- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Open Access

Stopping Cross Sections for Protons Across Different Phases of Water

F. Matias1, N. E. Koval2, P. de Vera3, R. Garcia-Molina3, I. Abril4, J. M. B. Shorto1, H. Yoriyaz1, J. J. N. Pereira1, T. F. Silva5 et al.

M. H. Tabacniks5, M. Vos6, and P. L. Grande7

PDF

Phys. Rev. Lett. 135, 148003 – Published 1 October, 2025

DOI: https://doi.org/10.1103/ksdx-mnd7

Abstract

Accurately quantifying the energy loss rate of proton beams in liquid water is crucial for the precise application and improvement of proton therapy, whereas the slowing down of protons in water ices also plays an important role in astrophysics. However, precisely determining the electronic stopping power, particularly for the liquid phase, has been elusive so far. Experimental techniques are difficult to apply to volatile liquids, and the availability of sufficient reliable measurements has been limited to the solid and vapor phases. The accuracy of current models is typically limited to proton energies just above the energy loss maximum, making it difficult to predict radiation effects at an energy range of special relevance. We elucidate the phase differences in proton energy loss in water in a wide energy range (0.001−10  MeV) by means of real-time time-dependent density functional theory combined with the Penn method. This nonperturbative model, more computationally efficient than current approaches, describes the phase effects in water in excellent agreement with available experimental data, revealing clear deviations around the maximum of the stopping power curve and below. As an important outcome, our calculations reveal that proton stopping quantities of liquid water and amorphous ice are identical, in agreement with recent similar observations for low-energy electrons, pointing out this equivalence for all charged particles. This could help to overcome the limitation in obtaining reliable experimental information for the biologically relevant liquid water target.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (65)

  1. M. Durante and H. Paganetti, Nuclear physics in particle therapy: A review, Rep. Prog. Phys. 79, 096702 (2016).
  2. D. Schardt, T. Elsässer, and D. Schulz-Ertner, Heavy-ion tumor therapy: Physical and radiobiological benefits, Rev. Mod. Phys. 82, 383 (2010).
  3. P. Andreo, On the clinical spatial resolution achievable with protons and heavier charged particle radiotherapy beams, Phys. Med. Biol. 54, N205 (2009).
  4. H. Paul, On the accuracy of stopping power codes and ion ranges used for hadron therapy, Adv. Quantum Chem. 65, 39 (2013).
  5. H. Nikjoo, S. Uehara, and D. Emfietzoglou, Interaction of Radiation with Matter (CRC Press, Boca Raton, FL, USA, 2012).
  6. H. Nikjoo, D. Emfietzoglou, T. Liamsuwan, R. Taleei, D. Liljequist, and S. Uehara, Radiation track, DNA damage and response—A review, Rep. Prog. Phys. 79, 116601 (2016).
  7. R. A. Baragiola, Water ice on outer solar system surfaces: Basic properties and radiation effects, Planet. Space Sci. 51, 953 (2003).
  8. L. S. Farenzena, P. Iza, R. Martinez, F. A. Fernandez-Lima, E. S. Duarte, G. S. Faraudo, C. R. Ponciano, M. G. P. Homem, A. N. de Brito, K. Wien, and E. F. da Silveira, Electronic sputtering analysis of astrophysical ices, Earth, Moon, Planets 97, 311 (2005).
  9. L. I. Cleeves, E. A. Bergin, C. M. O. Alexander, F. Du, D. Graninger, K. I. Öberg, and T. J. Harries, The ancient heritage of water ice in the solar system, Science 345, 1590 (2014).
  10. E. Dartois et al., Spectroscopic sizing of interstellar icy grains with JWST, Nat. Astron. 8, 359 (2024).
  11. S. Protopapa, U. Raut, I. Wong, J. Stansberry, G. L. Villanueva, J. Cook, B. Holler, W. M. Grundy, R. Brunetto, R. J. Cartwright, B. Mamo, J. P. Emery, A. H. Parker, A. Guilbert-Lepoutre, N. Pinilla-Alonso, S. N. Milam, and H. B. Hammel, Detection of carbon dioxide and hydrogen peroxide on the stratified surface of charon with JWST, Nat. Commun. 15, 8247 (2024).
  12. J. M. Sunshine et al., Exposed water ice deposits on the surface of comet 9P/Tempel 1, Science 311, 1453 (2006).
  13. B. D. Teolis, C. Plainaki, T. A. Cassidy, and U. Raut, Water ice radiolytic O2, H2, and H2O2 yields for any projectile species, energy, or temperature: A model for icy astrophysical bodies, J. Geophys. Res. 122, 1996 (2017).
  14. R. A. Baragiola, M. J. Loeffler, U. Raut, R. A. Vidal, and C. D. Wilson, Laboratory studies of radiation effects in water ice in the outer solar system, Radiat. Phys. Chem. 72, 187 (2005).
  15. P. Jenniskens and D. F. Blake, Structural transitions in amorphous water ice and astrophysical implications, Science 265, 753 (1994).
  16. C. Mejía, A. L. F. de Barros, A. Domaracka, P. Boduch, H. Rothard, and E. F. da Silveira, Modifications of astrophysical ices induced by cosmic rays—I. Water, carbon monoxide, and methanol mixtures, Astron. Astrophys. 687, A227 (2024).
  17. N. F. W. Ligterink, P. Pinilla, N. van der Marel, J. T. van Scheltinga, A. S. Booth, C. M. O. Alexander, and M. E. I. Riebe, The rapid formation of macromolecules in irradiated ice of protoplanetary disk dust traps, Nat. Astron. 8, 1257 (2024).
  18. M. Shimizu, M. Kaneda, T. Hayakawa, H. Tsuchida, and A. Itoh, Stopping cross sections of liquid water for MeV energy protons, Nucl. Instrum. Methods Phys. Res., Sect. B 267, 2667 (2009).
  19. M. Shimizu, T. Hayakawa, M. Kaneda, H. Tsuchida, and A. Itoh, Stopping cross-sections of liquid water for 0.3–2.0 MeV protons, Vacuum 84, 1002 (2010).
  20. T. Siiskonen, H. Kettunen, K. Peräjärvi, A. Javanainen, M. Rossi, W. H. Trzaska, J. Turunen, and A. Virtanen, Energy loss measurement of protons in liquid water, Phys. Med. Biol. 56, 2367 (2011).
  21. R. Garcia-Molina, I. Abril, P. de Vera, and H. Paul, Comments on recent measurements of the stopping power of liquid water, Nucl. Instrum. Methods Phys. Res., Sect. B 299, 51 (2013).
  22. P. de Vera, R. Garcia-Molina, and I. Abril, Simulation of the energy spectra of swift light ion beams after traversing cylindrical targets: A consistent interpretation of experimental data relevant for hadron therapy, Eur. Phys. J. D 73, 209 (2019).
  23. P. Bauer, W. Käferböck, and V. Nečas, Investigation of the electronic energy loss of hydrogen ions in H2O: Influence of the state of aggregation, Nucl. Instrum. Methods Phys. Res., Sect. B 93, 132 (1994).
  24. J. A. Phillips, The energy loss of low energy protons in some gases, Phys. Rev. 90, 532 (1953).
  25. H. K. Reynolds, D. N. F. Dunbar, W. A. Wenzel, and W. Whaling, The stopping cross section of gases for protons, 30–600 keV, Phys. Rev. 92, 742 (1953).
  26. C. Mitterschiffthaler and P. Bauer, Stopping cross section of water vapor for hydrogen ions, Nucl. Instrum. Methods Phys. Res., Sect. B 48, 58 (1990).
  27. W. Y. Baek, B. Grosswendt, and G. Willems, Ionization ranges of protons in water vapour in the energy range 1–100 keV, Radiat. Protect. Dosim. 122, 32 (2006).
  28. R. Signorell, Electron scattering in liquid water and amorphous ice: A striking resemblance, Phys. Rev. Lett. 124, 205501 (2020).
  29. D. R. Penn, Electron mean-free-path calculations using a model dielectric function, Phys. Rev. B 35. 482 (1987).
  30. R. Garcia-Molina, I. Abril, I. Kyriakou, and D. Emfietzoglou, Energy loss of swift protons in liquid water: Role of optical data input and extension algorithms, in Radiation Damage in Biomolecular Systems, edited by G. Garciá Gómez-Tejedor and M. C. Fuss (Springer Science + Business Media B.V., Dordrecht, 2012), Chap. 15, 10.1007/978-94-007-2564-5_15.
  31. M. Vos and P. L. Grande, Extension schemes of the dielectric function, and their implications for ion stopping calculations, J. Phys. Chem. Solids 133, 187 (2019).
  32. P. de Vera, I. Abril, and R. Garcia-Molina, Electronic cross section, stopping power and energy-loss straggling of metals for swift protons, alpha particles and electrons, Front. Mater. 10, 1249517 (2023).
  33. K. G. Reeves, Y. Yao, and Y. Kanai, Electronic stopping power in liquid water for protons and α particles from first principles, Phys. Rev. B 94, 041108(R) (2016).
  34. Y. Yao, D. C. Yost, and Y. Kanai, K-shell core-electron excitations in electronic stopping of protons in water from first principles, Phys. Rev. Lett. 123, 066401 (2019).
  35. B. Gu, B. Cunningham, D. M. Santiburcio, F. D. Pieve, E. Artacho, and J. Kohanoff, Efficient ab initio calculation of electronic stopping in disordered systems via geometry pre-sampling: Application to liquid water, J. Chem. Phys. 153, 034113 (2020).
  36. S. Taioli, P. E. Trevisanutto, P. de Vera, S. Simonucci, I. Abril, R. Garcia-Molina, and M. Dapor, Relative role of physical mechanisms on complex biodamage induced by carbon irradiation, J. Phys. Chem. Lett. 12, 487 (2021).
  37. N. E. Koval, P. Koval, F. Da Pieve, J. Kohanoff, E. Artacho, and D. Emfietzoglou, Inelastic scattering of electrons in water from first principles: Cross sections and inelastic mean free path for use in Monte Carlo track-structure simulations of biological damage, R. Soc. Open Sci. 9, 212011 (2022).
  38. B. Gu, D. Muñoz-Santiburcio, F. Da Pieve, F. Cleri, E. Artacho, and J. Kohanoff, Bragg’s additivity rule and core and bond model studied by real-time TDDFT electronic stopping simulations: The case of water vapor, Radiat. Phys. Chem. 193, 109961 (2022).
  39. C. Shepard and Y. Kanai, Ion-type dependence of DNA electronic excitation in water under proton, α-particle, and carbon ion irradiation: A first-principles simulation study, J. Phys. Chem. B 127, 10700 (2023).
  40. N. E. Koval, F. Da Pieve, B. Gu, D. Muñoz Santiburcio, J. Kohanoff, and E. Artacho, Nonlinear electronic stopping of negatively charged particles in liquid water, Phys. Rev. Res. 5, 033063 (2023).
  41. C. Shepard, D. C. Yost, and Y. Kanai, Electronic excitation response of DNA to high-energy proton radiation in water, Phys. Rev. Lett. 130, 118401 (2023).
  42. A. G. Borisov, D. Sánchez-Portal, R. Diez Muiño, and P. M. Echenique, Building up the screening below the femtosecond scale, Chem. Phys. Lett. 387, 95 (2004).
  43. M. Quijada, A. G. Borisov, I. Nagy, R. Diez Muiño, and P. M. Echenique, Time-dependent density-functional calculation of the stopping power for protons and antiprotons in metals, Phys. Rev. A 75, 042902 (2007).
  44. N. E. Koval, D. Sánchez-Portal, A. G. Borisov, and R. Diez Muiño, Dynamic screening and energy loss of antiprotons colliding with excited Al clusters, Nucl. Instrum. Methods Phys. Res., Sect. B 317, 56 (2013).
  45. F. Matias, R. C. Fadanelli, P. L. Grande, N. E. Koval, R. Díez Muiño, A. G. Borisov, N. R. Arista, and G. Schiwietz, Ground- and excited-state scattering potentials for the stopping of protons in an electron gas, J. Phys. B 50, 185201 (2017).
  46. F. Matias, T. F. Silva, N. E. Koval, J. J. N. Pereira, P. C. G. Antunes, P. T. D. Siqueira, M. H. Tabacniks, H. Yoriyaz, J. M. B. Shorto, and P. L. Grande, Efficient computational modeling of electronic stopping power of organic polymers for proton therapy optimization, Sci. Rep. 14, 9868 (2024).
  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ksdx-mnd7 for a detailed description of the TDDFT-Penn methodology.
  48. C. Auth, A. Mertens, H. Winter, and A. Borisov, Threshold in the stopping of slow protons scattered from the surface of a wide-band-gap insulator, Phys. Rev. Lett. 81, 4831 (1998).
  49. N. E. Koval, A. G. Borisov, L. F. S. Rosa, E. M. Stori, J. F. Dias, P. L. Grande, D. Sánchez-Portal, and R. Díez Muiño, Vicinage effect in the energy loss of H2 dimers: Experiment and calculations based on time-dependent density-functional theory, Phys. Rev. A 95, 062707 (2017).
  50. O. Gunnarsson and B. I. Lundqvist, Exchange and correlation in atoms, molecules, and solids by the spin-density-functional formalism, Phys. Rev. B 13, 4274 (1976).
  51. H. Hayashi, N. Watanabe, Y. Udagawa, and C. C. Kao, The complete optical spectrum of liquid water measured by inelastic x-ray scattering, Proc. Natl. Acad. Sci. U.S.A. 97, 6264 (2000).
  52. J. Daniels, Bestimmung der optischen konstanten von eis aus energie—verlustmessungen von schnellen elektronen, Opt. Commun. 3, 240 (1971).
  53. K. Kobayashi, Optical spectra and electronic structure of ice, J. Phys. Chem. 87, 4317 (1983).
  54. W. Chan, G. Cooper, and C. Brion, The electronic spectrum of water in the discrete and continuum regions. Absolute optical oscillator strengths for photoabsorption (6–200 eV), Chem. Phys. 178, 387 (1993).
  55. M. Vos and P. L. Grande, Dielectric functions, their properties and their relation to observables: Investigations using the Chapidif program for the case of aluminum, Comput. Phys. Commun. 314, 109657 (2025).
  56. D. Y. Smith, Dispersion theory, sum rules, and their application to the analysis of optical data, in Handbook of Optical Constants of Solids, edited by E. D. Palik (Academic Press, San Diego, 1998), Chap. 3, pp. 35–68.
  57. R. Garcia-Molina, I. Abril, C. D. Denton, S. Heredia-Avalos, I. Kyriakou, and D. Emfietzoglou, Calculated depth-dose distributions for H+ and He+ beams in liquid water, Nucl. Instrum. Methods Phys. Res., Sect. B 267, 2647 (2009).
  58. D. Emfietzoglou, H. Nikjoo, I. Petsalakis, and A. Pathak, A consistent dielectric response model for water ice over the whole energy-momentum plane, Nucl. Instrum. Methods Phys. Res., Sect. B 256, 141 (2007).
  59. S. Heredia-Avalos, R. Garcia-Molina, J. M. Fernández-Varea, and I. Abril, Calculated energy loss of swift He, Li, B, and N ions in SiO2, Al2O3, and ZrO2, Phys. Rev. A 72, 052902 (2005).
  60. N. D. Mermin, Lindhard dielectric function in the relaxation-time approximation, Phys. Rev. B 1, 2362 (1970).
  61. R. F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd ed. (Springer, New York, 2011).
  62. J. F. Ziegler, M. Ziegler, and J. Biersack, SRIM—The stopping and range of ions in matter (2010), Nucl. Instrum. Methods Phys. Res., Sect. B 268, 1818 (2010).
  63. H. Kamitsubo, Heavy ion science, Oyo Buturi (Japan) 43, 1019 (1974), https://www.jstage.jst.go.jp/article/oubutsu1932/43/10/43_10_1019/_article/-char/ja/.
  64. International Atomic Energy Agency/Nuclear Data Section, Electronic stopping power of matter for ions, Graphs, Data, Comments and Programs, https://www-nds.iaea.org/stopping (2021).
  65. R. Garcia-Molina, I. Abril, S. Heredia-Avalos, I. Kyriakou, and D. Emfietzoglou, A combined molecular dynamics and Monte Carlo simulation of the spatial distribution of energy deposition by proton beams in liquid water, Phys. Med. Biol. 56, 6475 (2011).

Outline

Information

Phys. Rev. Lett. 135, 148003– Published 1 October, 2025
  • Received 30 May 2025
  • Accepted 25 August 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation