Exact analytic formula for the correlation time of a single-domain ferromagnetic particle

Exact solutions for the longitudinal relaxation time ${\mathit{T}}_{\mathrm{\ensuremath{\parallel}}}$ and the complex susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\parallel}}}$(\ensuremath{\omega}) 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 ${\mathit{T}}_{\mathrm{\ensuremath{\parallel}}}$ as a series of confluent hypergeometric (Kummer) functions which is easily tabulated for all potential-barrier heights. The asymptotic formula for ${\mathit{T}}_{\mathrm{\ensuremath{\parallel}}}$ of Brown is recovered in the limit of high barriers. On conversion of the exact solution for ${\mathit{T}}_{\mathrm{\ensuremath{\parallel}}}$ 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 ${\mathit{T}}_{\mathrm{\ensuremath{\parallel}}}$ is discussed.