Mean-field cluster model for the critical behaviour of ferromagnets

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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.


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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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