Excitations of/on Fractal Networks

  • R. Orbach


Fractal symmetry (dilation invariance), as opposed to Euclidean symmetry (translation invariance), requires three dimensionalities to contain a physical description of the excitations of a fractal network: d, the Euclidean (or embedding) dimension; D, the Hausdorff (or fractal) dimension; and , the fracton (or spectral) dimension. The dynamical properties of percolarting networks are examined in this context. The vibrational density of states and the vigrational excitation dispersion law are calculated. The former is shown to be proportional to ωd−1 in the phonon or long length scale reggime. A crossover is found at frequency ωc, proportional to (p−pc) v[1+(θ/2)], where p is the bond occupancy probability and pc the critical percolation bond occupancy. Here, v is the correlation fength exponent, and θ is the exponent appropriate to the range dependence of the diffusion constant [D(r) ∝ r−θ]. At ωc, a scaling argument shows that a reasonably rapid rise occurs in the vibrational density of states, to which we shall refer as a “fracton edge.” For frequencies higher than ωc, the vibrational density of states continues to rise, but at a slower rate, proportional to . The excitations in this regime are termed “fractons.” It is shown that the electrical conductance on a fractal network also depends on D and . Use is made of Anderson localization scaling theory to show that, for < 2, the fracton eigenstates are localized. Very recent inelastic neutron diffraction measurements of the vibrational excitations of fused silica are shown to exhibit a density of states which agrees in form remarkably closely with a recent scaling model of phonon to fracton crossover.

The vibrational dispersion law also exhibits the effects of crossover. At long length scales, the dispersion law is linear, with a sound velocity proportional to (p−pc) vθ/2. Near crossover, the dispersion curve reduces its slope, exhibits an inflexion point, and then rises as the power of the inverse length scale.

These results suggest that conventional electron-phonon interactions may be strongly modified on a fractal network. We have analyzed the two cases of localized electron-one fracton interaction, and a localized electron-two fracton interaction. To do so, we have had to take explicit account of the spatial extent of the vibrational wave function. This has necessitated the introduction of a fourth dimensionality, dΦ, which determines the pythagorean range of the localized fracton wavefunction. The electronic relaxation rate differs from electronic site to electronic site. We are able to calculate the probability density for the electronic relaxation rate. The lack of a single relaxation rate results in a significant departure from an exponential time decay of the electronic departure from equilibrium. We find a time dependence which is faster than power law, but slower than exponential or stretched exponential. We suggest that these theoretical results call into question interpretations of phenomena involving electron-vibrational interactions in glassy materials which have been based on extended phonon states. We recalculate the energy and temperature dependence for the inelastic scattering time for extended electron states. Our results exhibit marked departure from those obtained with extended (and localized) phonon states.

We suggest a variety of calculations and experiments which can further elucidate the affect of fracton excitations on physical properties. We believe that fracton excitations may be relevant to some glassy materials above a crossover frequency. These excitations can have a profound influence on the vibrational density of states and the dynamics associated with electron-vibration interactions.


Relaxation Rate Fractal Network Glassy Material Cross Relaxation Vibrational Density 


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  1. 1.
    B. B. Mandelbrot, “The Fractal Geometry of Nature,” W. H. Freeman and Co., New York (1983); Ann. Isr. Phys. Soc. 5: 59 (1983).MathSciNetGoogle Scholar
  2. 2.
    S. Alexander, C. Laermans, R. Orbach, and H. M. Rosenberg, Phys. Rev. B28: 4615 (1983).CrossRefGoogle Scholar
  3. 3.
    S. Alexander and R. Orbach, J. Phys. (Paris) Lett. 43:L-625 (1982).Google Scholar
  4. 4.
    R. Rammal and G. Toulouse, J. Phys. (Paris) Lett. 44:L-13 (1983).Google Scholar
  5. 5.
    A. Kapitulnik and G. Deutscher, Phys. Rev. Lett. 49: 1444 (1982).CrossRefGoogle Scholar
  6. 6.
    R. F. Voss, R. B. Laibowitz, and E. I. Allessandrini, Phys. Rev. Lett. 49: 1441 (1982).CrossRefGoogle Scholar
  7. 7.
    A. Aharony, S. Alexander, O. Entin-Wohlman, and R. Orbach, Phys. Rev. B31: 2565 (1985).CrossRefGoogle Scholar
  8. 8.
    Y. Gefen, A. Aharony, and S. Alexander, Phys. Rev. Lett. 50: 77 (1983).CrossRefGoogle Scholar
  9. 9.
    P. G. de Gennes, Recherche 7: 919 (1976).Google Scholar
  10. 10.
    E. W. Montroll, “Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability,” Univ. of Calif., Berkeley, (1955), p. 209.Google Scholar
  11. 11.
    S. Alexander, J. Bernasconi, W. Schneider, and R. Orbach, Rev. Mod. Phys. 53: 175 (1981).MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Kelham and H. M. Rosenberg, J. Phys. C14:1737 (1981); C. I. Nichols and H. M. Rosenberg, J. Phys. C17: 1165 (1984);Google Scholar
  13. D. E. Farrell, J. E. de Oliveira, and H. M. Rosenberg, “Phonon Scattering in Condensed Matter,” ed. by W. Eisenmenger et al., Springer, Berlin (1984), p. 422.CrossRefGoogle Scholar
  14. 13.
    B. Derrida, R. Orbach, and K.-Wah Yu, Phys. Rev. B29: 6645 (1984).CrossRefGoogle Scholar
  15. 14.
    S. Feng and P. N. Sen, Phys. Rev. Lett. 52: 216 (1984).CrossRefGoogle Scholar
  16. 15.
    Y. Kantor and I. Webman, Phys. Rev. Lett. 52: 1891 (1984).CrossRefGoogle Scholar
  17. 16.
    D. J. Bergman and Y. Kantor, Phys. Rev. Lett. 53: 511 (1984).MathSciNetCrossRefGoogle Scholar
  18. 17.
    P. G. de Gennes, J. Phys. (Paris) Lett. 37: L1 (1976).CrossRefGoogle Scholar
  19. 18.
    K.-W. Yu, P. Chaikin, and R. Orbach, Phys. Rev. B28: 4831 (1983).CrossRefGoogle Scholar
  20. 19.
    I. Webman and G. S. Grest, Phys. Rev. B31: 1690 (1985).Google Scholar
  21. 20.
    E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42: 673 (1979).CrossRefGoogle Scholar
  22. 21.
    Entin-Wohlman, S. Alexander, R. Orbach, and K.-Wah Yu, Phys. Rev. B29: 4588 (1984).CrossRefGoogle Scholar
  23. 22.
    R. C. Zeller and R. O. Pohl, Phys. Rev. B4: 2029 (1971).CrossRefGoogle Scholar
  24. 23.
    M. P. Zaitlin and A. C. Anderson, Phys. Rev. B12: 4475 (1975).CrossRefGoogle Scholar
  25. 24.
    U. Strom, J. R. Hendrickson, R. J. Wagner, and P. C. Taylor, Sol. St. Comm. 15: 1871 (1974).CrossRefGoogle Scholar
  26. 25.
    R. Orbach, J. Stat. Phys. 36: 735 (1984).MathSciNetCrossRefGoogle Scholar
  27. 26.
    U. Buchenau, N. Nucker, and A. J. Dianoux, Phys. Rev. Lett. 53: 2316 (1984).CrossRefGoogle Scholar
  28. 27.
    A. F. Ioffe and A. R. Regel, Prog. Semicond. 4, 237 (1960).Google Scholar
  29. 28.
    H. M. Rosenberg, Phys. Rev. Lett. 54: 704 (1985).CrossRefGoogle Scholar
  30. 29.
    R. Orbach and H. M. Rosenberg, “LT-17,” ed. by U. Eckern, A. Schmid, W. Weber, and H. Wühl, Elsebier Science Publishers B. V., Amsterdam (1984), p, 375.Google Scholar
  31. 30.
    Y.-E. Levy and B. Souillard, unpublished.Google Scholar
  32. 31.
    K. M. Middlemiss, S. G. Whittington, and D. S. Gaunt, J. Phys. A: Math. Gen. 13: 183S (1980);Google Scholar
  33. R. Pike and H. E. Stanley, J. Phys. A: Math. Gen. 14: L169 (1981);CrossRefGoogle Scholar
  34. D. C. Hong and H. E. Stanley, J. Phys. A: Math. Gen. 16:L475 (1983); D. C. Hong and H. E. Stanley, J. Phys. A: Math. Gen. 16: L525 (1983);CrossRefGoogle Scholar
  35. H. J. Hermann, D. C. Hong, and H. E. Stanley, J. Phys. A: Math. Gen. 17: L261 (1984);CrossRefGoogle Scholar
  36. J. Vannimenus, J. P. Nodal, and C. Martin, J. Phys. A: Math. Gen. 17: L351 (1984);CrossRefGoogle Scholar
  37. S. Havlin and R. Nossal, J. Phys. A: Math. Gen. 17: L427 (1984);MathSciNetCrossRefGoogle Scholar
  38. S. Havlin, Z. V. Djordjevic, I. Majid, H. E. Stanley, and G. H. Weiss, Phys. Rev. Lett. 53: 178 (1984).CrossRefGoogle Scholar
  39. 32.
    H. J. Stapleton, J. P. Allen, C. P. Flynn, D. G. Stinson, and S. Kurtz, Phys. Rev. Lett. 45: 1456 (1980);MathSciNetCrossRefGoogle Scholar
  40. J. P. Allen, J. T. Colvin, D. G. Stinson, C. P. Flynn, and H. J. Stapleton, Biophys. J. 38: 299 (1982).CrossRefGoogle Scholar
  41. 33.
    J. Szeftal and H. Alloul, Phys. Rev. Lett. 34:657 (1975); J. Non-Cryst. Sol. 29: 253 (1978).CrossRefGoogle Scholar
  42. 34.
    S. Alexander, O. Entin-Wohlman, and R. Orbach, submitted to J. Phys. (Paris) Lett. for publication (1985).Google Scholar
  43. 35.
    S. Alexander, O. Entin-Wohlman, and R. Orbach, submitted to J. Phys. (Paris) Lett. for publication (1985).Google Scholar
  44. 36.
    S. Alexander, O. Entin-Wohlman, and R. Orbach, submitted to Phys. Rev. B-15 for publication (1985).Google Scholar
  45. 37.
    S. Alexander, O. Entin-Wohlman, and R. Orbach, submitted to Phys. Rev. B-15 for publication (1985).Google Scholar
  46. 38.
    S. Alexander, O. Entin-Wohlman, and R. Orbach, to be submitted for publication, 1985.Google Scholar
  47. 39.
    J. Tauc, private communication.Google Scholar
  48. 40.
    R. Orbach and H. J. Stapleton, “Electronic Paramagnetic Resonance,” ed. by S. Geschwind, Plenum Press, New York (1972), p. 121.Google Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • R. Orbach
    • 1
  1. 1.Department of PhysicsUniversity of CaliforniaLos AngelesUSA

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