Bridging the gap between discrete and continuous magnetic models in the scaling approach

The scaling technique, often used to study magnetic properties of nanostructures, is based on a reduction in the geometrical lengths of the system via factor x and a scaling [eta][Phys. Rev. Lett. 88, 237202 (2002)(http://dx.doi.org/10.1103/PhysRevLett.88.237202)]. At the same time, to keep competition among the energy terms involved, the exchange constant J is reduced throughout the system in accordance with the scaling factor x. Different values for the scaling exponent [eta] have been used in several studies, and the discussion of whether there is just one or several values is resolved here. In this study we revise the scaling method and propose a different approach than the above. In this generalized approach, the scaling scheme is applied to the energy term as a whole, instead of only to the exchange coupling J. We apply this proposed methodology to typical iron cylindrical samples and find that the application of the scaling technique to systems in the mesoscopic range does not depend on the selection of the value of a scaling exponent. We show that this generalized approach gives the same results when it is applied under both discrete and continuous frameworks, and, therefore, it represents a solution for this pending issue.