  We consider here the problem of a "giant spin", with spin quantum number
S>>1, interacting with a set of microscopic spins. Interactions between the
microscopic spins are ignored. This model describes the low-energy properties
of magnetic grains or magnetic macromolecules interacting with a surrounding
spin environment, such as nuclear spins. We describe a general method for
truncating the model to another one, valid at low energies, in which a
two-level system interacts with the environmental spins, and higher energy
terms are absorbed into a new set of couplings. This is done using an instanton
technique. We then verify the accuracy of this technique, by comparing the
results for the low energy effective Hamiltonian, with results derived for the
original giant spin, coupled to a microscopic spin, using exact diagonalisation
techniques.
