Theoretical and Mathematical Physics

, Volume 173, Issue 2, pp 1604–1619 | Cite as

Microscopic model of a non-Debye dielectric relaxation: The Cole-Cole law and its generalization

  • A. A. KhamzinEmail author
  • R. R. Nigmatullin
  • I. I. Popov


Based on a self-similar spatial-temporal structure of the relaxation process, we construct a microscopic model for a non-Debye (nonexponential) dielectric relaxation in complex systems. In this model, we derive the Cole-Cole expression for the complex dielectric permittivity and show that the exponent α involved in that expression is equal to the fractal dimension of the spatial-temporal self-similar ensemble characterizing the structure of the medium and the relaxation process occurring in it. We find a relation between the macroscopic relaxation time and the micro- and mesoparameters of the system. We obtain a generalized Cole-Cole expression for the complex dielectric permittivity involving log-periodic corrections that occur because of a discrete scaling invariance of the fractal structure generating the relaxation process on the mesoscopic scale. The found expression for the dielectric permittivity can be used to interpret dielectric spectra in disordered dielectrics.


dielectric relaxation complex dielectric permittivity non-Debye dielectric spectrum fractal discrete scaling invariance log-periodic oscillation 


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  1. 1.
    Y. Feldman, A. Puzenko, and Ya. Ryabov, “Dielectric relaxation phenomena in complex materials,” in: Advances in Chemical Physics (Y. P. Kalmykov, W. T. Coffey, and S. A. Rice, eds.), Vol. 133, Wiley, New York (2006), p. 1–125.Google Scholar
  2. 2.
    M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature, 363, 31–37 (1993).ADSCrossRefGoogle Scholar
  3. 3.
    A. Schönhals, F. Kremer, A. Hofmann, E. W. Fischer, and E. Schlosser, Phys. Rev. Lett., 70, 3459–3462 (1993).ADSCrossRefGoogle Scholar
  4. 4.
    P. Lunkenheimer, A. Pimenov, M. Dressel, Yu.G. Goncharov, R. Bohmer, and A. Loidl, Phys. Rev. Lett., 77, 318–321 (1996).ADSCrossRefGoogle Scholar
  5. 5.
    U. Scheider, P. Lunkenheimer, R. Brand, and A. Loidl, J. Non-Cryst. Solids, 235–237, 173–179 (1998).CrossRefGoogle Scholar
  6. 6.
    H. Fröhlich, Theory of Dielectrics: Dielectric Constant and Dielectric Loss, Oxford Univ. Press, Oxford (1958).zbMATHGoogle Scholar
  7. 7.
    C. J. F. Böttcher and P. Bordewijk, Theory of Electric Polarization, Vol. 2, Dielectrics in Time-Dependent Fields, Elsevier, Amsterdam (1992).Google Scholar
  8. 8.
    K. S. Cole and R. H. Cole, J. Chem. Phys., 9, 341–351 (1941).ADSCrossRefGoogle Scholar
  9. 9.
    G. Williams, Chem. Rev., 72, 55–69 (1972).CrossRefGoogle Scholar
  10. 10.
    R. H. Cole, “Dielectric polarization and relaxation,” in: Molecular Liquids: Dynamics and Interactions (NATO ASI Series C, Vol. 135, A. J. Barnes, W. J. Orville-Thomas, and J. Yarwood, eds.), Reidel, Dordrecht (1984), pp. 59–110.Google Scholar
  11. 11.
    R. Kohlrausch, Ann. Phys. (Leipzig), 12, 393 (1847).Google Scholar
  12. 12.
    G. Williams and D. C. Watts, Trans. Faraday Soc., 66, 80–85 (1970).CrossRefGoogle Scholar
  13. 13.
    A. K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectric, London (1983).Google Scholar
  14. 14.
    A. K. Jonscher, Universal Relaxation Law, Chelsea Dielectric, London (1996).Google Scholar
  15. 15.
    R. W. Zwanzig, “Statistical mechanics of irreversibility,” in: Lectures in Theoretical Physics (W. E. Brittin, B. W. Downs, and J. Downs, eds.), Vol. 3, Interscience, New York (1961), p. 106–141.Google Scholar
  16. 16.
    H. Mori, Prog. Theoret. Phys., 34, 399–416 (1965).ADSCrossRefGoogle Scholar
  17. 17.
    J. P. Boon and S. Yip, Molecular Hydrodynamics, McGraw-Hill, New York (1980).Google Scholar
  18. 18.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).zbMATHGoogle Scholar
  19. 19.
    R. R. Nigmatullin, Theor. Math. Phys., 90, 242–251 (1992).MathSciNetCrossRefGoogle Scholar
  20. 20.
    R. R. Nigmatullin and A. Le Mehaute, J. Non-Cryst. Solids, 351, 2888–2899 (2005).ADSCrossRefGoogle Scholar
  21. 21.
    E. Feder, Fractals, Plenum, New York (1988).zbMATHGoogle Scholar
  22. 22.
    V. V. Novikov and V. P. Privalko, Phys. Rev. E, 64, 031504 (2001).ADSCrossRefGoogle Scholar
  23. 23.
    G. P. Johari and M. Goldstein, J. Chem. Phys., 53, 2372–2388 (1970).ADSCrossRefGoogle Scholar
  24. 24.
    G. P. Johari, Ann. NY Acad. Sci., 279, 117–140 (1976).ADSCrossRefGoogle Scholar
  25. 25.
    R. R. Nigmatullin, Phys. B, 358, 201–205 (2005).ADSCrossRefGoogle Scholar
  26. 26.
    R. R. Nigmatullin, Phys. A, 363, 282–298 (2006).CrossRefGoogle Scholar
  27. 27.
    R. R. Nigmatullin, A. A. Arbuzov, F. Salehli, A. Giz, I. Bayrak, and H. Catalgil-Giz, Phys. B, 388, 418–434 (2007).ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • A. A. Khamzin
    • 1
    Email author
  • R. R. Nigmatullin
    • 1
  • I. I. Popov
    • 1
  1. 1.Institute for PhysicsKazan (Volga Region) Federal UniversityKazanRussia

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