# Magnetic Properties of Gd-Doped Fe\({}_{3}\)O\({}_{4}\) Nanoparticles
## 2 Model and [PERSON]'s Functions  
Magnetic Fe\({}_{3}\)O\({}_{4}\) is a ferrimagnetic mineral with a Curie temperature of \(T_{C}\) = 850 K [7]. Compounds with the structural formula Fe\({}_{3}\)O\({}_{4}\) can crystallize in an inverse spinel crystal structure where the tetrahedral A sites are occupied by Fe\({}^{3+}\)-ions, and the octahedral B sites by an equal amount of Fe\({}^{2+}\)- and Fe\({}^{3+}\)-ions. The magnetic ions in the A and B sites interact through super-exchange interactions through an intermediate oxygen ion A-O-A, A-O-B and B-O-B, with the following relation of the exchange interaction constants: \(\mid J_{A-B}\mid>\mid J_{B-B}\mid>>\mid J_{A-A}\mid\). The antiferromagnetic A-O-B interaction is the most important for the cooperative behaviour of the magnetic moments in ferrites and is of greatest intensity [16; 17]. In the A and B sublattices, the exchange interactions are ferromagnetic. By doping with RE ions, whose radius is larger than that of the iron ions (for example Gd\({}^{3+}\)), they can only substitute ions at the octahedral sites. The exchange interaction between the spins of the RE and the Fe ion is mediated by a nonmagnetic oxygen ion and preserves the interaction character between the tetrahedral and octahedral sites, i.e., antiferromagnetic collinear. The Hamiltonian, which describes the behavior of the spin subsystem of pure and Gd\({}^{+3}\)-doped Fe\({}_{3}\)O\({}_{4}\), is the modified Heisenberg one:  
\[H_{s} = -(1-x)\sum_{ij}J_{ij}^{FeA-FeB}\mathbf{S}_{i}^{FeA}\cdot\mathbf{S }_{j}^{FeB}-(1-x)\sum_{ij}J_{ij}^{FeB-FeB}\mathbf{S}_{i}^{FeB}\cdot\mathbf{S }_{j}^{FeB} \tag{1}\] \[- x\sum_{ij}J_{ij}^{FeA-GdB}\mathbf{S}_{i}^{FeA}\cdot\mathbf{S}_{ j}^{GdB}-x\sum_{ij}J_{ij}^{FeB-GdB}\mathbf{S}_{i}^{FeB}\cdot\mathbf{S}_{j}^{GdB}\] \[- \sum_{i}D_{i}^{FeA}(S_{i}^{FeAz})^{2}-(1-x)\sum_{i}D_{i}^{FeB}(S _{i}^{FeAz})^{2}-x\sum_{i}D_{i}^{GdB}(S_{i}^{FeAz})^{2}\] \[- g\mu_{B}\mathbf{h}\cdot\sum_{i}\mathbf{S}_{i}^{FeA}-(1-x)g\mu_{B }\mathbf{h}\cdot\sum_{i}\mathbf{S}_{i}^{FeB}-xg\mu_{B}\mathbf{h}\cdot\sum_{i} \mathbf{S}_{i}^{GdB}\]  
where \(\mathbf{S}_{i}^{Xa}\) (\(\alpha\) = A-, B-sublattices and X = Fe, Gd) is the Heisenberg spin operator at \(i\)-site. \(J_{ij}^{FeA-FeB}\), \(J_{ij}^{FeB-FeB}\), \(J_{ij}^{FeA-GdB}\), \(J_{ij}^{FeB-GdB}\) are the exchange interaction constants between the A-B and B-B magnetic ions at the \(i\) and \(j\) sites. \(D_{i}^{Xa}\) are the constants of the single-ion anisotropy, \(\mathbf{h}\) is a magnetic field along the \(z\)-axis, which is also the axis of the easy magnetization, \(x\) is the Gd doping concentration. The Hamiltonian of the magnetic system is built under the following assumptions:  
1. The Gd\({}^{3+}\)-ions substitute only the Fe\({}^{2+}\)-ions at the octahedral sites (B-sublattice). The tetrahedral sites have a volume smaller than the octahedral ones and the radius of the Gd\({}^{3+}\)-ions (1.07 \(\AA\)) is larger than that of the Fe\({}^{2+}\)-ions (0.70 \(\AA\)). Thus, the structural formula of the doped magnetite with Gd ions is as follows: [Fe\({}^{3+}_{A}\)][Fe\({}^{2+}_{1-x}\)Gd\({}^{3+}_{x}\)Fe\({}^{3+}\)]\({}_{B}\)O\({}_{4}\). This is valid for Gd concentrations up to \(x\) = 0.5. This is the concentration range in which the magnetic phase transition temperature in doped magnetite is biocompatible in the sense for medical applications. Above these concentrations, the Gd\({}^{3+}\) ions replace the Fe\({}^{3+}\) ions in the tetrahedral sites [13]. 2. The magnetic interaction between the magnetic ions in the tetrahedral sites (A-O-A) is neglected because it is shown experimentally to be 10 times smaller compared to the other interactions B-O-B and A-O-B [16]. 3. The substitution of the Fe-ion by the RE Gd\({}^{3+}\)-ion (B-sublattice) does not lead to a change in the interaction character of the Fe-ions between the A- and B-sublattices.