Precise helicity-dependent cross sections and the double-polarization observable $E$ were measured for $eta$ photoproduction from quasi-free protons and neutrons bound in the deuteron. The $etarightarrow 2gamma$ and $etarightarrow 3pi^0rightarrow 6gamma$ decay modes were used to optimize the statistical quality of the data and to estimate systematic uncertainties. The measurement used the A2 detector setup at the tagged photon beam of the electron accelerator MAMI in Mainz. A longitudinally polarized deuterated butanol target was used in combination with a circularly polarized photon beam from bremsstrahlung of a longitudinally polarized electron beam. The reaction products were detected with the electromagnetic calorimeters Crystal Ball and TAPS, which covered 98% of the full solid angle. The results show that the narrow structure observed earlier in the unpolarized excitation function of $eta$ photoproduction off the neutron appears only in reactions with antiparallel photon and nucleon spin ($sigma_{1/2}$). It is absent for reactions with parallel spin orientation ($sigma_{3/2}$) and thus very probably related to partial waves with total spin 1/2. The behavior of the angular distributions of the helicity-dependent cross sections was analyzed by fitting them with Legendre polynomials. The results are in good agreement with a model from the Bonn-Gatchina group, which uses an interference of $P_{11}$ and $S_{11}$ partial waves to explain the narrow structure.
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