Indirect exchange

Magnetic moments can interact through intermediate orbitals or itinerant carriers rather than direct overlap alone. Superexchange uses virtual charge transfer, the RKKY interaction uses conduction-electron spin response, and double exchange gains carrier kinetic energy through spin-dependent hopping. More than one mechanism can operate in the same solid, with different signs and ranges.

A local moment induces conduction-electron spin polarization, which mediates a second-order coupling to another moment. The static electronic spin susceptibility fixes the spatial interaction. For a three-dimensional spherical Fermi surface its leading large-distance form oscillates as $\cos (2k_{F}R)/R^3$, up to the overall sign convention and prefactor. The sign alternates with separation. Other band structures and dimensions change this form. ^[ref]