Bi 3 Mn 4 O 12 (NO 3 ) is a honeycomb bilayers antiferromagnet, each consists of Mn +4 (3d 3 , S=3/2) ions. The Curie-Weiss temperature has a large value of ? =-257K, this indicates strong AF interactions and the presence of LRO in this compound, however, there is no sign of LRO down to very low temperatures. This compound is the first honeycomb lattice system that shows no long-range ordering. The absence of magnetic LRO has been considered by theoretical and experimental scientists and is confirmed by different experiments such as Neutron Scattering experiments. Ab-initio methods based on DFT have also suggested an effective spin Hamiltonian for this substance. This Hamiltonian involves the Heisenberg antiferromagnetic spin interactions. The presence of frustration effects on this system is appropriate to compare different Monte Carlo methods. In this thesis, by considering a Heisenberg antiferromagnetic spin model, we compare the results of implementing two Monte Carlo methods, the metropolis algorithm(MC) and the parallel tempering algorithm(PT), in the calculation of spin correlation function and neutron structure factor. By this aim, we use ”MCARLO” code, to do simulate the behavior of this system. In order to discover the behavior of BMNO, we calculated the inter-layer spin-spin correlation function and also to investigate the nature of its phase, neutron structure factor has been calculated as a function of T in [hl0] plane. the computational cost of calculation and also ”seed” dependence of the magnitude of spin correlation function by using the metropolis algorithm lead to substitute a better algorithm. For this purpose, we used the PT algorithm. In this step, we re-calculate the thermodynamic quantities and compare them with the results of the metropolis algorithm and ensure that our assumptions are correct. We also calculate inter-layer spin-spin correlation function and neutron structure factor by use of PT algorithm and showed that without using different conditions, the correct results of BMNO behavior can be obtained.
