Since the "ab-initio" treatment of a magnetic nanoparticle remains a future challenge, in this chapter we consider a nanoparticle treated as a multi-spin problem but within a classical atomic spin approach. Even in this case, the investigation of the thermal magnetization switching of a multi-spin
nanoparticle is a real challenge. We are faced
with complex many-body aspects with the inherent difficulties
related with the analysis of the energy potential and its extrema.
This analysis is unavoidable since it is a crucial step in the calculation of the relaxation time and thereby in the study of the magnetization stability against thermal fluctuations.
As such, a question arises whether it is possible to map the
behavior of a multi-spin nanoparticle onto that of a simpler model
system as one effective magnetic moment, without loosing its main features such as surface
anisotropy, lattice structure, size and shape, and more
importantly the spin non-collinearities they entail.
A first answer to this question was given in Ref. [74] where it was shown that when the surface anisotropy is much smaller than the exchange parameter and in the absence of the core anisotropy, the single-site Néel surface anisotropy contribution to the particle's effective energy is of the fourth order in the net magnetization components, of the second order in the surface anisotropy constant, and is proportional to a surface integral. The latter accounts for the lattice structure and the particle's shape. Later it has been shown that in a more general situation with the core anisotropy, taken
as uniaxial, the energy of the multi-spin particle could be
modelled by that of an effective potential containing both uniaxial
and cubic anisotropy terms [87].
In this chapter we investigate this issue in a more extensive way by considering other lattice structures, particle's shapes and different anisotropies in the core.
For this purpose, we compute the energy potential of the
multi-spin particle using the Lagrangian multiplier method [74] and fit it to
the appropriate effective energy potential.

Rocio Yanes