4 Effective one spin problem (EOSP) approximation

Consequently, collecting all the contributions, one can model the energy of a multi-spin particle with an effective energy potential in the form:

$$\mathcal{E}_{\mathrm{EOSP}} = - k_\mathrm{ua}^\mathrm{eff} m_z^2 - \frac{1}{2}k_\mathrm{ca}^\mathrm{eff}\sum_{\alpha=x,y,z}m_{\alpha}^{4}.$$ (40)

The subscripts $\mathrm{ua/ca}$ stand for uniaxial/cubic anisotropy, respectively.

Now, we note that due to the contributions (2.16) and (2.20), even when the core anisotropy is not uniaxial, the effective energy contains two uniaxial contributions induced by the surface, one is due to elongation given by (2.16) and the other to the mixing between the core and the surface given by (2.20). Hence, the $2^{nd}-$order term $k_\mathrm {ua}^\mathrm {eff}$ in (2.23) takes into account these two contributions. Similarly, the $4^{th}-$order term $k_\mathrm{ca}^\mathrm{eff}$ is a result of the surface contributions (2.14) and part of (2.20), and may also contain a contribution from the core if the latter has a cubic anisotropy.

To clarify the dependence on the system size and the surface anisotropy constants of the different energy contributions to EOSP we summarized them in Tab. 2.1.

Although the analytical expressions are only established so far for sc lattices, they do provide us with a general form of the effective energy potential which allows us to investigate the different tendencies of the nanoparticle behavior, taking account of various contributions to its energy. These approximate expressions are also very useful in the interpretation of numerical results.

Table 2.1: Dependence of different contributions to the effective macrospin energy $E=\mathcal {E}\cdot (J\mathcal {N})$ on the surface anisotropy constant $k_{s}$ and its scaling with the system size.

Contribution	System size dependence	Dependence on ($k_{s}$)
${E}_{c}$	$Nc$	-
${E}_{1}$ (elongated particles only)	$Ns$	$\propto k_{s}$
${E}_{2}$	$\mathcal{N}$	$\propto k_{s}^{2}$
${E}_{21}$	$Ns$	$\propto k_{c} k_{s}^{2}$