The magnetic properties and thermodynamical description of the Fe-Mn-Al ternary alloy are studied using the spin 1 antiferromagnetic Blume-Capel (BC) model by the pair approximation based on the Gibbs-Feymann-Bogoliubov inequality for the free energy. The values of the spin operator are +1 for ferromagnetic interaction (Fe-Fe), -1 for the antiferromagnetic one (Mn-Mn, Mn-Fe), and 0 for the magnetic diluter (corresponding to Al). The BC model with antiferromagnetic (AF) next-nearest-neighbor coupling better accurately fit the mean hyperfine field experimental data obtained by Mössbauer spectroscopy. Considering the crystalline field, the predicted temperature as a function of the Al concentration phase diagram, for the fcc lattice, from the numerical solution is remarkably good and significantly improves the traditional Ising and random-site Ising models.