3 Mixed contribution to the energy $\mathcal {E}_{21}$

Taking into account the core anisotropy analytically to describe corrections to Eq. (2.14) due to the screening of the spin noncollinearities in the general case is difficult. However, one can consider this effects perturbatively, at least to clarify the validity limits of expression (2.14). In Ref. [97], the use of the general method outlined above (see section 2.3.1) led to a Helmholtz equation. In this case, there is no exact Green's function and as such the perturbation theory was used to write the Green's function $G$ in the presence of core anisotropy as the sum of the exact Green's function $G^{(0)}$, obtained in the absence of core anisotropy, and a correction $G^{(1)}$. The perturbation parameter:

Therefore, upon using $G=G^{(0)}+p_{\alpha}^{2}G^{(1)}$ in the energy, it was found that $G^{(1)}$ leads to a new contribution of the surface anisotropy which is also of the $2^{\mathrm{nd}}-$order in the surface anisotropy constant $k_s$ and the first order in $k_{c}$.

$$\mathcal{E}_{21}=k_{21} g(\mathbf{m})$$ (37)

with

$$k_{21}=\tilde{\kappa}\left(k_{c}N_s\right)k_s^{2}$$ (38)

where $\tilde{\kappa}$ is another surface integral whose integrand contains $G^{(1)}$. $g(\mathbf{m})$ is a function of $m_{\alpha}$ [97] which comprises, among other contributions, both the $2^\mathrm{nd}$- and the $4^\mathrm{th}$-order contributions in spin components. For example, if we work with spherical coordinates ( $\theta,\varphi$) for an sc lattice

$$g(\theta,\varphi=0)=-\cos^{2}\theta+3\cos^{4}\theta - 2 \cos^{6}\theta,$$

which is shown later to give an agreement with the numerical simulations, see Fig. 2.5.

The contribution (2.20), called here the core-surface mixing (CSM) contribution, should satisfy $k_{21}\lesssim k_{2}$ which requires:

$$N_{s}K_{c}/J\lesssim 1$$ (39)

This is exactly the condition that the screening length is still much greater than the linear size of the particle. For larger system sizes the perturbative treatment becomes invalid.