We report a neutron scattering study of the magnetic order and dynamics of the bilayer perovskite Sr$_3$Fe$_2$O$_7$, which exhibits a temperature-driven metal-insulator transition at 340 K. We show that the Fe$^{4+}$ moments adopt incommensurate spiral order below $T_text{N}=115$ K and provide a comprehensive description of the corresponding spin wave excitations. The observed magnetic order and excitation spectra can be well understood in terms of an effective spin Hamiltonian with interactions ranging up to third nearest-neighbor pairs. The results indicate that the helical magnetism in Sr$_3$Fe$_2$O$_7$ results from competition between ferromagnetic double-exchange and antiferromagnetic superexchange interactions whose strengths become comparable near the metal-insulator transition. They thus confirm a decades-old theoretical prediction and provide a firm experimental basis for models of magnetic correlations in strongly correlated metals.
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