The early 20th century:

Relativity and quantum mechanics bring understanding at last

With Maxwell's equations, classical electromagnetism was complete, but ferromagnetism remained a mystery. If there are Amperian currents of 1.76MA/m circulating in an iron bar, why does it not melt? These strange currents persist indefinitely, yet dissipate no heat.

The Molecular Field The first suggestion that there was an atomic dimension to magnetism lay in correspondence between Fresnel and Ampére found after Ampére's death, in which Fresnel suggested that the "Amperian currents" causing magnetism were microscopic in origin. However, it was not until 1907 that Pierre Weiss produced a theory of ferromagnetism based on the assumption that the interactions between magnetic molecules could be described empirically by an internal "molecular field". Hmol = nwM Combining this with the theory by Langevin describing paramagnetic solids gave a description of the phase transition at the Curie temperature, TC, where a ferromagnet loses its magnetization and becomes paramagnetic. But for iron (TC=943 K) the molecular field had to be huge, over 100 times greater than the biggest field that could be measured just outside an iron bar. The mystery of ferromagnetism was explained by the mystery of the molecular field! Demise of classical theory: Niels Bohr (below) in 1911, and J.H van Leeuwen independently in 1919 in her PhD thesis proved a famous theorem for classical nonrelativistic electrons using Maxwell's equations and statistical mechanics: "At any finite temperature, and in all finite applied electrical or thermal fields, the net magnetization of a collection of electrons in thermal equuilibrium vanishes identically." A starker conflict between theory and experiment would be hard to imagine: classical physics gives no ferromagnetism, no paramagnetism, no diamagnetism, in fact no magnetism at all!

The growth in understanding of magnetism was inextricably linked to that of the structure of the atom, shown above.

Quantum theory:

Resolution of this paradox came in the form of quantum theory. Bohr (left) postulated that the angular momentum of electrons is quantized, and that orbital magnetic moments are associated with the orbiting electron currents. In 1922 a famous experiment by Stern and Gerlach proved beyond all doubt that magnetic moments had a quantized character. Compton had suggested in 1921 that the electron also posessed a magnetic moment associated with an intrinsic spin angular momentum, and this was discovered by Goudsmit and Uhlenbeck in 1925. However they found a magnetic moment which, relative to the angular momentum, was twice the orbital value. In 1928, Dirac explained everything by writing down a relativistically invariant form of Schrodinger's equation where electron spin and the factor of two came naturally out of the calculation.

So both relativity and quantum mechanics, the twin pillars of modern physics, were essential to explain magnetism. From the revolutionary frenzy that established a completely new basis for physics in the period 1905-1930, there emerged an understanding of the persistent Amperian currents in terms of quantum mechanics. The Weiss field, which was shown by Dirac and Heisenberg (seen here meeting in 1929 in Chicago) to arise from the Pauli principle that no two electrons could occupy the same state. Together with the Coulomb interaction between electrons, this leads to a scalar isotropic interaction of two spins with a positive exchange constant J. The Heisenberg Hamiltonian was thus given by: