Density Oscillations in Nonuniform Systems

  • S. Lundqvist
Part of the Physics of Solids and Liquids book series (PSLI)


The dynamical properties of an inhomogeneous electron gas is a subject with a history almost as long as quantum mechanics. Because the subject is not widely known it will serve as a suitable introduction to this chapter to remind the reader about a few of the major steps. An important early problem was the theoretical understanding of the stopping of a fast charged particle in matter. A charged particle excites the medium with excitation energies covering a wide spectrum from far ultraviolet to soft X-ray frequencies. The stopping power itself does not depend on the details of the spectrum but only on an average excitation energy. The idea came up that one might replace the full dynamical theory by a simplified picture in which the medium was considered as an inhomogeneous electron gas. The charged particle would excite oscillations in the electron gas around its ground-state density and the particle would lose energy by exciting the various modes of excitation in the nonuniform electron gas. These ideas were developed in a classical paper by Bloch(1) in 1933. He developed a dynamical extension of the Thomas-Fermi theory treated in Chapter 1, considering the hydrodynamical oscillations of the density around the Thomas-Fermi ground-state density. Applications by Jensen(2) to a simplified model treating the atom as a small metallic sphere agreed with stopping power data in its dependence on atomic number and supported the hydrodynamical model. The Bloch equations were actually not fully solved until after the Second World War. Many extensions to include, e.g., exchange and correlations have been made. After the development of the density functional method presented in Chapter 2, a hydrodynamical approach based on the density-functional scheme was proposed by Ying et al.(3)


Hydrodynamical Model Free Oscillation Collective Motion Random Phase Approximation Photoabsorption Cross Section 
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  1. 1.
    F. Bloch, Z. Phys. 81, 363 (1933).ADSzbMATHCrossRefGoogle Scholar
  2. 2.
    H. Jensen, Z. Phys. 106, 620 (1937).ADSCrossRefGoogle Scholar
  3. 3.
    S. C. Ying, J.R. Smith, and W. Kohn, J. Vac. Sci. Technol. 9, 575 (1972).ADSCrossRefGoogle Scholar
  4. 4.
    J.A. Ball, J.A. Wheeler, and E. L. Fireman, Rev. Mod. Phys. 45, 333 (1973).ADSCrossRefGoogle Scholar
  5. 5.
    J. Lindhard, Dan. Mat. Fys. Medd. 28, 8 (1954).MathSciNetGoogle Scholar
  6. 6.
    P.A.M. Dirac, Proc. Cam. Phil. Soc. 26, 376 (1930).ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    W. Kohn and L. Sham, Phys. Rev. 140, Al 133 (1965).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    J.D. Walecka, Phys. Lett. 58A, 81 (1976).ADSGoogle Scholar
  9. 9.
    F.E. Serr, Phys. Lett. 62A, 325 (1977).ADSGoogle Scholar
  10. 10.
    P.B. Roulet and P. Noziéres, J. Phys. (Paris) 29, 167 (1968).CrossRefGoogle Scholar
  11. U.M. Ya. Amusia and N. A. Cherepkov, Case Studies in Atomic Physics 5, 47 (1976).Google Scholar
  12. 12.
    G. Wendin, in Photoionization and Other Probes of Many-Electron Interactions, F. Wuilleumier editor, NATO Advanced Study Institute series (Plenum Press, New York, 1976).Google Scholar
  13. 13.
    D.A. Kirzhnitz, Yu. E. Lozovik, and G.V. Shpatakovskaya, Sov. Phys. Usp. 18, 649 (1976).ADSCrossRefGoogle Scholar
  14. 14.
    S. Chakravarty, M.B. Vogel, and W. Kohn, Phys. Rev. Lett. 43, 775 (1979).ADSCrossRefGoogle Scholar
  15. 15.
    G.V. Gadiyak, D. A. Kirzhnits, and Yu. E. Lozovik, Sov. Phys.JETP 42, 62 (1976).ADSGoogle Scholar
  16. 16.
    W. Brandt and S. Lundqvist, Ark. Fys. 28, 399 (1965).Google Scholar
  17. 17.
    N.H. March and M.P. Tosi, Proc. R. Soc. (London) A330, 373 (1972).ADSGoogle Scholar
  18. 18.
    G. Mukhopadhyay and S. Lundqvist, Nuovo Cimento 27B, 1 (1975).ADSGoogle Scholar
  19. 19.
    G. Mukhopadhyay, Physica Scripta (in press).Google Scholar
  20. 20.
    O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976).ADSCrossRefGoogle Scholar
  21. 21.
    S.M. Valone and J.F. Capitani, Phys. Rev. A 23, 2127 (1981).ADSCrossRefGoogle Scholar
  22. 22.
    V. Peuckert, J. Phys. C: Solid State Phys. 11, 4945 (1978).ADSCrossRefGoogle Scholar
  23. 23.
    A. Zangwill and P. Soven, Phys. Rev. Lett. 45, 204 (1980).ADSCrossRefGoogle Scholar
  24. 24.
    A. Zangwill and P. Soven, Phys. Rev A 21, 1561 (1980).ADSCrossRefGoogle Scholar
  25. 25.
    A. Zangwill, Dissertation, University of Pennsylvania, 1981.Google Scholar
  26. 26.
    MJ. Stott and E. Zaremba, Phys. Rev. A 21, 12 (1980).ADSCrossRefGoogle Scholar
  27. 27.
    G.D. Mahan, Phys. Rev. A 22, 1780 (1980).MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    W. Brandt, L. Eder, and S. Lundqvist, J. Quant. Spectrosc. Radiat. Transfer 7, 185 (1967).ADSCrossRefGoogle Scholar
  29. 29.
    G. Barton, Rep. Progr. Phys. 42, 963 (1979).ADSCrossRefGoogle Scholar
  30. 30.
    PJ. Feibelman, Phys. Rev. B 9, 5077 (1974).ADSCrossRefGoogle Scholar
  31. 32.
    PJ. Feibelman, Phys. Rev. B 12, 1319 (1975).ADSCrossRefGoogle Scholar
  32. 33.
    P.J. Feibelman, Phys. Rev. B 14, 762 (1976).ADSCrossRefGoogle Scholar
  33. 34.
    J.E. Inglesfield and E. Wikborg, Solid State Commun. 14, 661 (1974).ADSCrossRefGoogle Scholar
  34. 35.
    N.D. Lang and W. Kohn, Phys. Rev. B 1, 4555 (1970).ADSCrossRefGoogle Scholar
  35. 36.
    G. Mukhopadhyay and S. Lundqvist, Phys. Scripta 17, 69 (1977).ADSCrossRefGoogle Scholar
  36. 37.
    P. Apell, Phys. Scripta 17, 535 (1977).ADSCrossRefGoogle Scholar
  37. 38.
    P. Apell, Phys. Scripta 25, 57 (1982).ADSCrossRefGoogle Scholar
  38. 39.
    J. Harris and A. Griffin, Phys. Lett. 34A, 51 (1971).ADSGoogle Scholar
  39. 40.
    F. Garcia-Moliner and F. Flores, Introduction to the Theory of Solid Surfaces (Cambridge University Press, 1979). The book contains references to the original papers.Google Scholar
  40. 41.
    P. Apell, Phys. Scripta 24, 795 (1981).ADSCrossRefGoogle Scholar
  41. 42.
    R.H. Ritchie, Phys. Rev. 106, 874 (1957).MathSciNetADSCrossRefGoogle Scholar
  42. 43.
    R.H. Ritchie and A. L. Marusak, Surf. Sci. 4, 234 (1966).ADSCrossRefGoogle Scholar
  43. 44.
    Ch. Heger and D. Wagner, Z. Phys. 244, 449 (1971).ADSCrossRefGoogle Scholar
  44. 45.
    E. Wikborg and J. E. Inglesfield, Phys. Scripta 15, 37 (1977).ADSCrossRefGoogle Scholar
  45. 46.
    A. Bagchi, C.B. Duke, P.J. Feibelman, and J.O. Porteus, Phys. Rev. Lett. 27, 998 (1971).ADSCrossRefGoogle Scholar
  46. 47.
    C.B. Duke and V. Landman, Phys. Rev. B 8, 505 (1973).ADSCrossRefGoogle Scholar
  47. 48.
    R. Ruppin, J. Phys. Chem. Solids 39, 233 (1978).ADSCrossRefGoogle Scholar
  48. 49.
    I. Lindgren and S. Lundqvist, editors, “Many-Body Theory of Atomic Systems,” Phys. Scripta 21, No. 3/4 (1980).Google Scholar
  49. 50.
    G. Mukhopadhyay and S. Lundqvist, J. Phys. B 12, 1297 (1979).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • S. Lundqvist
    • 1
  1. 1.Chalmers University of TechnologyGöteborgSweden

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