## 2 Spinel Structure  
Spinel ferrites possess the highly stable crystal structure of the natural spinel MgAl\({}_{2}\)O\({}_{4}\)[1]. An extremely large variety of oxides can adopt it, fulfilling the conditions of the overall cations/anions ratio of 3/4, a total cation valence of eight, and relatively small cationic radii. A universally known ferrite is magnetite, Fe\({}^{2+}\)Fe\({}^{3+}\)\({}_{2}\)O\({}_{4}\) (Fe\({}_{3}\)O\({}_{4}\)), probably the oldest magnetic solid with practical applications still currently studied due to the fascinating properties associated with the coexistence of ferrous and ferric cations. The oxygen network in the ferrite has an FCC (Face-Centered Cubic) symmetry, in which two types of interstitial sites, 64 tetrahedral and 32 octahedral, can be defined for a unit cell containing eight times the basic formula AB\({}_{2}\)O\({}_{4}\). Only one-eighth of tetrahedral and half of octahedral sites are occupied by cations and the resulting space group is \(Fd\overline{3}m\).  
It is defined as normal spinel the one in which the B\({}^{3+}\) and A\({}^{2+}\) cations occupy octahedral and tetrahedral sites, respectively (Figure 2, left), while the inverse spinel has a radically different cation distribution, where half of the trivalent cations occupy the A sites, while the B sites are shared by the divalent and the remaining trivalent cations (Figure 2, right). Complex spinels, in which bivalent and trivalent cations occupy in a random way the tetrahedral and octahedral sites, can be also stabilized.  
Figure 1: Scheme showing some applications of ferrites.  
The described polyhedral arrangement is consistent not only with the high-symmetry space group \(Fd\overline{3}m\) but also with several other space groups ranging from \(P4_{1}32\), \(P4_{1}2_{1}2\), \(I4_{1}/amd\), \(I4_{1}/a\), and \(P\overline{4}m2\) to \(R3\)[11].  
The cation distribution in the spinel structure is controlled by many factors: _elastic energy_, the lattice deformation produced by the differences of cationic radii, _electrostatic energy_, due to the electrical charge distribution in the crystal and the _crystal field_ stabilization energy, deriving from the geometry of the \(d\) orbitals. Magnetic ordering in ferrites produces antiferromagnetic arrangements even if, in most cases, a resulting magnetic moment is maintained due to the different numbers of cations hosted in the magnetic lattice. Some examples of the most studied and applied ferrites are represented by ZnFe\({}_{2}\)O\({}_{4}\), CoFe\({}_{2}\)O\({}_{4}\), Fe\({}_{3}\)O\({}_{4}\), NiFe\({}_{2}\)O\({}_{4}\), MgFe\({}_{2}\)O\({}_{4}\), and MnFe\({}_{2}\)O\({}_{4}\)[1; 8; 12; 13] with very different magnetic properties. For example, ZnFe\({}_{2}\)O\({}_{4}\) is a normal spinel with an anti-ferromagnetic character, [12; 14] which also exists in nature, known as Franklinite mineral, discovered in 1819 [15]. NiFe\({}_{2}\)O\({}_{4}\) is, instead, an inverse spinel with ferromagnetic ordering [16]. Almost all the spinels can form an extremely wide variety of total solid solutions due to the easiness of the spinel structure to host different kinds of cations in tetrahedral and octahedral sites, as well as in usually empty interstitial sites. This feature opens the possibility to strongly modify the chemical composition, maintaining practically unchanged the basic crystalline structure (see also the following paragraphs) [1]. The proper choices of the experimental synthesis conditions, as well as the stoichiometry and the kind of substituents, are the key parameters to produce spinels with a huge variety of functional properties, in particular, optical, magnetic, and electrochemical ones.