Past exam of the quantum condensed matter course of the University of Cambridge / 2005 / Section B / B4 / a

Exchange interaction arises from Coulomb interaction together with the antisymmetry required for identical electrons. Different spin states require different spatial symmetry and therefore different pair correlations and Coulomb energy; exchange is not the classical magnetic dipole force between spins. For electrons in distinct atomic orbitals, positive direct exchange favors parallel spins, underlying Hund's rules and the formation of local moments.

In a solid with localized moments, a common convention is $H=-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Then $J>0$ favors ferromagnetism and $J<0$ favors antiferromagnetism. Superexchange through a ligand and other virtual hopping processes can produce either effective sign depending on orbital occupancy and geometry; half-filled orbitals often give antiferromagnetic coupling. For itinerant electrons, the Stoner criterion describes a competition between exchange energy gain and the kinetic cost of splitting the spin populations.

Experiments distinguish these descriptions through both static and dynamic response. Magnetic neutron diffraction locates magnetic Bragg peaks and therefore the ordering wavevector; an antiferromagnet can have such peaks despite zero bulk moment. Inelastic neutron scattering measures spin-wave dispersion, from which an exchange scale can be fitted. A Curie-Weiss susceptibility intercept gives complementary information about dominant coupling, but its sign alone does not establish the actual ordered structure.