= Solution
<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 $J\simeq4|t|^2/U$: 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.<Interaction between the d-Shells in the Transition Metals. II. Ferromagnetic Compounds of Manganese with Perovskite Structure by Clarence Zener>[1951 abstract: correlation of electrical conduction with ferromagnetism and its double-exchange interpretation]{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 $\cos(2k_Fr)/r^3$. The original nuclear-moment calculation was motivated by resonance-line broadening in silver.<Indirect Exchange Coupling of Nuclear Magnetic Moments by Conduction Electrons by M. A. Ruderman and Charles Kittel>[1954 abstract and introduction: nuclear moments coupled through conduction-electron hyperfine interactions]{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.<Detection of Antiferromagnetism by Neutron Diffraction by Clifford Glenwood Shull and J. Samuel Smart>[1949, Fig. 1 and its room-temperature/80 K comparison]{ref}
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