# Magnetic Properties of Gd-Doped Fe\({}_{3}\)O\({}_{4}\) Nanoparticles
## 3 Results and Discussion  
In the numerical calculations, we will use a model of a spherical, cubooctahedral, structurally heterogeneous NP composed of core and surface shell. A nanoparticle is defined by fixing the origin at a certain Fe spin in the center of the particle and including all spins within the particle into shells numbered by \(n=1,\dots,N\), where \(n=1\) is the central spin and \(n=N\)--the surface shell. In our case, the the NPs offer an icosahedral symmetry, i.e., there are 12 spherical particles in the first shell, 42 in the second shell, 92 ones in the third shell, etc. The exchange interaction \(J_{ij}\equiv J(r_{i}-r_{j})\) depends inversely on the lattice parameters. The surface effects are included by the exchange interaction constant on the surface layer \(J_{s}\), which is different from the bulk one \(J_{b}\). The indices \(s\) and \(b\) are used for all model parameters. The calculations are made for non-interacting NPs with a diameter below 80 nm. The reason is that the dipole-dipole interaction between magnetic NPs is important. It modifies their magnetic anisotropy, which will affect the magnetic characteristics of the latter. The dipole-dipole interaction itself depends on the geometrical arrangement of particles. The inclusion of this effect, in our opinion, would lead to further complication from a theoretical point of view without significantly modifying the qualitative picture of the described processes. Since the magnetite is an inverse spinel, the super-exchange interaction between the magnetic A- and B-sublattices is antiferromagnetic. This is the most intense interaction compared to the others, so that it practically determines the magnetic structure of Fe\({}_{3}\)O\({}_{4}\). The interaction in the sublattices is ferromagnetic. As we mentioned in the introduction, the interaction between the magnetic moments in the tetrahedral sites is weakest and will be neglected. For the exchange interaction constants, we will use the following values: \(J^{FeB-FeB}=7.31\) K, \(J^{FeA-FeB}=-33.9\) K, \(J^{FeA-FeA}=0\)[20]. By doping, the Gd\({}^{3+}\) ions substitute the Fe\({}^{2+}\) ions from the octahedral sites. If we assume that the interaction between the spins of the Gd\({}^{3+}\) ions with the Fe\({}^{3+}\) ions in the octahedral sites is ferromagnetic and taking into account the inequality \(S(\text{Gd}^{3+})=7/2>S(\text{Fe}^{2+})=2\), then the saturation magnetization should increase with increasing the doping concentration. However, experimental results show that for \(0<x<0.5\) the saturation magnetization decreases [13]. This is only possible if \(J^{FeB-GdB}\) is antiferromagnetic, i.e., \(J^{FeB-GdB}<0\). Such an antiferromagnetic interaction has also been reported in Gd-Fe alloys [21] and, following the method of determining the exchange interaction between Gd and Fe ions in the same paper, we obtain \(J^{FeB-GdB}=-2.28\) K. For the interaction between the spins of the Gd ions at the octahedral sites and the spins of the iron ions at the tetrahedral sites, we will assume that it is also antiferromagnetic, but weaker than \(J^{FeA-FeB}\) because as the ion doping increases, the magnetic phase transition temperature strongly decreases. This could be due to the weaker overlap of the 4f orbitals and the p ones of the oxygen anion acting as a mediator in the double-exchange interaction. Based on these results for the numerical value of \(J^{FeA-GdB}\), we get the following: \(J^{FeA-GdB}=-10.5\) K. Let us emphasize that we investigate small doping concentrations so we can neglect the interaction between the Gd ions in the octahedral sites, i.e., \(J^{GdB-GdB}=0\) K. From experimental data [18] for the spin-phonon interaction constants, we get the following: \(F=27.3\) K and \(R=-23.5\) K. For the single ion anisotropy constant \(D\), we use for the bulk compound \(D^{Fe}\) (bulk) \(=-0.87\) K [22]; whereas for single domain NPs for the surface anisotropy, we use \(D^{Fe}\) (surface)= \(-0.28\) K [23].