  The oscillation of tunnel splitting in the biaxial spin system within
magnetic field along the anisotropy axis is analyzed within the particle
mapping approach, rather than in the (\theta-\phi) spin coherent-state
representation. In our mapping procedure, the spin system is transformed into a
particle moving in the restricted $S^1$ geometry whose wave function subjects
to the boundary condition involving additional phase shift. We obtain the new
topological phase that plays the same role as the Wess-Zumino action in spin
coherent-state representation. Considering the interference of two possible
trajectories, instanton and anti-instanton, we get the identical condition for
the field at which tunneling is quenched, with the previous result within spin
coherent-state representation.
