1. FKK 2010, St. Petersburg, 10.12.10 Precision calculations of the hyperfine structure in highly charged ions Andrey V. Volotka, Dmitry A. Glazov, Vladimir M. Shabaev, Ilya I. Tupitsyn, and Günter Plunien

2. FKK 2010, St. Petersburg, 10.12.10 Introduction and Motivation # Heavy few-electron ions provides possibility to test of QED at extremely strong electric fields Interelectronic interaction ~ 1 / Z QED ~ α => high-precision calculations are possible! However, in contrast to light atoms, the parameter αZ is not small => test of QED to all orders in αZ In U 92+ : αZ ≈ 0.7

3. FKK 2010, St. Petersburg, 10.12.10 Introduction and Motivation # Investigations of the hyperfine structure and g factor in heavy ions provide high-precision test of the magnetic sector of bound-state QED in the nonperturbative regime (hyperfine splitting and g factor of H-, Li-, and B-like heavy ions) Fundamental physics independent determination of the fine structure constant from QED at strong fields (g factor of H- and B-like heavy ions)

4. FKK 2010, St. Petersburg, 10.12.10 # Measurements of the ground-state hyperfine splitting in H-like ions Klaft et al., PRL 1994 209 Bi 82+ Crespo López-Urrutia et al., PRL 1996; PRA 1998 165 Ho 66+ 185 Re 74+ 187 Re 74+ Seelig et al., PRL 1998 207 Pb 81+ Beiersdorfer et al., PRA 2001 203 Tl 80+ 205 Tl 80+ Hyperfine structure in heavy ions

5. FKK 2010, St. Petersburg, 10.12.10 # Basic expression for the hyperfine splitting – relativistic factor – nuclear charge distribution correction – nuclear magnetization distribution correction – interelectronic-interaction correction of first-order in 1/Z – 1/Z 2 and higher-order interelectronic-interaction correction – screened QED correction – one-electron QED correction Hyperfine structure in heavy ions

6. FKK 2010, St. Petersburg, 10.12.10 Hyperfine structure in heavy ions # Ground-state hyperfine splitting in H-like ions

7. FKK 2010, St. Petersburg, 10.12.10 Hyperfine structure in heavy ions # Ground-state hyperfine splitting in Li-like ions

8. FKK 2010, St. Petersburg, 10.12.10 # Bohr-Weisskopf correction Bohr-Weisskopf correction depends linearly on the functions K S (r) and K L (r) [Shabaev et al., PRA 1998] Hyperfine structure in heavy ions

9. FKK 2010, St. Petersburg, 10.12.10 For a given κ the radial Dirac equations are the same in the nuclear region [Shabaev et al., PRL 2001] # Bohr-Weisskopf correction => the ratio of the Bohr-Weisskopf corrections is very stable with respect to variations of the nuclear models Hyperfine structure in heavy ions

10. FKK 2010, St. Petersburg, 10.12.10 Hyperfine structure in heavy ions

11. FKK 2010, St. Petersburg, 10.12.10 # Vacuum-polarization correction [Sunnergren, Persson, Salomonson, Schneider, Lindgren, and Soff, PRA 1998] [Schneider, Greiner, and Soff, PRA 1994] [Artemyev, Shabaev, Plunien, Soff, and Yerokhin, PRA 2001] [Sapirstein and Cheng, PRA 2001] Second-order terms in perturbation theory expansion Hyperfine structure in heavy ions

12. FKK 2010, St. Petersburg, 10.12.10 # Self-energy correction [Persson, Schneider, Greiner, Soff, and Lindgren, PRL 1996] [Shabaev, Tomaselli, Kühl, Artemyev, and Yerokhin, PRA 1997] [Blundell, Cheng, and Sapirstein, PRA 1997] Second-order terms in perturbation theory expansion Hyperfine structure in heavy ions [Yerokhin and Shabaev, PRA 2001]

13. FKK 2010, St. Petersburg, 10.12.10 Hyperfine structure in heavy ions # Screened QED correction: effective potential approach [Glazov, Volotka, Shabaev, Tupitsyn, and Plunien, PLA 2006] [Volotka, Glazov, Tupitsyn, Oreshkina, Plunien, and Shabaev, PRA 2008]

14. FKK 2010, St. Petersburg, 10.12.10 Hyperfine structure in heavy ions # Screened self-energy correction: effective potential approach Different screening potential have been employed core-Hartree potential Kohn-Sham potential – density of the core electrons – total electron density

15. FKK 2010, St. Petersburg, 10.12.10 # Screened vacuum-polarization correction Third-order terms in perturbation theory expansion 32 diagrams Hyperfine structure in heavy ions

16. FKK 2010, St. Petersburg, 10.12.10 # Screened self-energy correction Third-order terms in perturbation theory expansion 36 diagrams Hyperfine structure in heavy ions

17. FKK 2010, St. Petersburg, 10.12.10 # Screened self-energy correction Derivation of the formal expressions Regularizations of the divergences Ultraviolet divergences: diagrams (A), (B), (C), (E), and (F) Infrared divergences: diagrams (C), (D), and (F) Calculation Angular integrations Evaluation of regularized zero- and one-potential terms in momentum-space Contour rotation: identification of the poles structure Integration over the electron coordinates and the virtual photon energy Verification Angular integrations: analytical and numerical 2 different contours for the integration over the virtual photon energy Different gauges: Feynman and Coulomb Comparison with results obtained within screening potential approx. Hyperfine structure in heavy ions

18. FKK 2010, St. Petersburg, 10.12.10 Numerical results # Screened self-energy correction x SQED (SE) in the Feynman and Coulomb gauges for the Li-like 209 Bi 80+

19. FKK 2010, St. Petersburg, 10.12.10 Numerical results # Specific difference between hyperfine splitting in H- and Li-like bismuth in meV for Z=83 we obtain ξ=0.16886 => possibility for a test of screened QED on the level of few percent Remaining uncertainty ≈ 0.005 – 0.010 meV [Volotka, Glazov, Shabaev, Tupitsyn, and Plunien, PRL 2009] [Glazov, Volotka, Shabaev, Tupitsyn, and Plunien, PRA 2010]

20. FKK 2010, St. Petersburg, 10.12.10 Numerical results # Bohr-Weisskopf corrections for H-, Li-, and B-like bismuth The ratio of the Bohr-Weisskopf corrections Knowing 1s hyperfine splitting from experiment, the Bohr-Weisskopf correction can be obtained

21. FKK 2010, St. Petersburg, 10.12.10 Numerical results # Hyperfine splitting in Li- and B-like bismuth in meV *Beiersdorfer et al., PRL 1998 **Beiersdorfer et al., unpublished

22. FKK 2010, St. Petersburg, 10.12.10 # Two-photon exchange correction Outlook

23. FKK 2010, St. Petersburg, 10.12.10 Summary # Conclusion the most accurate theoretical prediction for the specific difference between hyperfine structure values in H- and Li-like Bi has been obtained rigorous evaluation of the complete gauge-invariant set of the screened QED corrections has been performed