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On the vacuum-polarization Uehling potential for a Fermi charge distribution

  • Jean-Christophe Pain
Regular Article
  • 44 Downloads

Abstract

We present analytical formulas for the vacuum-polarization Uehling potential in the case where the finite size of the nucleus is modeled by a Fermi charge distribution. Using a Sommerfeld-type development, the potential is expressed in terms of multiple derivatives of a particular integral. The latter and its derivatives can be evaluated exactly in terms of Bickley-Naylor functions, whose connection to the Uehling potential was already pointed out in the pure Coulomb case, and of usual Bessel functions of the second kind. The cusp and asymptotic expressions for the Uehling potential with a Fermi charge distribution are also provided. Analytical results for the higher-order-contribution Källén-Sabry potential are given.

Graphical abstract

Keywords

Atomic Physics 

References

  1. 1.
    A.I. Akhiezer, V.B. Beresteskii, Quantum electrodynamics (Nauka Science, Moscow, 1981) in Russian; Interscience, New York, 1965 Google Scholar
  2. 2.
    W. Greiner, J. Reinhardt, Quantum electrodynamics, 4th edn. (Springer Verlag, Berlin, 2010) Google Scholar
  3. 3.
    E.A. Uehling, Phys. Rev. 48, 55 (1935) ADSCrossRefGoogle Scholar
  4. 4.
    E.H. Wichmann, N.H. Kroll, Phys. Rev. 101, 843 (1956) ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    P. Pyykkö, M. Tokman, L.N. Labzowsky, Phys. Rev. A 57, R689 (1998) ADSCrossRefGoogle Scholar
  6. 6.
    P. Pyykkö, L.-B. Zhao, J. Phys. B: At. Mol. Opt. Phys. 36, 1469 (2003) ADSCrossRefGoogle Scholar
  7. 7.
    M. Abramowitz, I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1972) Google Scholar
  8. 8.
    I. Mező, An alternative form for the Uehling potential, http://sites.google.com/site/istvanmezo81/others
  9. 9.
    V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, in Quantum electrodynamics (Pergamon Press, Oxford, 1982), Vol. 4 Google Scholar
  10. 10.
    K.-N.Huang, Phys. Rev. A14, 1311 (1976) ADSCrossRefGoogle Scholar
  11. 11.
    L.W. Fullerton, G.A. Rinker Jr., Phys. Rev. A 13, 1283 (1976) ADSCrossRefGoogle Scholar
  12. 12.
    S. Klarsfeld, Phys. Lett. 66B, 86 (1977) ADSCrossRefGoogle Scholar
  13. 13.
    W. Pauli, M.E. Rose, Phys. Rev. 49, 462 (1936) ADSCrossRefGoogle Scholar
  14. 14.
    W.G. Bickley, Philos. Mag. 20, 322 (1935) CrossRefGoogle Scholar
  15. 15.
    W.G. Bickley, J. Naylor, Philos. Mag. 20, 343 (1935) CrossRefGoogle Scholar
  16. 16.
    J.M. Blair, C.A. Edwards, J.H. Johnson, Math. Comput. 32, 876 (1978) Google Scholar
  17. 17.
    A.M. Frolov, D.M. Wardlaw, Eur. Phys. J. B 85, 348 (2012) ADSCrossRefGoogle Scholar
  18. 18.
    A.M. Frolov, Can. J. Phys. 92, 1094 (2014) ADSCrossRefGoogle Scholar
  19. 19.
    V. Hnizdo, Comput. Phys. Commun. 83, 95 (1994) ADSCrossRefGoogle Scholar
  20. 20.
    A.M. Frolov, arXiv:1210.6737v8 (2013)
  21. 21.
    J.S.M. Ginges, J.C. Berengut, J. Phys. B: At. Mol. Opt. Phys. 49, 095001 (2016) ADSCrossRefGoogle Scholar
  22. 22.
    R.D. Woods, D.S. Saxon, Phys. Rev. 95, 577 (1954) ADSCrossRefGoogle Scholar
  23. 23.
    W.R. Johnson, Note on the Uehling potential, http://www3.nd.edu/johnson/Publications/uehling.pdf
  24. 24.
    B. Fricke, W. Greiner, J.T. Waber, Theor. Chim. Acta 21, 235 (1971) CrossRefGoogle Scholar
  25. 25.
    D. Andrae, Phys. Rep. 336, 413 (2000) ADSCrossRefGoogle Scholar
  26. 26.
    G.D. Mahan, Many particle physics (Plenum, New York, 1981) Google Scholar
  27. 27.
    S. Goedecker, Phys. Rev. B 48, 17573 (1993) ADSCrossRefGoogle Scholar
  28. 28.
    M. Grypeos, C. Koutroulos, V. Lukyanov, A. Shebeko, J. Phys. G: Nucl. Part. Phys. 24, 1913 (1998) ADSCrossRefGoogle Scholar
  29. 29.
    N.W. Ashcroft, N.D. Mermin, Solid state physics (Saunders, Philadelphia, 1976) Google Scholar
  30. 30.
    R.J. McKee, Phys. Rev. 180, 1139 (1969) ADSCrossRefGoogle Scholar
  31. 31.
    R. Glauber, W. Rarita, P. Schwed, Phys. Rev. 120, 609 (1960) ADSCrossRefGoogle Scholar
  32. 32.
    F. Roesel, D. Trautman, R.D. Viollier, Nucl. Phys. A292, 523 (1977) ADSCrossRefGoogle Scholar
  33. 33.
    Y.L. Luke, Integrals of Bessel functions (McGraw-Hill Book Co. Inc., New York, 1962) Google Scholar
  34. 34.
    R. Hem Prabha, R.D.S. Yadav, Ann. Nucl. Energy 23, 1021 (1996) CrossRefGoogle Scholar
  35. 35.
    T.H. Schucan, Nucl. Phys. 61, 417 (1965) CrossRefGoogle Scholar
  36. 36.
    G. Källén, A. Sabry, Det Kongelige Danske Videnskabernes Selskab Matematisk-Fysiske Meddelelser 29, 3 (1955) Google Scholar
  37. 37.
    S.M. Schneider, W. Greiner, G. Soff, J. Phys. B: At. Mol. Opt. Phys. 26, L529 (1993) ADSCrossRefGoogle Scholar
  38. 38.
    P. Indelicato, Phys. Rev. A 87, 022501 (2013) ADSCrossRefGoogle Scholar
  39. 39.
    J. Blomqvist, Nucl. Phys. B 48, 95 (1972) ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CEA, DAM, DIFArpajonFrance

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