  We consider here the problem of a "central spin", with spin quantum number $S
\gg 1$, interacting with a set of microscopic spins. Interactions between the
microscopic spins are ignored. This model describes magnetic grains or magnetic
macromolecules (ferromagnetically or antiferromagnetically ordered) interacting
with nuclear spins and with any surrounding paramagnetic electronic spins. It
has also been used to describe $Si:P$ near the metal-insulator transition, and
quantum spin glasses.
  In this paper we consider the possible coherent motion of the central spin.
We reduce the model to an effective low-energy Hamiltonian, and then calculate
the correlation function <S(t)S(0)> using instanton methods. The general
solution is given for the unbiased case, for all relevant values of the
couplings. Under certain conditions, nuclear spin diffusion also plays a role.
The results are then used to calculate the spectral absorption for 2
experimental systems, viz., $TbFe_3$ grains, and ferritin molecules. Our
results have general implications for the observation of mesoscopic and
macroscopic quantum coherence, and for the foundations of quantum mechanics.
They show that a spin environment usually has a very destructive effect on
coherence, which cannot be understood using the conventional "oscillator bath"
models of quantum environments.
