= Solution
<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,
$$
\boxed{H_{\rm eff}=J_{\rm AF}\mathbf S_i\cdot\mathbf S_j,\qquad J_{\rm AF}\simeq\frac{4|t_{\rm eff}|^2}{U}>0.}
$$
This positive sign denotes antiferromagnetism in this convention, unlike the $-2J$ convention in Question1. A metal–ligand–metal path has $t_{\rm eff}\sim t_{pd}^2/\Delta$ and hence the scale $J_{\rm AF}\sim t_{pd}^4/(\Delta^2U)$. 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 as
$$
\boxed{J_{\rm RKKY}(R)\propto J_K^2\frac{\cos(2k_{\rm F}R)}{R^3}.}
$$
The 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. <Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons by M. A. Ruderman and Charles Kittel>[1954, conduction-electron-mediated indirect magnetic coupling]{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 is
$$
\boxed{|t_{\rm eff}(\theta)|=|t|\cos(\theta/2),\qquad0\leq\theta\leq\pi.}
$$
Parallel 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. <Interaction between the d-Shells in the Transition Metals. II. Ferromagnetic Compounds of Manganese with Perovskite Structure by Clarence Zener>[1951, connection between conduction and ferromagnetism underlying double exchange]{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.
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