Magnetic-field dependence of macroscopic quantum tunnelingand coherence of ferromagnetic particles

We calculate the quantum tunneling rate of a ferromagnetic particle of ∼100 Å diameter in a magnetic field of arbitrary angle. We consider the magnetocrystalline anisotropy with the biaxial symmetry and that with the tetragonal symmetry. Using the spin-coherent-state path integral, we obtain approximate analytic formulas of the tunneling rates in the small ε(=1-H/${\mathrm{H}}_{\mathrm{c}}$) limit for the magnetic field normal to the easy axis (${\mathrm{\theta }}_{\mathrm{H}}$=π/2), for the field opposite to the initial easy axis (${\mathrm{\theta }}_{\mathrm{H}}$=π), and for the field at an angle between these two orientations (π/2≪${\mathrm{\theta }}_{\mathrm{H}}$≪π). In addition, we obtain numerically the tunneling rates for the biaxial symmetry in the full range of the angle ${\mathrm{\theta }}_{\mathrm{H}}$ of the magnetic field (π/2<${\mathrm{\theta }}_{\mathrm{H}}$⩽π), for the values of ε=0.01 and 0.001.