**4.4 Crystalline field term**

Other observable interaction in EPR is the crystalline field [9–17], this

interaction is represented by tensors given by *S* ! ∙ *D* \$ ∙ *S* ! . This is the interaction of electron spin with the electric field of the charges of the neighboring ions placed in specific symmetries. The expression for crystalline field is given by [9–12]:

$$D\left[\mathbf{S}\_{\mathbf{x}}^{2} - \frac{\mathbf{1}}{3}\mathbf{S}(\mathbf{S} + \mathbf{1})\right] + E\left(\mathbf{S}\_{\mathbf{x}}^{2} - \mathbf{S}\_{\mathbf{y}}^{2}\right) \tag{12}$$

The Zero-field splitting parameters *D* and *E* split the energies of *ms* levels at zero applied magnetic field according to the magnitude of *ms* for each level. Thus the *D* and *E* ensure the singularity of each energy gap between *ms* levels under a non-zero magnetic field. The magnitudes of *D* and *E* are dependent on the ligant field theory (or from the Crystal Field Theory point view), and therefore are easily tunable by coordination geometry [12, 18]. Even the *g* factor could give in terms of crystal splitting factor Δ [11, 12, 18].

The **Figure 5** shows how the splitting of electronic levels depends on the ion interaction with electric field of the neighboring charge ions placed in specific symmetries [9–12].

The crystal field splitting energy *Δ* is the energy of repulsion between electrons of the ligands and the central metal ion and their bounding in complex ions such as octahedral, square planar and tetrahedral structural symmetries [9, 12]. If the *Δ* is greater than electron spin pairing energy the greater stability would be obtained if the fourth and fifth electrons get paired with the ones in lower level. If the *Δ* is less than the pairing energy, greater stability is obtained by keeping the electrons unpaired. So, if *Δ* is weak then the spin *S* is high and this yields a strongly

#### **Figure 5.**

*Crystalline field interaction for a 3d electron with neighboring placed in specific symmetries, below are the corresponding splits electronic levels. The splitting levels have crystal field splitting energy* Δ *or in terms of D and E zero-field splitting parameters [9–17].*

paramagnetic complexes, and if *Δ* is high then the spin S is weak and this yields low spin complexes and weakly paramagnetic or sometimes even diamagnetic. Through microwave excitation the electronic transition energy levels are possible when this obey the rules for allowed transitions [11, 12].

For example for octahedral symmetry, the tetragonal distortion could provide a high **Δ** for *d***<sup>3</sup>** electrons them could be arrangement in levels *B***2***<sup>g</sup>* and *Eg* resulting spin S = 1/2, **Figure 6(A)**. For example for tetrahedral symmetry, the tetragonal distortion could provide a high **Δ** for *d***<sup>3</sup>** electrons them could be arrangement in levels *A***1***<sup>g</sup>* and *B***1***<sup>g</sup>* resulting spin S = 1/2, **Figure 6(B)**.

For the tetragonal symmetry, the degeneracy of the *Eg* (*dz***<sup>2</sup>** and *dx***<sup>2</sup>***<sup>y</sup>***<sup>2</sup>** orbitals) term is no affected by the spin orbit coupling and by Jahn-Teller theorem applies. The orbital degeneracy is lifted and the energy of the system lowered by a displacement of the ligands on the z-axis [11, 12]. An elongated or compressed of the coordination tetrahedron (or tetragonal distortion) leads to the energy level scheme shown in **Figure 6(B)** with unpaired electron in the *dx***<sup>2</sup>***<sup>y</sup>***<sup>2</sup>** orbital. The measure of EPR spectra is limited to the Zeeman splitting imposed by an external field on the unpaired electron in the non-degenerated *dx***<sup>2</sup>***<sup>y</sup>***<sup>2</sup>** orbital and one could think that the spin system can now be described by the spin Hamiltonian with S = 1/2 since the ground state is non-degenerate and has only associated spin angular momentum [9–15].

#### **Figure 6.**

*Splittings and degeneracies of orbital levels d<sup>1</sup> or d6 ions in two types of electric field caused by negative charges for (A) octahedral field (***Δ > 0***) plus tetragonal distortion and (B) tetrahedral field (***Δ < 0***) plus tetragonal distortion. For d4 and d9 ions applieds to octahedral and tetrahedral fields. Shifting of the center of gravity of the set of levels is ignored [9–15]. If* **Δ** *is high, then the spin S is weak, and this yields low spin complexes and weakly paramagnetic. For example, for tetrahedral field, if* **Δ** *is high for d***<sup>3</sup>** *electrons them could be arrangement in levels A***<sup>1</sup>***<sup>g</sup> and B***<sup>1</sup>***<sup>g</sup> yield a spin S = 1/2.*

*Paramagnetic Transitions Ions as Structural Modifiers in Ferroelectrics DOI: http://dx.doi.org/10.5772/intechopen.95983*
