A Generalized Approach to Relaxation Time of Magnetic Nanoparticles With Interactions: From Superparamagnetic Behavior to Spin-Glass Transition

TL;DR

This work addresses how dipolar interactions modify magnetic-nanoparticle relaxation by extending Néel–Brown theory with Tsallis nonextensive statistics, replacing Boltzmann–Gibbs energy distributions with a $q$-distribution. The authors derive a Brownian/Landau–Gilbert–Brown framework on the unit sphere with a Tsallis stationary solution, producing a generalized reversal time and an asymptotic form that reveals three regimes controlled by the entropic index $q$ and a spin-glass–like transition at $T_{cut-off}$. The key contributions are a unified description of relaxation from weak to strong coupling, the identification of a nonergodic glassy state below $T_{cut-off}$, and the interpretation of $T_{cut-off}$ as a spin-glass transition, supported by fits to experimental data showing $q\neq1$ as interactions strengthen. The results offer a consistent theoretical basis for interpreting aging and freezing in dense MNP assemblies and provide a framework for reinterpreting experimental cut-off temperatures in terms of nonextensive thermodynamics with potential practical implications for nanoparticle design. Overall, the Tsallis-based approach extends the scope of relaxation theory beyond Boltzmann–Gibbs statistics to capture complex collective dynamics in interacting magnetic nanoparticle systems.

Abstract

A novel theoretical expression for the relaxation time of magnetic nanoparticles with dipolar interactions is derived from Kramers' theory, extending the Boltzmann-Gibbs framework to incorporate Tsallis statistics. The model provides, for the first time, a unified description of magnetic relaxation from weakly to strongly interacting regimes, culminating in a spin-glass transition. It accounts for both the decrease and increase of the relaxation time with growing dipolar coupling, a long-standing problem in nanoparticle magnetism, as classical phenomenological models fail to elucidate this transition. This result also offers an innovative interpretation of the cut-off temperature, $T_{cut-off}$, as a spin glass transition under the Tsallis distribution framework within the context of Néel-Brown's relaxation theory, thereby contributing to ongoing scientific discussions regarding this phenomenon.

Figures (5)

• Figure 1: (first) $T^{*}$ vs temperature $T$ ; (second) $T_{\mathrm{cut-off}}$ vs temperature for an energy barrier $\Delta V/k_B =555K$ and different values of $q$.
• Figure 2: Temperatures $T$ (solid line) and $T_{\mathrm{cut-off}}$ (dashed line) as a function of $1/\beta'_q$ for q=0.7, indicating the transition temperature where $T=T_{cut-off}$.
• Figure 3: A comparative analysis between the numerically calculated relaxation time, Eq \ref{['relaxtheta']} (symbols) and the asymptotic approximation, Eq. \ref{['model']} (lines), for various values of the parameter $q$. The approximation is more accurate for lower temperatures in the high barrier limit.
• Figure 4: Relaxation time $\tau$ as a function of inverse temperature $1/T$ for samples with increasing concentrations IF, IN, Floc for the conglomerated sample and for a Powder sample. The lines indicate the fit from Eq \ref{['model']}. The relaxation time increases with interaction strength, indicating slower dynamics in more strongly interacting assemblies. An inset displays a fit to the spin-glass equation for the Powder sample yielding $\tau^{*}=10^{-10\pm2}$ s, $z\nu=7\pm3$ and $T_{g} = 130\pm 4\,$K. Data reproduced from Fig. 1 of Dormann et al.Dormann1998.
• Figure 5: Relaxation time versus inverse temperature for the samples C5% and C0.6%. The main plot shows fits to Eq. \ref{['model']}. The inset displays a fit to the spin-glass equation for the C5% sample, yielding the parameters $\tau^{*} = 10^{-6\pm1}$ s, $T_g = 40\pm2$ K, and $z\nu = 12\pm3$. Data reproduced from Fig. 2 of Djurberg et al.Djurberg1997.