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  • Letter
  • Open Access

Periodic table for highly charged ions

Chunhai Lyu*, Christoph H. Keitel, and Zoltán Harman

  • *Contact author: chunhai.lyu@mpi-hd.mpg.de
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Phys. Rev. Research 7, L042071 – Published 31 December, 2025

DOI: https://doi.org/10.1103/bppp-53g1

Abstract

Mendeleev's periodic table successfully groups atomic elements according to their chemical and spectroscopic properties. However, it becomes less sufficient in describing the electronic properties of highly charged ions (HCIs) in which many of the outermost electrons are ionized. In this work, we put forward a periodic table particularly suitable for HCIs. It is constructed purely based on the successive electron occupation of relativistic orbitals. While providing a much simplified description of the level structure of highly charged isoelectronic ions—essential for laboratory and astrophysical plasma spectroscopies, such a periodic table predicts a large family of highly forbidden transitions suitable for the development of next-generation optical atomic clocks. Furthermore, we also identify universal linear Z scaling laws (Z is the nuclear charge) in the so-called “Coulomb splittings” between angular momentum multiplets along isoelectronic sequences, complementing the physics of electron-electron interactions in multielectron atomic systems.

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References (61)

  1. C. J. Foot, Atomic Physics (Oxford University Press, Oxford, 2005).
  2. P. Schwerdtfeger, O. R. Smits, and P. Pyykkö, The periodic table and the physics that drives it, Nat. Rev. Chem. 4, 359 (2020).
  3. P. Beiersdorfer, K. Boyce, G. Brown, H. Chen, S. Kahn, R. Kelley, M. May, R. Olson, F. Porter, C. Stahle, et al., Laboratory simulation of charge exchange-produced x-ray emission from comets, Science 300, 1558 (2003).
  4. S. Bernitt, G. V. Brown, J. K. Rudolph, R. Steinbrügge, A. Graf, M. Leutenegger, S. W. Epp, S. Eberle, K. Kubiček, V. Mäckel, et al., An unexpectedly low oscillator strength as the origin of the Fe XVII emission problem, Nature (London) 492, 225 (2012).
  5. L. Schmöger, O. Versolato, M. Schwarz, M. Kohnen, A. Windberger, B. Piest, S. Feuchtenbeiner, J. Pedregosa-Gutierrez, T. Leopold, P. Micke, et al., Coulomb crystallization of highly charged ions, Science 347, 1233 (2015).
  6. J. Zhang, A. MacPhee, J. Lin, E. Wolfrum, R. Smith, C. Danson, M. Key, C. Lewis, D. Neely, J. Nilsen, et al., A saturated x-ray laser beam at 7 nanometers, Science 276, 1097 (1997).
  7. A. Depresseux, E. Oliva, J. Gautier, F. Tissandier, J. Nejdl, M. Kozlova, G. Maynard, J. Goddet, A. Tafzi, A. Lifschitz, et al., Table-top femtosecond soft-x-ray laser by collisional ionization gating, Nat. Photon. 9, 817 (2015).
  8. C. Lyu, S. M. Cavaletto, C. H. Keitel, and Z. Harman, Narrow-band hard-x-ray lasing with highly charged ions, Sci. Rep. 10, 9439 (2020).
  9. J. Sheil, L. Poirier, A. C. Lassise, D. J. Hemminga, S. Schouwenaars, N. Braaksma, A. Frenzel, R. Hoekstra, and O. O. Versolato, Power-law scaling relating the average charge state and kinetic energy in expanding laser-driven plasmas, Phys. Rev. Lett. 133, 125101 (2024).
  10. D. Schardt, T. Elsässer, and D. Schulz-Ertner, Heavy-ion tumor therapy: Physical and radiobiological benefits, Rev. Mod. Phys. 82, 383 (2010).
  11. S. A. King, L. J. Spieß, P. Micke, A. Wilzewski, T. Leopold, E. Benkler, R. Lange, N. Huntemann, A. Surzhykov, V. A. Yerokhin, et al., An optical atomic clock based on a highly charged ion, Nature (London) 611, 43 (2022).
  12. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).
  13. M. G. Kozlov, M. S. Safronova, J. R. Crespo López-Urrutia, and P. O. Schmidt, Highly charged ions: Optical clocks and applications in fundamental physics, Rev. Mod. Phys. 90, 045005 (2018).
  14. J. Morgner, B. Tu, C. König, T. Sailer, F. Heiße, H. Bekker, B. Sikora, C. Lyu, V. Yerokhin, Z. Harman, et al., Stringent test of QED with hydrogen-like tin, Nature (London) 622, 53 (2023).
  15. R. Loetzsch, H. F. Beyer, L. Duval, U. Spillmann, D. Banaś, P. Dergham, F. Kröger, J. Glorius, R. E. Grisenti, M. Guerra, et al., Testing quantum electrodynamics in extreme fields using helium-like uranium, Nature (London) 625, 673 (2024).
  16. J. C. Berengut, V. A. Dzuba, and V. V. Flambaum, Enhanced laboratory sensitivity to variation of the fine-structure constant using highly charged ions, Phys. Rev. Lett. 105, 120801 (2010).
  17. S. Goudsmit and P. I. Richards, The order of electron shells in ionized atoms, Proc. Natl. Acad. Sci. USA 51, 664 (1964).
  18. I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation (Springer, New York, 2007).
  19. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, 1981).
  20. H. Gomes, pgf-PeriodicTable—Create custom periodic tables of elements, version 2.1.1 (2024), https://ctan.org/pkg/pgf-periodictable?lang=en, accessed 7 July 2024.
  21. P. W. Atkins, J. De Paula, and J. Keeler, Physical Chemistry (Oxford University Press, Oxford, 2023).
  22. G. C. Rodrigues, P. Indelicato, J. P. Santos, P. Patté, and F. Parente, Systematic calculation of total atomic energies of ground state configurations, At. Data Nucl. Data Tables 86, 117 (2004).
  23. C. Froese-Fischer, G. Gaigalas, P. Jönsson, and J. Bieroń, GRASP2018—A Fortran 95 version of the general relativistic atomic structure package, Comput. Phys. Commun. 237, 184 (2019).
  24. See Supplemental Material at http://link.aps.org/supplemental/10.1103/bppp-53g1 for the description of the jj coupling of multiply occupied orbitals, the level structures of 4fm uranium ions, and the 2D plots of the transition energies, instabilities, and differential E1 and M1 polarizabilities of the clock transitions.
  25. H. K. Chung, M. H. Chen, W. L. Morgan, Y. Ralchenko, and R. W. Lee, FLYCHK: Generalized population kinetics and spectral model for rapid spectroscopic analysis for all elements, High Energy Density Phys. 1, 3 (2005).
  26. F. B. Rosmej, V. A. Astapenko, and V. S. Lisitsa, Plasma Atomic Physics (Springer, Cham, 2021).
  27. J. S. Kaastra, A. J. J. Raassen, J. de Plaa, and L. Gu, SPEX x-ray spectral fitting package, Zenodo (2024), doi:10.5281/zenodo.10822753.
  28. L. Mercadier, A. Benediktovitch, Š. Krušič, J. J. Kas, J. Schlappa, M. Agåker, R. Carley, G. Fazio, N. Gerasimova, Y. Y. Kim, et al., Transient absorption of warm dense matter created by an x-ray free-electron laser, Nat. Phys. 20, 1564 (2024).
  29. T. Pütterich, R. Neu, R. Dux, A. Whiteford, M. O’Mullane, and ASDEX Upgrade Team, Modelling of measured tungsten spectra from ASDEX Upgrade and predictions for ITER, Plasma Phys. Control. Fusion 50, 085016 (2008).
  30. A. Kramida and T. Shirai, Energy levels and spectral lines of tungsten, W III through W LXXIV, At. Data Nucl. Data Tables 95, 305 (2009).
  31. P. Beiersdorfer, Highly charged ions in magnetic fusion plasmas: Research opportunities and diagnostic necessities, J. Phys. B 48, 144017 (2015).
  32. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997).
  33. C. Brif, R. Chakrabarti, and H. Rabitz, Control of quantum phenomena: Past, present and future, New J. Phys. 12, 075008 (2010).
  34. B. W. Adams, C. Buth, S. M. Cavaletto, J. Evers, Z. Harman, C. H. Keitel, A. Pálffy, A. Picón, R. Röhlsberger, Y. Rostovtsev, et al., X-ray quantum optics, J. Mod. Opt. 60, 2 (2013).
  35. K. P. Heeg, A. Kaldun, C. Strohm, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, R. Rüffer, et al., Coherent x-ray–optical control of nuclear excitons, Nature (London) 590, 401 (2021).
  36. K. P. Heeg, A. Kaldun, C. Strohm, P. Reiser, C. Ott, R. Subramanian, D. Lentrodt, J. Haber, H.-C. Wille, S. Goerttler, et al., Spectral narrowing of x-ray pulses for precision spectroscopy with nuclear resonances, Science 357, 375 (2017).
  37. J. Haber, X. Kong, C. Strohm, S. Willing, J. Gollwitzer, L. Bocklage, R. Rüffer, A. Pálffy, and R. Röhlsberger, Rabi oscillations of x-ray radiation between two nuclear ensembles, Nat. Photon. 11, 720 (2017).
  38. S. M. Cavaletto, Z. Harman, C. Ott, C. Buth, T. Pfeifer, and C. H. Keitel, Broadband high-resolution x-ray frequency combs, Nat. Photon. 8, 520 (2014).
  39. C. Lyu, S. M. Cavaletto, C. H. Keitel, and Z. Harman, Interrogating the temporal coherence of EUV frequency combs with highly charged ions, Phys. Rev. Lett. 125, 093201 (2020).
  40. A. Derevianko, V. A. Dzuba, and V. V. Flambaum, Highly charged ions as a basis of optical atomic clockwork of exceptional accuracy, Phys. Rev. Lett. 109, 180801 (2012).
  41. C. Lyu, C. H. Keitel, and Z. Harman, Ultrastable and ultra-accurate clock transitions in open-shell highly charged ions, Commun. Phys. 8, 3 (2025).
  42. K. Kromer, C. Lyu, M. Door, P. Filianin, Z. Harman, J. Herkenhoff, P. Indelicato, C. H. Keitel, D. Lange, Y. N. Novikov, C. Schweiger, S. Eliseev, and K. Blaum, Observation of a low-lying metastable electronic state in highly charged lead by penning-trap mass spectrometry, Phys. Rev. Lett. 131, 223002 (2023).
  43. K. J. Arnold, R. Kaewuam, A. Roy, T. R. Tan, and M. D. Barrett, Blackbody radiation shift assessment for a lutetium ion clock, Nat. Commun. 9, 1650 (2018).
  44. H. Bekker, A. Borschevsky, Z. Harman, C. H. Keitel, T. Pfeifer, P. O. Schmidt, J. R. Crespo López-Urrutia, and J. C. Berengut, Detection of the 5p−4f orbital crossing and its optical clock transition in Pr9+, Nat. Commun. 10, 5651 (2019).
  45. J. C. Berengut, V. A. Dzuba, V. V. Flambaum, and A. Ong, Optical transitions in highly charged californium ions with high sensitivity to variation of the fine-structure constant, Phys. Rev. Lett. 109, 070802 (2012).
  46. K. Beloy, Trap-induced ac Zeeman shift of the thorium-229 nuclear clock frequency, Phys. Rev. Lett. 130, 103201 (2023).
  47. T. Rosenband, D. Hume, P. Schmidt, C.-W. Chou, A. Brusch, L. Lorini, W. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, et al., Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place, Science 319, 1808 (2008).
  48. S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, Al+27 quantum-logic clock with a systematic uncertainty below 10−18, Phys. Rev. Lett. 123, 033201 (2019).
  49. E. Peik and C. Tamm, Nuclear laser spectroscopy of the 3.5 eV transition in Th-229, Europhys. Lett. 61, 181 (2003).
  50. P. O. Schmidt, T. Rosenband, C. Langer, W. M. Itano, J. C. Bergquist, and D. J. Wineland, Spectroscopy using quantum logic, Science 309, 749 (2005).
  51. J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum, C. Frugiuele, E. Fuchs, C. Grojean, R. Harnik, R. Ozeri, G. Perez, and Y. Soreq, Probing new long-range interactions by isotope shift spectroscopy, Phys. Rev. Lett. 120, 091801 (2018).
  52. I. Counts, J. Hur, D. P. L. Aude Craik, H. Jeon, C. Leung, J. C. Berengut, A. Geddes, A. Kawasaki, W. Jhe, and V. Vuletić, Evidence for nonlinear isotope shift in Yb+ search for new boson, Phys. Rev. Lett. 125, 123002 (2020).
  53. C. Solaro, S. Meyer, K. Fisher, J. C. Berengut, E. Fuchs, and M. Drewsen, Improved isotope-shift-based bounds on bosons beyond the standard model through measurements of the D23/2−D25/2 interval in Ca+, Phys. Rev. Lett. 125, 123003 (2020).
  54. N. L. Figueroa, J. C. Berengut, V. A. Dzuba, V. V. Flambaum, D. Budker, and D. Antypas, Precision determination of isotope shifts in ytterbium and implications for new physics, Phys. Rev. Lett. 128, 073001 (2022).
  55. J. Hur, D. P. L. Aude Craik, I. Counts, E. Knyazev, L. Caldwell, C. Leung, S. Pandey, J. C. Berengut, A. Geddes, W. Nazarewicz, P.-G. Reinhard, A. Kawasaki, H. Jeon, W. Jhe, and V. Vuletić, Evidence of two-source King plot nonlinearity in spectroscopic search for new boson, Phys. Rev. Lett. 128, 163201 (2022).
  56. K. Ono, Y. Saito, T. Ishiyama, T. Higomoto, T. Takano, Y. Takasu, Y. Yamamoto, M. Tanaka, and Y. Takahashi, Observation of nonlinearity of generalized King plot in the search for new boson, Phys. Rev. X 12, 021033 (2022).
  57. N.-H. Rehbehn, M. K. Rosner, J. C. Berengut, P. O. Schmidt, T. Pfeifer, M. F. Gu, and J. R. C. López-Urrutia, Narrow and ultranarrow transitions in highly charged Xe ions as probes of fifth forces, Phys. Rev. Lett. 131, 161803 (2023).
  58. M. Door, C.-H. Yeh, M. Heinz, F. Kirk, C. Lyu, T. Miyagi, J. C. Berengut, J. Bieroń, K. Blaum, L. S. Dreissen, et al., Probing new bosons and nuclear structure with ytterbium isotope shifts, Phys. Rev. Lett. 134, 063002 (2025).
  59. A. Wilzewski, L. Huber, M. Door, J. Richter, A. Mariotti, L. Spieß, M. Wehrheim, S. Chen, S. King, P. Micke, et al., Nonlinear calcium King plot constrains new bosons and nuclear properties, Phys. Rev. Lett. 134, 233002 (2025).
  60. M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Search for new physics with atoms and molecules, Rev. Mod. Phys. 90, 025008 (2018).
  61. W. R. Johnson, Atomic Structure Theory (Springer, New York, 2007).

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Phys. Rev. Research 7, L042071– Published 31 December, 2025
  • Received 27 February 2025
  • Revised 6 May 2025
  • Accepted 6 December 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 Max Planck Society.

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