Advertisement

Theory of Atom Scattering from Surface Phonons: Basic Concepts and Temperature Effects

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

Abstract

The reflection coefficient determining the inelastic scattered intensity is defined and analysed theoretically. Various approximations including the Born approximation, the Distorted-Wave Born approximation (DWBA), the GR method and the eikonal approximation are discussed and illustrated with examples. From an expression for the DWBA inelastic reflection coefficient for n-phonon processes, factors determining the relative role of temperature for multiphonon processes and one-phonon scattering are analysed. Several approximate expressions for the Debye-Waller factor are derived and used to define experimental criteria favouring one-phonon scattering. Most of the examples dealt with are for insulator surfaces. Chapter  8 then describes the scattering theory for metal and semiconductor surfaces.

References

  1. 1.
    L. Miglio, F. Quasso, G. Benedek, Surf. Sci. 136, L9 (1984)CrossRefADSGoogle Scholar
  2. 2.
    L. Miglio, F. Quasso, G. Benedek, J. Chem. Phys. 83, 913 (1985)CrossRefGoogle Scholar
  3. 3.
    V. Bortolani, A. Franchini, F. Nizzoli, G. Santoro, Phys. Rev. Lett. 52, 429 (1984)CrossRefADSGoogle Scholar
  4. 4.
    A. Liebsch, J. Harris, B. Salanon, J. Lapujoulade, Surf. Sci. 123, 338 (1982)CrossRefADSGoogle Scholar
  5. 5.
    M.S. Daw, M.I. Baskes, Phys. Rev. Lett. 50, 1285 (1983)CrossRefADSGoogle Scholar
  6. 6.
    M.S. Daw, M.I. Baskes, Phys. Rev. B 29, 6443 (1984)Google Scholar
  7. 7.
    J.S. Nelson, E.C. Sowa, M.S. Daw, Phys. Rev. Lett. 61, 1977 (1988)Google Scholar
  8. 8.
    P.R. Underhill, Surf. Sci. 200, L441 (1988)Google Scholar
  9. 9.
    W.K. Jacobsen, J.K. Nørskov, M.J. Puska, Phys. Rev. B 35, 7423 (1987)Google Scholar
  10. 10.
    P.D. Ditlevsen, J.K. Nørskov, J. El. Spectr. Rel. Phenom. 54(55), 237 (1990)CrossRefGoogle Scholar
  11. 11.
    P.D. Ditlevsen, J.K. Nørskov, Surf. Sci. 254, 261 (1991)CrossRefADSGoogle Scholar
  12. 12.
    F. Montalenti, M.I. Trioni, G.P. Brivio, S. Crampin, Surf. Sci. Lett. 364, L595 (1996)CrossRefADSGoogle Scholar
  13. 13.
    M.I. Trioni, F. Montalenti, G.P. Brivio, Surf. Sci. Lett. 401, L383 (1998)CrossRefADSGoogle Scholar
  14. 14.
    M.I. Trioni, S. Marcotulio, G. Santoro, V. Bortolani, G. Palumbo, G.P. Brivio, Phys. Rev. B 58, 11043 (1998)CrossRefADSGoogle Scholar
  15. 15.
    G.P. Brivio, M.I. Trioni, Rev. Mod. Phys. 71, 231 (1999)CrossRefADSGoogle Scholar
  16. 16.
    N.D. Lang, Phys. Rev. Lett. 46, 842 (1981)CrossRefADSGoogle Scholar
  17. 17.
    E. Zaremba, W. Kohn, Phys. Rev. B 13, 2270 (1976)CrossRefADSGoogle Scholar
  18. 18.
    E. Zaremba, W. Kohn, Phys. Rev. B 15, 12769 (1977)CrossRefGoogle Scholar
  19. 19.
    N. Esbjerg, J.K. Norskov, Phys. Rev. Lett. 45, 807 (1980)CrossRefADSGoogle Scholar
  20. 20.
    J. Harris, A. Liebsch, Phys. Rev. Lett. 49, 341 (1982)CrossRefADSGoogle Scholar
  21. 21.
    A. Liebsch, J. Harris, Surface Sci. 111, 2721 (1981)CrossRefGoogle Scholar
  22. 22.
    M.W. Cole, F. Toigo, Phys. Rev. B 31, 727 (1985)CrossRefADSGoogle Scholar
  23. 23.
    V. Celli in Helium Atom Scattering from Surfaces, E. Hulpke (Ed.), (Springer, Berlin, 1992), p. 25Google Scholar
  24. 24.
    P. Nordlander, J. Harris, J. Phys. C 17, 1141 (1984)CrossRefADSGoogle Scholar
  25. 25.
    Y. Chen, S.Y. Tong, K.P. Bohnen, T. Rodach, K.M. Ho, Phys. Rev. Lett. 70, 603 (1993)CrossRefADSGoogle Scholar
  26. 26.
    S.Y. Tong, Y. Chen, K.P. Bohnen, T. Rodach, K.M. Ho, Surf. Rev. Lett. 1, 97 (1994)CrossRefADSGoogle Scholar
  27. 27.
    K.P. Bohnen, K.M. Ho, Surf. Sci. Reports 19, 99 (1993)CrossRefADSGoogle Scholar
  28. 28.
    J.P. Senet, G. Benedek, J.P. Toennies, Europhys. Lett. 57, 430 (2002)CrossRefADSGoogle Scholar
  29. 29.
    G. Benedek, M. Bernasconi, V. Chis, E. Chulkov, P.M. Echenique, B. Hellsing, J. Peter, Toennies. J. Phys.: Cond. Matter 22, 084020 (2010)ADSGoogle Scholar
  30. 30.
    IYu. Sklyadneva, G. Benedek, E.V. Chulkov, P.M. Echenique, R. Heid, K.-P. Bohnen, J.P. Toennies, Phys. Rev. Lett. 107, 095502 (2011)CrossRefADSGoogle Scholar
  31. 31.
    G. Benedek, M. Bernasconi, K.-P. Bohnen, D. Campi, E.V. Chulkov, P.M. Echenique, R. Heid, IYu. Sklyadneva, J.P. Toennies, Phys. Chem. Chem. Phys. 16, 7159 (2014)CrossRefGoogle Scholar
  32. 32.
    L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968)Google Scholar
  33. 33.
    B.H. Brandsen, C.J. Joachain, Introduction to Quantum Mechanics (Longman, New York)Google Scholar
  34. 34.
    N. Cabrera, V. Celli, J.R. Manson, Phys. Rev. Lett. 22, 346 (1969)CrossRefADSGoogle Scholar
  35. 35.
    J.R. Manson, V. Celli, Surf. Sci. 24, 495 (1971)CrossRefADSGoogle Scholar
  36. 36.
    V. Bortolani, A.C. Levi, Riv. Nuovo Cim. 9, 1 (1986)CrossRefADSGoogle Scholar
  37. 37.
    L. van Hove, Phys. Rev. 95, 249 (1954)CrossRefADSMathSciNetGoogle Scholar
  38. 38.
    V. Celli, N. Garcia, J. Hutchison, Surf. Sci. 87, 112 (1979)CrossRefADSGoogle Scholar
  39. 39.
    A.A. Maradudin, E.W. Montroll, G.H. Weiss, I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Solid State Physics, Suppl. 3 (Academic Press, New York, 1971)Google Scholar
  40. 40.
    A.A. Maradudin, R.F. Wallis, L. Dobrzynski, Handbook of Surfaces and Interfaces, vol. 3 (Garland, New York, 1980)Google Scholar
  41. 41.
    D. Eichenauer, J.P. Toennies, J. Chem. Phys. 85, 532 (1986)CrossRefADSGoogle Scholar
  42. 42.
    C. Zener, Proc. Roy. Soc. (London) 40 (1932) 178 and 335Google Scholar
  43. 43.
    J.M. Jackson, N.F. Mott, Proc. Roy. Soc. (London) A 137, 703 (1932)Google Scholar
  44. 44.
    G. Benedek, G. Seriani, Japan J. Appl. Phys. Suppl 2(Pt. 2), 545 (1974)CrossRefADSGoogle Scholar
  45. 45.
    G. Benedek, G. Boato, Europhys. News 8, 5 (1977)CrossRefGoogle Scholar
  46. 46.
    Y. Wang, B.H. Choi, N.L. Liu, Phys. Rev. B 43, 7458 (1991)CrossRefADSGoogle Scholar
  47. 47.
    B.H. Choi, R.T. Poe, J. Chem. Phys. 83, 1330 (1985); ibidem 83,1344 (1985)Google Scholar
  48. 48.
    H.-J. Ernst, E. Hulpke, J.P. Toennies, Phys. Rev. Lett. 58, 1941 (1987)CrossRefADSGoogle Scholar
  49. 49.
    N. Garcia, J. Ibanez, J. Solana, N. Cabrera, Surf. Sci. 60, 385 (1976)CrossRefADSGoogle Scholar
  50. 50.
    N. Garcia, Phys. Rev. Lett. 37, 912 (1976)CrossRefADSGoogle Scholar
  51. 51.
    N. Garcia, J. Chem. Phys. 67, 897 (1977)CrossRefADSGoogle Scholar
  52. 52.
    G. Benedek, N. Garcia, Ned. Tijdsch. Vacuumtechniek 16, 344 (1978)Google Scholar
  53. 53.
    G. Benedek, N. Garcia Surf. Sci. 80, 543 (1979). This work is very similar to that of Armand and Manson (AM), quoted at [53]. Here however the first order correction to the source function (δf) is included while AM produced an argument to disregard it. Indeed δf contributes two additional terms in the one-phonon cross section, the first being an inelastic correction to the diffraction amplitudes (required by unitarity), the second an incoherent contribution to the elastic diffuse scattering due to the thermal motion. Both terms, though physically meaningful, do not affect the shape of the one-phonon HAS spectrum and were therefore disregarded in AM theory as well as in all subsequent calculations based on the GR methodGoogle Scholar
  54. 54.
    G. Benedek, in Dynamics of Gas-Surface Interaction, ed. by G. Benedek, U. Valbusa (Springer, Berlin, Heidelberg, 1982), p. 227Google Scholar
  55. 55.
    G. Benedek, N. Garcia, Surf. Sci. 103, L143 (1981)CrossRefADSGoogle Scholar
  56. 56.
    G. Armand, J.R. Manson, Surf. Sci. 80, 532 (1979)CrossRefADSGoogle Scholar
  57. 57.
    G. Benedek, R.B. Doak, J.P. Toennies, Phys. Rev. B 28, 7277 (1983)CrossRefADSGoogle Scholar
  58. 58.
    G. Benedek, N. Garcia, Le Vide Suppl. 201, 818 (1980)Google Scholar
  59. 59.
    G. Benedek, G. Brusdeylins, R.B. Doak, J.P. Toennies, J. Physique (Paris) 42, C6–793 (1981)Google Scholar
  60. 60.
    J.H. Weare, J. Chem. Phys. 61, 2900 (1974)CrossRefADSGoogle Scholar
  61. 61.
    G. Brusdeylins, R.B. Doak, J.P. Toennies, Phys. Rev. Lett. 44, 1417 (1980)CrossRefADSGoogle Scholar
  62. 62.
    A.C. Levi, Nuovo Cim. 54 B, 357 (1979)Google Scholar
  63. 63.
    A.C. Levi, G. Benedek, L. Miglio, G. Platero, V.R. Velasco, F. Garcia-Moliner, Surf. Sci. 143, 253 (1984)CrossRefADSGoogle Scholar
  64. 64.
    G. Platero, V.R. Velasco, F. Garcia-Moliner, G. Benedek, L. Miglio, Surf. Sci. 143, 244 (1984)CrossRefADSGoogle Scholar
  65. 65.
    G. Brusdeylins, R. Rechsteiner, J.G. Skofronick, J.P. Toennies, G. Benedek, L. Miglio, Phys. Rev. Lett. 54, 466 (1985)CrossRefADSGoogle Scholar
  66. 66.
    B. Dorner, Coherent Inelastic Neutron Scattering in Lattice Dynamics (Springer, Berlin, Heidelberg, 1982)CrossRefGoogle Scholar
  67. 67.
    R.S. Leigh, B. Szigeti, V.K. Tewari, Proc. R. Soc. London Ser. A 320, 505 (1971)CrossRefADSGoogle Scholar
  68. 68.
    W. Cochran, Acta Crystallogr. Sec. A 27, 556 (1971)CrossRefADSGoogle Scholar
  69. 69.
    N.J. Chesser, J.D. Axe, Phys. Rev. B 9, 4060 (1974)CrossRefADSGoogle Scholar
  70. 70.
    H. Ibach, D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982)Google Scholar
  71. 71.
    J.E. Yater, A.D. Kulkarni, F.W. de Wette, J.L. Erskine, J. Electr. Spectr. Rel. Phenom. 54/55, 395 (1990)Google Scholar
  72. 72.
    J.L. Erskine, E.-J. Jeong, J. Yater, Y. Chen, S.Y. Tong, J. Vac. Sci. Technol., A 8, 2649 (1990)CrossRefADSGoogle Scholar
  73. 73.
    A. Glebov, W. Silvestri, J.P. Toennies, G. Benedek, J.G. Skofronick, Phys. Rev. B 54, 17866 (1996)CrossRefADSGoogle Scholar
  74. 74.
    A. Schuller, S. Wethekam, H. Winter, Phys. Rev. Lett. 98, 016103 (2007)CrossRefADSGoogle Scholar
  75. 75.
    J.R. Manson, H. Khemliche, P. Roncin, Phys. Rev. B 78, 155408 (2008)CrossRefADSGoogle Scholar
  76. 76.
    N. Bundaleski, H. Khemliche, P. Soulisse, P. Roncin, Phys. Rev. Lett. 101, 177601 (2008)CrossRefADSGoogle Scholar
  77. 77.
    W. Brenig, Z. Phys. B 36, 81 (1979)Google Scholar
  78. 78.
    J. Böheim, W. Brenig, Z. Phys. B 41, 243 (1981)Google Scholar
  79. 79.
    K. Burke, B. Gumhalter, D.C. Langreth, Phys. Rev. B 47, 12852 (1993)CrossRefADSGoogle Scholar
  80. 80.
    B. Gumhalter, K. Burke, D.C. Langreth, Surf. Rev. Lett. 1, 133 (1994)CrossRefADSGoogle Scholar
  81. 81.
    A. Bilić, B. Gumhalter, Phys. Rev. B 52, 12307 (1995)CrossRefADSGoogle Scholar
  82. 82.
    B. Gumhalter, Phys. Rep. 351, 1 (2001)CrossRefADSGoogle Scholar
  83. 83.
    W. Brenig, B. Gumhalter, J. Phys. Chem. B 108, 14549 (2004)CrossRefGoogle Scholar
  84. 84.
    J.L. Beeby, J. Phys. C 5, 3438 (1972)CrossRefADSGoogle Scholar
  85. 85.
    J.L. Beeby, J. Phys. C 5, 3457 (1972)CrossRefADSGoogle Scholar
  86. 86.
    R. Brako, D.M. Newns, Phys. Rev. Lett. 48, 1859 (1982)CrossRefADSGoogle Scholar
  87. 87.
    R. Brako, D.M. Newns, Surf. Sci. 117, 42 (1982)CrossRefADSGoogle Scholar
  88. 88.
    R. Brako, D.M. Newns, Surf. Sci. 123, 439 (1982)CrossRefADSGoogle Scholar
  89. 89.
    V. Celli, D. Himes, P. Tran, J.P. Toennies, Ch. Wöll, G. Zhang, Phys. Rev. Lett. 66, 3160 (1991)CrossRefADSGoogle Scholar
  90. 90.
    D. Himes, V. Celli, Surf. Sci. 272, 139 (1992)CrossRefADSGoogle Scholar
  91. 91.
    J.R. Manson, Comp. Phys. Comm. 80, 145 (1994)CrossRefADSGoogle Scholar
  92. 92.
    R.J. Glauber, Phys. Rev. 98, 1692 (1955); Phys. Rev. 131, 2766 (1963)Google Scholar
  93. 93.
    N. Terzi, in Collective Excitations in Solids, ed. by. B. Di Bartolo (Plenum, New York, 1983) pp. 149–181; G.M. Mazzucchelli, N. Terzi, Solid State Comm. 48, 679 (1983)Google Scholar
  94. 94.
    L. van Hove, Phys. Rev. 95, 249 (1954)CrossRefADSMathSciNetGoogle Scholar
  95. 95.
    M.H.L. Pryce, in Phonons in Perfect Lattices and in Lattices with Point Imperfections, ed. by R.W.H. Stevenson (Oliver and Boyd, Edimburgh, 1966), p. 403Google Scholar
  96. 96.
    A. Giorgetti, N. Terzi, Solid State Comm. 39, 635 (1981)CrossRefADSGoogle Scholar
  97. 97.
    A. Šiber, B. Gumhalter, A.P. Graham, J.P. Toennies, Phys. Rev. B 63, 115411 (2001)CrossRefADSGoogle Scholar
  98. 98.
    K. Kern, Dissertation (University of Bonn, 1986)Google Scholar
  99. 99.
    A.C. Levi, H.G. Suhl, Surf. Sci. 88, 221 (1979)CrossRefADSGoogle Scholar
  100. 100.
    H.-D. Meyer, Surf. Sci. 104, 117 (1981)CrossRefADSGoogle Scholar
  101. 101.
    T. Rahman, private communicationGoogle Scholar
  102. 102.
    S.K.S. Ma, F.W. de Wette, G.P. Alldredge, Surf. Sci. 78, 598 (1978)CrossRefADSGoogle Scholar
  103. 103.
    L. Yang, T.S. Raman, Phys. Rev. Lett. 67, 2327 (1991)CrossRefADSGoogle Scholar
  104. 104.
    L. Yang, T.S. Rahman, M.S. Daw, Phys. Rev. B 44, 13725 (1991-II)Google Scholar
  105. 105.
    Y.W. Lin, S.A. Adelman, G. Wolken, Surf. Sci. 66, 376 (1977)CrossRefADSGoogle Scholar
  106. 106.
    B. Baule, Ann. der Physik 44, 145 (1914)CrossRefADSGoogle Scholar
  107. 107.
    For a derivation of the Baule formula see R.E. Stickney, Adv. Atomic and Mol. Physics 3 (1967) 143 AppendixGoogle Scholar
  108. 108.
    J.W.M. Frenken, P.M.J. Maree, J.F. van der Veen, Phys. Rev. B. 34, 7506 (1986)CrossRefADSGoogle Scholar
  109. 109.
    F. Lindemann, Phys. Z. 11, 609 (1910)Google Scholar
  110. 110.
    H. Lowey, Phys. Rep. 237, 249 (1994)CrossRefADSGoogle Scholar
  111. 111.
    T.S. Rahman, J.D. Spangler, A. Al-Rawi, J. Phys. Cond. Matt. 14, 4903 (2002)CrossRefGoogle Scholar
  112. 112.
    G. Brusdeylins, R.B. Doak, J.P. Toennies, Phys. Rev. B 27, 3662 (1983)CrossRefADSGoogle Scholar
  113. 113.
    D. Eichenauer, U. Harten, J.P. Toennies, V. Celli, J. Chem. Phys. 86, 3693 (1987)CrossRefADSGoogle Scholar
  114. 114.
    J.R. Manson, G. Benedek, S. Miret-Artès, Chem. Phys. Lett. 7, 1016–1021 (2016)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

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

Personalised recommendations