Density Functional Theory of Stopping Power

  • P. M. Echenique
  • M. E. Uranga
Chapter
Part of the Nato ASI Series book series (NSSB, volume 271)

Abstract

The density-functional (DF) formalism provides a very useful tool to tackle many-body problems in a simplified manner. In this approach, the many-body problem is replaced by a Hartree-like equation, such as

$$ \left[ { - \frac{{{\nabla ^{\text{2}}}}}{2} + {{\text{V}}_{{\text{eff}}}}\left( {\text{r}} \right)} \right]{\text{ }}{\psi _{\text{i}}}\left( {\text{r}} \right) = {\varepsilon _{\text{i}}}{\psi _{\text{i}}}\left( {\text{r}} \right). $$
(1)

(We use atomic units, e2 = h = m = 1 throughout except where otherwise indicated. The unit of energy is the Hartree (27.2 eV) and the Bohr’s radius a0 = 0.529 Å is the unit of length).

Keywords

Effective Charge Yukawa Potential Bear Approximation Scatter Phase Shift Transport Cross Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. 1.
    P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864.MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A11113.MathSciNetCrossRefGoogle Scholar
  3. 3.
    O. Gunnarsson, The density Functional Theory of Metallic Surfaces, P. Phariseau, B.L. Györffy, and L. Scheire, eds., NATO Advance Institutes Series, vol. 41, 1, Plenum (1979).Google Scholar
  4. 4.
    L.H. Thomas, Proc. Cam. Philos. Soc. 23 (1926) 542ADSCrossRefGoogle Scholar
  5. E. Fermi, Rc. Acc. Lincei, 6 (1927) 602.Google Scholar
  6. 5.
    N.H. March, Theory of the Inhomogeneous Electron Gas, S. Lundqvist and N.H. March, eds., Plenum (1983) 7.Google Scholar
  7. 6.
    P.A.M. Dirac, Proc. Camb. Philos. Soc.26 (1930) 376.ADSMATHCrossRefGoogle Scholar
  8. 7.
    J.C. Salter, Phys. Rev. 81 (1951) 385.ADSCrossRefGoogle Scholar
  9. 8.
    R. Caspar, Acta Phys. Hung. 3 (1954) 263.CrossRefGoogle Scholar
  10. 9.
    L. Hedin and S. Lundqvist, Solid State Physics, H. Ehrenreich, F. Seitz, and D. Turnbull, eds., 23 (1969) 1Google Scholar
  11. L. Hedin and B.I. Lundqvist, J. Phys. C 4 (1971) 2064.Google Scholar
  12. 10.
    O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B 13 (1976) 4274.ADSCrossRefGoogle Scholar
  13. 11.
    E. Bonderup, Lecture notes on the Penetration of Charged Particles Through Matter, University of Aarhus, Denmark (1978), unpublished.Google Scholar
  14. 12.
    P.M. Echenique, I. Nagy, and A. Arnau, Int. J. of Quantum Chem. 23 (1989) 521.Google Scholar
  15. 13.
    B.A. Trubnikov and Yu N. Yavlinski, Sov. Phys. JETP 21 (1965) 167.ADSGoogle Scholar
  16. 14.
    H.J. Davis and R. Dagonnier, J. Chem. Phys. 44 (1966) 4030.ADSCrossRefGoogle Scholar
  17. 15.
    J. Finneman, M. Sc. Dissertation, The Institute of Physics, Aarhus University, Denmark (1968), unpublished.Google Scholar
  18. 16.
    B.D. Josephson and J. Lekner, Phys. Rev. Lett. 23 (1969) 111.ADSCrossRefGoogle Scholar
  19. 17.
    E.G. d’Agliano, P. Kumar, W. Schaich, and M. Suhl, Phys. Rev. B 11 (1975) 2122.ADSCrossRefGoogle Scholar
  20. 18.
    A. Blandin, A. Nourtier, and D.W. Hone, J. Phys (Paris) 37 (1976) 369.CrossRefGoogle Scholar
  21. 19.
    P. Minnhagen, J. Phys. C 15 (1982) 2293.ADSCrossRefGoogle Scholar
  22. 20.
    P. Sigmund, Phys. Rev A 26 (1982) 24–97.CrossRefGoogle Scholar
  23. 21.
    K. Schonhammer, Phys. Rev. B 37 (1988) 7735.ADSCrossRefGoogle Scholar
  24. 22.
    L.D. Landau and E.M. Lifshitz, Physical Kinetics, Pergamon (1981) 41.Google Scholar
  25. 23.
    C. Kittel, Quantum Theory of Solids, Wiley, New York (1963).Google Scholar
  26. 24.
    J. Friedel, Phil. Mag. 43 (1952) 153MATHGoogle Scholar
  27. J. Friedel, Nuovo Cimento Suppl. 7 (1958) 287.CrossRefGoogle Scholar
  28. 25.
    J.S. Langer and V. Ambegaokar, Phys. Rev. 121 (1961) 1090.ADSMATHCrossRefGoogle Scholar
  29. 26.
    D.C. Langreth, Phys. Rev. 150 (1966) 516.ADSCrossRefGoogle Scholar
  30. 27.
    N. Levinson, Kgl. Dan. Videns K. Selsk. Mat.-Fys. Medd. 25 (1949) No.9.Google Scholar
  31. 28.
    A. Galindo and P. Pascual,Mecánica Cuántical, II, Eudema Universidad, Madrid (1989).Google Scholar
  32. 29.
    L.D. Landau and E.M. Lifshitz, Quantum Mechanism, Pergamon (1977).Google Scholar
  33. 30.
    G.D. Mahan, Many Particle Physics, Plenum, New York (1981).Google Scholar
  34. 31.
    C.O. Almbladh, U. von Barth, Z.D. Popovic, and M.J. Stott, Phys. Rev. B 14 (1976) 2250ADSCrossRefGoogle Scholar
  35. P. Jena and K.S. Singwi, Phys. Rev. B 17 (1978) 3518.ADSCrossRefGoogle Scholar
  36. 32.
    N. Barberan and P.M. Echenique, J. Phys. B 19 (1986) L81.ADSCrossRefGoogle Scholar
  37. 33.
    P.M. Echenique, R.M. Nieminen, and R.H. Ritchie, Solid State Commun. 37 (1981) 779.ADSCrossRefGoogle Scholar
  38. 34.
    M.J. Puska and R.M. Nieminen, Phys. Rev. B27 (1983) 6121.ADSCrossRefGoogle Scholar
  39. 35.
    T.L. Ferrell and R.H. Ritchie, Phys. Rev. B16 (1977) 115.ADSCrossRefGoogle Scholar
  40. 36.
    A. Mann and W. Brandt, Phys. Rev. B24 (1981) 4999.ADSCrossRefGoogle Scholar
  41. 37.
    P.M. Echenique, F. Flores, and R.H. Ritchie,Dynamic Screening of Ions in Condensed Waiter, Solid State Physics series, Erhenreich and Tumbu 11 eds., Academic 43 (1990) 229.Google Scholar
  42. 38.
    J. Lindhard and A. Winther, K. Dan. Vidensk. Selsk, Mat. Pys. Medd. 34 (1964) No.4.Google Scholar
  43. 39.
    J. Ziegler, J.P. Biersack, and U. Littmark, The Stopping and Range of Ions in Solids, Pergamon (1985).Google Scholar
  44. 40.
    I. Nagy, A. Arnau, P.M. Echenique, and E. Zaremba, Phys. Rev. B 40 (1989) 11 983.CrossRefGoogle Scholar
  45. 41.
    F. Flores, in this volumen.Google Scholar
  46. 42.
    A.H. Sorensen, Nucl. Instr. and Meth. B 48 (1990) 10.ADSCrossRefGoogle Scholar
  47. 43.
    I. Nagy and P.M. Echenique, Proceedings of the 12 W. Brandt Symposium, Donostia, Spain (1990).Google Scholar
  48. 44.
    W. Barkas, W. Birnbaum, and F.M. Smith, Phys. Rev. 101 (1956) 778.ADSCrossRefGoogle Scholar
  49. 45.
    W.H. Barkas, N.J. Dyer, and H.H. Heckman, Phys. Rev. Lett 11 (1963) 26.ADSCrossRefGoogle Scholar
  50. 46.
    H.H. Andersen, in Semiclassical Descriptions of Atomic and Nuclear Collisions, J. Bang and J. De Boer, eds., Elsevier, Amsterdam, (1985) 409.Google Scholar
  51. 47.
    G. Basbas, Nucl. Instr. and Meth. B 4 (1984) 227.ADSCrossRefGoogle Scholar
  52. 48.
    C.D. Hu and E. Zaremba, Phys. Rev. B 37 (1988) 9268.ADSCrossRefGoogle Scholar
  53. 49.
    S.H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58 (1980) 1200.ADSCrossRefGoogle Scholar
  54. 50.
    A. Cherubini and A. Ventura, Lett. Nuovo Cimento 44 (1985) 503.CrossRefGoogle Scholar
  55. 51.
    B. Apagyi and I Nagy, J. Phys. C 20 (1987) 1465.ADSCrossRefGoogle Scholar
  56. 52.
    B. Apagyi and I Nagy, J. Phys. C 21 (1988) 3845.ADSCrossRefGoogle Scholar
  57. 53.
    I. Nagy, in this book.Google Scholar
  58. 54.
    E. Steiner, J. Chem. Phys. 39 (1963) 2365.ADSCrossRefGoogle Scholar
  59. 55.
    A. Kallio, P. Pietiläinen and L. Lantto, Physica Scripta 25 (1982) 943.ADSCrossRefGoogle Scholar
  60. 56.
    P.M. Echenique, R.M. Nieminen, J.C. Ashley, and R.H. Ritchie, Phys. Rev. A33 (1986) 897.ADSCrossRefGoogle Scholar
  61. 57.
    F. Shulz and W. Brandt, Phys. Rev. B 26 (1982) 4864.ADSCrossRefGoogle Scholar
  62. 58.
    W. Brandt, Nucl. Instr. and Meth. 194 (1982) 13.ADSCrossRefGoogle Scholar
  63. 59.
    W. Brandt and M. Kitagawa, Phys. Rev. B 25 (1982) 5631.ADSCrossRefGoogle Scholar
  64. 60.
    J.C. Ashley, R.H. Ritchie, P.M. Echenique, and R.M. Nieminen, Nucl. Instr. and Meth. B 15 (1986) 11.ADSCrossRefGoogle Scholar
  65. 61.
    P.M. Echenique, Nucl. Instr. and Meth. B 27 (1987) 256.ADSCrossRefGoogle Scholar
  66. 62.
    F.H. Eisen, Can. J. Phys. 46 (1968) 561.ADSCrossRefGoogle Scholar
  67. 63.
    J.S. Briggs and A. Pathak, J. Phys. C 7 (1974) 1929, display F.H. Eisen’s previously unpublished data for the <111> channel in Si.Google Scholar
  68. 64.
    J.A. Golovchenko, D.E. Cox, and A.N. Goland, Phys. Rev. B 26 (1982) 2335.ADSCrossRefGoogle Scholar
  69. 65.
    I. Nagy, A. Arnau, and P.M. Echenique, Phys. Rev. B 38 (1988) 9191.ADSCrossRefGoogle Scholar
  70. 66.
    J.C. Ashley, A. Gras-Marti, and P.M. Echenique, Phys. Rev. A 34 (1986) 2495.ADSCrossRefGoogle Scholar
  71. 67.
    A. Arnau, P.M. Echenique, and R.H. Ritchie, Nucl. Instr. and Meth. B 33 (1988) 138.ADSCrossRefGoogle Scholar
  72. 68.
    E. Muller-Hartmann, T.V. Ramakrishnan, and G. Toulouse, Phys. Rev. B 3 (1971) 1102.ADSCrossRefGoogle Scholar
  73. 69.
    I. Nagy, A. Arnau, and P.M. Echenique, Phys. Rev. A 40 (1989) 987.ADSCrossRefGoogle Scholar
  74. 70.
    A.A. Kugler, J. St. Phys. 12 (1975) 35.ADSCrossRefGoogle Scholar
  75. 71.
    S. Ichimaru, Rev. Mod. Phys. 54 (1982) 1017.ADSCrossRefGoogle Scholar
  76. 72.
    R.G. Dandrea, N.W. Ashcroft, and A.E. Carlson, Phys. Rev. B 34 (1986) 2097.ADSCrossRefGoogle Scholar
  77. 73.
    E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 55 (1985) 2850ADSCrossRefGoogle Scholar
  78. E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 57 (1986) 923(E).ADSGoogle Scholar
  79. 74.
    N. Iwamoto and E.K.U. Gross, Phys. Rev. B 35 (1987) 3003.ADSCrossRefGoogle Scholar
  80. 75.
    N. Iwamoto, Phys. Rev. A 30 (1984) 3289.ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • P. M. Echenique
    • 1
  • M. E. Uranga
    • 2
  1. 1.Dpto. de Física de Materiales, Facultad de QuímicaUniversidad del País Vasco/Euskal Herriko UnibertsitateaSan SebastiánSpain
  2. 2.Dpto. de Matemática Aplicada, E.T.S.I. IndustrialesUniversidad del País Vasco/Euskal Herriko UnibertsitateaBilbaoSpain

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