## 3 Results and Discussion
### Magnetic Properties  
Magnetization versus applied magnetic field (\(M\)-\(H\)) measurements were carried out under zero field cooling (ZFC) and field cooling (FC) conditions. The measurements were conducted at several temperatures (2, 10, 30, 50, 70 and 100 K) and several field cooling values, \(H_{\text{FC}}\) (0.5, 1, 2 and 3 T). In each hysteresis loop cycle, the magnetization data was recorded while the applied field was swapped between \(-\)3 to 3 T. In the ZFC protocol, the sample was cooled down from room temperature to a particular temperature in the absence of any applied magnetic field. After that, a magnetic field was applied and ramped between \(\pm\)T. The magnetic moment was measured during the change of the applied magnetic field. The \(M\)-\(H\) hysteresis loops under ZFC at \(T\) = 2 K, for samples S2 and S3 are shown as examples in Figure 3a,b. Typical ZFC and FC hysteresis loops were obtained for all the samples at the above mention temperatures and field cooling values. As can be seen in Figure 3, there are openings in the hysteresis loops indicating the absence of superparamagnetic behavior at 2 K. Also, saturation magnetization was not completely achieved in both samples, even at the maximum applied field of 3 T (here we show up to 1 T only so that the openings in the hysteresis loops become visible). From these hysteresis loops we obtained the saturation magnetization, \(M_{\text{S}}\) at different temperatures and field cooling values, \(H_{\text{FC}}\). The role of field cooling on \(M_{\text{S}}\) in each sample was analyzed at each temperature. The \(M_{\text{S}}\) for all samples did not saturate up to maximum applied field of 3 T at all temperatures and all \(H_{\text{FC}}\) values. This could be attributed to surface spin effects and inter-particle interactions. The saturation magnetization (\(M_{\text{S}}\)) was determined by extrapolating the high field region (at \(H\) = 3 T) of the \(M\)-\(H\) curve to the infinity field. Figure 4 shows the saturation magnetization versus the shell thickness at several temperatures (from 2 to 300 K) in the ZFC state as well as in the FC state at several \(H_{\text{FC}}\) values. Figure 5 shows the saturation magnetization versus \(H_{\text{FC}}\) for each sample at several temperatures (from 2 to 300 K). Figure 2: The transmission electron microscopy (TEM) images of samples (**a**) S1; (**b**) S2; and (**c**) S3. The particle size histograms of the samples (**d**) S1; (**e**) S2; and (**f**) S3. (Reproduced from [37]. Copyright [PERSON], et al., 2017.). Figure 4: The saturation magnetization versus the shell thickness at several temperatures (from 2 to 300 K) (**a**) in the ZFC state as well as in the field-cooling (FC) state at several field-cooling values of (**b**) 0.5 T, (**c**) 1 T, (**d**) 2 T, and (**e**) 3 T.  
Figure 3: The applied magnetic field (\(H\)) versus the magnetization (\(M\)) at 2 K under zero field-cooling condition for (**a**) sample S2 and (**b**) sample S3. ZFC, zero field-cooling. These two figures show some key results. (a) Saturation magnetization, with large magnitudes, has been detected in all samples at all temperatures in the ZFC state; and (b) as function of the shell thickness, in the ZFC state, the saturation magnetization has a systematic and linear increase with the increase in shell thickness at all temperatures. This reflects the significant enhancing role of shell thickens on saturation magnetization in the ZFC state. This could be due to the larger ferrimagnetic \(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) shell thickness (and thus volume) in sample S2. Also, this increase in \(M_{\rm S}\) may be induced by the lattice strain between the Fe\({}_{3}\)O\({}_{4}\)/\(\gamma\)-Fe\({}_{2}\)O\({}_{3}\) core/shell [38].