This manuscript investigates the magnetocaloric effect and magnetic properties of the Gd \(_3\) In compound as functions of magnetic field and temperature using a mean-field approximation. We first introduce the model and methodology, followed by the determination of the transition temperature through the analysis of magnetization, susceptibility, and magnetic entropy change ( \(-\Delta S_m\) ). Our temperature-dependent magnetization analysis reveals a second-order phase transition from the ferromagnetic (FM) to the paramagnetic (PM) phase at 208 K, consistent with previous theoretical and experimental findings. The maximum magnetic entropy change ( \(-\Delta S_m^{\text {max}}\) ) is 6.46 J/kg.K at 208 K under a 5 T magnetic field, with a corresponding relative cooling power (RCP) of 680.11 J/kg. Additionally, hysteresis behavior is observed at various temperatures. These findings suggest that Gd \(_3\) In is a promising candidate for magnetic refrigeration applications.
