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

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An Erratum to this article was published on 13 February 2020

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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.

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  • 13 February 2020

    Some typographical errors remain after the publication of my paper ���On the vacuum-polarization Uehling potential for a Fermi charge distribution���, Eur. Phys. J. D 72, 61 (2018). In order to improve its clarity and accuracy, the corrections are itemized below:

References

  1. A.I. Akhiezer, V.B. Beresteskii, Quantum electrodynamics (Nauka Science, Moscow, 1981) in Russian; Interscience, New York, 1965

  2. W. Greiner, J. Reinhardt, Quantum electrodynamics, 4th edn. (Springer Verlag, Berlin, 2010)

  3. E.A. Uehling, Phys. Rev. 48, 55 (1935)

    Article  ADS  Google Scholar 

  4. E.H. Wichmann, N.H. Kroll, Phys. Rev. 101, 843 (1956)

    Article  ADS  MathSciNet  Google Scholar 

  5. P. Pyykkö, M. Tokman, L.N. Labzowsky, Phys. Rev. A 57, R689 (1998)

    Article  ADS  Google Scholar 

  6. P. Pyykkö, L.-B. Zhao, J. Phys. B: At. Mol. Opt. Phys. 36, 1469 (2003)

    Article  ADS  Google Scholar 

  7. M. Abramowitz, I.A. Stegun, Handbook of mathematical functions (Dover, New York, 1972)

  8. I. Mező, An alternative form for the Uehling potential, http://sites.google.com/site/istvanmezo81/others

  9. V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, in Quantum electrodynamics (Pergamon Press, Oxford, 1982), Vol. 4

  10. K.-N.Huang, Phys. Rev. A14, 1311 (1976)

    Article  ADS  Google Scholar 

  11. L.W. Fullerton, G.A. Rinker Jr., Phys. Rev. A 13, 1283 (1976)

    Article  ADS  Google Scholar 

  12. S. Klarsfeld, Phys. Lett. 66B, 86 (1977)

    Article  ADS  Google Scholar 

  13. W. Pauli, M.E. Rose, Phys. Rev. 49, 462 (1936)

    Article  ADS  Google Scholar 

  14. W.G. Bickley, Philos. Mag. 20, 322 (1935)

    Article  Google Scholar 

  15. W.G. Bickley, J. Naylor, Philos. Mag. 20, 343 (1935)

    Article  Google Scholar 

  16. J.M. Blair, C.A. Edwards, J.H. Johnson, Math. Comput. 32, 876 (1978)

    Google Scholar 

  17. A.M. Frolov, D.M. Wardlaw, Eur. Phys. J. B 85, 348 (2012)

    Article  ADS  Google Scholar 

  18. A.M. Frolov, Can. J. Phys. 92, 1094 (2014)

    Article  ADS  Google Scholar 

  19. V. Hnizdo, Comput. Phys. Commun. 83, 95 (1994)

    Article  ADS  Google Scholar 

  20. A.M. Frolov, arXiv:1210.6737v8 (2013)

  21. J.S.M. Ginges, J.C. Berengut, J. Phys. B: At. Mol. Opt. Phys. 49, 095001 (2016)

    Article  ADS  Google Scholar 

  22. R.D. Woods, D.S. Saxon, Phys. Rev. 95, 577 (1954)

    Article  ADS  Google Scholar 

  23. W.R. Johnson, Note on the Uehling potential, http://www3.nd.edu/johnson/Publications/uehling.pdf

  24. B. Fricke, W. Greiner, J.T. Waber, Theor. Chim. Acta 21, 235 (1971)

    Article  Google Scholar 

  25. D. Andrae, Phys. Rep. 336, 413 (2000)

    Article  ADS  Google Scholar 

  26. G.D. Mahan, Many particle physics (Plenum, New York, 1981)

  27. S. Goedecker, Phys. Rev. B 48, 17573 (1993)

    Article  ADS  Google Scholar 

  28. M. Grypeos, C. Koutroulos, V. Lukyanov, A. Shebeko, J. Phys. G: Nucl. Part. Phys. 24, 1913 (1998)

    Article  ADS  Google Scholar 

  29. N.W. Ashcroft, N.D. Mermin, Solid state physics (Saunders, Philadelphia, 1976)

  30. R.J. McKee, Phys. Rev. 180, 1139 (1969)

    Article  ADS  Google Scholar 

  31. R. Glauber, W. Rarita, P. Schwed, Phys. Rev. 120, 609 (1960)

    Article  ADS  Google Scholar 

  32. F. Roesel, D. Trautman, R.D. Viollier, Nucl. Phys. A292, 523 (1977)

    Article  ADS  Google Scholar 

  33. Y.L. Luke, Integrals of Bessel functions (McGraw-Hill Book Co. Inc., New York, 1962)

  34. R. Hem Prabha, R.D.S. Yadav, Ann. Nucl. Energy 23, 1021 (1996)

    Article  Google Scholar 

  35. T.H. Schucan, Nucl. Phys. 61, 417 (1965)

    Article  Google Scholar 

  36. G. Källén, A. Sabry, Det Kongelige Danske Videnskabernes Selskab Matematisk-Fysiske Meddelelser 29, 3 (1955)

    Google Scholar 

  37. S.M. Schneider, W. Greiner, G. Soff, J. Phys. B: At. Mol. Opt. Phys. 26, L529 (1993)

    Article  ADS  Google Scholar 

  38. P. Indelicato, Phys. Rev. A 87, 022501 (2013)

    Article  ADS  Google Scholar 

  39. J. Blomqvist, Nucl. Phys. B 48, 95 (1972)

    Article  ADS  Google Scholar 

Download references

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  1. CEA, DAM, DIF, 91297, Arpajon, France

    Jean-Christophe Pain

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Pain, JC. On the vacuum-polarization Uehling potential for a Fermi charge distribution. Eur. Phys. J. D 72, 61 (2018). https://doi.org/10.1140/epjd/e2018-80457-8

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