Differential and total cross sections for the quasifree reactions $gamma prightarroweta p$ and $gamma nrightarroweta n$ have been determined at the MAMI-C electron accelerator using a liquid deuterium target. Photons were produced via bremsstrahlung from the 1.5 GeV incident electron beam and energy-tagged with the Glasgow photon tagger. Decay photons of the neutral decay modes $etarightarrow 2gamma$ and $etarightarrow 3pi^0 rightarrow 6gamma$ and coincident recoil nucleons were detected in a combined setup of the Crystal Ball and the TAPS calorimeters. The $eta$-production cross sections were measured in coincidence with recoil protons, recoil neutrons, and in an inclusive mode without a condition on recoil nucleons, which allowed a check of the internal consistency of the data. The effects from nuclear Fermi motion were removed by a kinematic reconstruction of the final-state invariant mass and possible nuclear effects on the quasifree cross section were investigated by a comparison of free and quasifree proton data. The results, which represent a significant improvement in statistical quality compared to previous measurements, agree with the known neutron-to-proton cross-section ratio in the peak of the $S_{11}(1535)$ resonance and confirm a peak in the neutron cross section, which is absent for the proton, at a center-of-mass energy $W = (1670pm 5)$ MeV with an intrinsic width of $Gammaapprox 30$ MeV.
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