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Using the finite cluster approximation (FCA) based on a single-site cluster theory and Monte Carlo simulation (MCS) under the Metropolis algorithm, we study the critical properties of two-dimensional (2D) spins-½ system by analyzing the effect of the anisotropic parameter [Formula: see text] and next-nearest-neighbor (NNN) interaction [Formula: see text] on the critical temperature [Formula: see text] of the 2D anisotropic square lattice. It is found that the MCS is more reliable than the FCA method in the resolution of the Ising model. Moreover, MC calculations reveal that [Formula: see text] is strongly correlated with the external magnetic field [Formula: see text], [Formula: see text], and [Formula: see text], where the competition between these parameters can generate several rich phase diagrams.

Owing to their useful properties, the ferromagnetic mixed spins Ising have been extensively investigated. In this context, the paper aims to study the phase diagram of 3d mixed spin-1 and spin-3/2 Ising system and its critical properties. The adapted Hamiltonian of the considered mixed spin-(1, 3/2) Ising model is parameterized by three exchange interaction between first nearest neighbors spins Ji (with i = 1, 2 and 3), two crystal field interactions and the external magnetic field. The competition between these parameters can generate rich critical behavior with the coexistence of several phases in the ground state phase diagram. In this work, Monte Carlo simulation based on Metropolis algorithm, was used to study the critical properties in different phases of ground state. The analysis of the behavior of total magnetization and susceptibility as a function of temperature allowed us to study the behavior of the critical temperature of the ferromagnetic – paramagnetic transition TC as a function of different parameters. The obtained results shows that the critical temperature TC has linear Ji dependency in strong spin–spin interaction regime.

The magnetic properties of anisotropic 3D Ising model on a cubic lattice are studied by Monte Carlo simulation. In particular, we have considered an extended 3D Ising model with spatially uniaxial anisotropic bond randomness on the simple cubic lattice parameterized by exchange interaction parameter [Formula: see text], anisotropy parameter [Formula: see text] and external longitudinal magnetic field [Formula: see text]. The obtained numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model at critical temperature [Formula: see text] that is strongly correlated to [Formula: see text] and [Formula: see text]. Especially, in the limit, [Formula: see text], the spin ½ cubic lattice becomes a collection of noncorrelated Ising chains, whereas in the other limit, [Formula: see text], the system becomes a stack of noncorrelated Ising square lattice.

Magnetization, susceptibility and Curie temperature of 3D anisotropic Ising model for ½ spins system have been investigated by making use of the Monte Carlo simulation. The considered model is parametrized by exchange interaction parameter J, anisotropy parameter γ. The obtained results allowed us to show that for strong coupling case in z axis (γ > 1), the critical temperature is widely to the conventional value of the 3D materials (Tc = 6,32J).Thus, an optimal integration of interplanar atoms in z-axis can generate materials that retain the ferromagnetic property even at high temperature.