  Low-temperature magnetization process of the nanoscale iron cluster in
linearly sweeped fields is investigated by a numerical analysis of
time-dependent Schr$\ddot{\rm o}$dinger equation and the quantum master
equation. We introduce an effective basis method extracting important states,
by which we can obtain the magnetization process effectively. We investigate
the structure of the field derivative of the magnetization. We find out that
the antisymmetric interaction determined from the lattice structure reproduces
well the experimental results of the iron magnets and that this interaction
plays an important role in the iron cluster. Deviations from the adiabatic
process are also studied. In the fast sweeping case, our calculations indicate
that the nonadiabatic transition dominantly occurs at the level crossing for
the lowest field. In slow sweeping case, due to the influence of the thermal
environment to the spin system, the field derivative of the magnetization shows
an asymmetric behavior, the magnetic F$\ddot{\rm o}$hn effect, which explains
the substructure of the experimental results in the pulsed field.
