## 3 Results and Discussions
In addition, we observed that in the range of temperatures studied, ncalculated from a positive and a negative saturation showed a difference, which could be due to the dynamic state of the system (Figure S3 in the Supplementary Information). The difference in nfrom the positive and the negative saturation was higher in the NRs compared with the NTs. This was also observed in the difference between the ZFC and FC magnetization curves. In Figure 3d, we represented the thermal evolution of the maximum change in TS (\(\Delta\chi_{\text{T}}/\chi_{\text{T}}\))\({}_{\text{max}}\) curves. This parameter, which was sensitive to changes in the dynamic state of the system, showed a similar behavior at a low temperature with a decrease below \(\sim\)100 K ascribed to the Verwey transition of Fe\({}_{3}\)O\({}_{4}\). However, we observed several differences at higher temperatures with a sudden increase in the (\(\Delta\chi_{\text{T}}/\chi_{\text{T}}\))\({}_{\text{max}}\) above 200 K for the NRs and 250 K for the NTs. This could be related to the thermal disorder overcoming the magnetic order with an increasing temperature [57]. The fact that this happened at lower temperatures for the NRs than for the NTs supported the weaker effective anisotropy of the NRs in comparison with the NTs. Therefore, the magnetic measurements indicated that both NRs and NTs presented a similar magnetic behavior with the expected structurally-coupled magnetic transitions intrinsic to Fe\({}_{3}\)O\({}_{4}\) but there were a few quantitative differences related to the higher effective anisotropy of the NTs, which could be attributed to the stronger role of intraparticle interactions in these samples. In order to ascertain how these differences affected their efficiency as magnetic hyperthermia mediators, we carried out AC magnetometry measurements in these samples and obtained their SAR vs. H curves. The AC magnetometry measurements allowed us to gain an insight into the differences of the hysteresis losses and, hence, the heating efficiency obtained for both samples. In addition, AC hysteresis loops can allow us to better understand the nature and origin of the magnetic interactions [25]. We have represented the AC hysteresis loops for the NRs and NTs in Figure 4a,b. As depicted, in both cases, for fields up to 400 Oe we obtained minor loops with a maximum magnetization (\(\approx\)15 emu/g), far from the M\({}_{\text{S}}\) values reached in the DC magnetic measurements. The fact that even at 400 Oe we obtained narrow minor loops made sense considering the high anisotropy field values exhibited near room temperature by both NRs(H\({}_{\rm K}\) = 450 Oe) and NTs (H\({}_{\rm K}\) = 510 Oe). As has been described in the literature, we could distinguish two regimes in the heating efficiency of MNPs under AC magnetic fields: at low fields, H \(<<\) H\({}_{\rm K}\), the power absorption is mainly caused by viscous losses in the system and this regime is characterized by small hysteresis loop areas [17; 60]. On the other hand, at higher fields, H \(>>\) H\({}_{\rm K}\), hysteresis losses dominate and the area of the hysteresis loops appreciably increases, eventually tending to saturate. In our case, in the range of AC fields applied, we were mainly working in the first regime. There were, nevertheless, several differences in the shape of the AC hysteresis loops for both samples with those of the NTs being slightly narrower and less squared. This could be related to the effect of intraparticle interactions: in the NTs, intraparticle interactions played a more important role than in the NRs and this gave rise to stronger demagnetizing effects that made the AC hysteresis loops narrower, as has been reported before [17; 25; 61]. As depicted in Figure 4c, we plotted the SAR vs. H curves computed from the area of the AC hysteresis loops [62].