Using a pure quantum calculations and magnons approach, we investigated the magnetic properties of SMMs with crystal-field anisotropy. Our analysis introduces a clear physical mechanism for the appearance of the magnetic plateaus in the system which is related to tunnel and relaxation magnetization effects. We consider two cases of interest: when the SMM are completely independent and when there are inter-cluster magnetic couplings, especially super exchange and dipolar interactions. In such case, we give a special attention to single-ion anisotropies and dipole-dipole interactions. The dispersion relations and magnetization behaviours are explored and analyzed in the quantum magnons formalism taking into account the experimental reality and using commonly parameters values.