  Equilibrium magnetization curve of a rigid finite-size spherical cluster of
single-domain particles is investigated both numerically and analytically. The
spatial distribution of particles within the cluster is random. Dipole-dipole
interactions between particles are taken into account. The particles are
monodisperse. It is shown, using the stochastic Landau-Lifshitz-Gilbert
equation that the magnetization of such clusters is generally lower than
predicted by the classical Langevin model. In a broad range of dipolar coupling
parameters and particle volume fractions, the cluster magnetization in the weak
field limit can be successfully described by the modified mean-field theory,
which was originally proposed for the description of concentrated ferrofluids.
In moderate and strong fields, the theory overestimates the cluster
magnetization. However, predictions of the theory can be improved by adjusting
the corresponding mean-field parameter. If magnetic anisotropy of particles is
additionally taken into account and if the distribution of the particles' easy
axes is random and uniform, then the cluster equilibrium response is even
weaker. The decrease of the magnetization with increasing anisotropy constant
is more pronounced at large applied fields. The phenomenological generalization
of the modified mean-field theory, that correctly describes this effect for
small coupling parameters, is proposed.
