Distribution of magnetization of a cold ferromagnetic cluster beam

Above a critical temperature, a supported single domain ferromagnetic particle responds to an applied magnetic field as if it were a paramagnet with a very large spin. Its average magnetization is given by the Langevin equation as expected from simple thermodynamic considerations. The average magnetization of an ensemble of unsupported ferromagnetic clusters also approximately follows the Langevin equation even for small clusters in a low-temperature ensemble. The reason is not obvious because there is no heat bath for low-energy clusters so that elementary thermodynamic requirements for the Langevin equation are not satisfied. We investigated the magnetic deflections of cobalt clusters (${\text{Co}}_{N}$, $12\ensuremath{\le}N\ensuremath{\le}200$) using molecular-beam methods over a wide range of temperatures $(20\ensuremath{\le}T\ensuremath{\le}100\text{ }\text{K})$ and magnetic fields $(0\ensuremath{\le}B\ensuremath{\le}2\text{ }\text{T})$. A distribution of magnetization is observed for the cluster beams. Previously, we showed that the average magnetization of the cluster beam follows Langevin function closely for all temperatures and magnetic fields investigated, and proposed an avoided-crossing model that takes into account interacting spin-rotational states. In this paper, we report a comprehensive study of the magnetization distribution and present in depth the avoided-crossing model. The model explains both the average and the width of the magnetization distributions of the cluster beam in terms of the ensemble temperature without requiring that individual clusters have defined temperatures. We also show that the spin-relaxation model is the high-temperature limit of the avoided-crossing model.