofer
, and
W.
Rühm
, “
Assessment of neutrons from secondary cosmic rays at mountain altitudes—Geant4 simulations of environmental parameters including soil moisture and snow cover
,”
Cryosphere
15
,
4769
4780
(
2021
).
15.
A.
Mishev
,
S.
Kodaira
,
H.
Kitamura
,
O.
Ploc
,
I.
Ambrožová
,
R.
Tolochek
,
I.
Kartsev
,
V.
Shurshakov
,
A.
Artamonov
, and
K.
Inozemtsev
, “
Radiation environment in high-altitude antarctic plateau: Recent measurements and model studies
,”
Sci. Total Environ.
890
,
164304
(
2023
).
16.
R.
Ramirez
and
C. P.
Herrero
, “
Kinetic energy of protons in ice Ih and water: A path integral study
,”
Phys. Rev. B
84
,
064130
(
2011
).
17.
L.
Lin
,
J. A.
Morrone
,
R.
Car
, and
M.
Parrinello
, “
Momentum distribution, vibrational dynamics, and the potential of mean force in ice
,”
Phys. Rev. B
83
,
220302
(
2011
).
18.
M.
Ceriotti
and
D. E.
Manolopoulos
, “
Efficient first-principles calculation of the quantum kinetic energy and momentum distribution of nuclei
,”
Phys. Rev. Lett.
109
,
100604
(
2012
).
19.
G.
Romanelli
,
M.
Ceriotti
,
D. E.
Manolopoulos
,
C.
Pantalei
,
R.
Senesi
, and
C.
Andreani
, “
Direct measurement of competing quantum effects on the kinetic energy of heavy water upon melting
,”
J. Phys. Chem. Lett.
4
,
3251
3256
(
2013
).
20.
C.
Andreani
,
G.
Romanelli
, and
R.
Senesi
, “
A combined INS and DINS study of proton quantum dynamics of ice and water across the triple point and in the supercritical phase
,”
Chem. Phys.
427
,
106
110
(
2013
).
21.
Y.
Finkelstein
and
R.
Moreh
, “
Temperature dependence of the proton kinetic energy in water between 5 and 673K
,”
Chem. Phys.
431-432
,
58
63
(
2014
).
22.
M.
Vos
,
E.
Weigold
, and
R.
Moreh
, “
Elastic electron scattering from water vapor and ice at high momentum transfer
,”
J. Chem. Phys.
138
,
044307
(
2013
).
23.
C.
Andreani
,
M.
Krzystyniak
,
G.
Romanelli
,
R.
Senesi
, and
F.
Fernandez-Alonso
, “
Electron-volt neutron spectroscopy: Beyond fundamental systems
,”
Adv. Phys.
66
,
1
73
(
2017
).
24.
G. K.
Ivanov
and
Y. S.
Sayasov
, “
Theory of the vibrational excitation of a molecule in the impulse approximation
,”
Sov. Phys.-Dokl.
9
,
171
(
1964
).
25.
V.
Sears
, “
Slow-neutron multiple scattering
,”
Adv. Phys.
24
,
1
45
(
1975
).
26.
H. A.
Gersch
,
L. J.
Rodriguez
, and
P. N.
Smith
, “
Corrections to the impulse approximation for high-energy neutron scattering from liquid helium
,”
Phys. Rev. A
5
,
1547
1558
(
1972
).
27.
G. B.
West
, “
Electron scattering from atoms, nuclei and nucleons
,”
Phys. Rep.
18
,
263
323
(
1975
).
28.
J.
Mayers
and
G.
Reiter
, “
The vesuvio electron volt neutron spectrometer
,”
Meas. Sci. Technol.
23
,
045902
(
2012
).
29.
G.
Romanelli
,
M.
Krzystyniak
,
R.
Senesi
,
D.
Raspino
,
J.
Boxall
,
D.
Pooley
,
S.
Moorby
,
E.
Schooneveld
,
N. J.
Rhodes
,
C.
Andreani
, and
F.
Fernandez-Alonso
, “
Characterisation of the incident beam and current diffraction capabilities on the vesuvio spectrometer
,”
Meas. Sci. Technol.
28
,
095501
(
2017
).
30.
See http://www.isis.stfc.ac.uk for information about the ISIS Pulsed Neutron and Muon Source (last accessed on May 2024).
31.
E. M.
Schooneveld
,
J.
Mayers
,
N. J.
Rhodes
,
A.
Pietropaolo
,
C.
Andreani
,
R.
Senesi
,
G.
Gorini
,
E.
Perelli-Cippo
, and
M.
Tardocchi
, “
Foil cycling technique for the VESUVIO spectrometer operating in the resonance detector configuration
,”
Rev. Sci. Instrum.
77
,
5103
(
2006
).
32.
C.
Andreani
,
D.
Colognesi
,
E.
Degiorgi
,
A.
Filabozzi
,
M.
Nardone
,
E.
Pace
,
A.
Pietropaolo
, and
R.
Senesi
, “
Double difference method in deep inelastic neutron scattering on the VESUVIO spectrometer
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
497
,
535
549
(
2003
).
33.
V. F.
Sears
, “
Neutron scattering lengths and cross sections
,”
Neutron News
3
,
26
37
(
1992
).
34.
J.
Mayers
,
A. L.
Fielding
, and
R.
Senesi
,
Nucl. Instrum. Methods Phys. Res., Sect. A
481
,
454
(
2002
).
35.
G.
Romanelli
,
B.
Hewer
,
M.
Krzystyniak
,
M.
Gigg
,
R.
Tolchenov
,
S.
Mukhopadhyay
, and
F.
Fernandez-Alonso
, “
Data analysis of neutron Compton scattering experiments using MANTID
,”
J. Phys.: Conf. Ser.
1055
,
012016
(
2018
).
36.
G.
Romanelli
and
M.
Krzystyniak
, “
On the line-shape analysis of compton profiles and its application to neutron scattering
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
819
,
84
88
(
2016
).
37.
S.
Imberti
,
C.
Andreani
,
V.
Garbuio
,
G.
Gorini
,
A.
Pietropaolo
,
R.
Senesi
, and
M.
Tardocchi
, “
Resolution of the VESUVIO spectrometer for high-energy inelastic neutron scattering experiments
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
552
,
463
(
2005
).
38.
C.
Andreani
,
R.
Senesi
,
M.
Krzystyniak
,
G.
Romanelli
, and
F.
Fernandez-Alonso
, “
Experimental studies of nuclear quantum effects in condensed matter: The case of water
,”
Riv. Nuovo Cimento
41
,
291
340
(
2018
).
39.
C.
Andreani
,
G.
Romanelli
, and
R.
Senesi
, “
Direct measurements of quantum kinetic energy tensor in stable and metastable water near the triple point: An experimental benchmark
,”
J. Phys. Chem. Lett.
7
,
2216
2220
(
2016
).
40.
B.
Cheng
,
J.
Behler
, and
M.
Ceriotti
, “
Nuclear quantum effects in water at the triple point: Using theory as a link between experiments
,”
J. Phys. Chem. Lett.
7
,
2210
2215
(
2016
).
41.
R.
Senesi
,
D.
Flammini
,
A. I.
Kolesnikov
,
E. D.
Murray
,
G.
Galli
, and
C.
Andreani
, “
The quantum nature of the oh stretching mode in ice and water probed by neutron scattering experiments
,”
J. Chem. Phys.
139
,
074504
(
2013
).
42.
G.
Romanelli
,
F.
Fernandez-Alonso
, and
C.
Andreani
, “
The harmonic picture of nuclear mean kinetic energies in heavy water
,”
J. Phys.: Conf. Ser.
571
,
012003
(
2014
).
43.
P.
Ulpiani
,
G.
Romanelli
,
D.
Onorati
,
M.
Krzystyniak
,
C.
Andreani
, and
R.
Senesi
, “
The effective isotropy of the hydrogen local potential in biphenyl and other hydrocarbons
,”
J. Chem. Phys.
153
,
234306
(
2020
).
44.
C. P.
Herrero
and
R.
Ramirez
, “
Isotope effects in ice Ih: A path-integral simulation
,”
J. Chem. Phys.
134
,
094510
(
2011
), arXiv:1108.1923 [cond-mat.mtrl-sci].
45.
H.
McMurry
,
I. U.
National Reactor Testing Station
, and
P. P. C. A. E.
Division
, “
Differential scattering cross sections of moderators in the short collision time approximation: Theoretical
,” in
AEC Research and Development Report No. Pt. 1
(
U.S. Atomic Energy Commission, Idaho Operations Office
,
1962
).
46.
J.
Robledo
,
J.
Dawidowski
,
J. M.
Damián
,
G.
Škoro
,
C.
Bovo
, and
G.
Romanelli
, “
Measurement of neutron total cross sections at the vesuvio spectrometer
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
971
,
164096
(
2020
).
47.
A.
Sjolander
, “
Multi-phonon processes in slow neutron scattering by crystals
,”
Ark. Fys.
14
,
1
(
1958
).
48.
G.
Romanelli
,
D.
Onorati
,
P.
Ulpiani
,
S.
Cancelli
,
E.
Perelli-Cippo
,
J. I.
Márquez Damián
,
S. C.
Capelli
,
G.
Croci
,
A.
Muraro
,
M.
Tardocchi
et al, “
Thermal neutron cross sections of amino acids from average contributions of functional groups
,”
J. Phys.: Condens. Matter
33
,
285901
(
2021
).
49.
J.
Dawidowski
,
L.
Rodríguez Palomino
,
G.
Romanelli
,
G.
Cuello
,
J.
Márquez Damián
,
J.
Robledo
, and
M.
Krzystyniak
, “
Determination of effective temperatures of hydrogenated and deuterated alcohols using the vesuvio spectrometer
,”
Nucl. Instrum. Methods Phys. Res., Sect. A
989
,
164948
(
2021
).
50.
A.
Bocedi
,
G.
Romanelli
,
C.
Andreani
, and
R.
Senesi
, “
Hydrogen nuclear mean kinetic energy in water down the Mariana Trench: Competition of pressure and salinity
,”
J. Chem. Phys.
153
,
134306
(
2020
).
51.
J.
Jakobi
,
J. A.
Huisman
,
M.
Köhli
,
D.
Rasche
,
H.
Vereecken
, and
H. R.
Bogena
, “
The footprint characteristics of cosmic ray thermal neutrons
,”
Geophys. Res. Lett.
48
,
e2021GL094281
, (
2021
).
52.
M.
Köhli
,
M.
Schrön
,
S.
Zacharias
, and
U.
Schmidt
, “
Uranos v1.0—The ultra rapid adaptable neutron-only simulation for environmental research
,”
Geosci. Model Dev.
16
,
449
477
(
2023
).
53.
C.
Weulersse
,
M.
Mazurek
,
C. D.
Pinas
,
R.
Mills
, and
B.
Guerard
, “
Thermal neutron measurements inside an aircraft for see assessment
,”
IEEE Trans. Nucl. Sci.
70
,
1548
1554
(
2023
).
54.
G.
Hubert
, “
Analyses of continuous measurements of cosmic ray induced-neutrons spectra at the concordia antarctic station from 2016 to 2024
,”
Astropart. Phys.
159
,
102949
(
2024
).
55.
R.
Baatz
,
H. R.
Bogena
,
H.-J.
Hendricks Franssen
,
J. A.
Huisman
,
C.
Montzka
, and
H.
Vereecken
, “
An empirical vegetation correction for soil water content quantification using cosmic ray probes
,”
Water Resour. Res.
51
,
2030
2046
, (
2015
).
56.
S.
Vernetto
,
M.
Laurenza
,
M.
Storini
,
A.
Zanini
,
P.
Diego
,
S.
Massetti
,
A.
Liberatore
,
J.
Terrazas
,
C.
Vigorito
,
P.
Vallania
, and
S.
Cirilli
, “
Long term measurements of neutron dose rates at Testa Grigia high altitude research station (3480 m. a.s.l.)
,”
Radiat. Phys. Chem.
193
,
109972
(
2022
).
57.
J. D.
Wille
,
S. P.
Alexander
,
C.
Amory
,
R.
Baiman
,
L.
Barthélemy
,
D. M.
Bergstrom
,
A.
Berne
,
H.
Binder
,
J.
Blanchet
,
D.
Bozkurt
,
T. J.
Bracegirdle
,
M.
Casado
,
T.
Choi
,
K. R.
Clem
,
F.
Codron
,
R.
Datta
,
S. D.
Battista
,
V.
Favier
,
D.
Francis
,
A. D.
Fraser
,
E.
Fourré
,
R. D.
Garreaud
,
C.
Genthon
,
I. V.
Gorodetskaya
,
S.
González-Herrero
,
V. J.
Heinrich
,
G.
Hubert
,
H.
Joos
,
S.-J.
Kim
,
J. C.
King
,
C.
Kittel
,
A.
Landais
,
M.
Lazzara
,
G. H.
Leonard
,
J. L.
Lieser
,
M.
Maclennan
,
D.
Mikolajczyk
,
P.
Neff
,
I.
Ollivier
,
G.
Picard
,
B.
Pohl
,
F. M.
Ralph
,
P.
Rowe
,
E.
Schlosser
,
C. A.
Shields
,
I. J.
Smith
,
M.
Sprenger
,
L.
Trusel
,
D.
Udy
,
T.
Vance
,
É.
Vignon
,
C.
Walker
,
N.
Wever
, and
X.
Zou
, “
The extraordinary march 2022 East Antarctica “heat” wave. Part II: Impacts on the Antarctic ice sheet
,”
J. Clim.
37
,
779
799
(
2024
).