The magnetic properties, magnetocaloric effect and hysteresis behavior of the Barium-Strontium ferrites compound Ba \( _{0.8} \) Sr \( _{0.2} \) FeO \( _{3} \) have been studied using the mean-field approximation based on the Gibbs-Bogoliubov inequality. The influence of an external magnetic field and temperature on magnetization, magnetic susceptibility, magnetic entropy change and hysteresis behavior was analyzed in detail. The results show that magnetization decreases progressively with increasing temperature until it becomes zero at the critical temperature \( T_c=60 \) K, while the application of an external magnetic field increases the critical temperature \( T_{c} \) by strengthening the alignment of magnetic moments. The variation in magnetic entropy - \( \Delta S_m \) reveals a characteristic peak at \( T_{c} \) , indicating a strong magnetocaloric effect. Furthermore, the relative cooling power RCP increases linearly with the external magnetic field strength. Under a magnetic field of \( h=5 \) T, the maximum values obtained for the variation in magnetic entropy \( \Delta S_m \) and for the relative cooling power RCP are 4.8 J.Kg \( ^{-1} \) .K \( ^{-1} \) and 176 J.Kg \( ^{-1} \) respectively. These results confirm that the compound Ba \( _{0.8} \) Sr \( _{0.2} \) FeO \( _{3} \) is a promising material for magnetic refrigeration applications.
