## 3 Results and Discussion
### Magnetic Properties
Figure 6: The temperature (\(T\)) dependent \(M_{\rm S}\) for samples (**a**) S1; (**b**) S2; and (**c**) S3, under zero field-cooling condition. The solid lines represent the fittings of the experimental data to a modified [PERSON]’s law. Field-cooling shows a significant effect on \(\alpha_{B}\) where the values are considerably larger than that for the ZFC state (displayed in Table 2) and thus reflect the significant role of field-cooling on \(M_{\rm S}(T)\). The \(M_{\rm S}(T)\) for nanoparticles were reported to deviate from the [PERSON]'s law for bulk materials where the Bloch's exponent \(\alpha_{B}\) could have values larger or smaller than 3/2 [46; 54; 55]. Various suggestions were made to explain the deviation from the [PERSON]'s law in nanoparticle systems [49; 56; 57; 58; 59; 60; 61; 62]. For nickel ferrite nanoparticles, the modified [PERSON]'s model was found to be valid for temperatures between 50 and 300 K [39]. At temperatures lower than 50 K, the authors reported a deviation from the modified Bloch law where a sharp increase in saturation magnetization occurred [39]. This anomaly was suggested to be due to some extra contribution of the shell-spins moment to the total magnetization at low temperatures [39; 60] or due to the presence of some paramagnetic impurities in the nanoparticles which would be activated at low temperatures [39; 61]. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{\(H_{\rm FC}\) (T)} & \multicolumn{2}{c}{**Fitted Parameters**} & \multirow{2}{*}{\(\alpha_{B}\)} \\ \cline{2-3} \cline{5-5}  & \(M_{\rm 0}\) **(emu/g)** & & \(T_{\rm C}\) **(K)** & \\ \hline
0.5 & 78.0 \(\pm\) 0.1 & 443 \(\pm\) 5 & 4.0 \\
1 & 78.7 \(\pm\) 0.14 & 452 \(\pm\) 2.6 & 3.8 \\
2 & 78.1 \(\pm\) 0.6 & 412 \(\pm\) 9.9 & 4.9 \\
3 & 78.3 \(\pm\) 0.2 & 427 \(\pm\) 3 & 4.6 \\ \hline \end{tabular}
\end{table} 
Table 2: The fitting parameters of the experimental data to a modified Bloch’s law in sample S1. Figure 7: Temperature dependence of the saturation magnetization (in sample S1) measured under (**a**) the zero field-cooling (ZFC) condition and under the field-cooling (FC) condition at several field-cooling values of (**b**) 0.5 T, (**c**) 1 T, (**d**) 2 T, and (**e**) 3 T. The dashed lines represent the fittings of the experimental data to modified Bloch’s laws. The effect of core/shell on different magnetic properties has been reported in many studies [30; 31; 32; 33; 34; 35]. However, we are not aware of a research study on the effect of shell thickness and field cooling on \(M_{\rm S}\) and \(M_{\rm S}(T)\). In our system of Fe\({}_{3}\)O\({}_{4}\)/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) core/shell nanoparticles, it is very evident that the shell thickness and field cooling have significant influence on \(M_{\rm S}\) and \(M_{\rm S}(T)\). It is observed that as the shell thickness increases, the temperature dependence of saturation magnetization deviates nonmonotonically from the original Bloch's law and the deviation is maximum for the nanoparticle with shell thickness of 5 nm. These results could be attributed to the existence of interface spin-glass clusters (SGCs) which occurred due to structural defects at the core/shell interface. The existence of such (SGCs) was investigated and reported in our recently published paper [37]. The spins in these SGCs are not aligned in a particular direction due to different spin interactions among the spins of each cluster and the interactions with the core and shell spins. In our core/shell particle system, we suggest that, as the shell thickness varies, the SGCs with randomly oriented spins vary, leading to a varied magnetic anisotropy at the interface and hence resulting in deviations from the original Bloch's law.