3 Analytical background

In Refs. [74,87,97,73] analytical as well as numerical calculations showed that a multi-spin particle, cut from a sc lattice, and when its surface anisotropy is small with respect to the exchange coupling, may be modeled by an effective one-spin particle (EOSP), i.e., a single macroscopic magnetic moment $\mathbf{m}$ representing the net magnetic moment of the multi-spin particle. The energy of this EOSP (normalized to $J{\cal N}$) may be written as

$$\mathcal{E}_{\mathrm{EOSP}}=\bigl(\mathcal{E}_{c}+\mathcal{E}_{1}+\mathcal{E}_{2}+\mathcal{E}_{21}\bigr).$$ (23)

The $\mathcal {E}_{1}$ is the first-order anisotropy energy with surface contribution, $\mathcal {E}_{2}$ is the second-order anisotropy energy with surface contribution, $\mathcal {E}_{21}$ is a mixed contribution to the energy that is second order in surface anisotropy and first order in core anisotropy and $\mathcal{E}_{c}$ is the core anisotropy energy (per spin).

The $\mathcal{E}_{c}$ energy has the following form:

$$\mathcal{E}_{c}=\frac{N_{c}}{\mathcal{N}}k_{c}\left\{\begin{array... ... & \ \frac{1}{2}(m_{x}^{4}+ m_{y}^{4}+ m_{z}^{4})& & cubic, \end{array}\right.$$ (24)

The other three contributions ( $\mathcal {E}_{1}$, $\mathcal {E}_{2}$ and $\mathcal {E}_{21}$) stem from the surface, which we discuss now.