A topic in the Open Knowledge Graph — a free, open map of 15,290 topics and the order to learn them in.
Ferromagnetism — the spontaneous alignment of magnetic moments below a Curie temperature T_C — arises from the quantum mechanical exchange interaction, not from magnetic dipole forces (which are far too weak). The Heisenberg model H = -J sum_{<ij>} S_i · S_j with J > 0 captures this: the exchange coupling J favoring parallel spins originates from the Pauli exclusion principle and Coulomb repulsion. Mean-field theory predicts T_C = zJS(S+1)/(3k_B), spontaneous magnetization below T_C, and Curie-Weiss susceptibility chi = C/(T - T_C) above T_C. Real ferromagnets (Fe, Co, Ni) have T_C values of 600-1400 K, confirming that exchange is an electronic energy scale, not a magnetic one.
Ferromagnetism — the phenomenon behind permanent magnets — is one of the oldest known physical effects and one of the most striking demonstrations of quantum mechanics at macroscopic scales. Below the Curie temperature T_C, a ferromagnetic material develops a spontaneous magnetization even in zero applied field. The moments of billions of atoms align cooperatively, producing a macroscopic magnetic field. The driving force is the exchange interaction: a purely quantum mechanical effect arising from the interplay of the Pauli exclusion principle and Coulomb repulsion.
The Heisenberg model H = -J sum_{<ij>} S_i · S_j captures the essential physics. Each lattice site i carries a spin operator S_i, and the coupling J between nearest neighbors <ij> determines whether parallel alignment (J > 0, ferromagnetic) or antiparallel alignment (J < 0, antiferromagnetic) is favored. The exchange constant J is not a magnetic interaction — it is electrostatic in origin and typically 104 times larger than magnetic dipole energies. For two electrons, the triplet state (parallel spins, antisymmetric spatial wavefunction) and singlet state (antiparallel spins, symmetric spatial wavefunction) have different Coulomb energies because of their different spatial correlations. The energy difference is J.
Mean-field theory provides the simplest analysis: replace the fluctuating exchange field from neighboring spins with its thermal average, giving an effective field B_eff = zJ<S>/g mu_B, where z is the coordination number. Self-consistently solving the resulting Brillouin function equation yields the Curie temperature T_C = zJS(S+1)/(3k_B) and the Curie-Weiss susceptibility chi = C/(T - T_C) above T_C. Below T_C, the spontaneous magnetization grows continuously from zero — a second-order phase transition with the magnetization as the order parameter.
The limitations of mean-field theory become apparent near T_C, where critical fluctuations dominate and the actual critical exponents differ from mean-field predictions. The renormalization group treatment shows that the critical behavior depends only on dimension and symmetry (universality class), not on microscopic details. Away from T_C, the elementary excitations of the ordered state are spin waves (magnons): collective precession modes where the magnetization direction varies smoothly in space, with a characteristic omega proportional to k2 dispersion for ferromagnets. Magnons reduce the magnetization at finite temperature, contributing to the Bloch T3/2 law for the spontaneous magnetization: M(T) = M(0)[1 - (T/T_C)3/2] at low T.