Past exam of the condensed matter physics course of the University of Oxford / 2013 / 4 / e / Solution

Room-temperature iron is ferromagnetic: interactions among its electronic moments produce spontaneous local order. The independent-ion Curie law therefore needs an interaction and an ordered-state description. In a Weiss mean-field approximation, replace the applied field by $H_{\mathrm{eff}}=H+\Lambda M$. Above the ordering temperature, linearizing the single-ion response yieldsThis is the Curie–Weiss approximation on the paramagnetic side. Merely substituting room temperature below $T_C$ into that high-temperature formula would produce a negative response and is not a valid model of stable iron.

Below $T_C$, use the full nonlinear moment response to the effective field; it has nonzero zero-field solutions. On a chosen ordered branch the local small-field response has the form $M\simeq M_s(T)+{\chi }_{\mathrm{diff}}(T)H$, rather than $M\propto H/T$ through the origin. Macroscopic magnetic domains and hysteresis further determine the measured net moment. Exchange interactions, arising from electronic quantum statistics and Coulomb correlations, supply the alignment energy; bare dipolar fields alone do not explain ordinary iron's strong ferromagnetism. A localized-ion Weiss model describes the qualitative ordering, while iron's itinerant electronic structure limits a literal independent-fixed-ion interpretation.