The magnetic properties of an anti-ferromagnetic and ferrimagnetic mixed spin-1/2 and spin-5/2 Ising model with a crystal field in a longitudinal magnetic field on the honeycomb (z = 3) and square lattice (z = 4) are studied by using the effective-field theory with correlations. The ground state phase diagram of the model is obtained in the longitudinal magnetic Field (h) and a single-ion potential or crystal-field interaction (Delta) plane. We also investigate the thermal variations of the sublattice and total magnetizations, and present the phase diagrams in the (Delta/vertical bar J vertical bar, k(B)T/vertical bar J vertical bar) plane. The phase diagrams have one, two or even three compensation temperatures depending on the values of the crystal-field interaction. Moreover, the Susceptibility, internal energy and specific beat of the system are numerically examined, and some interesting phenomena in these quantities are found due to the applied longitudinal magnetic field. (C) 2009 Elsevier B.V. All rights reserved.