Model for the FC and ZFC Ferrimagnetic Spinel

There are two methods of preparation of ferrimagnetic spinel. If, during the preparation, an external magnetic field as high as 300 O\"{e} is applied upon cooling the material is named field-cooled (FC). If the applied field is about 1O\"{e} the material is zero-field cooled (ZFC). To explore the magnetic and thermodynamic properties of these materials we consider two-sublattice spin system, defined on the bcc lattice, with spin-$s^A$ operators $\bf{S_{i}^A}$ at the sublattice $A$ site and spin-$s^B$ operators $\bf{S_{i}^B}$ at the sublattice $B$ site, where $s^A>s^B$. The subtle point is the exchange between sublattice A and B spins, which is antiferromanetic. Applying magnetic field along the sublattice A magnetization, during preparation of the material, one compensates the Zeeman splitting, due to the exchange, of sublattice B electrons. This effectively leads to a decrease of the $s^B$ spin. We consider a model with $s^B$ varying parameter which accounts for the applied, during the preparation, magnetic field. It is shown that the model agrees well with the observed magnetization-temperature curves of zero field cooled (ZFC) and non-zero field cooled (FC) spinel ferrimagnetic spinel and explains the anomalous temperature dependence of the specific heat.