Comparison of methods for the calculation of superparamagnetic relaxation times

A general expression for the correlation time of the decay of the magnetization of an assembly of single-domain noninteracting ferromagnetic particles is given in terms of the inverse of the Fokker-Planck operator, The results of Moro and Nordio [G. Moro and P. L. Nordio, Mol. Phys. 56. 255 (1985)], given in the context of dielectric relaxation, are recovered when the Fokker-Planck operator is axially symmetric. Their result is a particular example of Szabo's calculation of the correlation times of the autocorrelation functions of the Legendre polynomials by means of a generalization of the theory of first-passage times [A. Szabo, J. Chem. Phys. 12, 4620 (1980)]. Likewise, the results of Garanin, Ischenko, and Panina (D. A. Garanin, V. V. Ischenko, and L. V. Panina, Teor. Mat. Fiz. 82, 242 (1990) [Theor. Math. Phys. 82, 169 (1990)]) for the integral relaxation time, i.e., the area under the curve of the normalized decay of the magnetization, are regained in the axially symmetric case where it is possible to integrate the Fokker-Planck equation directly. It is shown by manipulation of Kummer's functions that the exact integral expression for the correlation time for simple uniaxial anisotropy derived by Coffey et al. [W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, E. S. Massawe, and J. T. Waldron. Phys. Rev. E 49, 1869 (1994)] by representing the Fokker-Planck equation as a differential-recurrence relation is identical to the integral relaxation time originally derived by Garanin et al. by direct integration of the Fokker-Planck equation.