Order from Quantum Mechanics
Ferromagnetism arises from the quantum mechanical exchange interaction between neighboring electron spins. This interaction, far stronger than classical magnetic dipole coupling, energetically favors parallel spin alignment in iron, cobalt, nickel, and their alloys. At low temperatures, exchange wins decisively — atomic moments lock into long-range parallel order, producing the spontaneous magnetization that makes permanent magnets possible.
The Thermal Battle
Temperature is the enemy of magnetic order. As temperature rises, thermal energy (kT) increasingly competes with exchange energy, randomizing spin orientations. The Curie temperature marks the critical point where thermal disorder wins: above T_C, the time-averaged magnetization vanishes and the material becomes paramagnetic. This transition is not abrupt at the atomic level — near T_C, the material shows enormous fluctuations, with regions of correlated spins spanning all length scales.
Mean-Field Theory
Pierre-Ernest Weiss proposed in 1907 that each atomic moment experiences an effective 'molecular field' proportional to the average magnetization. This mean-field approximation predicts the Curie temperature, the shape of the M(T) curve, and the Curie-Weiss susceptibility law above T_C. While quantitatively approximate (it overestimates T_C by 10-20% and gets the wrong critical exponents), mean-field theory captures the essential physics of cooperative magnetic ordering and remains the starting point for understanding magnetic phase transitions.
Critical Phenomena
The Curie transition belongs to the universality class of the 3D Heisenberg (or Ising) model. Near T_C, physical quantities follow power laws with universal critical exponents independent of material details: magnetization vanishes as (T_C − T)^0.326, susceptibility diverges as (T − T_C)^(−1.24), and the correlation length diverges as (T − T_C)^(−0.63). This universality — identical behavior in systems as different as iron and EuO — is one of the deep triumphs of statistical mechanics and the renormalization group.