  The quantum-classical transitions of the escape rates in a uniaxial spin
model relevant to the molecular magnet Mn$_{12}$Ac and a biaxial anisotropic
ferromagnetic particle are investigated by applying the periodic instanton
method. The effective free energies are expanded around the top of the
potential barrier in analogy to Landau theory of phase transitions. We show
that the first-order transitions occur below the critical external magnetic
field $h_x = 1/4$ for the uniaxial spin model and beyond the critical
anisotropy constant ratio $\lambda = 1/2$ for the biaxial ferromagnetic grains,
which are in good agreement with earlier works.
