  The helicity-dependent single $\pi^{0}$ photoproduction cross section on the
deuteron and the angular dependence of the double polarisation observable $E$
for the quasi-free single $\pi^0$ production off the proton and the neutron
have been measured for the first time from the threshold region up to the
photon energy 1.4 GeV. The experiment was performed at the tagged photon
facility of the MAMI accelerator and used a circularly polarised photon beam
and longitudinally polarised deuteron target. The reaction products were
detected using the large acceptance Crystal Ball/TAPS calorimeter, which
covered 97% of the full solid angle.
  Comparing the cross section from the deuteron with the sum of free nucleon
cross sections provides a quantitative estimate of the effects of the nuclear
medium on pion production. In contrast, comparison of $E$ helicity asymmetry
data from quasi-free protons off deuterium with data from a free proton target
indicates that nuclear effects do not significantly affect this observable. As
a consequence, it is deduced that the helicity asymmetry $E$ on a free neutron
can be reliably extracted from measurements on a deuteron in quasi-free
kinematics.
