  The magnetocrystalline anisotropy energy $E_{anis}$ for a monolayer of Fe and
Ni is determined using a fully convergent tight-binding calculation including
$s$-$d$ hybridization. The spin-orbit interaction $\lambda_{so}$ is treated
non-perturbatively. Remarkably, we find $E_{anis}\propto\lambda_{so}^2$ and
important contributions to $E_{anis}$ due to the lifting of degeneracies near
the Fermi-level. This is supported by the calculated decrease of the anisotropy
energy with increasing temperature on a scale of several hundred K. Our results
clarify the present debate on the origin of $E_{anis}$.
