Novel oscillation for the indirect coupling between magnetic   nanoparticles
  We study the prospect of using magnetic nanoparticles for the diluted magnetic seminconductors. The Ruderman-Kittel-Kasuya-Yosida formula is modified by explicitly taking into account the role of charge carriers inside the nanoparticles in addition to those in the medium. Calculations are done analytically. The final form of the coupling between nanoparticles is similar to the original formula except for additional phase terms which render a novel oscillatory feature with respect to the particle size - an interesting analogy to the same behavior when we vary their distance. This is to be contrasted to the previous approach which did not include the inner carriers and only favored a ferromagnetic coupling. The effect of inevitable deviations from a perfect sphere is estimated by the Born approximation. Our derivations can be readily generalized to finite temperatures. 
C. H. Chang and T. M. Hong