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We consider here the problem of a "giant spin", with spin quantum numberS>>1, interacting with a set of microscopic spins. Interactions between themicroscopic spins are ignored. This model describes the low-energy propertiesof magnetic grains or magnetic macromolecules interacting with a surroundingspin environment, such as nuclear spins. We describe a general method fortruncating the model to another one, valid at low energies, in which atwo-level system interacts with the environmental spins, and higher energyterms are absorbed into a new set of couplings. This is done using an instantontechnique. We then verify the accuracy of this technique, by comparing theresults for the low energy effective Hamiltonian, with results derived for theoriginal giant spin, coupled to a microscopic spin, using exact diagonalisationtechniques.Comment: 15 pages, Latex, with 9 ps figure

Effective Hamiltonian in the Problem of a "Central Spin" Coupled to a Spin Environment