Scaling form of zero-field-cooled and field-cooled susceptibility in superparamagnet

The scaling form of the normalized ZFC and FC susceptibility of superparamagnets (SPM's) is presented as a function of the normalized temperature $y$ ($=k_{B}T/K_{u}< V>$), normalized magnetic field $h$ ($=H/H_{K}$), and the width $\sigma$ of the log-normal distribution of the volumes of nanoparticles, based on the superparamagnetic blocking model with no interaction between the nanoparticles. Here $<V>$ is the average volume, $K_{u}$ is the anisotropy energy, and $H_{K}$ is the anisotropy field. Main features of the experimental results reported in many SPM's can be well explained in terms of the present model. The normalized FC susceptibility increases monotonically increases as the normalized temperature $y$ decreases. The normalized ZFC susceptibility exhibits a peak at the normalized blocking temperature $y_{b}$ ($=k_{B}T_{b}/K_{u}< V>$), forming the $y_{b}$ vs $h$ diagram. For large $\sigma$ ($\sigma >0.4$), $y_{b}$ starts to increase with increasing $h$, showing a peak at $h=h_{b}$, and decreases with further increasing $h$. The maximum of $y_{b}$ at $h=h_{b}$ is due to the nonlinearity of the Langevin function. For small $\sigma$, $y_{b}$ monotonically decreases with increasing $h$. The derivative of the normalized FC magnetization with respect to $h$ shows a peak at $h$ = 0 for small $y$. This is closely related to the pinched form of $M_{FC}$ vs $H$ curve around $H$ = 0 observed in SPM's.