Longitudinal complex magnetic susceptibility and relaxation times of superparamagnetic particles with triaxial anisotropy

The longitudinal relaxation time and spectrum of the complex magnetic susceptibility of single domain ferromagnetic particles with triaxial (orthorhombic) anisotropy are calculated by averaging the Gilbert-Langevin equation for the magnetization of an individual particle and by reducing the problem to that of solving a system of linear differential-recurrence relations for the appropriate equilibrium correlation functions. The solution of this system is obtained in terms of matrix continued fractions. It is shown that in contrast to the linear magnetic response of particles with uniaxial anisotropy, there is an inherent geometric dependence of the complex susceptibility and the relaxation time on the damping parameter arising from coupling of longitudinal and transverse relaxation modes. Simple analytic equations, which allow one to understand the qualitative behavior of the system and to accurately predict the spectrum of the longitudinal complex susceptibility in wide ranges of the barrier height and dissipation parameters, are proposed.

Figure 1

${\tau }_{\Vert }∕{\tau }_0$ vs $\alpha$ for $\Delta =10$ and various values of $\sigma$. Solid lines: exact matrix continued fraction solution; dashed lines: the VLD Eq. (19); dotted lines: the IHD Eq. (18); symbols: the universal Eq. (17).

Figure 2

${\tau }_{\Vert }∕{\tau }_0$ vs $\alpha$ for $\sigma =10$ and various values of $\Delta$. Solid lines: matrix continued fraction solution; dashed lines: the VLD Eq. (19); dotted lines: the IHD Eq. (18); symbols: the universal Eq. (17).

Figure 3

${\tau }_{\Vert }∕{\tau }_0$ vs $\sigma$ for $\alpha =0.1$ and various values of $\Delta$. Solid lines: matrix continued fraction solution; symbols: the universal Eq. (17); dotted line: Eq. (26).

Figure 4

$-\mathrm{Im}\left[{\chi }_{\Vert }\left(\omega \right)\right]$ vs $\omega {\tau }_N$ for $\sigma =10$, $\Delta =10$, and various values of $\alpha$. Solid lines 1–3: matrix continued fraction solution. Symbols: Eqs. (21, 23, 24, 25); dotted and dashed lines: Eq. (22).

Figure 5

$-\mathrm{Im}\left[{\chi }_{\Vert }\left(\omega \right)\right]$ vs $\omega {\tau }_N$ for $\Delta =10$, $\alpha =0.01$, and various values of $\sigma$. Solid lines 1–3: matrix continued fraction solution. Symbols: Eqs. (21, 23, 24, 25); dotted and dashed lines: Eq. (22).

Figure 6

$-\mathrm{Im}\left[{\chi }_{\Vert }\left(\omega \right)\right]$ vs $\omega {\tau }_N$ for $\sigma =10$, $\alpha =0.01$, and various values of $\Delta$. Solid lines 1–4: matrix continued fraction solution. Filled circles: Eqs. (21, 23, 24, 25); dotted and dashed lines: Eq. (22).