# Spinel Iron Oxide by the Co-Precipitation Method: Effect of the Reaction Atmosphere
## 3 Results and Discussions
### Magnetic Properties

Table 3: Temperature Corresponding to the Maximum in ZFC Curve (\(\mathrm{T_{max}}\)), Irreversibility Temperature (\(\mathrm{T_{irr}}\)), and Blocking Temperature (\(\mathrm{T_{B-TRM}}\)) From TRM measurement and the blocking temperature from Hansen and Morup approach (\(\mathrm{T_{B,H,M}}\)) for FO\(-\)CA and FO\(-\)NA samples, uncertainties on the last digit are given in parentheses. The phenomenological approach of Hansen and Morup (H.M) has also been applied as a second estimation of the distribution of energy barriers in small particles from ZFC-FC magnetization curves using the equation [53]:  
\[\langle\mathrm{T_{B.H.M}}\rangle=\mathrm{T_{max}}[1.792+0.186\ln\left(\tau_{ \mathrm{ir}}/\tau_{\mathrm{max}}-0.918\right)]-1+0.0039\mathrm{T_{irr}} \tag{7}\]  
Similarly to the previous results, samples also show equal \(\mathrm{T_{B}}\) values according to the H.M. approach (Table 3). The dependence of magnetization on an external magnetic field has been investigated at 300 K and 5 K. At 300 K, the results show superparamagnetic behavior for both samples (Figure 6a). Typically, for a material with superparamagnetic behavior, magnetization curves can also be used in order to extract the mean size of the magnetic core (\(\mathrm{D_{m}}\)) (see supporting information for further details) using the Langevin function [54]. A small deviation of the fit from the experimental data has been observed, which can be related to the non-negligible interparticle interactions within the samples [55]. However, the value of \(\mathrm{D_{m}}\) is in good agreement with experimental \(\mathrm{D_{TEM}}\) and \(\mathrm{D_{XRD}}\). At 5 K (Figure 6b), the hysteresis loops show low values of reduced remanence magnetization (\(\mathrm{M_{r}}/\mathrm{M_{s}}\)) and coercive field (\(\mathrm{H_{C}}\)) (Table 4) in agreement with that is reported in literature [9]. The obtained high \(\mathrm{M_{s}}\) value (close to the bulk one \(\mathrm{Ms}\) = 92 \(\mathrm{Am^{2}Kg^{-1}}\)) indicates high crystallinity and magnetic ordering of the particles [56]. Figure 5: \(\mathrm{M_{TRM}}\) curve (full circles), measured after cooling at 2.5 mT and anisotropy energy barrier distribution (empty circles) for (**a**) FO\(-\)CA and (**b**) FO\(-\)NA samples. Figure 6: Hysteresis loops recorded at (**a**) 300 K and at (**b**) at 5 K, Inset (**b**): details around zero field. To evaluate the type and the strength of interparticle interactions, \(\delta\)m(H) plots at 5 K (Figure 7) have been investigated using direct current demagnetization (DCD) and isothermal remnant magnetization (IRM) protocols (see supporting information for further details). Generally, a positive peak in the \(\delta\)m vs-\(\mu_{0}\)H curve is an indication of magnetizing interactions(e.g., exchange interactions) among nanoparticles while a negative peak suggests the existence of demagnetizing interactions (e.g., dipole-dipole interactions) [57]. \(\delta\)m plots confirming the presence of strong dipolar interactions, with equal strength in both samples. Since no significant difference between the two samples can be detected in terms of magnetic measurements, this gives us more evidence that the reaction atmosphere does not affect the magnetic properties of FO\(-\)CA and FO\(-\)NA samples in our case. The more explanations could be found in Figures S1-S6, and Table S1.