  We examine an exactly solvable model of decoherence -- a spin-system
interacting with a collection of environment spins. We show that in this simple
model (introduced some time ago to illustrate environment--induced
superselection) generic assumptions about the coupling strengths lead to a
universal (Gaussian) suppression of coherence between pointer states. We
explore the regime of validity of this result and discuss its relation to
spectral features of the environment. We also consider its relevance to the
experiments on the so-called Loschmidt echo (which measures, in effect, the
fidelity between the initial and time-reversed or "echo" signal). In
particular, we show that for partial reversals (e.g., when of only a part of
the total Hamiltonian changes sign) fidelity will exhibit a Gaussian dependence
on the time of reversal. In such cases echo may become independent of the
details of the reversal procedure or the specifics of the coupling to the
environment. This puzzling behavior was observed in several NMR experiments.
Natural candidates for such two environments (one of which is easily reversed,
while the other is ``irreversible'') are suggested for the experiment involving
ferrocene.
