  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 $\eta\rightarrow 2\gamma$ and
$\eta\rightarrow 3\pi^0\rightarrow 6\gamma$ 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.
