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The Atom-Surface Potential

  • Giorgio Benedek
  • Jan Peter Toennies
Chapter
Part of the Springer Series in Surface Sciences book series (SSSUR, volume 63)

Abstract

In this chapter the various models and approximations used for describing the static atom-surface potential are reviewed and analyzed. The potentials are of fundamental importance for the calculation of the elastic scattering of atoms from surfaces. These potentials also are the basis for the calculations of the inelastic scattering from surface phonons on insulator surfaces, discussed in detail in Chap.  7. In the case of metal and semiconductor surfaces the semi-empirical parameters used to describe the modifications of the potential resulting from the spill-out of the conduction electrons are discussed. The theory of inelastic surface phonon scattering from metals and semiconductors is the subject of Chap.  8.

References

  1. 1.
    J.P. Toennies, Appl. Phys. 3, 91 (1974)ADSCrossRefGoogle Scholar
  2. 2.
    V. Celli, D. Evans, in Dynamics of Gas-Surface Interaction, ed. by G. Benedek, U. Valbusa (Springer, Berlin Heidelberg, 1982), p. 2Google Scholar
  3. 3.
    V. Bortolani, A.C. Levi, Riv. Nuovo Cim. 9, 1 (1986)ADSCrossRefGoogle Scholar
  4. 4.
    J.L. Beeby, J. Phys. Part C Solid State Phys. 4(18), L359–000 (1971)ADSCrossRefGoogle Scholar
  5. 5.
    H. Legge, J.R. Manson, J.P. Toennies, J. Chem. Phy. 110(17), 8767–8785 (1999)ADSCrossRefGoogle Scholar
  6. 6.
    U. Garibaldi, A.C. Levi, R. Spadacini, G.E. Tommei, Surf. Sci. 55(1), 40–60 (1976)ADSCrossRefGoogle Scholar
  7. 7.
    G. Boato, P. Cantini, L. Matera, Surf. Sci. 55(1), 141–178 (1976)ADSCrossRefGoogle Scholar
  8. 8.
    G. Benedek, G. Brusdeylins, R.B. Doak, J.G. Skofronik, J.P. Toennies, Phys. Rev. B 28(4), 2104–2113 (1983)ADSCrossRefGoogle Scholar
  9. 9.
    P. Cantini, R. Tatarek, G.P. Felcher, Phys. Rev. B 19(2), 1161–1171 (1979)ADSCrossRefGoogle Scholar
  10. 10.
    H. Hoinkes, Rev. Mod. Phys. 52(4), 933–970 (1980)ADSCrossRefGoogle Scholar
  11. 11.
    R. Martinez-Casado, B. Meyer, S. Miret-Artes, F. Traeger, C. Woll, J. Phys.-Condensed Matter 19(30), (2007)Google Scholar
  12. 12.
    W.E. Carlos, M.W. Cole, Surf. Sci. 77(1), L173–L176 (1978)ADSCrossRefGoogle Scholar
  13. 13.
    L. Mattera, R. Musenich, C. Salvo, S. Terreni, Faraday Discuss. 80, 115–126 (1985)CrossRefGoogle Scholar
  14. 14.
    A. Luntz, L. Mattera, M. Rocca, S. Terreni, F. Tommasini, U. Valbusa, Surf. Sci. 126(1–3), 695–701 (1983)Google Scholar
  15. 15.
    J. Lapujoulade, J. Perreau, Phys. Scr. T4, 138–140 (1983)ADSCrossRefGoogle Scholar
  16. 16.
    K.H. Rieder, W. Stocker, Phys. Rev. B 31(6), 3392–3497 (1985)ADSCrossRefGoogle Scholar
  17. 17.
    A.L. Glebov, J.P. Toennies, F. Träger, Phys. Rev. Lett. 82, 4492 (1999)ADSCrossRefGoogle Scholar
  18. 18.
    K.H. Rieder, Surf. Sci. 117(1–3), 13–22 (1982)ADSCrossRefGoogle Scholar
  19. 19.
    D.O. Hayward, A.O. Taylor, J. Phy. C-Solid State Phys. 19(15), L309–L314 (1986)ADSCrossRefGoogle Scholar
  20. 20.
    G. Parschau, E. Kirsten, A. Bischof, K.H. Rieder, Physi. Rev. B 40(9), 6012–6017 (1989)ADSCrossRefGoogle Scholar
  21. 21.
    K.H. Rieder, W. Stocker, J. Phys. C-Solid State Phys. 16(22), L783–L788 (1983)ADSCrossRefGoogle Scholar
  22. 22.
    D. Cvetko, A. Morgante, A. Santaniello, F. Tommasini, J. Chem. Phy. 104(19), 7778–7783 (1996)ADSCrossRefGoogle Scholar
  23. 23.
    A. Tamtögl, M. Mayrhofer-Reinhartshuber, N. Balak, W.E. Ernst, K.H. Rieder, J. Phys.: Condens. Matter 22, 304019 (2010)Google Scholar
  24. 24.
    M. Mayrhofer-Reinhartshuber, A. Tamtögl, P. Kraus, K.H. Rieder, W.E. Ernst, J. Phys.: Condens. Matter 24, 104008 (2012)ADSGoogle Scholar
  25. 25.
    A. Tamtögl, P. Kraus, M. Mayrhofer-Reinhartshuber, D. Campi, M. Bernasconi, G. Benedek, W.E. Ernst, Phys. Rev. B 87, 035410 (2013)ADSCrossRefGoogle Scholar
  26. 26.
    U. Buck, Adv. Chem. Phys. 36, 313 (1975)Google Scholar
  27. 27.
    K.T. Tang, J.P. Toennies, J. Chem. Phys. 118, 4976 (2003)ADSCrossRefGoogle Scholar
  28. 28.
    W.A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon Press, Oxford, 1974), Chap. 2Google Scholar
  29. 29.
    V. Celli, in Surface Phonons, ed. by W. Kress, F.W. de Wette (Springer, Berlin, 1991), p. 167Google Scholar
  30. 30.
    V. Celli, in Helium Atom Scattering from Surfaces, ed. by E. Hulpke (Springer, Berlin, 1992), p. 25Google Scholar
  31. 31.
    L.W. Bruch, M.W. Cole, E. Zaremba, Int. Series of Monographs on Chemistry, vol. 33 (Clarendon, New York, 1997), Chap. 2Google Scholar
  32. 32.
    A.J. Stone, The Theory of Intermolecular Forces (Clarendon, Oxford, 1996)Google Scholar
  33. 33.
    I.G. Kaplan, Intermolecular Interactions (Wiley, Hoboken, NJ, 2006); Theory of Molecular Interactions (Elsevier, Amsterdam 1986).Google Scholar
  34. 34.
    F. London, Z. Physik, 63, 245 (1930) and Z. Physik. Chemie B 11, 222 (1930); English translations in H. Hettema, Quantum Chemistry, Classic Scientific Papers, World Scientific, Singapore (2000)Google Scholar
  35. 35.
    H.B.G. Casimir, D. Polder, Phys. Rev. 73, 360 (1948)ADSCrossRefGoogle Scholar
  36. 36.
    N. Esbjerg, J.K. Nørskov, Phys. Rev. Lett. 45, 807 (1980)ADSCrossRefGoogle Scholar
  37. 37.
    K.T. Tang, J.M. Norbeck, P.R. Certain, J. Chem. Phys. 64, 3063 (1976)ADSCrossRefGoogle Scholar
  38. 38.
    A.A. Radzig, B.M. Smirnov, Reference Data on Atoms, Molecules, and Ions (Springer, Berlin, Heidelberg, New York, Tokyo, 1980)Google Scholar
  39. 39.
    S.H. Patil, K.T. Tang. Asymptotic Methods in Quantum Mechanics (Springer, 2008)Google Scholar
  40. 40.
    K. Rosciszewski, B. Paulus, P. Fulde, H. Stoll, Phys. Rev. B 60, 7905 (1999)ADSCrossRefGoogle Scholar
  41. 41.
    P.L. Silvestrelli, Phys. Rev. Lett. 100, 053002 (2008)ADSCrossRefGoogle Scholar
  42. 42.
    A. Tkatchenko, M. Scheffler, Phys. Rev. Lett. 102, 073005 (2009)ADSCrossRefGoogle Scholar
  43. 43.
    T. Sato, H. Nakai, J. Chem Phys. 131, 224104 (2009)ADSCrossRefGoogle Scholar
  44. 44.
    N.A. de Lima, J. Chem. Phys. 132(1) (2010)Google Scholar
  45. 45.
    J. Jiang, J. Mitroy, Y. Cheng, M.W.J. Bromley, At. Data Nucl. Data Tables 101, 158 (2015)ADSCrossRefGoogle Scholar
  46. 46.
    L. Miglio, F. Quasso, G. Benedek, Surf. Sci. 136, L9 (1984)ADSCrossRefGoogle Scholar
  47. 47.
    L. Miglio, F. Quasso, G. Benedek, J. Chem. Phys. 83, 913 (1985)CrossRefGoogle Scholar
  48. 48.
    M. Born, J.E. Mayer, Z. Physik (1932) 1Google Scholar
  49. 49.
    E.S. Rittner, J. Chem. Phys. 19, 1030 (1951)ADSCrossRefGoogle Scholar
  50. 50.
    M.P. Tosi, F.G. Fumi, J. Phys. Chem. Solids 25, 45 (1964)ADSCrossRefGoogle Scholar
  51. 51.
    M.P. Tosi, Solid State Physics 16, 1–120 (1964)CrossRefGoogle Scholar
  52. 52.
    C. Zener, Phys. Rev. 40, 178 and 335 (1932)ADSzbMATHCrossRefGoogle Scholar
  53. 53.
    J.M. Jackson, N.F. Mott, Proc. Roy. Soc. (London) A 137, 703 (1932)Google Scholar
  54. 54.
    G. Mie, Ann. Phys. 11, 657 (1903)CrossRefGoogle Scholar
  55. 55.
    The 12–6 Lennard-Jones potential is a special case of the general n-m potential first introduced by Mie [Ann. der Physik 11 (1903) 657] and thoroughly discussed in E.A. Moelwyn-Hughes, Physical Chemistry (Pergamon, Oxford, 1978), p. 311Google Scholar
  56. 56.
    J.E. Lennard-Jones, Proc. Royal Soc. Lond. A 106, 463 (1924)ADSCrossRefGoogle Scholar
  57. 57.
    P.M. Morse, Phys. Rev. 34, 57–64 (1929)ADSCrossRefGoogle Scholar
  58. 58.
    I.G. Kaplan, in Handbook of Molecular Physics and Quantum Chemistry (Wiley, 2003), p. 207Google Scholar
  59. 59.
    R.E. Grisenti, W. Schollkopf, J.P. Toennies, G.C. Hegerfeldt, T. Köhler, Phys. Rev. Lett. 83(9), 1755–1758 (1999)ADSCrossRefGoogle Scholar
  60. 60.
    R. Bruhl, P. Fouquet, R.E. Grisenti, J.P. Toennies, G.C. Hegerfeldt, T. Köhler, M. Stoll, C. Walter, Europhys. Lett. 59(3), 357–363 (2002)ADSCrossRefGoogle Scholar
  61. 61.
    K.T. Tang, J.P. Toennies, J. Chem. Phys. 118(11), 4976 (2003)ADSCrossRefGoogle Scholar
  62. 62.
    H. Patil, K.T. Tang, J.P. Toennies, J. Chem. Phys. 116, 8118 (2002)ADSCrossRefGoogle Scholar
  63. 63.
    R. Eisenschitz, F. London, Zeitschrift für Physik, vol. 60, p. 491 (1930). English translation in H. Hettema, Quantum Chemistry, Classic Scientific Papers, World Scientific, Singapore (2000)Google Scholar
  64. 64.
    Note that the classical Clausius-Mossotti (CM) relation \( 4\pi \,\alpha (i\omega )/3v_{c} = [\varepsilon (i\omega ) - 1]/[\varepsilon (i\omega ) + 2] \) is obtained by integrating over a spherical distribution of dipoles around a given probe dipole. This holds for gases and homogeneous bulk media, whereas for the surface geometry the integration gives the CM relation in the form \( 2\pi \,\alpha (i\omega )/v_{c} = [\varepsilon (i\omega ) - 1]/[\varepsilon (i\omega ) + 1] \). Substitution of this equation into (6.19) yields the Casimir-Polder formula for \( C_{6} = (6v_{c} /\pi ){\kern 1pt} c_{3} \), (6.13)Google Scholar
  65. 65.
    E.M. Lifshitz, Sov. Phys. 2, 73 (1956)Google Scholar
  66. 66.
    G. Vidali, M.W. Cole, Surf. Sci. 110, 10 (1981)ADSCrossRefGoogle Scholar
  67. 67.
    J. Jiang, J. Mitroy, Y. Cheng, M.W.J. Bromley, At. Data Nucl. Data Tables 101, 158 (2015)Google Scholar
  68. 68.
    X.-P. Jiang, F. Toigo, M.W. Cole, Surf. Sci. 145, 281 (1984)ADSCrossRefGoogle Scholar
  69. 69.
    J.M. Hutson, P.W. Fowler, E. Zaremba, Surf. Sci. 175(2), L775–L781 (1986)ADSCrossRefGoogle Scholar
  70. 70.
    M. Dion, H. Rydberg, E. Schröder, D.C. Langreth, B.I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004)ADSCrossRefGoogle Scholar
  71. 71.
    D.C. Langreth, M. Dion, H. Rydberg, E. Schröder, P. Hyldgaard, B.I. Lundqvist, Int. J. Quantum Chem. 101, 599 (2005)CrossRefGoogle Scholar
  72. 72.
    S. Grimme, J. Comput. Chem. 27, 1787 (2006)CrossRefGoogle Scholar
  73. 73.
    A. Tkatchenko, M. Scheffler, Phys. Rev. Lett. 102, 073005 (2009)ADSCrossRefGoogle Scholar
  74. 74.
    R. Martinez-Casado, G. Mallia, D. Usvyat, L. Maschio, S. Cassasa, M. Schutz, N.M. Harrison, J. Chem. Phys. 134(1), (2011)Google Scholar
  75. 75.
    J. Klimes, A. Michaelides, J. Chem. Phys. 137(12) (2012)Google Scholar
  76. 76.
    K. Lee, K. Berland, M. Yoon, S. Andersson, E. Schroder, P. Hyldgaard, B.I. Lundqvist, J. Phys.-Condensed Matter 24(42), 424213 (2012)ADSCrossRefGoogle Scholar
  77. 77.
    P.L. Silvestrelli, A. Ambrosetti, S. Grubisic, F. Ancilotto, Phys. Rev. B 85(16) (2012)Google Scholar
  78. 78.
    K. Berland, V.R. Cooper, K. Lee, E. Schröder, T. Thonhauser, P. Hyldgaard, B.I. Lundqvist, Rep. Prog. Phys. 78, 066501 (2015)ADSCrossRefGoogle Scholar
  79. 79.
    J. Tao, A.M. Rappe, Phys. Rev. Lett. 111, 106101 (2014)ADSCrossRefGoogle Scholar
  80. 80.
    J. Tao, A.M. Rappe, J. Chem. Phys. 144, 031102 (2016)ADSCrossRefGoogle Scholar
  81. 81.
    A. Erdélyi (ed.), Higher Trascendental Functions (McGraw Hill, New York, 1953)Google Scholar
  82. 82.
    A. Tsuchida, Sci. Papers Inst. Phys. Chem. Res. (Tokyo) 64, 75 (1970); J. Cryst. Soc. Japan, 14, 17 (1972); Surf. Sci. 46, 611 (1974).Google Scholar
  83. 83.
    V. Celli, Interaction of Atoms with Surfaces. in Landolt-Börnstein ed. by G. Chiarotti, vol. 24 (Springer, Berlin, Heidelberg, 1994/1995), pp. 278–328,Google Scholar
  84. 84.
    E. Zaremba, W. Kohn, Phys. Rev. B 13, 2270 (1976)ADSCrossRefGoogle Scholar
  85. 85.
    N.D. Lang, W. Kohn, Phys. Rev. B 7, 3541 (1973)ADSCrossRefGoogle Scholar
  86. 86.
    A. Chizmeshya, E. Zaremba, Surf. Sci. 220(2–3), 443–470 (1989)ADSCrossRefGoogle Scholar
  87. 87.
    A. Liebsch, Phys. Rev. B 33(10), 7249–7251 (1986)ADSCrossRefGoogle Scholar
  88. 88.
    O.A. von Lilienfeld, A. Tkatchenko, J. Chem. Phys. 132, 234109 (2010)ADSCrossRefGoogle Scholar
  89. 89.
    V.V. Gobre, A. Tkatchenko, Nat. Commun. 4, 2341 (2013)ADSCrossRefGoogle Scholar
  90. 90.
    G. Vidali, G. Ihm, H.-Y. Kim, M.W. Cole, Surf. Sci. Rep. 12, 133 (1991)ADSCrossRefGoogle Scholar
  91. 91.
    S. Lepoutre, V.P.A. Lomij, H. Jelassi, J. Trenec, M. Buchner, A.D. Cronin, J. Vigue, Eur. Phys. J. D 62, 309 (2011)ADSCrossRefGoogle Scholar
  92. 92.
    V.P.A. Lomij, E.K. Catherine, W.F. Holmgren, A.D. CroninJ, Phys. Chem A 115, 71234 (2011)Google Scholar
  93. 93.
    P.W. Fowler, J.M. Hutson, Phys. Rev. B 33, 3724 (1986) (for He-LiF)ADSCrossRefGoogle Scholar
  94. 94.
    D.J. Riley, A.P. Jardine, S. Dworski, G. Alexandrowicz, P. Fouquet, J. Ellis, W. Allison, J. Chem. Phys. 126, 104702 (2007)ADSCrossRefGoogle Scholar
  95. 95.
    V. Celli, D. Eichenauer, A. Kaufhold, J.P. Toennies, J. Chem. Phys. 86, 2504 (1985)ADSCrossRefGoogle Scholar
  96. 96.
    G. Benedek, G. Brusdeylins, V. Senzet, J.G. Skofronik, J.P. Tonnies, F. TRaeger, F. Vollmer, Phys. Rev. B 64(12) (2001)Google Scholar
  97. 97.
    R.E. Grisenti, W. Schöllkopf, J.P. Toennies, G.C. Hegerfeldt, T. Köhler, Phys. Rev. Lett. 83, 1755 (1999)ADSCrossRefGoogle Scholar
  98. 98.
    J.A. Kunc, D.E. Shemansky, Surf. Sci. 163(1), 237–248 (1985)ADSCrossRefGoogle Scholar
  99. 99.
    A. Siber, B. Gumhalter, A.P. Graham, J.P. Toennies, Phys. Rev. B 63(11) (2001)Google Scholar
  100. 100.
    R.B. Laughlin, Phys. Rev. B 25(4), 2222–2247 (1982)ADSCrossRefGoogle Scholar
  101. 101.
    W.R. Lambert, P.L. Trevor, M.J. Cardillo, A. Sakai, D.R. Hamann, Phys. Rev. B 35(15), 8055–8064 (1987)ADSCrossRefGoogle Scholar
  102. 102.
    J.R. Buckland, W. Allison, J. Chem. Phy. 112(2), 970–978 (2000)ADSCrossRefGoogle Scholar
  103. 103.
    D. Eichenauer, U. Harten, J.P. Toennies, V. Celli, J. Chem. Phys. 86, 3693 (1987)ADSCrossRefGoogle Scholar
  104. 104.
    K. Lenarcicpoljanec, M. Hodoscek, D. Lovrik, B. Gumhalter, Surf. Sci. 251, 706–711 (1991)ADSCrossRefGoogle Scholar
  105. 105.
    G. Lach, M. Dekieviet, U.D. Jentschura, Int. J. Mod. Phys. A 25(11), 2337–2344 (2010)ADSCrossRefGoogle Scholar
  106. 106.
    T. Andersson, P. Linde, M. Hassel, S. Andersson, J. Chem. Phys. 124(11) (2006)Google Scholar
  107. 107.
    G.-K. Kroes, R.C. Mowrey, J. Chem. Phys. 103, 2186 (1995)ADSCrossRefGoogle Scholar
  108. 108.
    S. Andersson, M. Persson, J. Harris, Surf. Sci. 360, L499–L504 (1996)ADSCrossRefGoogle Scholar
  109. 109.
    W.A. Steele, Surf. Sci. 36, 317 (1973)ADSCrossRefGoogle Scholar
  110. 110.
    M.W. Cole, T.T. Tsong, Surf. Sci. 69(1), 325–335 (1977)ADSCrossRefGoogle Scholar
  111. 111.
    C. Kittel, Introduction to Solid State Physics (John Wiley and Sons, Inc. 2005), Chap. 2Google Scholar
  112. 112.
    A. Erdélyi, Asymptotic expansions (Dover Publications Inc., New York, 1956)zbMATHGoogle Scholar
  113. 113.
    V. Celli, N. Garcia, J. Hutchison, Surf. Sci. 87, 112 (1979)ADSCrossRefGoogle Scholar
  114. 114.
    D. Farias, K.-H. Rieder, Rep. Prog. Phys. 61, 1575 (1998)ADSCrossRefGoogle Scholar
  115. 115.
    H. Jónnson, J.H. Weare, Surf. Sci. 181, 495 (1987)ADSCrossRefGoogle Scholar
  116. 116.
    F.G. Fumi, M.P. Tosi, J. Phys. Chem. Solids 25, 31 (1964)ADSCrossRefGoogle Scholar
  117. 117.
    K.T. Tang, J.P. Toennies, J. Chem. Phys. 80, 3726 (1984)ADSCrossRefGoogle Scholar
  118. 118.
    J.M. Hutson, P.W. Fowler, Surf. Sci. 173, 337 (1986) (for He-NaCl)Google Scholar
  119. 119.
    D. Eichenauer, J.P. Toennies, Surf. Sci. 197, 267 (1988)ADSCrossRefGoogle Scholar
  120. 120.
    A.P. Jardine, S. Dworski, P. Fouquet, G. Alexandrowicz, D.J. Riley, G.Y.H. Lee, J. Ellis, W. Allison, Science 304, 1790 (2004)ADSCrossRefGoogle Scholar
  121. 121.
    D. Campi, M. Bernasconi, G. Benedek, J.P. Toennies, J. Phys. Chem. C 119, 14579 (2015)CrossRefGoogle Scholar
  122. 122.
    G. Boato, P. Cantini, R. Tatarek, J. Phys. F: Metal Phys. 6, L237 (1976)ADSCrossRefGoogle Scholar
  123. 123.
    J.M. Horne, D.R. Miller, Surf. Sci. 66, 365 (1977)ADSCrossRefGoogle Scholar
  124. 124.
    R. Smoluchowski, Phys. Rev. 60, 661 (1941)ADSCrossRefGoogle Scholar
  125. 125.
    M.W. Finnis, V. Heine, J. Phys. F (Metal Phys). 4, L37 (1974)ADSCrossRefGoogle Scholar
  126. 126.
    N. Esbjerg, J.K. Nørskov, Phys. Rev. Lett. 45, 807 (1980)ADSCrossRefGoogle Scholar
  127. 127.
    A.C. Levi, H. Suhl, Surf. Sci. 88, 221–254 (1979)ADSCrossRefGoogle Scholar
  128. 128.
    J.F. Annett, R. Haydock, Phys. Rev. Lett. 53, 838 (1984)ADSCrossRefGoogle Scholar
  129. 129.
    N. Garcia, J.M. Soler, Surface Sci. 126, 689 (1983)ADSCrossRefGoogle Scholar
  130. 130.
    Y. Takeda, W. Kohn, Phys. Rev. Lett. 54, 470 (1985)ADSCrossRefGoogle Scholar
  131. 131.
    I.P. Batra, P.S. Bagus, J.A. Barker, Phys. Rev. B 31, 1737 (1985)ADSCrossRefGoogle Scholar
  132. 132.
    G.P. Brivio, M.I. Trioni, Rev. Mod. Phys. 71, 231 (1999)ADSCrossRefGoogle Scholar
  133. 133.
    P. Nordlander, J. Harris, J. Phys. C-Solid State Phys. 17(6), 1141–1152 (1984)ADSCrossRefGoogle Scholar
  134. 134.
    J.L.F. Da Silva, C. Stampfl, M. Scheffler, Phys. Rev. Lett. 90, 066104 (2003)ADSCrossRefGoogle Scholar
  135. 135.
    P.L. Silvestrelli, Phys. Rev. Lett. 100, 053002 (2008); J. Phys. Chem. A 113, 5224 (2009)Google Scholar
  136. 136.
    J.P. Senet, G. Benedek, J.P. Toennies, Europhys. Lett. 57, 430 (2002)ADSCrossRefGoogle Scholar
  137. 137.
    J. Harris, A. Liebsch, Phys. Rev. Lett. 49, 341 (1982)ADSCrossRefGoogle Scholar
  138. 138.
    A. Liebsch, J. Harris, Surface Sci. 111, 2721 (1981)CrossRefGoogle Scholar
  139. 139.
    A. Liebsch, Phys. Rev. B 33, 7249 (1986)ADSCrossRefGoogle Scholar
  140. 140.
    N.D. Lang, Phys. Rev. Lett. 46, 842 (1981)ADSCrossRefGoogle Scholar
  141. 141.
    E. Zaremba, W. Kohn, Phys. Rev. B 15, 12769 (1977)CrossRefGoogle Scholar
  142. 142.
    H. Hoinkes, H. Nahr, H. Wilsch, Surf. Sci. 33, 516–524 (1972)ADSCrossRefGoogle Scholar
  143. 143.
    G. Armand, J. Lapujoulade, Y. Lejay, Surf. Sci. 63, 143–152 (1977)ADSCrossRefGoogle Scholar
  144. 144.
    B. Feuerbacher, R.F. Willis, Phys. Rev. Lett. 47, 526 (1981)ADSCrossRefGoogle Scholar
  145. 145.
    V. Bortolani, A. Franchini, F. Nizzoli, G. Santoro, G. Benedek, V. Celli, Surf. Sci. 128, 249 (1983)ADSCrossRefGoogle Scholar
  146. 146.
    F. Hofmann, J.P. Toennies, J.R. Manson, J. Chem. Phys. 101, 20155 (1994)CrossRefGoogle Scholar
  147. 147.
    V. Bortolani, A. Franchini, N. Garcia, F. Nizzoli, G. Santoro, Phys. Rev. B 28, 7358 (1983)ADSCrossRefGoogle Scholar
  148. 148.
    A. Lock, Diplom Thesis, University of Göttingen, (1987)Google Scholar
  149. 149.
    N. Bunjes, N.S. Luo, P. Ruggerone, J.P. Toennies, G. Witte, Phys. Rev. B 50, 8897 (1994-II)Google Scholar
  150. 150.
    V. Celli, G. Benedek, U. Harten, J.P. Toennies, R.B. Doak, V. Bortolani, Surf. Sci. 143, L376 (1984)ADSCrossRefGoogle Scholar
  151. 151.
    D.R. Hamann, Phys. Rev. Lett. 46, 1227 (1981)ADSCrossRefGoogle Scholar
  152. 152.
    K. Kern, Ph.D. Thesis, University of Bonn, (1986)Google Scholar
  153. 153.
    J.P. Toennies, G. Witte, Ch. Wöll, Surf. Sci. 323, 1228 (1995)Google Scholar
  154. 154.
    K.H. Rieder, G. Parschau, B. Burg, Phys. Rev. Lett. 71, 1059 (1993)ADSCrossRefGoogle Scholar
  155. 155.
    M. Petersen, S. Wilke, P. Ruggerone, B. Kohler, M. Scheffler, Phys. Rev. Lett. 76, 995 (1996)ADSCrossRefGoogle Scholar
  156. 156.
    F. Montalenti, M.I. Trioni, G.P. Brivio, S. Crampin, Surf. Sci. Lett. 364, L595 (1996); M.I. Trioni, F. Montalenti and G.P. Brivio, ibidem 401, L383 (1998)Google Scholar
  157. 157.
    M.I. Trioni, S. Marcotulio, G. Santoro, V. Bortolani, G. Palumbo, G.P. Brivio, Phys. Rev. B 58, 11043 (1998)ADSCrossRefGoogle Scholar
  158. 158.
    N. Jean, M.I. Trioni, G.P. Brivio, V. Bortolani, Phys. Rev. Lett. 92, 013201 (2004)ADSCrossRefGoogle Scholar
  159. 159.
    G. Santoro, A. Franchini, V. Bortolani, Phys. Rev. Lett. 80, 2378 (1998)ADSCrossRefGoogle Scholar
  160. 160.
    G. Santoro, A. Franchini, V. Bortolani, D.L. Mills, R.F. Wallis, Surf. Sci. 478, 99 (2001)ADSCrossRefGoogle Scholar
  161. 161.
    A. Franchini, G. Santoro, J. Phys.-Cond. Matter 14, 5881 (2002)Google Scholar
  162. 162.
    K.T. Tang, J.P. Toennies, Surf. Sci. Lett. 279, L203 (1992)ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Università di Milano-BicoccaMilanItaly
  2. 2.Max Planck Institute for Dynamics and Self-OrganizationGöttingenGermany
  3. 3.Donostia International Physics CenterDonostia/San SebastianSpain

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