We report high-resolution, bulk Compton scattering measurements unveiling the Fermi surface of an optimally-doped iron-arsenide superconductor, Ba(Fe$_{0.93}$Co$_{0.07}$)$_2$As$_2$. Our measurements are in agreement with first-principles calculations of the electronic structure, revealing both the $X$-centered electron pockets and the $Gamma$-centered hole pockets. Moreover, our data are consistent with the strong three-dimensionality of one of these sheets that has been predicted by electronic structure calculations at the local-density-approximation-minimum As position. Complementary calculations of the noninteracting susceptibility, $chi_0({bf q}, omega)$, suggest that the broad peak that develops due to interband Fermi-surface nesting, and which has motivated several theories of superconductivity in this class of material, survives the measured three dimensionality of the Fermi surface in this family.
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