## 3 Results and Discussion
### Magnetic Properties
[PERSON] proposed the \(T^{3/2}\) law by considering the magnon excitation of long wave-length spin-waves at low temperatures [39]. Figure 6 shows the temperature dependence of \(M_{\rm S}\) measured under the ZFC protocol for all the samples. We found that the experimental \(M_{\rm S}\) values of our core/shell nanoparticles deviate from original [PERSON]'s law. Hence, we have fitted the experimental data to a modified [PERSON]'s law for each sample. The fitted data for each sample is shown as a solid line in Figure 6. The experimental data was fitted using Equation (1) and the fitted magnetic parameters are shown in Table 1. For nanoparticles, the finite size effects are expected to be significant resulting in deviations from the ideal Bloch's equation. The values of \(\alpha_{B}\) for nanoparticles were reported to vary within the range of 1.5 to 2.0 [43; 44; 45; 46; 47; 48]. In our samples, it is observed from the fitting parameters that the modified exponent (\(\alpha_{B}\)) is larger than the exponent for bulk materials. The values of the modified Bloch exponent inferred from the fitting data for S1, S2 and S3 samples are 2.0, 2.5 and 3.0, respectively. As can be seen in Table 1, \(\alpha_{B}\) increases monotonically (and linearly) with shell thickness. This indicates that the larger the shell thickness, the larger is the deviation from the original Bloch's law. For normal nanoparticles, it is expected that as the size of the particle increases, \(\alpha_{B}\) should approach that of the bulk materials. However, for our core/shell structured nanoparticles, the surface and interface have contributed effectively such that the opposite behavior is obtained. The deviation from the original Bloch's law could be attributed to the surface and interface effects [49; 50; 51; 52]. Due to the broken exchange bonds [53] at the core/shell interface, the interface spins are suggested to be in disordered states. We suggest that (as shown in [37]) at the core/shell interface, spin-glass like structures [49; 50] occur with amounts that increase with shell thickness. Hence, we attribute the increase of \(\alpha_{B}\) (increase of deviation from the original [PERSON]'s law) with shell thickness to the increase of spin-glass structures as the shell thickness increases. The surface spins also could contribute to this deviation where the role of surface spins increases with decreasing the total size of the particle. The role of field cooling on \(M_{\rm S}(T)\) was investigated on all samples. Here we present the results on the sample S1. Typical results were obtained on the other samples. Figure 7 shows the temperature dependence of \(M_{\rm S}\) in sample S1. The \(M_{\rm S}\) values were obtained from the magnetic hysteresis measurements under both the ZFC and the FC protocols at the field cooling values: \(H_{\rm FC}\) = 0.5, 1, 2 and 3 T. We found that the experimental \(M_{\rm S}\) values for our core/shell sample in all conditions deviate considerably from the original [PERSON]'s law. Thus, we fitted the temperature dependence of \(M_{\rm S}\) in each case to a modified [PERSON]'s law (shown as dashed lines in Figure 7) by varying the parameters \(M_{0}\), \(T_{0}\) and \(\alpha_{B}\). For the ZFC state, \(\alpha_{B}\) was found to be nearly 3.1, which is a large deviation from the value of 1.5 for the original [PERSON]'s law. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{**Shell Thickness (nm)**} & \multicolumn{2}{c}{**Fitted Parameters at ZFC**} & \multirow{2}{*}{\(\alpha B\)} \\ \cline{3-3} \cline{5-5}  & & \(M_{0}\) **(emu/g)** & & \(T_{0}\) **(K)** & \\ \hline S1 & 1 & 77.5 (\(\pm\)0.4) & 852 (\(\pm\)15) & 2.0 \\ S2 & 3 & 79.9 (\(\pm\)0.1) & 689 (\(\pm\)10) & 2.5 \\ S3 & 5 & 82.8 (\(\pm\)0.2) & 580 (\(\pm\)5) & 3.0 \\ \hline \end{tabular} \({}^{1}\) ZFC, zero field-cooling.  
\end{table} 
Table 1: The fitting parameters of the experimental data to a modified [PERSON]’s law \({}^{1}\).