# Magnetic Properties of Gd-Doped Fe\({}_{3}\)O\({}_{4}\) Nanoparticles
## 3 Results and Discussion
### Gd Doping Dependence of the Magnetization and Curie Temperature  
Next, we will study the influence of RE ion doping on the spontaneous magnetization \(M_{s}\) and the Curie temperature \(T_{C}\). It must be mentioned that the Gd ions substitute the Fe ions in the whole NP, in the surface shell and in the bulk, in the core. Moreover, due to the different ionic radii of the Gd and the Fe ions, there appear a strain which changes the interaction constants at the doped state noted as \(J_{d}\) compared to the undoped one \(J_{b}\). The results for Gd doping for \(0\leq x\leq 0.2\) on the spontaneous magnetization \(M_{s}\) and the Curie temperature \(T_{C}\) are shown in Figures 3 and 4, respectively. It is clearly seen that the Gd ion doping leads to decrease of \(M_{s}\) and \(T_{C}\). [PERSON] et al. [14] reported also a slightly decreased magnetization at 260 K in Gd-doped Fe\({}_{3}\)O\({}_{4}\) NPs in contrast to the undoped ones. A similar decrease of \(M_{s}\) is observed also in Ce [29] or Dy, Nd, La doped Fe\({}_{3}\)O\({}_{4}\) NPs [12]. We observe for Gd-doped Fe\({}_{3}\)O\({}_{4}\) NPs a \(T_{C}\) value of 312 K for \(x=0.2\). [PERSON] et al. [30] and [PERSON] et al. [31] have obtained also a decrease of the critical temperature \(T_{C}\) of Gd-doped Fe\({}_{3}\)O\({}_{4}\) NPs. Moreover, a decrease of the magnetic coercive field, the remanent magnetization and the Curie temperature with decreasing the thickness of a Fe\({}_{3}\)O\({}_{4}\) thin film is reported recently by [PERSON] et al. [20].  
The decrease of the magnetization and magnetic phase transition temperature with doping qualitatively is explained as follows: The larger ionic radius of Gd\({}^{3+}\) compared to the Fe\({}^{2+}\) radius leads to an increase in the volume of the elementary cell. As a consequence, the distance between the interacting magnetic moments increases, which leads to a decrease in the exchange interaction constants and to a decrease of \(T_{C}\). Although the net magnetic moment of the Gd ion (\(7\mu_{B}\)) is larger compared with that of the Fe\({}^{2+}\) ion (\(5\mu_{B}\)), \(M_{s}\) and the saturation magnetization decrease. This is a consequence of the antiferromagnetic arrangement of the Gd and Fe spins at the octahedral sites. The interaction between the  
Figure 2: Size dependence of the Curie temperature \(T_{C}\) for a pure Fe\({}_{3}\)O\({}_{4}\) NP, \(J_{s}=0.9J_{b}\).  
magnetic moments of the RE and the Fe\({}^{3+}\) ions in the tetrahedral sites is antiferromagnetic, which means that, in the A sublattice, antiparallel iron spins will appear, leading to an additional reduction of \(M_{s}\) and the saturation magnetization. Speculation suggests that competing magnetic interactions arise due to Gd doping, leading to magnetic system frustration and the potential emergence of non-collinear structures. However, there is a lack of experimental results either confirming or refuting this hypothesis.  
Let us emphasize that [PERSON] [9] have studied Co-doped Fe\({}_{3}\)O\({}_{4}\) NPs with \(d\approx 30\) nm and reported a decrease of \(M_{s}\) and \(T_{C}\). This result can be confirmed from our model. The ionic radius of Co is larger compared to that of the Fe ion, i.e., the tensile strain that appears leads to a decrease of the magnetization and the Curie temperature. The transition metal ion-doped Fe\({}_{3}\)O\({}_{4}\) NPs will be investigated in a future paper.