Magnetic relaxation time for an ensemble of nanoparticles with randomly aligned easy axes: A simple expression

A critical parameter in characterizing the properties of single-domain nanoparticles is their magnetic relaxation time. It must be known, for example, to estimate the anisotropy from magnetization versus temperature measurements. The time it takes for the magnetization to relax also determines the behavior of particles in various oscillating applied fields, which is critically important for their application in magnetic particle imaging and hyperthermia treatment. However, an analytic expression for this relaxation time has been generally missing. Brown's [Phys. Rev. 130, 1677 (1963)] famous result is only valid for the easy anisotropy axes of each particle in the ensemble aligned along the external field direction and overestimates the relaxation time. Despite this overestimation, this expression is most commonly used to extract magnetic nanoparticle parameters such as anisotropy energy from magnetometry data. Here, we use Brown's formalism to derive a different, simple, approximate relaxation time expression that is valid for randomly aligned easy axes. Using parameters appropriate for magnetite, we compare our results to other results in the literature and with stochastic Landau-Lifshitz-Gilbert simulations to show that our result is more accurate across a range of applied field strengths and temperatures. We note that several analytic expressions for the relaxation time do a reasonable job, as long as one uses a full calculation for the attempt time, rather than the commonly used estimate ${\tau }_0\sim 1$ ns.

Figure 1

(a) The nanoparticle geometry. The easy axis (denoted $\mathbf{K}$) for each particle is aligned along the $z$ direction, and the external applied field $\mathbf{H}$ is oriented in the $xz$ plane at an angle $\psi$ from the easy axis. The magnetic moment $\mathbf{M}$ is allowed to rotate freely with polar angle $\theta$ and azimuthal angle $\varphi$. (b) The probability density for particles in a random ensemble to have a polar angle between $\psi$ and $\psi +d\psi$. The line shows the $sin\left(\theta \right)$ prediction, and the bars represent binned orientations from a simulation of MNPs. (c) The energy landscape for parallel alignment, given by Eq. (9), as a function of polar angle $\theta$. Note that for MNPs aligned with the applied field there are two barriers shown by the orange circle and green diamond. (d) The energy landscape for perpendicular alignment, given by Eq. (11). The dashed line is for $\varphi =\pi$ (maximum barrier), and the solid line is for $\varphi =0$ (minimum energy barrier, star). (e) Comparing the energy barrier heights for MNPs that have various orientation angles $\psi$ with the applied field. The symbols correspond to the limiting barriers indicated in (c) and (d).

Figure 2

Normalized magnetic moment along the applied field direction of 5000 particles (randomly aligned easy axes) as a function of time, simulated by the RK4 spherical polar stochastic LLG equation (19). Here, $H=200$ Oe, $\alpha =0.01$, and $T=150$ K (blue dots). The trajectory is fitted by the red dashed line starting from the vertical orange line to find a value of the relaxation time $\tau$. The fit has a square mean error less than 0.57%.

Figure 3

The relaxation time for an ensemble of MNPs with easy axes all aligned along the applied field ${\tau }_{\left|\right|}$ as a function of temperature. Symbols represent results from LLG simulations, and the lines are the predictions of Eq. (3) from Brown. Damping parameters $\alpha =0.01$ (red/orange palette) and $\alpha =0.1$ (green/blue palette) are considered. The applied field strengths are 50 and 200 Oe.

Figure 4

The relaxation time for parallel (diamonds) and randomly aligned MNPs (dots) as a function of temperature, with results taken from simulations. Here, $\alpha =0.01$, and $H=200$ Oe. The lines represent analytic expressions for ${\tau }_{\left|\right|}$ [solid line, Eq. (3)] and ${\tau }_{\perp }$ [dashed line, Eq. (17)].

Figure 5

The relaxation times ${\tau }_{\text{rand}}$ (from simulations, dots and squares) and ${\tau }_{\perp }$ [Eq. (17), lines] as a function of temperature, this time for $\alpha =0.1$. Two applied field values, namely, 50 Oe (solid line) and 200 (dashed line), are shown.

Figure 6

Simulated relaxation times (diamonds) as a function of temperature, with $\alpha =0.01$ and an applied field strength of 200 Oe, are compared to various analytic expressions for relaxation time. (a) The Néel-Arrhenius equation with constant prefactor ${\tau }_0=1$ ns (dashed line) and Brown's temperature-dependent prefactor (solid line). (b) The Néel barrier equation with (i) constant prefactor ${\tau }_0=1$ ns (long-dashed line), (ii) Brown's temperature-dependent prefactor (short-dashed line), and (iii) our Eq. (17) (solid line). (c) Literature results, including those by Coffey and Kalmykov [26] (pink dotted line), an integration of Pfeiffer's result [24] (mauve dashed line), and Victora's expression (solid purple line) [16]. See the main text for details.