  In this paper we define a non-dynamical phase for a spin-1/2 particle in a
rotating magnetic field in the non-adiabatic non-cyclic case, and this phase
can be considered as a generalized Berry phase. We show that this phase reduces
to the geometric Berry phase, in the adiabatic limit, up to a factor
independent of the parameters of the system. We could add an arbitrary phase to
the eigenstates of the Hamiltonian due to the gauge freedom. Then, we fix this
arbitrary phase by comparing our Berry phase in the adiabatic limit with the
Berry's result for the same system. Also, in the extreme non-adiabatic limit
our Berry phase vanishes, modulo $2\pi$, as expected. Although, our Berry phase
is in general complex, it becomes real in the expected cases: the adiabatic
limit, the extreme non-adiabatic limit, and the points at which the state of
the system returns to its initial form, up to a phase factor. Therefore, this
phase can be considered as a generalization of the Berry phase. Moreover, we
investigate the relation between the value of the generalized Berry phase, the
period of the states and the period of the Hamiltonian.
