  The magnetocrystalline anisotropy energy E(anis) of free-standing monolayers
and thin films of Fe and Ni is determined using two different semi-empirical
schemes. Within a tight-binding calculation for the 3d bands alone, we analyze
in detail the relation between bandstructure and E(anis), treating spin-orbit
coupling non-pertubatively. We find important contributions to E(anis) due to
the lifting of band degeneracies near the Fermi level by SOC. The important
role of degeneracies is supported by the calculation of the electron
temperature dependence of the magnetocrystalline anisotropy energy, which
decreases with the temperature increasing on a scale of several hundred K. In
general, E(anis) scales with the square of the SOC constant. Including 4s bands
and s-d hybridization, the combined interpolation scheme yields anisotropy
energies that quantitatively agree well with experiments for Fe and Ni
monolayers on Cu(001). Finally, the anisotropy energy is calculated for systems
of up to 14 layers. Even after including s-bands and for multilayers, the
importance of degeneracies persists. Considering a fixed fct-Fe structure, we
find a reorientation of the magnetization from perpendicular to in-plane at
about 4 layers. For Ni, we find the correct in-plane easy-axis for the
monolayer. However, since the anisotropy energy remains nearly constant, we do
not find the experimentally observed reorientation.
