The effect of a uniform magnetic field applied at an oblique angle to the easy axis of magnetization on the superparamagnetic ͑longitudinal or Néel͒ relaxation time is investigated by numerically solving the FokkerPlanck equation for the smallest nonvanishing eigenvalue. It is demonstrated that the reciprocal of the asymptotic formula for the Kramers escape rate in the intermediate to high damping limit for general nonaxially symmetric potentials when applied to the present problem, yields an acceptable asymptotic approximation to the Néel time for moderate to high values of the damping. Alternatively the corresponding Kramers low dissipation formula ͑energy controlled diffusion͒ provides an acceptable approximation for very small values of the damping. The effect of the gyromagnetic term which gives rise to coupling between the longitudinal and transverse modes of motion generally corresponds to an increase of the smallest nonvanishing eigenvalue and so to a decrease of the Néel relaxation time. The integral relaxation time or area under the slope of the curve of the decay of the magnetization is also evaluated. It is demonstrated that for sufficiently high values of the uniform field ͑much less, however, than that required to destroy the bistable nature of the potential͒ the reciprocal of the lowest nonvanishing eigenvalue ͑proportional to the Néel time, or the time of reversal of the magnetization͒ and the integral relaxation time may differ exponentially from one another signifying the contributions of modes other than that associated with the overbarrier ͑Néel͒ relaxation process to the overall relaxation process. The overall behavior is qualitatively similar ͑apart from the azimuthal dependence͒ to that of the axially symmetric case which arises due to the depletion of the shallower of the two potential wells by the uniform field, so that the fast processes in the deeper of the two wells may come to dominate the relaxation process at sufficiently high values of the uniform field. ͓S0163-1829͑98͒04229-5͔
The effect of a constant magnetic field, applied at an angle P to the easy axis of magnetization, on the Neel relaxation time r of a single domain ferromagnetic particle (with uniaxial anisotropy) is studied by calculating the lowest nonvanishing eigenvalue k, (the escape rate) of the appropriate Fokker-Planck equation using matrix methods. The effect is investigated by plotting kl versus the anisotropy parameter n for various values of P, and the ratio h =g/2a t where g is the external field parameter and k& versus P for various h values (for rotation of the magnetization vector M both in a plane and in three dimensions). If M rotates in a plane the curve of k& versus 1/I is symmetric about t/t=vrl4 in the range 0(t/1(m/2 and significant decrease in r with increasing 1/I is predicted for large ( and n The m. aximum decrease in r occurs at t/mI/4 whereupon r increases again to the /=0 value at P=rr/2. For rotation of M in three dimensions, the curve of X, versus P (0(~7r) is symmetric about~7r/2. Thus the maximum decrease in r again occurs at P= 7r/4 with maximum increase to a value exceeding that at /=0 (i.e., with the field applied along the polar axis with that axis taken as the easy axis), at t/r=rr/2 (field applied along the equator), the /=0 value being again attained at $=7r The.results are shown to be consistent with the behavior predicted by the Kramers theory of the rate of escape of particles over potential barriers. This theory when applied to the potential barriers for the equatorial orientation of the field for rotation in three dimensions yields a simple approximate formula for the escape rate which is in reasonable agreement with the exact k& calculated from the Fokker-Planck equation. Pfeiffer's approximate formula for the barrier height as a function of tr [H. Pfeiffer, Phys. Status Solidi 122, 377 (1990)]is shown to be in reasonable agreement with our results.
The range of validity of the Kramers escape rate for non-axially symmetric problems of superparamagnetic relaxation as a function of friction is investigated. A comparison of the exact smallest non-vanishing eigenvalue of the Fokker-Planck equation with the asymptotic expressions for the Kramers escape rate in very low-damping and intermediate- to high-damping regimes is made. It is demonstrated, by calculating the smallest non-vanishing eigenvalue for the particular non-axially symmetric problem of a uniform magnetic field applied at an oblique angle to the easy axis of a particle having simple uniaxial anisotropy, that the asymptotic formulae provide an acceptable approximation in the ranges of damping for which they are expected to be valid. The range of validity of the non-axially symmetric intermediate- to high-damping formula as a function of the field angle (which is effectively a measure of the departure from axial symmetry) is also investigated.
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