## 3 Results
### Magnetic Properties
#### 3.3.2 Size of the Magnetic Domains  
By analysing magnetization vs. magnetic field intensity curves of superparamagnetic NPs above T\({}_{\text{b}}\), it is possible to determine the average size of magnetic domains and their size distribution function [28]. If the NPs are single domain, the size of the magnetic domains coincides with that of the single NPs. In the case of non-interacting magnetic particles, the magnetization vs. field intensity curves can be modelled by using this function:  
\[M(H)=M_{\text{s}}\int_{0}^{\infty}L(H,D)P_{V}(D)dD \tag{1}\]  
where \(D\) is the diameter of the particles, \(L(H,D)\) is the Langevin function which depends on the magnetic dipole moment (and thus on D) and on the magnetic field intensity and \(P_{V}(D)\) is the volume-weighted diameter distribution function [27; 29; 43; 44]. Equation (1) was used to fit the experimental magnetization vs. magnetic field intensity curves to derive the average size of the magnetic domains and the standard deviation of the size distribution (see the Supplementary Materials). The lognormal distribution function was used for  
Figure 6: ZFC and FC curves measured for the three samples.  
\(P_{\text{p}}(D)\)[27; 44]. The experimental magnetization curves and their best fit curves are shown in Figure 7 for all the three samples. For a better comparison the curves are shown on a double log scale.  
The quality of the fit is rather good with only some residual discrepancies in the region of low magnetic field. The results of the analysis, summarized in Table 5, show that the sizes of the magnetic domains are very close to the sizes of the coherence domains as determined by XRD (see Table 2), that is the crystalline and magnetic order have the same extension. The distribution curves of the magnetic domains for the three samples are shown in Figure S6, Supplementary Materials. The curves for samples A and B are broader than that of sample C. However, the relative standard deviations are the same for all the samples (see Table 5).  
Although the NPs are interacting magnetically (as indicated by the ZFC-FC measurements), the good correspondence between the size of the magnetic domains and the size of the individual NPs for these samples shows that magnetization curves can be analysed in terms of Langevin equation, which, on the other hand, consider non-interacting particles. This can be explained taking into account that the relaxation time of the magnetization for superparamagnetic NPs depends exponentially on the effective anisotropy constant which contains an additional contribution due to magnetic interactions among the NPs [40; 45]. Hence, even relatively weak interactions are able to produce the observed shift of T\({}_{\text{b}}\) but do not affect significantly the magnetization curves which can be analysed in the framework of the Langevin model.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **\textless{}D\textgreater{} (nm)** & **\textless{} (nm)** & **\textless{} (nm)** & **\textless{}D\textgreater{} (cm\textgreater{}g)** \\ \hline A & 8.7 \(\pm\) 0.6 & 5.1 \(\pm\) 0.1 & 0.6 \(\pm\) 0.1 & 67 \(\pm\) 1 \\ B & 9.9 \(\pm\) 0.5 & 4.9 \(\pm\) 0.2 & 0.5 \(\pm\) 0.1 & 58 \(\pm\) 1 \\ C & 5.1 \(\pm\) 0.6 & 2.8 \(\pm\) 0.1 & 0.5 \(\pm\) 0.1 & 46 \(\pm\) 1 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Results of the analysis of the magnetization curves for the three samples. \(\textless{}\)D\textgreater{} is the average diameter of the magnetic domains, \(\alpha\) the standard deviation of the distribution, \(\alpha/\textless{}\)D\textgreater{} the relative standard deviation. M\({}_{\text{s}}\) is the saturation magnetization obtained by fitting the experimental magnetization curves.  
Figure 7: Fitting (solid lines) of the experimental magnetization vs. magnetic field intensity curves measured at RT for the three samples (filled circles) by using Equation (1).