  Non-adiabatic transitions are studied in a spin-boson model with multiple
scattering points. In order to generalize the Landau-Zener formula, which
describes the case of a single scattering point, we define an ``effective gap''
for a set of scattering points. The generalized formula agrees very well with
numerical results of the non-adiabatic dynamics, which we obtained by a direct
numerical method. This will make the Landau-Zener formula yet more useful in
analyzing experimental data of magnetic-moment inversion.
