- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

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
  • Open Access

QED calculations of the 2p–2s transition energies in Li-like ions

V. A. Yerokhin*, Z. Harman, and C. H. Keitel

  • *Contact author: vladimir.yerokhin@mpi-hd.mpg.de
PDF

Phys. Rev. A 112, 042801 – Published 2 October, 2025

DOI: https://doi.org/10.1103/pwmr-r9gc

Abstract

Systematic QED calculations of ionization energies of the 2s, 2p1/2, and 2p3/2 states as well as the 2p1/2–2s and 2p3/2–2s transition energies are performed for Li-like ions with the nuclear charge numbers Z=10–100. The convergence of QED perturbative expansion is improved by using the extended Furry picture, which starts from the Dirac equation with a local screening potential. An ab initio treatment is accomplished for one- and two-photon electron-structure QED effects and the one-photon screening of the self-energy and vacuum-polarization corrections. This is complemented with an approximate treatment of the two-photon QED screening and higher-order (three or more photon) electron-structure effects. As a result, the obtained theoretical predictions improve upon the accuracy achieved in previous calculations. Comparison with available experimental data shows a good agreement between theory and experiment. In most cases, the theoretical values surpass the experimental results in precision with only a few exceptions. In the case of uranium and bismuth, the comparison provides one of the most stringent tests of bound-state QED in the strong-field regime. Alternatively, the obtained results can be employed for high-precision determinations of nuclear charge radii.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (97)

  1. A. Gumberidze, T. Stöhlker, D. Banaś, K. Beckert, P. Beller, H. F. Beyer, F. Bosch, S. Hagmann, C. Kozhuharov, D. Liesen et al., Phys. Rev. Lett. 94, 223001 (2005).
  2. R. Loetzsch, H. Beyer, L. Duval, U. Spillmann, D. Banaś, P. Dergham, F. Kröger, J. Glorius, R. Grisenti, M. Guerra et al., Nature (London) 625, 673 (2024).
  3. P. Pfäfflein, G. Weber, S. Allgeier, Z. Andelkovic, S. Bernitt, A. I. Bondarev, A. Borovik, L. Duval, A. Fleischmann, O. Forstner et al., Phys. Rev. Lett. 134, 153001 (2025).
  4. P. Beiersdorfer, D. Knapp, R. E. Marrs, S. R. Elliott, and M. H. Chen, Phys. Rev. Lett. 71, 3939 (1993).
  5. P. Beiersdorfer, A. L. Osterheld, J. H. Scofield, J. R. Crespo López-Urrutia, and K. Widmann, Phys. Rev. Lett. 80, 3022 (1998).
  6. P. Beiersdorfer, H. Chen, D. B. Thorn, and E. Träbert, Phys. Rev. Lett. 95, 233003 (2005).
  7. J. Nauta, A. Borodin, H. B. Ledwa, J. Stark, M. Schwarz, L. Schmöger, P. Micke, J. R. Crespo López-Urrutia, and T. Pfeifer, Nucl. Instrum. Methods Phys. Res. B 408, 285 (2017).
  8. J. Nauta, J.-H. Oelmann, A. Borodin, A. Ackermann, P. Knauer, I. S. Muhammad, R. Pappenberger, T. Pfeifer, and J. R. C. López-Urrutia, Opt. Express 29, 2624 (2021).
  9. D. Budker, J. R. Crespo López-Urrutia, A. Derevianko, V. V. Flambaum, M. W. Krasny, A. Petrenko, S. Pustelny, A. Surzhykov, V. A. Yerokhin, and M. Zolotorev, Ann. Phys. (NY) 532, 2000204 (2020).
  10. M. Puchalski and K. Pachucki, Phys. Rev. A 78, 052511 (2008).
  11. M. Puchalski and K. Pachucki, Phys. Rev. Lett. 113, 073004 (2014).
  12. M. Puchalski and K. Pachucki, Phys. Rev. A 92, 012513 (2015).
  13. V. A. Yerokhin, A. N. Artemyev, V. M. Shabaev, M. M. Sysak, O. M. Zherebtsov, and G. Soff, Phys. Rev. A 64, 032109 (2001).
  14. V. A. Yerokhin, P. Indelicato, and V. M. Shabaev, Phys. Rev. Lett. 97, 253004 (2006).
  15. V. A. Yerokhin, A. N. Artemyev, and V. M. Shabaev, Phys. Rev. A 75, 062501 (2007).
  16. Y. S. Kozhedub, A. V. Volotka, A. N. Artemyev, D. A. Glazov, G. Plunien, V. M. Shabaev, I. I. Tupitsyn, and T. Stöhlker, Phys. Rev. A 81, 042513 (2010).
  17. J. Sapirstein and K. T. Cheng, Phys. Rev. A 64, 022502 (2001).
  18. J. Sapirstein and K. T. Cheng, Phys. Rev. A 83, 012504 (2011).
  19. V. P. Kosheleva, A. V. Volotka, D. A. Glazov, D. V. Zinenko, and S. Fritzsche, Phys. Rev. Lett. 128, 103001 (2022).
  20. A. V. Malyshev, Y. S. Kozhedub, and V. M. Shabaev, Phys. Rev. A 107, 042806 (2023).
  21. A. V. Malyshev, Y. S. Kozhedub, V. M. Shabaev, and I. I. Tupitsyn, Phys. Rev. A 110, 062824 (2024).
  22. J. Morgner, V. A. Yerokhin, C. M. König, F. Heiße, B. Tu, T. Sailer, B. Sikora, Z. Harman, J. R. Crespo López-Urrutia, C. H. Keitel et al., Science 388, 945 (2025).
  23. J. Sapirstein, K. T. Cheng, and M. H. Chen, Phys. Rev. A 59, 259 (1999).
  24. J. Sapirstein and K. T. Cheng, Phys. Rev. A 66, 042501 (2002).
  25. W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  26. The Theory of Atomic Spectra, edited by R. Cowan (University of California Press, Berkeley, 1981).
  27. R. Latter, Phys. Rev. 99, 510 (1955).
  28. V. M. Shabaev, I. I. Tupitsyn, K. Pachucki, G. Plunien, and V. A. Yerokhin, Phys. Rev. A 72, 062105 (2005).
  29. V. A. Yerokhin, Phys. Rev. A 78, 012513 (2008).
  30. S. A. Blundell, D. S. Guo, W. R. Johnson, and J. Sapirstein, At. Data Nucl. Data Tables 37, 103 (1987).
  31. W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev. A 37, 2764 (1988).
  32. V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. Lett. 93, 130405 (2004).
  33. V. A. Yerokhin and A. Surzhykov, Phys. Rev. A 86, 042507 (2012).
  34. V. A. Yerokhin, A. N. Artemyev, V. M. Shabaev, M. M. Sysak, O. M. Zherebtsov, and G. Soff, Phys. Rev. Lett. 85, 4699 (2000).
  35. A. N. Artemyev, V. M. Shabaev, M. M. Sysak, V. A. Yerokhin, T. Beier, G. Plunien, and G. Soff, Phys. Rev. A 67, 062506 (2003).
  36. V. A. Yerokhin and V. M. Shabaev, J. Phys. Chem. Ref. Data 44, 033103 (2015).
  37. V. A. Yerokhin, Z. Harman, and C. H. Keitel, Phys. Rev. A 111, 042820 (2025).
  38. S. Volkov, V. A. Yerokhin, Z. Harman, and C. H. Keitel, arXiv:2509.03284.
  39. V. A. Yerokhin, A. N. Artemyev, T. Beier, G. Plunien, V. M. Shabaev, and G. Soff, Phys. Rev. A 60, 3522 (1999).
  40. A. N. Artemyev, V. M. Shabaev, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. A 71, 062104 (2005).
  41. V. A. Yerokhin, Phys. Rev. A 83, 012507 (2011).
  42. V. A. Yerokhin and V. M. Shabaev, Phys. Rev. A 60, 800 (1999).
  43. V. A. Yerokhin, K. Pachucki, and V. M. Shabaev, Phys. Rev. A 72, 042502 (2005).
  44. V. A. Yerokhin, Z. Harman, and C. H. Keitel, Phys. Rev. A 111, 012802 (2025).
  45. J. Sapirstein and K. T. Cheng, Phys. Rev. A 108, 042804 (2023).
  46. A. V. Malyshev, E. A. Prokhorchuk, and V. M. Shabaev, Phys. Rev. A 109, 062802 (2024).
  47. A. N. Artemyev, T. Beier, G. Plunien, V. M. Shabaev, G. Soff, and V. A. Yerokhin, Phys. Rev. A 60, 45 (1999).
  48. A. Artemyev (private communication).
  49. V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Phys. Rev. A 88, 012513 (2013).
  50. V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Comput. Phys. Commun. 189, 175 (2015).
  51. V. M. Shabaev, Theor. Math. Phys. 63, 588 (1985).
  52. V. M. Shabaev, Sov. J. Nucl. Phys. 47, 69 (1988).
  53. V. M. Shabaev, Phys. Rev. A 57, 59 (1998).
  54. N. A. Zubova, Y. S. Kozhedub, V. M. Shabaev, I. I. Tupitsyn, A. V. Volotka, G. Plunien, C. Brandau, and T. Stöhlker, Phys. Rev. A 90, 062512 (2014).
  55. A. N. Artemyev, V. M. Shabaev, and V. A. Yerokhin, Phys. Rev. A 52, 1884 (1995).
  56. A. N. Artemyev, V. M. Shabaev, and V. A. Yerokhin, J. Phys. B: At. Mol. Opt. Phys. 28, 5201 (1995).
  57. A. V. Malyshev, I. S. Anisimova, D. V. Mironova, V. M. Shabaev, and G. Plunien, Phys. Rev. A 100, 012510 (2019).
  58. A. V. Malyshev, I. S. Anisimova, D. A. Glazov, M. Y. Kaygorodov, D. V. Mironova, G. Plunien, and V. M. Shabaev, Phys. Rev. A 101, 052506 (2020).
  59. V. A. Yerokhin and N. S. Oreshkina, Phys. Rev. A 108, 052824 (2023).
  60. K. Pachucki and V. A. Yerokhin, Phys. Rev. Lett. 130, 053002 (2023).
  61. I. Angeli and K. Marinova, At. Data Nucl. Data Tables 99, 69 (2013).
  62. W. R. Johnson and G. Soff, At. Data Nucl. Data Tables 33, 405 (1985).
  63. B. Ohayon, At. Data Nucl. Data Tables 165, 101732 (2025).
  64. K. A. Beyer, T. E. Cocolios, C. Costache, M. Deseyn, P. Demol, A. Doinaki, O. Eizenberg, M. Gorshteyn, M. Heines, A. Herzáň et al., arXiv:2506.08804.
  65. Z. Sun, K. Beyer, Z. A. Mandrykina, I. A. Valuev, C. H. Keitel, and N. S. Oreshkina, arXiv:2504.19977.
  66. I. A. Valuev, Z. Harman, C. H. Keitel, and N. S. Oreshkina, Phys. Rev. A 101, 062502 (2020).
  67. Y. S. Kozhedub, O. V. Andreev, V. M. Shabaev, I. I. Tupitsyn, C. Brandau, C. Kozhuharov, G. Plunien, and T. Stöhlker, Phys. Rev. A 77, 032501 (2008).
  68. G. Plunien, B. Müller, W. Greiner, and G. Soff, Phys. Rev. A 43, 5853 (1991).
  69. G. Plunien and G. Soff, Phys. Rev. A 51, 1119 (1995); 53, 4614 (1996).
  70. A. V. Nefiodov, L. N. Labzowsky, G. Plunien, and G. Soff, Phys. Lett. A 222, 227 (1996).
  71. A. V. Nefiodov, G. Plunien, and G. Soff, Phys. Rev. Lett. 89, 081802 (2002).
  72. K. Bockasten, R. Hallin, and T. Hughes, Proc. Phys. Soc. 81, 522 (1963).
  73. B. Edlén, Phys. Scr. 28, 51 (1983).
  74. N. Peacock, M. Stamp, and J. Silver, Phys. Scr. 1984, 10 (1984).
  75. M. Lestinsky, E. Lindroth, D. A. Orlov, E. W. Schmidt, S. Schippers, S. Böhm, C. Brandau, F. Sprenger, A. S. Terekhov, A. Müller et al., Phys. Rev. Lett. 100, 033001 (2008).
  76. R. J. Knize, Phys. Rev. A 43, 1637 (1991).
  77. E. Hinnov and the TFTR Operating Team, B. Denne and the JET Operating Team, Phys. Rev. A 40, 4357 (1989).
  78. S. W. Epp, J. R. C. López-Urrutia, G. Brenner, V. Mäckel, P. H. Mokler, R. Treusch, M. Kuhlmann, M. V. Yurkov, J. Feldhaus, J. R. Schneider et al., Phys. Rev. Lett. 98, 183001 (2007).
  79. J. Reader, J. Sugar, N. Acquista, and R. Bahr, J. Opt. Soc. Am. B 11, 1930 (1994).
  80. J. Sugar, V. Kaufman, and W. L. Rowan, J. Opt. Soc. Am. B 9, 344 (1992).
  81. J. Sugar, V. Kaufman, and W. L. Rowan, J. Opt. Soc. Am. B 10, 13 (1993).
  82. U. Staude, P. Bosselmann, R. Büttner, D. Horn, K.-H. Schartner, F. Folkmann, A. E. Livingston, T. Ludziejewski, and P. H. Mokler, Phys. Rev. A 58, 3516 (1998).
  83. S. Madzunkov, E. Lindroth, N. Eklow, M. Tokman, A. Paal, and R. Schuch, Phys. Rev. A 65, 032505 (2002).
  84. B. Denne and E. Hinnov, Phys. Scr. 35, 811 (1987).
  85. R. Silwal, E. Takacs, J. M. Dreiling, J. D. Gillaspy, and Y. Ralchenko, Atoms 5, 30 (2017).
  86. P. Bosselmann, U. Staude, D. Horn, K.-H. Schartner, F. Folkmann, A. E. Livingston, and P. H. Mokler, Phys. Rev. A 59, 1874 (1999).
  87. D. Feili, P. Bosselmann, K.-H. Schartner, F. Folkmann, A. E. Livingston, E. Träbert, X. Ma, and P. H. Mokler, Phys. Rev. A 62, 022501 (2000).
  88. D. Bernhardt, C. Brandau, Z. Harman, C. Kozhuharov, S. Böhm, F. Bosch, S. Fritzsche, J. Jacobi, S. Kieslich, H. Knopp et al., Phys. Rev. A 91, 012710 (2015).
  89. J. Reader, J. D. Gillaspy, D. Osin, and Y. Ralchenko, J. Phys. B: At. Mol. Opt. Phys. 47, 145003 (2014).
  90. J. Clementson, P. Beiersdorfer, G. Brown, M. Gu, H. Lundberg, Y. Podpaly, and E. Träbert, Can. J. Phys. 89, 571 (2011).
  91. C. Brandau, C. Kozhuharov, A. Müller, W. Shi, S. Schippers, T. Bartsch, S. Böhm, C. Böhme, A. Hoffknecht, H. Knopp et al., Phys. Rev. Lett. 91, 073202 (2003).
  92. X. Zhang, N. Nakamura, C. Chen, M. Andersson, Y. Liu, and S. Ohtani, Phys. Rev. A 78, 032504 (2008).
  93. P. Beiersdorfer, A. Osterheld, S. R. Elliott, M. H. Chen, D. Knapp, and K. Reed, Phys. Rev. A 52, 2693 (1995).
  94. L. Wang, T. Liu, W. Yang, and Z.-C. Yan, Chin. Phys. B 32, 033102 (2023).
  95. V. A. Yerokhin, A. Surzhykov, and S. Fritzsche, Phys. Rev. A 90, 022509 (2014).
  96. V. A. Yerokhin and A. Surzhykov, J. Phys. Chem. Ref. Dat. 47, 023105 (2018).
  97. V. A. Yerokhin, M. Puchalski, and K. Pachucki, Phys. Rev. A 102, 042816 (2020).

Outline

Information

Phys. Rev. A 112, 042801– Published 2 October, 2025
  • Received 29 July 2025
  • Accepted 12 September 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. Open access publication funded by Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation