Spin tunneling of trigonal and hexagonal ferromagnets in an arbitrarily directed magnetic field

The quantum tunneling of the magnetization vector are studied theoretically in single-domain ferromagnetic nanoparticles placed in an external magnetic field at an arbitrarily directed angle in the $ZX$ plane. We consider the magnetocrystalline anisotropy with trigonal and hexagonal crystal symmetry, respectively. By applying the instanton technique in the spin-coherent-state path-integral representation, we calculate the tunnel splittings, the tunneling rates and the crossover temperatures in the low barrier limit for different angle ranges of the external magnetic field ($\theta_{H}=\pi/2$, $\pi/2\ll\theta_{H}\ll\pi$, and $\theta_{H}=\pi$). Our results show that the tunnel splittings, the tunneling rates and the crossover temperatures depend on the orientation of the external magnetic field distinctly, which provides a possible experimental test for magnetic quantum tunneling in nanometer-scale single-domain ferromagnets.