Mean-field cluster model for the critical behaviour of ferromagnets

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Abstract

Two separate theories are often used to characterize the paramagnetic properties of ferromagnetic materials. At temperatures T well above the Curie temperature, TC (where the transition from paramagnetic to ferromagnetic behaviour occurs), classical mean-field theory1 yields the Curie–Weiss law for the magnetic susceptibility: χ( T) 1/(T - Θ), where Θ is the Weiss constant. Close to TC, however, the standard mean-field approach breaks down so that better agreement with experimental data is provided by critical scaling theory2,3: χ(T) 1/(T - TC)γ, where γ is a scaling exponent. But there is no known model capable of predicting the measured values of γ nor its variation among different substances4. Here I use a mean-field cluster model5 based on finite-size thermostatistics6,7 to extend the range of mean-field theory, thereby eliminating the need for a separate scaling regime. The mean-field approximation is justified by using a kinetic-energy term to maintain the microcanonical ensemble8. The model reproduces the Curie–Weiss law at high temperatures, but the classical Weiss transition at TC = Θ is suppressed by finite-size effects. Instead, the fraction of clusters with a specific amount of order diverges at T C, yielding a transition that is mathematically similar to Bose–Einstein condensation. At all temperatures above TC, the model matches the measured magnetic susceptibilities of crystalline EuO, Gd, Co and Ni, thus providing a unified picture for both the critical-scaling and Curie–Weiss regimes.

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Figure 1: Scaling plot of the magnetic susceptibilities of crystalline europium oxide23,24, gadolinium25–27, cobalt28,29 and nickel30,31.
Figure 2: Landau-like plot of the reduced grand potential per particle, Ω( L)/kBT, as a function of the scaled alignment variable L, at four different temperatures.

References

  1. 1

    Weiss, P. L'hypothèse du champ moleculaire et la propriéte ferromagnetique. J. Phys. (Paris) 6, 661– 690 (1907).

  2. 2

    Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, New York, 1971).

  3. 3

    Stanley, H. E. Scaling universality, and renormalization. Rev. Mod. Phys. 71, S358–S366 (1999).

  4. 4

    Collins, M. F. Magnetic Critical Scattering (Oxford Univ. Press, New York, 1989).

  5. 5

    Chamberlin, R. V. Mesoscopic mean-field theory for supercooled liquids and the glass transition. Phys. Rev. Lett. 82, 2520– 2523 (1999).

  6. 6

    Hill, T. L. Thermodynamics of Small Systems Parts I and II (Dover, New York, 1994).

  7. 7

    Hill, T. L. & Chamberlin, R. V. Extension of the thermodynamics of small systems to open metastable states: An example. Proc. Natl Acad. Sci. USA 95, 12779–12782 (1998).

  8. 8

    Creutz, M. Deterministic Ising dynamics. Ann. Phys. 167, 62–72 (1986).

  9. 9

    Rowlinson, J. S. Legacy of van der Waals. Nature 244, 414 –417 (1973).

  10. 10

    Onsager, L. Crystal statistics I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944).

  11. 11

    Guggenheim, E. A. The principle of corresponding states. J. Chem. Phys. 13, 253–261 (1945).

  12. 12

    Wegner, F. J. Corrections to scaling laws. Phys. Rev. B 5, 4529–4536 (1972).

  13. 13

    Souletie, J. & Tholence, J. L. Critical behavior of nickel between TC and 3TC. Solid State Commun. 48, 407–410 (1983).

  14. 14

    Smart, J. S. Effective Field Theories of Magnetism (Saunders, Philadelphia, 1966).

  15. 15

    Huang, K. Statistical mechanics 2nd edn (Wiley, New York, 1987 ).

  16. 16

    Chamberlin, R. V. & Holtzberg, F. Remanent magnetization of a simple ferromagnet. Phys. Rev. Lett. 67, 1606–1609 (1991).

  17. 17

    Korenman, V. Theories of itinerant magnetism. J. Appl. Phys. 57, 3000–3005 (1985).

  18. 18

    Mezei, F., Farago, B., Hayden, S. M. & Stirling, W. G. Breakdown of conventional dynamic scaling at the ferromagnetic Curie point in EuO. Physica B 156&157, 226– 228 (1989).

  19. 19

    Schiener, B., Böhmer, R., Loidl, A. & Chamberlin, R. V. Nonresonant spectral hole burning in the slow dielectric response of supercooled liquids. Science 274, 752– 754 (1996).

  20. 20

    Chamberlin, R. V. Nonresonant spectral hole burning in a spin glass. Phys. Rev. Lett. 83, 5134–5137 ( 1999).

  21. 21

    Chamberlin, R. V. Experiments and theory of the nonexponential relaxation in liquids, glasses, polymers and crystals. Phase Transitions 65, 169–209 (1998).

  22. 22

    Bally, D., Popovici, M., Totia, M., Grabcev, B. & Lungu, A. M. Small-angle critical magnetic scattering of neutrons in Co. Neutron Inelastic Scattering Vol. II, 75–82 (IAEA, Vienna, 1968).

  23. 23

    Menyuk, N., Dwight, K. & Reed, T. B. Critical magnetic properties and exchange interactions in EuO. Phys. Rev. B 3, 1689– 1698 (1976).

  24. 24

    Huang, C. C. & Ho, J. T. Faraday rotation near the Curie point of EuO. Phys. Rev. B 12, 5255– 5260 (1975).

  25. 25

    Arajs, S. & Colvin, R. V. Paramagnetism of polycrystalline gadolinium, terbium, and dysprosium metals. J. Appl. Phys. 32, 336S–337S (1961).

  26. 26

    Nigh, H. E., Legvold, S. & Spedding, F. H. Magnetization and electrical resistivity of gadolinium single crystals. Phys. Rev. 132, 1092– 1097 (1963).

  27. 27

    Geldart, D. J. W., Hargraves, P., Fujiki, N. M. & Dunlap, R. A. Anisotropy of the critical magnetic susceptibility of gadolinium. Phys. Rev. Lett. 62, 2728–2731 (1989).

  28. 28

    Nakagawa, Y. Change of magnetic susceptibility of transition metals and alloys at their melting points. J. Phys. Soc. Jpn. 11, 855 –863 (1956).

  29. 29

    Colvin, R. V. & Arajs, S. Magnetic susceptibility of face-centred cubic cobalt just above the ferromagnetic Curie temperature. J. Phys. Chem. Solids 26, 435–437 (1965).

  30. 30

    Fallot, M. Paramagnétisme des èléments ferromagnétiques. J. Phys. (Paris) 8, 153– 163 (1944).

  31. 31

    Arajs, S. Paramagnetic behavior of nickel just above the ferromagnetic Curie temperature. J. Appl. Phys. 36, 1136– 1137 (1965).

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Acknowledgements

I thank K. E. Schmidt and G. H. Wolf for several insights into the subject reported here. I also thank B. Geil, T. L. Hill, S. M. Lindsay and R. Richert for discussions. This work was supported by the NSF.

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Correspondence to Ralph V. Chamberlin.

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