Manifestation of intermediate spin for Fe$_{8}$

Intermediate spin, which occurs in the theory of anyons, can also be exhibited by mesoscopic magnetic particles. The necessary broken time reversal symmetry is due to a suitably directed magnetic field. As a function of this field a system passes periodically through points which correspond to whole or half-integer spin. Intermediate spin is defined by fields which lie between these points. Since the tunnel splitting in the ground doublet vanishes for half-integer spin, this splitting becomes periodic. The doubly periodic oscillations observed in the magnetic molecular cluster Fe$_{8}$ represent the first unequivocal observation of this phenomena. Here the whole or half-integer nature of the system, i.e., the parity, is periodic for a field which is along either the easy or hard axis or a suitable combination of the two. A detailed theory is presented.