Surface-supported nanoparticles of magnetic atoms raised and still raise a lot of interest, because of potential applications in non-volatile magnetic storage media. In the last few years, magnetic nanostructures were investigated experimentally in terms of various experimental methods [1, 2, 3, 4, 5, 6] such as Scanning Tunnelling Microscopy (STM), X-ray Magnetic Circular Dichroism (XMCD) and the Magneto-Optical Kerr Eﬀect (MOKE). In interpreting these techniques in terms of phenomenological models  and sum-rules , e.g., high anisotropies and orbital moments of single magnetic adatoms on a non-magnetic substrate were found or predicted. Eventually it was and is the ambition of experimental methods to produce and manipulate nanostructures on an atom-by-atom level using for example magnetic tunnel tips . Furthermore, by combining magnetic and non-magnetic materials like Co and Pt , or two diﬀerent magnetic species like Fe and Co , see also the next chapter, to form nanoclusters, tunable magnetic properties seem to be in reach. A theoretical description of systems nanostructured in two dimensions is
even more complex than those in one dimension, since in principle twodimensional translational symmetry no longer helps to reduce the number of possible magnetic configurations. In systems lacking any translational symmetry every single atom has to be characterized by an individual orientation of the magnetization. In order to be still able to describe such systems one has to go back to Section 4.13 in which a scheme was introduced to embed an ensemble of adatoms on top of an otherwise perfect substrate (with twodimensional translational invariance). Suppose that as indicated in Fig. 13.1 a certain section ("cluster") of a solid
system consisting of a substrate and adatoms is selected that contains the actual adatoms, perturbed and unperturbed substrate atoms and of course vacuum sites. Now let m be the total number of adatoms and perturbed
FIGURE 13.1: Cluster section of a semi-infinite system.