  We present the results of Monte Carlo simulations of a model of a
$\gamma$-Fe$_2$ O$_3$ (maghemite) single particle of spherical shape. The
magnetic Fe$^{3+}$ ions are represented by Ising spins on a spinel lattice that
consists on two sublattices with octhaedral and tetrahedral coordination with
exchange interactions among them and with an external magnetic field. By
varying the particle diameter, we have studied the influence of the finite size
of the particle on the equilibrium properties, field cooling magnetization and
hysteresis loops. The simulations allow to distinguish the different roles
played by the surface and the core spins of the particle on its magnetic
properties. We show that for small particle sizes the core is uncoupled from
the surface, that behaves as a quasi-independent layer, whereas for bigger
particles the surface and the core are coupled and follow the behaviour of the
bulk.
