  A characteristic of spin reversal in the presence of phonon-bottleneck is the
deviation of the magnetization cycle from a reversible function into an opened
hysterezis cycle. In recent experiments on molecular magnets (e.g. V$_{15}$ and
Ru$_2$), the zero-field level repulsion was sufficiently large to ensure an
otherwise adiabatic passage through zero-field and the magnetization curves can
be described by using only a phonon-bottleneck model. Here, we generalize the
phonon-bottleneck model into a model able to blend the non-adiabatic dynamics
of spins with the presence of a non-equilibrium phonon bath. In this simple
phenomenological model, Bloch equations are written in the eigenbasis of the
effective spin Hamiltonian, considered to be a two-level system at low
temperatures. The relaxation term is given by the phonon-bottleneck mechanism.
To the expense of calculus time, the method can be generalized to multi-level
systems, where the notion of Bloch sphere does not apply but the density matrix
formalism is still applicable.
