Effective Magnetic Susceptibility of Dust Grains with Superparamagnetic Inclusions and Implications

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 $\omega$, we identify a characteristic resonance size of SPIs that dominates the magnetic response, $N_{\rm res} = (T_{d}/T_{\rm act}) \ln (\nu_{0}/\omega)$ with $T_{\rm act}$ activation temperature and $\nu_{0}$ the characteristic attempt frequency of SPIs. The frequency-dependent effective susceptibility is well described by the maximum susceptibility $\chi_{\rm eff}^{\rm max}(\omega)$ 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 $\nu_{0}$, 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.