Damping dependence of the magnetization relaxation time of single-domain ferromagnetic particles

The relaxation time of the magnetization of single-domain ferromagnetic particles with cubic and triaxial (i.e., orthorhombic) magnetocrystalline anisotropy is estimated for all dissipation regimes, i.e., very low damping (VLD), intermediate-to-high damping (IHD), and turnover, using the method of Coffey et al. (Adv. Chem. Phys. 117 (2001) 483; Phys. Rev. E 63 (2001) 021102). Their method generalizes the Mel'nikov-Meshkov approach (J. Chem. Phys. 85 (1986) 1018) for bridging the VLD and IHD Kramers' escape rates for mechanical Brownian particles (the Kramers' turnover problem) to the analogous magnetic turnover problem. It is shown that the simple asymptotic formulae for the greatest relaxation time so obtained are in complete agreement with the relaxation time calculated from the infinite hierarchy of linear differential-recurrence equations for the statistical moments. This hierarchy, which governs the relaxation of the magnetization of an individual particle, is derived by averaging the underlying stochastic Landau-Lifshitz-Gilbert equation over its realizations. The exact solution of the system of moment equations is obtained by matrix continued fractions.