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Precision calculations of the hyperfine structure in highly charged ions

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. Introduction and Motivation. # Heavy few-electron ions provides possibility to test of QED

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Precision calculations of the hyperfine structure in highly charged ions

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  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 αZis not small In U92+: αZ ≈ 0.7 => test of QED to all orders in αZ

  3. FKK 2010, St. Petersburg, 10.12.10 Introduction and Motivation # Investigations of the hyperfine structure and g factor in heavy ions provide Fundamental physics 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) 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 Hyperfine structure in heavy ions # Measurements of the ground-state hyperfine splitting in H-like ions Klaft et al., PRL 1994 209Bi82+ Crespo López-Urrutia et al., PRL 1996; PRA 1998 165Ho66+ 185Re74+ 187Re74+ Seelig et al., PRL 1998 207Pb81+ Beiersdorfer et al., PRA 2001 203Tl80+ 205Tl80+

  5. FKK 2010, St. Petersburg, 10.12.10 Hyperfine structure in heavy ions # Basic expression for the hyperfine splitting – relativistic factor – nuclear charge distribution correction – nuclear magnetization distribution correction – one-electron QED correction – interelectronic-interaction correction of first-order in 1/Z – 1/Z2 and higher-order interelectronic-interaction correction – screened QED correction

  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 Hyperfine structure in heavy ions # Bohr-Weisskopf correction Bohr-Weisskopf correction depends linearly on the functions KS(r) and KL(r) [Shabaev et al., PRA 1998]

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

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

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

  12. FKK 2010, St. Petersburg, 10.12.10 Hyperfine structure in heavy ions # Self-energy correction Second-order terms in perturbation theory expansion [Persson, Schneider, Greiner, Soff, and Lindgren, PRL 1996] [Blundell, Cheng, and Sapirstein, PRA 1997] [Shabaev, Tomaselli, Kühl, Artemyev, and Yerokhin, PRA 1997] [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 – density of the core electrons Kohn-Sham potential – total electron density

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

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

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

  18. FKK 2010, St. Petersburg, 10.12.10 Numerical results # Screened self-energy correction xSQED(SE) in the Feynman and Coulomb gauges for the Li-like 209Bi80+

  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 obtainξ=0.16886 Remaining uncertainty ≈ 0.005 – 0.010 meV => possibility for a test of screened QED on the level of few percent [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 Knowing 1s hyperfine splitting from experiment, the Bohr-Weisskopf correction can be obtained The ratio of the Bohr-Weisskopf corrections

  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 Outlook # Two-photon exchange correction

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

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