Differential Recurrence Relations for Non‐Axially Symmetric Rotational Fokker‐Planck Equations

Historical Background and Introductory Concepts

“…Then as a result of thermal fluctuations, on a noisy trajectory in the vicinity of the saddle energy the spin may have is the TST reversal time. In the VLD regime, the system is only very lightly coupled to the bath so that the energy loss per cycle of the almost-periodic noisy motion of the magnetization on the saddle-point energy (escape) trajectory is much less than the thermal energy, The application of Kramers' escape theory to superparamagnetic relaxation in the IHD limit has been given in detail by Smith and de Rozario [126], Brown [112], Klik and Gunther [125], and Geoghegan et al [78]. Klik and Gunther [125] used Langer's method [73] (described in Section 1.13.5) and realized that the various Kramers damping regimes also applied to magnetic relaxation of single-domain ferromagnetic particles.…”

“…Equation (1.13.5.6) constitutes the paraboloidal approximation to the potential in the vicinity of the saddle point. For example, in magnetic relaxation in a uniform field with uniaxial anisotropy, the energy surface in the vicinity of the saddle point will be a hyperbolic paraboloid [78].…”

Effects of dipolar interactions on the zero-field-cooled magnetization of a nanoparticle assembly

“…8 it was shown that the bell-like shape of T max (H) is not very sensitive to the intrinsic properties of the particles, of course in the OSP approximation. Exact numerical calculations [12,13,14] of the smallest eigenvalue of the Fokker-Planck operator invariably led to a monotonic decrease in the blocking temperature, and thereby in the temperature T max , as a function of the magnetic field. Indeed, it was shown that the expression of the single-particle relaxation time does not play a crucial role and that even the (relatively) simple Néel-Brown expression for the relaxation time in a longitudinal field leads to a maximum in T max (H).…”

Reversal time of the magnetization of single-domain ferromagnetic particles with mixed uniaxial and cubic anisotropy

“…By substituting crossover function ͑A3͒ into Eq. ͑A1͒, we obtain a nonlinear partial differential equation for ͑n 1 28 the well boundary is parametrized by n 1 = 0, therefore putting dn 1 =0 in the contour integral ͓Eq. ͑18͔͒; and retaining the parabolic approximation only in the factor e −␤V͑0,n 2 ͒ , Eq.…”