Coffey, W.T., Crothers, D.S.F., Kalmykov, Yu.P., Massawe, E.S., Waldron, J.T., 'Exact analytic formula for the correlation time of a single-domain ferromagnetic particle' in Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 49, 3, (1994), pp 1869-1882
Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics 49 3
Exact solutions for the longitudinal relaxation time T and the complex susceptibility χ(ω) of a thermally agitated single-domain ferromagnetic particle are presented for the simple uniaxial potential of the crystalline anisotropy considered by Brown [Phys. Rev. 130, 1677 (1963)]. This is accomplished by expanding the spatial part of the distribution function of magnetic-moment orientations on the unit sphere in the Fokker-Planck equation in Legendre polynomials. This leads to the three-term recurrence relation for the Laplace transform of the decay functions. The recurrence relation may be solved exactly in terms of continued fractions. The zero-frequency limit of the solution yields an analytic formula for T as a series of confluent hypergeometric (Kummer) functions which is easily tabulated for all potential-barrier heights. The asymptotic formula for T of Brown is recovered in the limit of high barriers. On conversion of the exact solution for T to integral form, it is shown using the method of steepest descents that an asymptotic correction to Brown's high-barrier result is necessary. The inadequacy of the effective-eigenvalue method as applied to the calculation of T is discussed.
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