We study the magnetic state of a thin film undergoing a reorientation
transition, in the plane defined by the temperature T and the thicknes
s N. In going beyond free spin wave theory, three distinct regions are
found: one with perpendicular magnetization; one with in-plane magnet
ization and a third (high T)-region with no magnetization. We are able
to determine the boundaries between these regions and the resulting p
hase diagram agrees with the existing experimental data. A possible ma
ximum in the reorientation transition T(R)(N) is predicted. Further ex
perimental work is necessary to check this possibility.