Non-Kramers degeneracy and oscillatory tunnel splittings in the biaxial Spin System

We have investigated analytically quantum tunneling of large spin in the biaxial spin system with the magnetic field applied along the hard and medium anisotropy axes by using a purely quantum-mechanical approach. When the magnetic field parallels the hard axis, the tunnel splittings of all the energy level pairs are oscillatory as a function of the magnetic field. The quenching points are completely determined by the coexistence of solutions of the Ince's equation. When the magnetic field points the medium axis, the tunnel splitting oscillations disappear due to no coexistence of solutions. These results coincide the recent experimental observations in the nanomagnet Fe$_8$.