Advertisement

The effective single particle potential and the tadpole

  • P. F. Bortignon
  • E. E. Saperstein
  • M. BaldoEmail author
Regular Article - Theoretical Physics
  • 17 Downloads
Part of the following topical collections:
  1. Giant, Pygmy, Pairing Resonances and Related Topics

Abstract.

The Energy Density Functional (EDF) theory is extremely successful within the effective force approach, noticeably the Skyrme or Gogny forces, in reproducing the nuclear binding energies and other nuclear properties along the whole mass table. The EDF is in this case represented formally as the Hartree-Fock (HF) mean field of an effective force, and the associated single particle states can be considered the eigenstates of the corresponding effective mean field. In general, the phenomenological single particle spectrum is not accurately reproduced. To overcome this difficulty one can improve the functional in order to incorporate in the mean field additional correlations in an effective way. In an alternative viable scheme one can introduce explicitly many-body correlations which affect the single particle motion at both dynamic and static levels. In particular, the particle-vibration coupling scheme modifies the positions of the individual single particle energies. In this case one also introduce the fragmentation of the single particle states, a feature that cannot be described at the mean field level. At the same time the so-introduced correlations modify the static single particle potential, and the single particle states that diagonalize the corresponding density matrix are not the orbitals of a mean field with occupation numbers 1 (occupied orbitals) and 0 (unoccupied orbitals) anymore. In this paper we show that both static and dynamic effects on the single particle motion can be introduced within a previously developed scheme, based on the conserving approximations, and that the static part can be identified with the so-called tadpole term introduced by Khodel in the self-consistent theory of finite Fermi systems. The treatment remains at the formal level, but we hope that several aspects of the particle-vibration coupling scheme will be clarified.

Notes

References

  1. 1.
    A. Bohr, B. Mottelson, Nuclear Structure, Vol. 1 (Benjamin, New York, 1969)Google Scholar
  2. 2.
    B.R. Mottelson, Nucl. Phys. A 649, 45c (1999)ADSCrossRefGoogle Scholar
  3. 3.
    A. de-Shalit, I. Talmi, Nuclear Shell Theory (Academic Press, New York, 1963)Google Scholar
  4. 4.
    A. Bohr, B. Mottelson, Nuclear Structure, Vol. 2 (Benjamin, New York, 1974)Google Scholar
  5. 5.
    A.B. Migdal, J. Exp. Theor. Phys. 32, 399 (1957)Google Scholar
  6. 6.
    A.A. Abricosov, L.P. Gorkov, I.E. Dzialoshinski, Quantum Field Theoretical Methods in Statistical Physics (Pergamon Press, 1965)Google Scholar
  7. 7.
    L.D. Landau, J. Exp. Theor. Phys. 35, 97 (1958)Google Scholar
  8. 8.
    A.B. Migdal, Theory of Finite Fermi System and Applications to Atomic Nuclei (Nauka Moskow 1965, Wiley New York, 1967)Google Scholar
  9. 9.
    S.A. Fayans, V.A. Khodel, J. Exp. Theor. Phys. Lett. 17, 444 (1973)Google Scholar
  10. 10.
    V.A. Khodel, E.E. Saperstein, Phys. Rep. 92, 183 (1982)ADSCrossRefGoogle Scholar
  11. 11.
    V.A. Khodel, E.E. Saperstein, M.V. Zverev, Nucl. Phys. A 465, 397 (1987)ADSCrossRefGoogle Scholar
  12. 12.
    D. Vautherin, D.M. Brink, Phys. Rev. C 5, 626 (1972)ADSCrossRefGoogle Scholar
  13. 13.
    J. Speth, S. Krewald, F. Grümmer, P.-G. Reinhard, N. Lyutorovich, V. Tselyaev, Nucl. Phys. A 928, 14 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    E.E. Saperstein, S.V. Tolokonnikov, EPJ Web of Conferences 107, 02001 (2016)CrossRefGoogle Scholar
  15. 15.
    E.E. Saperstein, S.V. Tolokonnikov, Phys. At. Nucl. 79, 1030 (2016)CrossRefGoogle Scholar
  16. 16.
    M. Honma, T. Otsuka, B.A. Brown, T. Mizusaki, Phys. Rev. C 69, 034335 (2004)ADSCrossRefGoogle Scholar
  17. 17.
    E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, A.P. Zuker, Rev. Mod. Phys. 77, 427 (2005)ADSCrossRefGoogle Scholar
  18. 18.
    M. Baldo, P.F. Bortignon, G. Coló, D. Rizzo, L. Schiaccitano, J. Phys. G: Nucl. Part. Phys. 42, 085109 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    G. Baym, L.P. Kadanoff, Phys. Rev. 124, 287 (1961)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    G. Baym, Phys. Rev. 127, 1391 (1962)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    V.A. Khodel, Sov. J. Nucl. Phys. 24, 282 (1976)Google Scholar
  22. 22.
    P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer, N.Y. 1980)CrossRefGoogle Scholar
  23. 23.
    M. Baldo, P.F. Bortignon, Lett. Nuovo Cimento 14, 587 (1975)CrossRefGoogle Scholar
  24. 24.
    O.-P. Löwdin, Phys. Rev. 97, 1474 (1955)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    J.C. Greer, Mol. Phys. 106, 1363 (2008)ADSCrossRefGoogle Scholar
  26. 26.
    L. Hedin, S. Lundqvist, Solid State Phys. 23, 1 (1969)Google Scholar
  27. 27.
    L. Hedin, Phys. Rev. A 139, 796 (1965)ADSCrossRefGoogle Scholar
  28. 28.
    G. Onida, L. Reining, A. Rubio, Rev. Mod. Phys. 74, 601 (2002)ADSCrossRefGoogle Scholar
  29. 29.
    S. Fracasso, PhD Thesis, University of Milano (unpublished)Google Scholar
  30. 30.
    S. Kamerdzhiev, E.E. Saperstein, Eur. Phys. J. A 37, 333 (2008)ADSCrossRefGoogle Scholar
  31. 31.
    V.A. Khodel, E.E. Saperstein, Phys. Rep. 92, 183 (1982)ADSCrossRefGoogle Scholar
  32. 32.
    A. Bohr, B.R. Mottelson, Nuclear Structure, Vol. II (W.A. Benjamin Inc., 1975)Google Scholar
  33. 33.
    N.V. Gnezdilov, I.N. Borzov, E.E. Saperstein, S.V. Tolokonnikov, Phys. Rev. C 89, 034304 (2014)ADSCrossRefGoogle Scholar
  34. 34.
    E.E. Saperstein, M. Baldo, N.V. Gnezdilov, S.V. Tolokonnikov, Phys. Rev. C 93, 034302 (2016)ADSCrossRefGoogle Scholar
  35. 35.
    E.E. Saperstein, M. Baldo, S.S. Pankratov, S.V. Tolokonnikov, J. Exp. Theor. Phys. Lett. 104, 609 (2016)CrossRefGoogle Scholar
  36. 36.
    J.P. Jeukenne, A. Lejeune, C. Mahaux, Phys. Rep. 25C, 83 (1976)ADSCrossRefGoogle Scholar
  37. 37.
    M. Baldo, I. Bombaci, G. Giansiracusa, U. Lombardo, C. Mahaux, R. Sartor, Phys. Rev. C 41, 1748 (1990)ADSCrossRefGoogle Scholar
  38. 38.
    C. Mahaux, P.F. Bortignon, R.A. Broglia, C.H. Dasso, Phys. Rep. 120, 1 (1985)ADSCrossRefGoogle Scholar
  39. 39.
    R.D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem (Dover Publications, INC., New York, 1976)Google Scholar
  40. 40.
    J.M. Luttinger, J.C. Ward, Phys. Rev. 118, 1417 (1960)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    E.E. Saperstein, M. Baldo, N.V. Gnedzilov, S.V. Tolokonnikov, Phys. Rev. C 93, 034312 (2016)ADSCrossRefGoogle Scholar
  42. 42.
    E.E. Saperstein, M. Baldo, S.S. Pankratov, S.V. Tolokonnikov, J. Exp. Theor. Phys. Lett. 104, 609 (2016)CrossRefGoogle Scholar
  43. 43.
    E.E. Saperstein, M. Baldo, S.S. Pankratov, S.V. Tolokonnikov, J. Exp. Theor. Phys. Lett. 104, 743 (2016)CrossRefGoogle Scholar
  44. 44.
    E.E. Saperstein, M. Baldo, S.S. Pankratov, S.V. Tolokonnikov, Phys. Rev. C 97, 054324 (2018)ADSCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • P. F. Bortignon
    • 1
  • E. E. Saperstein
    • 2
    • 3
  • M. Baldo
    • 4
    Email author
  1. 1.Dipartimento di Fisica, Università degli Studi di Milano and INFN, Sezione di MilanoMilanoItaly
  2. 2.Kurchatov InstituteMoscowRussia
  3. 3.National Research Nuclear University MEPhIMoscowRussia
  4. 4.INFN, Sezione di CataniaCataniaItaly

Personalised recommendations