Magnetic Phase Transitions and Mixed Spin in Double Perovskite $Sr_{2}FeMoO_{6}$

The magnetic properties of the double perovskite oxide $Sr_{2}$FeMo$O_{6}$ are analyzed using a mixed-spin Ising model with spins $\left( \frac{1}{2},\frac{5}{2}\right) $ in the presence of a random crystal field $\Delta$ and exchange interactions $ J $ on a three-dimensional (3D) cubic lattice. The study employs both the Mean-Field Approximation (MFA) based on the Bogoliubov inequality for Gibbs free energy and Monte Carlo (MC) simulations using the Metropolis algorithm to provide a comprehensive analysis of the system's phase transitions and magnetization behavior, with focusing on the role of Fe and Mo sublattices. We establish the ground-state phase diagram, identifying multiple stable magnetic configurations and first-order transitions at low temperatures. Indicate compensation temperature $T_{comp}$. This work provides deeper insight into the thermodynamic, physics statistic and magnetic properties of $Sr_{2}$FeMo$O_{6}$, with implications for future applications in spintronics and magnetic storage technologies.