Mean-field cluster model for the critical behaviour of ferromagnets

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.