  The magnetocrystalline anisotropy energy $E_{anis}$ for free-standing chains
(quantum wires) and rings (quantum corrals) of Fe-adatoms $N=$(2...48) is
determined using an electronic tight-binding theory. Treating spin-orbit
coupling non-perturbatively, we analyze the relationship between the electronic
structure of the Fe $d$-electrons and $E_{anis}(n_{d})$, for both the chain and
ring conformations. We find that $E_{anis}(N)$ is larger for wires than for
rings or infinite monolayers. Generally $E_{anis}(n_{d})$ decreases in chains
upon increasing $N$, while for rings $E_{anis}(n_{d})$ is essentially
independent of $N$. For increasing $N$, $E_{anis}(n_{d})$ in corrals approaches
the results for freestanding monolayers. Small rings exhibit clear odd-even
oscillations of $E_{anis}(N)$. Within our theoretical framework we are able to
explain the experimentally observed oscillations of $E_{anis}(n_{d})$ during
film growth with a period of one monolayer. Finally, a generalization of Hund's
third rule on spin-orbit coupling to itinerant ferromagnets is proposed.
