Effective Magnetic Susceptibility of Dust Grains with Superparamagnetic Inclusions and Implications

Abstract

Magnetic properties of dust grains play a fundamental role in their alignment with ambient magnetic fields and magnetic dipole emission. In the radiative torque (RAT) paradigm, superparamagnetic inclusions (SPIs) embedded within dust grains are expected to significantly enhance magnetic susceptibility and alignment efficiency. Previous studies have generally assumed SPIs of a single characteristic size. In this work, we develop an effective superparamagnetism model that explicitly accounts for a power-law size distribution of SPIs. We show that the effective zero-frequency susceptibility can be described by the superparamagnetic susceptibility of uniform-sized inclusions evaluated at the critical blocking size, reduced by a factor $F_{\rm eff}\sim 0.1$. It exhibits a slight increase with dust temperature $T_{d}$, in contrast to the rapid decrease for the case of single-size SPIs. For rotating grains at angular frequency $ω$, we identify a characteristic resonance size of SPIs that dominates the magnetic response, $N_{\rm res} = (T_{d}/T_{\rm act}) \ln (ν_{0}/ω)$ with $T_{\rm act}$ activation temperature and $ν_{0}$ the characteristic attempt frequency of SPIs. The frequency-dependent effective susceptibility is well described by the maximum susceptibility $χ_{\rm eff}^{\rm max}(ω)$ at $N_{\rm res}$, reduced by a factor $G_{\rm eff}\sim 0.1$. Unlike models assuming uniform-sized inclusions, we find that the effective susceptibility exhibits a nearly flat spectrum for frequency below $ν_{0}$, arising from the progressive activation of larger inclusions at lower frequencies. This effective superparamagnetism model based on the SPI size distrbution has important implications for magnetic grain alignment, dust polarization, and magnetic dipole emission across diverse environments.

Figures (5)

• Figure 1: Left panel: The variation of the critical blocking size of SPIs and the effective reduction factor with $T_{d}$ for the standard slope of $q=11/6, 13/6, 15/7$. Right panel: The variation of the zero-frequency susceptibility with the dust temperature for the single-size SPIs compared to the effective susceptibility and the maximum susceptibility at the blocking size $N_{\rm cri}$. The maximum and effective susceptibility increase with $T_{d}$, in contrast to the decrease as $1/T_{d}$ for the single-size SPIs.
• Figure 2: Variation of the resonance SPI size ($N_{\rm res}$) and reduction factor $G_{\rm eff}$ as a function of the grain rotation rate for the different dust temperature. For realistic rotation rates of $\omega<10^{6}\,{\rm s}^{-1}$, $N_{\rm res}$ increases from $10^{4}$ to the maximum SPI size of $N_{\rm max}=5\times 10^{5}$ for $T_{d}$ increases from $15\,{\rm K}$ to $500\,{\rm K}$.
• Figure 3: The imaginary part of the magnetic susceptibility, $\chi_{2}(\omega, N_{\rm cl})$, for superparamagnetic grains with uniformly sized SPIs as a function of grain rotation rate for different values of $N_{\rm cl}$ and dust temperatures $T_d = 20,50, 200,500\,{\rm K}$ (thin solid lines). For comparison, the maximum susceptibility at the resonance size (thin dotted line) and the effective susceptibility (thick solid line) are also plotted. As the rotation rate decreases from the characteristic $\nu_0=10^{9}\,{\rm s}^{-1}$, the peak of $\chi_{2}(N_{\rm cl}, \omega)$ shifts toward larger $N_{\rm cl}$ due to the resonance effect, while both the maximum and effective susceptibilities slightly increase.
• Figure 4: Variation of $\chi_{2}(\omega, N_{\rm cl})$ and effective susceptibility as functions of $T_{d}$ for different iron clusters, assuming the grain rotation rate of $\omega=10^{4}\,{\rm s}^{-1}$ (left panel) and $10^{6}\,{\rm s}^{-1}$ (right panel). As the temperature increases, the peak of $\chi_{2}(N_{\rm cl})$ shifts toward larger $N_{\rm cl}$ due to stronger thermal fluctuations. The effective susceptibility slightly increases with $T_{d}$, in contrast to the resonant behavior at $T_{d}=T_{\rm res}$ in the case of the uniform-sized SPIs.
• Figure 5: Magnetic dissipation strength $K=\chi_{2}(\omega,N_{\rm cl})/\omega$ as function of the grain angular frequencies for different dust temperatures from $20-500\,{\rm K}$. The effective strength in the thick solid line is shown for comparison.