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Splitting isotope shift in the 1s2p3P0,1,2 fine-structure triplet in C4+12,13,14: Experiment and theory

Patrick Müller1, Kristian König1,2,*, Emily Burbach1, Gordon W. F. Drake3, Phillip Imgram1, Bernhard Maass1,†, Titamarie M. Maggio3, Wilfried Nörtershäuser1,2, and Julien Spahn1

  • 1Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
  • 2Helmholtz Research Academy Hesse for FAIR, GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany
  • 3Department of Physics, University of Windsor, Windsor, Ontario, Canada N9B 3P4

  • *Contact author: kkoenig@ikp.tu-darmstadt.de
  • †Current address: Physics Division, Argonne National Laboratory, 60439 Lemont, Illinois USA.
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Phys. Rev. A 113, 012806 – Published 2 January, 2026

DOI: https://doi.org/10.1103/24k3-r2gp

Abstract

We report measurements and theoretical calculations of the fine-structure splittings in all three 1s2sS13→1s2pP0,1,23 transitions in the heliumlike systems of the isotopes C12,13,14. The metastable triplet state was efficiently populated in an electron beam ion source and the C4+ ions were electrostatically accelerated to 50 keV to perform collinear laser spectroscopy. From the determined transition frequencies, the splitting isotope shift (SIS), i.e., the difference in fine-structure splittings between different isotopes of the same element, was extracted. In the SIS, theoretical uncertainties due to higher-order quantum electrodynamic corrections are strongly suppressed since they are independent of both nuclear mass and the fine-structure quantum number J in lowest order. Comparison with theory provides an important test of experimental accuracy, particularly in the C4+13 case, for which the nuclear spin leads to hyperfine-induced fine-structure mixing. At the same time, the even-even isotopes C4+12,14 without nuclear spin can be used to confirm theory. Theoretical values of the SIS are given for all the heliumlike ions with 2≤Z≤10.

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References (62)

  1. G. W. F. Drake, Accuracy in atomic and molecular data, J. Phys. B 53, 223001 (2020).
  2. M. Haidar, Z.-X. Zhong, V. I. Korobov, and J.-P. Karr, Nonrelativistic QED approach to the fine- and hyperfine-structure corrections of order mα6 and mα6(m/M): Application to the hydrogen atom, Phys. Rev. A 101, 022501 (2020).
  3. K. Pachucki and V. A. Yerokhin, Fine structure of heliumlike ions and determination of the fine structure constant, Phys. Rev. Lett. 104, 070403 (2010).
  4. H. Persson, S. Salomonson, P. Sunnergren, and I. Lindgren, Two-Electron Lamb-shift calculations on heliumlike ions, Phys. Rev. Lett. 76, 204 (1996).
  5. O. Smits, P. Indelicato, W. Nazarewicz, M. Piibeleht, and P. Schwerdtfeger, Pushing the limits of the periodic table — a review on atomic relativistic electronic structure theory and calculations for the superheavy elements, Phys. Rep. 1035, 1 (2023).
  6. P. C. Pastor, G. Giusfredi, P. De Natale, G. Hagel, C. de Mauro, and M. Inguscio, Absolute frequency measurements of the 2S13−2P0,1,23 atomic Helium transitions around 1083 nm, Phys. Rev. Lett. 92, 023001 (2004).
  7. J. S. Borbely, M. C. George, L. D. Lombardi, M. Weel, D. W. Fitzakerley, and E. A. Hessels, Separated oscillatory-field microwave measurement of the 2P13−2P23 - fine-structure interval of atomic helium, Phys. Rev. A 79, 060503(R) (2009).
  8. F. Heydarizadmotlagh, T. D. G. Skinner, K. Kato, M. C. George, and E. A. Hessels, Precision measurement of the n=2 triplet PJ=1 to J=0 fine structure of atomic Helium using frequency-offset separated oscillatory fields, Phys. Rev. Lett. 132, 163001 (2024).
  9. L.-B. Wang, P. Mueller, K. Bailey, G. W. F. Drake, J. P. Greene, D. Henderson, R. J. Holt, R. V. F. Janssens, C. L. Jiang, Z.-T. Lu, T. P. O'Connor, R. C. Pardo, K. E. Rehm, J. P. Schiffer, and X. D. Tang, Laser spectroscopic determination of the He6 nuclear charge radius, Phys. Rev. Lett. 93, 142501 (2004).
  10. P. Mueller, I. A. Sulai, A. C. C. Villari, J. A. Alcántara-Núñez, R. Alves-Condé, K. Bailey, G. W. F. Drake, M. Dubois, C. Eléon, G. Gaubert, R. J. Holt, R. V. F. Janssens, N. Lecesne, Z.-T. Lu, T. P. O'Connor, M.-G. Saint-Laurent, J.-C. Thomas, and L.-B. Wang, Nuclear charge radius of He8, Phys. Rev. Lett. 99, 252501 (2007).
  11. Z. T. Lu, P. Mueller, G. W. F. Drake, W. Nörtershäuser, S. C. Pieper, and Z. C. Yan, Colloquium: Laser probing of neutron-rich nuclei in light atoms, Rev. Mod. Phys. 85, 1383 (2013).
  12. R. Loetzsch, H. F. Beyer, L. Duval, U. Spillmann, D. Banaś, P. Dergham, F. M. 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).
  13. P. Pfäfflein, G. Weber, S. Allgeier, Z. Andelkovic, S. Bernitt, A. I. Bondarev, A. Borovik, L. Duval, A. Fleischmann, O. Forstner, et al., Quantum electrodynamics in strong electromagnetic fields: Substate resolved Kα transition energies in heliumlike uranium, Phys. Rev. Lett. 134, 153001 (2025).
  14. V. Patkóš, V. A. Yerokhin, and K. Pachucki, Complete α7m Lamb shift of helium triplet states, Phys. Rev. A 103, 042809 (2021).
  15. V. A. Yerokhin, V. Patkóš, and K. Pachucki, Relativistic corrections to the Bethe logarithm for the 2S3 and 2P3 states of He, Phys. Rev. A 98, 032503 (2018).
  16. V. A. Yerokhin, V. Patkóš, and K. Pachucki, QED calculations of energy levels of heliumlike ions with 5≤Z≤30, Phys. Rev. A 106, 022815 (2022).
  17. V. A. Yerokhin and V. M. Shabaev, Lamb shift of n=1 and n=2 states of hydrogen-like atoms, 1≤Z≤110, J. Phys. Chem. Ref. Data 44, 033103 (2015).
  18. P. Indelicato, QED tests with highly charged ions, J. Phys. B 52, 232001 (2019).
  19. A. V. Malyshev, Y. S. Kozhedub, and V. M. Shabaev, Ab initio calculations of the 2p3/2→2s transition in He-, Li-, and Be-like uranium, Phys. Rev. A 107, 042806 (2023).
  20. E. G. Myers, H. S. Margolis, J. K. Thompson, M. A. Farmer, J. D. Silver, and M. R. Tarbutt, Precision measurement of the 1s2pP23−P13 fine structure interval in heliumlike fluorine, Phys. Rev. Lett. 82, 4200 (1999).
  21. J. K. Thompson, D. J. H. Howie, and E. G. Myers, Measurements of the 1s2sS01−1s2pP1,03 transitions in heliumlike nitrogen, Phys. Rev. A 57, 180 (1998).
  22. A. Ozawa, T. Kobayashi, T. Suzuki, K. Yoshida, and I. Tanihata, New magic number, N=16, near the neutron drip line, Phys. Rev. Lett. 84, 5493 (2000).
  23. M. Puchalski and K. Pachucki, Ground state hyperfine splitting in Li6,7 atoms and the nuclear structure, Phys. Rev. Lett. 111, 243001 (2013).
  24. X.-Q. Qi, P.-P. Zhang, Z.-C. Yan, G. W. F. Drake, Z.-X. Zhong, T.-Y. Shi, S.-L. Chen, Y. Huang, H. Guan, and K.-L. Gao, Precision calculation of hyperfine structure and the Zemach radii of Li+6,7 ions, Phys. Rev. Lett. 125, 183002 (2020).
  25. G. W. F. Drake, Unified relativistic theory for 1s2pP13−1s2S01 and 1s2pP11−1s2S01 frequencies and transition rates in heliumlike ions, Phys. Rev. A 19, 1387 (1979).
  26. G. W. Drake, Theoretical energies for the n=1 and 2 states of the helium isoelectronic sequence up to Z=100, Can. J. Phys. 66, 586 (1988).
  27. P. Imgram, K. König, B. Maaß, P. Müller, and W. Nörtershäuser, Collinear laser spectroscopy of 2S13→2PJ3 transitions in helium-like C4+12, Phys. Rev. Lett. 131, 243001 (2023).
  28. L. Cardman, J. Lightbody, S. Penner, S. Fivozinsky, X. Maruyama, W. Trower, and S. Williamson, The charge distribution of C12, Phys. Lett. B 91, 203 (1980).
  29. I. Sick, Precise nuclear radii from electron scattering, Phys. Lett. B 116, 212 (1982).
  30. W. Reuter, G. Fricke, K. Merle, and H. Miska, Nuclear charge distribution and rms radius of C12 from absolute elastic electron scattering measurements, Phys. Rev. C 26, 806 (1982).
  31. E. A. J. M. Offermann, L. S. Cardman, C. W. de Jager, H. Miska, C. de Vries, and H. de Vries, Energy dependence of the form factor for elastic electron scattering from C12, Phys. Rev. C 44, 1096 (1991).
  32. L. Schaller, L. Schellenberg, T. Phan, G. Piller, A. Ruetschi, and H. Schneuwly, Nuclear charge radii of the carbon isotopes C12, C13, and C14, Nucl. Phys. A 379, 523 (1982).
  33. W. Ruckstuhl, B. Aas, W. Beer, I. Beltrami, K. Bos, P. Goudsmit, H. Leisi, G. Strassner, A. Vacchi, F. De Boer, U. Kiebele, and R. Weber, Precision measurement of the 2p−1s transition in muonic C12: Search for new muon-nucleon interactions or accurate determination of the rms nuclear charge radius, Nucl. Phys. A 430, 685 (1984).
  34. P. Müller, M. Heinz, P. Imgram, K. König, B. Maass, T. Miyagi, W. Nörtershäuser, R. Roth, and A. Schwenk, The nuclear charge radius of C13, Nat. Commun. 16, 6234 (2025).
  35. G. W. Drake, W. Nörtershäuser, and Z. C. Yan, Isotope shifts and nuclear radius measurements for helium and lithium, Can. J. Phys. 83, 311 (2005).
  36. L. M. Wang, C. Li, Z.-C. Yan, and G. W. F. Drake, Isotope shifts and transition frequencies for the S and P states of lithium: Bethe logarithms and second-order relativistic recoil, Phys. Rev. A 95, 032504 (2017).
  37. W. Nörtershäuser, C. Geppert, A. Krieger, K. Pachucki, M. Puchalski, K. Blaum, M. L. Bissell, N. Frömmgen, M. Hammen, M. Kowalska, et al., Precision test of many-body QED in the Be+2p fine structure doublet using short-lived isotopes, Phys. Rev. Lett. 115, 033002 (2015).
  38. W. Nörtershäuser, R. Sánchez, G. Ewald, A. Dax, J. Behr, P. Bricault, B. A. Bushaw, J. Dilling, M. Dombsky, G. W. F. Drake, S. Götte, et al., Isotope-shift measurements of stable and short-lived lithium isotopes for nuclear-charge-radii determination, Phys. Rev. A 83, 012516 (2011).
  39. C. J. Sansonetti, C. E. Simien, J. D. Gillaspy, J. N. Tan, S. M. Brewer, R. C. Brown, S. Wu, and J. V. Porto, Absolute transition frequencies and quantum interference in a frequency comb based measurement of the Li6,7D lines, Phys. Rev. Lett. 107, 023001 (2011).
  40. K. König, J. Krämer, C. Geppert, P. Imgram, B. Maaß, T. Ratajczyk, and W. Nörtershäuser, A new collinear apparatus for laser spectroscopy and applied science (COALA), Rev. Sci. Instrum. 91, 081301 (2020).
  41. P. Imgram, K. König, B. Maaß, P. Müller, and W. Nörtershäuser, Collinear laser spectroscopy of highly charged ions produced with an electron-beam ion source, Phys. Rev. A 108, 062809 (2023).
  42. P. M. Müller, Laser spectroscopic determination of the nuclear charge radius of C13, Ph.D. thesis, Technische Universität Darmstadt, Darmstadt, 2024.
  43. Z.-C. Yan, W. Nörtershäuser, and G. W. F. Drake, High precision atomic theory for Li and Be+: QED shifts and isotope shifts, Phys. Rev. Lett. 100, 243002 (2008).
  44. A. P. Stone, Nuclear and relativistic effects in atomic spectra: II, Proc. Phys. Soc. 81, 868 (1963).
  45. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer-Verlag, Berlin, 1957), sect. 39.
  46. M. Douglas and N. M. Kroll, Quantum electrodynamical corrections to the fine structure of helium, Ann. Phys. 82, 89 (1974).
  47. G. W. Drake, Progress in helium fine-structure calculations and the fine-structure constant, Can. J. Phys. 80, 1195 (2002).
  48. P. K. Kabir and E. E. Salpeter, Radiative corrections to the ground-state energy of the helium atom, Phys. Rev. 108, 1256 (1957).
  49. H. Araki, Quantum-electrodynamical corrections to energy-levels of helium, Prog. Theor. Phys. 17, 619 (1957).
  50. J. Sucher, Energy levels of the two-electron atom to order αRy3, Ionization energy of helium, Phys. Rev. 109, 1010 (1958).
  51. G. Drake and S. P. Goldman, Bethe logarithms for Ps−, H−, and heliumlike atoms, Can. J. Phys. 77, 835 (2000).
  52. M. I. Eides, H. Grotch, and V. A. Shelyuto, Theory of light hydrogenlike atoms, Phys. Rep. 342, 63 (2001).
  53. Z.-C. Yan and G. W. F. Drake, Lithium transition energies and isotope shifts: QED recoil corrections, Phys. Rev. A 66, 042504 (2002).
  54. K. Pachucki, Simple derivation of helium Lamb shift, J. Phys. B 31, 5123 (1998).
  55. K. Pachucki and J. Sapirstein, Recoil corrections to the Lamb shift in helium, J. Phys. B 33, 455 (2000).
  56. G. W. F. Drake and Z.-C. Yan, Energies and relativistic corrections for the Rydberg states of helium: Variational results and asymptotic analysis, Phys. Rev. A 46, 2378 (1992).
  57. G. Drake and A. Makowski, High-precision eigenvalues for the 1s2pP1 and P3 states of helium, J. Opt. Soc. Am. B 5, 2207 (1988).
  58. A. Kono and S. Hattori, Variational calculations for excited states in He I: Improved estimation of the ionization energy from accurate energies for the S3,n D1, nD3 series, Phys. Rev. A 31, 1199 (1985).
  59. P. Zhao, J. R. Lawall, and F. M. Pipkin, High-precision isotope-shift measurement of 2S3−2P3 transition in helium, Phys. Rev. Lett. 66, 592 (1991).
  60. E. Riis, A. G. Sinclair, O. Poulsen, G. W. F. Drake, W. R. C. Rowley, and A. P. Levick, Lamb shifts and hyperfine structure in Li+6 and Li+7: Theory and experiment, Phys. Rev. A 49, 207 (1994).
  61. W. R. Johnson, K. T. Cheng, and D. R. Plante, Hyperfine structure of 2P3 levels of heliumlike ions, Phys. Rev. A 55, 2728 (1997).
  62. Data for C124+ can be found at: https://doi.org/10.48328/tudatalib-1235, Data for C134+ can be found at: https://doi.org/10.48328/tudatalib-1500, and Data for C144+ can be found at: https://doi.org/10.48328/tudatalib-2034.

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Phys. Rev. A 113, 012806– Published 2 January, 2026
  • Received 16 October 2025
  • Accepted 25 November 2025
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