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Pb208 Nuclear Charge Radius Revisited: Closing the Fine-Structure-Anomaly Gap

Zewen Sun, Konstantin A. Beyer, Zoia A. Mandrykina, Igor A. Valuev, Christoph H. Keitel, and Natalia S. Oreshkina*

  • *Contact author: Natalia.Oreshkina@mpi-hd.mpg.de
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Phys. Rev. Lett. 135, 163002 – Published 15 October, 2025

DOI: https://doi.org/10.1103/h3xz-xdxr

Abstract

A comprehensive reevaluation of the root-mean-square nuclear charge radius is presented for the doubly magic Pb208 extracted from muonic spectroscopy measurements. By integrating rigorous theoretical quantum electrodynamics calculations, state-of-the-art numerical methods, and a systematic reanalysis of the uncertainties, we reduced the long-standing muonic fine-structure anomaly and improved the goodness of fit by a factor of 20. The resulting value of 5.5062(5) fm for a Fermi distribution is fairly consistent with the previously reported muonic spectroscopy value, and 3 standard deviations larger than the commonly used compilation data, which indicates that the current value and its uncertainty could be significantly underestimated. Attributing the remaining discrepancy to theory errors which cannot be rigorously calculated, we suggest the rms charge radius with reduced model dependence to be 5.5062(17) fm. This Letter sets an improved benchmark for charge radius extraction in heavy nuclei and paves a path for systematic reevaluations across the nuclear chart.

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References (41)

  1. J. A. Wheeler, Some consequences of the electromagnetic interaction between μ−-mesons and nuclei, Rev. Mod. Phys. 21, 133 (1949).
  2. E. Borie and G. A. Rinker, The energy levels of muonic atoms, Rev. Mod. Phys. 54, 67 (1982).
  3. C. G. Parthey, A. Matveev, J. Alnis, R. Pohl, T. Udem, U. D. Jentschura, N. Kolachevsky, and T. W. Hänsch, Precision measurement of the hydrogen-deuterium 1s−2s isotope shift, Phys. Rev. Lett. 104, 233001 (2010).
  4. R. Pohl, F. Nez, G. S. King, A. Antognini, F. Biraben, A. P. Brandt, F. D. Colsher, M. D. Deslattes, A. E. Eilbeck, C. R. Gentry et al., Laser spectroscopy of muonic deuterium, Science 330, 1509 (2010).
  5. K. Schuhmann, L. M. P. Fernandes, F. Nez, M. A. Ahmed, F. D. Amaro, P. Amaro, F. Biraben, T.-L. Chen, D. S. Covita, A. J. Dax et al., The helion charge radius from laser spectroscopy of muonic helium-3 ions, Science 388, 854 (2025).
  6. A. Antognini, N. Berger, T. E. Cocolios, R. Dressler, R. Eichler, A. Eggenberger, P. Indelicato, K. Jungmann, C. H. Keitel, K. Kirch et al., Measurement of the quadrupole moment of Re185 and Re187 from the hyperfine structure of muonic x rays, Phys. Rev. C 101, 054313 (2020).
  7. Y. Yamazaki, H. D. Wohlfahrt, E. B. Shera, M. V. Hoehn, and R. M. Steffen, Discrepancy between theory and experiment in nuclear polarization corrections of muonic Pb208, Phys. Rev. Lett. 42, 1470 (1979).
  8. T. Q. Phan, P. Bergem, A. Rüetschi, L. A. Schaller, and L. Schellenberg, Nuclear polarization in muonic Zr90, Phys. Rev. C 32, 609 (1985).
  9. P. Bergem, G. Piller, A. Rueetschi, L. A. Schaller, L. Schellenberg, and H. Schneuwly, Nuclear polarization and charge moments of Pb208 from muonic x rays, Phys. Rev. C 37, 2821 (1988).
  10. C. Piller, C. Gugler, R. Jacot-Guillarmod, L. A. Schaller, L. Schellenberg, H. Schneuwly, G. Fricke, T. Hennemann, and J. Herberz, Nuclear charge radii of the tin isotopes from muonic atoms, Phys. Rev. C 42, 182 (1990).
  11. I. A. Valuev, G. Colò, X. Roca-Maza, C. H. Keitel, and N. S. Oreshkina, Evidence against nuclear polarization as source of fine-structure anomalies in muonic atoms, Phys. Rev. Lett. 128, 203001 (2022).
  12. I. Angeli and K. Marinova, Table of experimental nuclear ground state charge radii: An update, At. Data Nucl. Data Tables 99, 69 (2013).
  13. G. Fricke, C. Bernhardt, K. Heilig, L. Schaller, L. Schellenberg, E. Shera, and C. Dejager, Nuclear ground state charge radii from electromagnetic interactions, At. Data Nucl. Data Tables 60, 177 (1995).
  14. P. Mazanek, Gemeinsame Auswertung von Messungen myonischer Atome, optischer Isotopieverschiebungen und elastischer Elektronenstreuung zur Festlegung von Radien stabiler und instabiler Kerne unterhalb von N=126 sowie Bestimmung von Kernradien aus μ-Atom-Messungen für Erbium- und Wolframisotope, Ph.D. thesis, Johannes Gutenberg University, 1992.
  15. B. Dreher, J. Friedrich, K. Merle, H. Rothhaas, and G. Lührs, The determination of the nuclear ground state and transition charge density from measured electron scattering data, Nucl. Phys. A235, 219 (1974).
  16. G. Fricke and K. Heilig, Nuclear Charge Radii (Springer-Verlag, Berlin Heidelberg, 2004).
  17. See Supplemental Material at http://link.aps.org/supplemental/10.1103/h3xz-xdxr for details of the calculations and fitting procedure, which includes Refs. [18–29].
  18. Z. Sun, I. A. Valuev, and N. S. Oreshkina, Nuclear deformation effects in the spectra of highly charged ions, Phys. Rev. Res. 6, 023327 (2024).
  19. P. J. Mohr, E. Tiesinga, D. B. Newell, and B. N. Taylor, Codata Internationally recommended 2022 values of the fundamental physical constants, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=958002 (2024).
  20. M. J. Martin, Nuclear data sheets for A=208, Nucl. Data Sheets 108, 1583 (2007).
  21. G. Audi, A. H. Wapstra, and C. Thibault, The Ame2003 atomic mass evaluation: (II). Tables, graphs and references, Nucl. Phys. A729, 337 (2003).
  22. National Nuclear Data Center, Reduced transition probabilities or B(E2; 2+→0+) values (National Nuclear Data Center, 2025), https://www.nndc.bnl.gov/be2/help.html (Accessed: 2025-03-18).
  23. J. B. Bellicard and K. J. van Oostrum, Elastic electron scattering from lead-208 at 175 and 250 MeV, Phys. Rev. Lett. 19, 242 (1967).
  24. J. Heisenberg, R. Hofstadter, J. S. McCarthy, I. Sick, B. C. Clark, R. Herman, and D. G. Ravenhall, Elastic electron scattering by Pb208 and new information about the nuclear charge distribution, Phys. Rev. Lett. 23, 1402 (1969).
  25. I. Sick, The charge density difference between Pb208 and Bi209, Nucl. Phys., A208, 557 (1973).
  26. H. Euteneuer, J. Friedrich, and N. Voegler, Charge distribution of Pb208 and the difference in ρ(r) for Pb and Tl investigated by elastic electron scattering, Phys. Rev. Lett. 36, 129 (1976).
  27. H. Euteneuer, J. Friedrich, and N. Voegler, The charge-distribution differences of Pb209, Pb208,207,206,204, Tl205,203 investigated by elastic electron scattering and muonic x-ray data, Nucl. Phys. A298, 452 (1978).
  28. H. L. Anderson, C. K. Hargrove, E. P. Hincks, J. D. McAndrew, R. J. McKee, R. D. Barton, and D. Kessler, Precise measurement of the muonic x-rays in the lead isotopes, Phys. Rev. 187, 1565 (1969).
  29. D. Kessler, H. Mes, A. C. Thompson, H. L. Anderson, M. S. Dixit, C. K. Hargrove, and R. J. McKee, Muonic x rays in lead isotopes, Phys. Rev. C 11, 1719 (1975).
  30. S. Klarsfeld, Analytical expressions for the evaluation of vacuum-polarization potentials in muonic atoms, Phys. Lett. 66B, 86 (1977).
  31. V. K. Ivanov, S. S. Baturin, D. A. Glazov, and A. V. Volotka, Vacuum-polarization Wichmann-Kroll correction in the finite-basis-set approach, Phys. Rev. A 110, 032815 (2024).
  32. Z. Mandrykina, Z. Sun, and N. S. Oreshkina, Wichmann-Kroll correction including finite-nuclear-size effect for muonic atoms (to be published).
  33. P. Indelicato, Nonperturbative evaluation of some QED contributions to the muonic hydrogen n=2 Lamb shift and hyperfine structure, Phys. Rev. A 87, 022501 (2013).
  34. S. Breidenbach, E. Dizer, H. Cakir, and Z. Harman, Hadronic vacuum polarization correction to atomic energy levels, Phys. Rev. A 106, 042805 (2022).
  35. Z. A. Mandrykina, M. Thierfeld, and N. S. Oreshkina, Hadronic vacuum polarization effect in muonic atoms, arXiv:2509.01311.
  36. N. S. Oreshkina, Self-energy correction to the energy levels of heavy muonic atoms, Phys. Rev. Res. 4, L042040 (2022).
  37. V. A. Yerokhin and N. S. Oreshkina, QED calculations of the nuclear recoil effect in muonic atoms, Phys. Rev. A 108, 052824 (2023).
  38. C. M. Shakin and M. S. Weiss, Nuclear polarization and the muonic x-ray spectrum of Pb, Phys. Rev. C 8, 411 (1973).
  39. N. Michel, N. S. Oreshkina, and C. H. Keitel, Theoretical prediction of the fine and hyperfine structure of heavy muonic atoms, Phys. Rev. A 96, 032510 (2017).
  40. A. Knecht (private communication).
  41. J. Henderson, J. Heery, M. Rocchini, M. Siciliano, N. Sensharma, A. D. Ayangeakaa, R. V. F. Janssens, T. M. Kowalewski, Abhishek, P. D. Stevenson et al., Deformation and collectivity in doubly magic Pb208, Phys. Rev. Lett. 134, 062502 (2025).

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Phys. Rev. Lett. 135, 163002– Published 15 October, 2025
  • Received 29 April 2025
  • Accepted 5 September 2025
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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. Open access publication funded by the Max Planck Society.

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