A semi-empirical analysis of the paramagnetic susceptibility of solid state magnetic clusters

Abstract: Recent developments in the synthesis of new magnetic materials lead to the discovery of new quantum paramagnets. Many of these materials, such as the perovskites Ba${4}$LnMn${4}$O${12}$ (Ln = Sc or Nb), Ba${3}$Mn${2}$O${8}$, and Sr${3}$Cr${2}$O${8}$ present isolated magnetic clusters with strong intracluster interactions but weak intercluster interactions, which delays the onset of order to lower temperatures ($T$). This offset between the local energy scale and the magnetic ordering temperature is the hallmark of magnetic frustration. At sufficient high-$T$, the paramagnetic susceptibility ($\chi$) of frustrated cluster magnets can be fit to a Curie-Weiss law, but the derived microscopic parameters cannot in general be reconciled with those obtained from other methods. In this work, we present an analytical microscopic theory to obtain $\chi$ of dimer and trimer cluster magnets, the two most commonly found in literature, making use of suitable Heisenberg-type Hamiltonians. We also add intercluster interactions in a mean-field level, thus obtaining an expression to the critical temperature of the system and defining a new effective frustration parameter $f{\text{eff}}$. Our method is exemplified by treating the $\chi$ data of some selected materials.