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A localized magnetic moment can polarize a metal's conduction electrons, whose response changes the energy of a second moment. This indirect exchange interaction can be long ranged and oscillate in sign with separation because of the Fermi surface. In the three-dimensional isotropic free-electron asymptotic limit its range dependence is proportional to $\cos (2k_{F}r)/r^3$, with coupling-dependent prefactor and sign convention. The original calculation for nuclear moments used their contact hyperfine interaction with conduction electrons; it was motivated by nuclear magnetic resonance line widths in metallic silver.^[ref]

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.

Exchange interactions arise from quantum electronic motion, Coulomb interactions and fermionic antisymmetry. A localized-spin description can be written $H={\sum }_{i<j}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. In this explicit plus-sign convention $J<0$ favors parallel spins and $J>0$ favors antiparallel spins; many texts put a minus sign into the Hamiltonian and reverse that convention.

Direct exchange involves overlapping orbitals and their Coulomb exchange matrix elements. The Pauli exclusion principle correlates spin symmetry with spatial symmetry, changing the electronic energy. Orbital occupation and competing kinetic processes determine which spin state is favored; the preference is not universally ferromagnetic.

Superexchange couples moments through virtual hopping, often via a nonmagnetic ligand. For two half-filled orbitals in the simple repulsive one-band model, the antiferromagnetic coupling is $J\simeq 4|t{|}^2/U$: virtual double occupation lowers the antiparallel-spin state more effectively. Orbital geometry and intermediate-state interactions can change the sign in other configurations. Real charge transport can remain insulating while this magnetic coupling acts.

Double exchange instead uses actual carrier motion between mixed-valence sites. Local Hund's coupling ties the mobile electron's spin to a core moment. Parallel neighboring cores allow a larger effective hopping amplitude and a lower carrier kinetic energy, favoring ferromagnetism. The original model explicitly connected conduction and magnetic order in manganese perovskites.^[ref]

Conduction-electron-mediated exchange couples localized moments through a metal's electronic spin response. Its strength and sign depend on separation and the Fermi surface; in a simple three-dimensional asymptotic model the range factor oscillates as $\cos (2k_{F}r)/r^3$. The original nuclear-moment calculation was motivated by resonance-line broadening in silver.^[ref] In an itinerant metal, an alternative description compares the kinetic-energy cost of unequal spin populations with the exchange-energy gain; the Stoner criterion expresses when spontaneous polarization is favorable.

The resulting order also depends on the lattice and competing couplings. In ferromagnetism, moments align to give a nonzero moment within a domain. In antiferromagnetism, opposing sublattices cancel their moments. In ferrimagnetism, oppositely ordered sublattices have unequal moments and leave a net one. A bulk zero magnetization alone cannot distinguish antiferromagnetic order from a disordered state or canceling magnetic domains.

The manganese monoxide neutron diffraction experiment makes that distinction experimentally. A monochromatic neutron beam scatters from powdered material, and a counter records angular intensity. Comparing the same compound at room temperature and $80\mathrm{K}$ reveals additional low-temperature magnetic reflections, indexed by a larger magnetic unit cell, while nuclear reflections persist. That spatial periodicity is evidence of alternating order even though the net moment cancels.^[ref]

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]