We consider a magnetic nanoparticle of spins in the
many-spin approach, i.e., taking account of its intrinsic
properties such as the lattice structure, shape, and size. This also includes the
(nearest-neighbor) exchange interactions, single-site core and
surface anisotropy. The magnetic properties of such a multi-spin particle (MSP)
can be described by the anisotropic Heisenberg model of classical
2 Localized spin (atomistic or Heisenberg) model
where J is the exchange parameter.
The anisotropy energy
will be different if we are working with core spins or surface spins. For core spins, i.e., those spins with full coordination, the
is taken either as
uniaxial with easy axis along and a constant (per
atom), that is
where is the number of core spins in the particle.
For surface spins the anisotropy is taken according to the Néel's surface anisotropy
model (referred to in the sequel as NSA), expressed as:
where is the number of surface spins, the number of
nearest neighbors of site , and
- a unit vector
connecting this site to its nearest neighbors labeled by .
Dipolar interactions are known to produce an additional "shape"
anisotropy. However, in the atomistic description, their role in
describing the spin non-collinearities is negligible as compared
to that of all other contributions. In order to compare particles
with the same strength of anisotropy in the core, we assume that
the shape anisotropy is included in the core uniaxial anisotropy
and neglect the dipolar energy contribution to the spin non-collinearities. We also assume that in the ellipsoidal nanoparticles the anisotropy easy axis is parallel to the elongation direction.
All physical constants will be measured with respect to the
exchange coupling (unless explicitly stated otherwise), so we define the reduced constants,
The core anisotropy
constant will be taken as
. The latter constant in real units corresponds to
erg/atom and is similar to cobalt fcc
value. On the other hand, the surface anisotropy constant is unknown "a priori" and
will be varied.