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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.

Indirect exchange couples magnetic moments through intervening degrees of freedom. Its mechanism and sign depend on whether the mediators are virtual charge excitations or mobile carriers.

In superexchange, electrons hop virtually through intermediate ligand orbitals in an insulating solid and return to the original charge sector. For two half-filled orbitals, virtual double occupation lowers the antiparallel-spin state more strongly. In a one-band reduction,This positive sign denotes antiferromagnetism in this convention, unlike the $-2J$ convention in Question1. A metal–ligand–metal path has $t_{\mathrm{eff}}\sim t_{pd}^2/\Delta$ and hence the scale $J_{\mathrm{AF}}\sim t_{pd}^4/({\Delta }^{2}U)$. Orbital geometry, occupation and Hund's coupling can instead favour ferromagnetic channels; superexchange is not universally antiferromagnetic.

In the RKKY interaction, a localized moment polarizes conduction-electron spins through a local exchange, and another moment samples that polarization. Second-order response gives an effective coupling governed by the static spin susceptibility. For an ideal three-dimensional spherical Fermi surface, its long-distance form oscillates approximately asThe overall sign convention and precise prefactor must be specified; the alternating sign and power-law range are the main features. This mechanism is useful for localized moments in metals, for example rare-earth systems. The Fermi surface and dimensionality change the range and oscillation pattern. ^[ref]

In double exchange, carriers move between sites of different valence while strongly aligned by Hund's coupling to local core spins. In the strong-coupling classical-spin limit, the magnitude of their hopping isParallel moments maximize the carrier bandwidth and lower its kinetic energy, favouring ferromagnetism. Doped mixed-valence manganites are the standard setting. The process involves real carrier motion and is distinct from a virtual insulating superexchange or weak-coupling conduction-electron response. ^[ref]

In a real material several mechanisms compete. Ligand geometry, charge-transfer and repulsion energies, carrier concentration and the band susceptibility are therefore needed before predicting the net ordering.

Eu${}^{2+}$ has the half-filled $4f^7$ term with $L=0$. First-order orbital magnetism is therefore quenched already at the ion level, making the dominant moment approximately spin-only. Spin-orbit-induced magnetocrystalline anisotropy arises only through weaker admixtures and other small interactions; magnetic dipole interactions can also contribute. This explains why a cubic anisotropy term is allowed by symmetry but numerically weak.

An appropriate insulating local-moment model is a nearly isotropic Heisenberg model,with dominant effective ferromagnetic couplings in the sign convention $J_{ij}>0$. The Eu sublattice in rocksalt EuO is face-centered cubic; nearest- and further-neighbor effective exchanges can be retained as needed. Because the solid is not metallic, an itinerant-electron Stoner model or a generic free-carrier RKKY interaction mechanism is not the natural starting point. Exchange can still arise through virtual hopping and ligand-mediated processes in an insulator; it does not require mobile dc carriers.