We demonstrate that the reorientation transition from out-of-plane to
in-plane magnetization with decreasing temperature as observed experim
entally in Ni films on Cu(001) can be explained on a microscopic basis
. Using a combination of mean-field theory and perturbation theory, we
derive an analytic expression for the temperature-dependent anisotrop
y. The reduced magnetization in the film surface at finite temperature
s plays a crucial role for this transition as with increasing temperat
ure the influence of the uniaxial anisotropies is reduced at the surfa
ce and is enhanced inside the film.