Magnetization dynamics of two interacting spins in an external magnetic field

The longitudinal relaxation time of the magnetization of a system of two exchange coupled spins subjected to a strong magnetic field is calculated exactly by averaging the stochastic Gilbert-Landau-Lifshitz equation for the magnetization, i.e., the Langevin equation of the process, over its realizations, so reducing the problem to a system of linear differential-recurrence relations for the statistical moments (averaged spherical harmonics). The system is solved in the frequency domain by matrix-continued fractions, yielding the complete solution of the two-spin problem in external fields for all values of the damping and barrier height parameters. The magnetization relaxation time extracted from the exact solution is compared with the inverse relaxation rate from Langer’s theory of the decay of metastable states [J. S. Langer, Ann. Phys. (N.Y.) 54, 258 (1969)], which yields in the high barrier and intermediate-to-high damping limits the asymptotic behavior of the greatest relaxation time.

Figure 1

Relaxation time $\tau ∕{\tau }_N$ as a function of the interaction parameter $\varsigma$ for various values of the external field parameter $h_{\mathrm{II}}$ in the linear response condition $h_{\mathrm{I}}-h_{\mathrm{II}}=0.001$.

Figure 2

Relaxation time $\tau ∕{\tau }_N$ as a function of the anisotropy parameter $\sigma$ for various values of the interaction parameter $\varsigma$. Exact matrix-continued fraction solution for the relaxation time $\tau ∕{\tau }_N$ [solid lines: Eq. (24)] is compared with the inverse reaction rate rendered by Langer’s theory of the decay of metastable states [filled circles and triangles: Eqs. (B1, B2)].

Figure 3

Relaxation time $\tau ∕{\tau }_N$ as a function of the interaction parameter $\varsigma$ for various values of the anisotropy parameter $\sigma$. Exact solution [solid lines: Eq. (24)] is compared with the inverse reaction rate rendered by Langer’s theory of the decay of metastable states [filled circles and triangles: Eqs. (B1, B2)].

Figure 4

Relaxation time $\tau ∕{\tau }_N$ as a function of the anisotropy parameter $\sigma$ for various values of the interaction parameter $\varsigma$.

Figure 5

The imaginary part of the complex susceptibility ${\chi }^{″}\left(\omega \right)$ as a function of $\omega {\tau }_N$ for various values of the interaction parameter $\varsigma$. Solid lines: the matrix-continued fraction solution; filled circles: the approximate Eq. (32); dotted and dashed lines: the low- and high-frequency asymptotes, Eq. (8), respectively. In the calculation, the following numerical values of $\tau ∕{\tau }_N$ [Eq. (24)], ${\tau }^{\mathit{ef}}∕{\tau }_N$ [Eq. (30)], and $\Delta$ (best fit) were used in Eq. (32) : $\tau ∕{\tau }_N=61.1$, 143.8, and 3348; ${\tau }^{\mathit{ef}}∕{\tau }_N=10.52$, 18.65, and 29.14; $\Delta =0.01$, 0.005, and 0.003 for $\varsigma =0.01$, 1.0, and 5.0, respectively.