**1. Introduction**

The ferroelectric materials are very important for technological applications in general, like sensor and actuators they are important part of electronic devices. The temperature, pressure, electric and electromagnetic field nonlinear response of ferroelectrics make them ideals like active elements due to pyroelectric and piezoelectric effects [1–4]. Ferroelectrics are used like ultrasonic generator; high voltage transforms and accelerators [1–4], also, they are using in optical components, piezoelectric transducers, and pyroelectric sensors [1–4]. By fabrication ferroelectric materials born with electric dipole different to zero, *P* 6¼ 0; this electric property is by asymmetry crystalline center structure. The **Figure 1** shows two crystalline structures, first is a symmetry perovskite structure with *P* ¼ 0, and second is an asymmetry perovskite structure with *P* 6¼ 0 typical to lead titanate (*PbTiO*3) [5]. The lead titanate (*PbTiO*3) is used intensively in electronic devices [5–7]. The *PbTiO*3 ceramic had perovskite structure (tetragonal structure) with ratio *c=a* ¼ 1*:*064, where *c* ¼ 4*:*154Å, *a* ¼ 3*:*899Å, the **Figure 1a** shows the lead ion ð Þ *Pb* occupies places A, the titanium ion (Ti) occupies the places B and the oxygen ion is on the parallelepiped faces occupies the places O [8].

**Figure 1.**

*The perovskite structure for PbTiO*<sup>3</sup> *a) Centre symmetry and b) non Centre with ratio c=a* ¼ 1*:*064*, where c* ¼ 4*:*154̊A*, a* ¼ 3*:*899̊A*.*

The ceramic fragility is reduced for inserting rare earths or transitions metal atoms [6–8]. The partial or total substitution of Pb or Ti produces modified compounds with new electrical polarization and modifies their structure depending of doping atoms and the interactions material [8]. The changes in the structure could be small or big depending to the kind of dopand and the percentage of them [1–8]. If the dopand is a paramagnetic ion it could be detected by paramagnetic resonance (EPR) and it is capable to sense structure changes principally by spin-orbit interactions [9–11].

The paramagnetic resonance technique (EPR) detects the spin-orbital magnetic interaction in doped ferroelectrics when the dopant is a rare earth with a paramagnetic ion. If dopant is chromium (Cr), it can give a paramagnetic state, in this case the electron paramagnetic resonance EPR technique is capable to detect paramagnetic ions, and the EPR-technique is high sensitivity to spin interactions due crystalline structure, nuclear interaction (hyperfine), or anisotropies (orientations) [9–11]. The nondestructive EPR-technique is applied to organic and inorganic molecules, ions, and atoms that have unpaired electrons that have information about oxidation state and spin state due to electron unpaired spin [9–11]. The EPR technique use microwave energy to induce resonances or electron transitions between electronic energy levels separated by applying magnetic field for paramagnetic compounds or substances. The EPR typical spectra for Cr, Fe, Mn, Co, Ni, Cu, V, etc. [11–15] contained information about the *g*-factor, the *A*-hyperfine couple and crystalline field factors, *D* and *E*.

#### **2. Atoms into magnetic field** *H* !

The energy interaction between magnetic moment *μ* ! and extern magnetic field *H* ! is given by *W* ¼ *μ* ! ∙ *H* ! . The electronic orbital movement is the cause of atomic momentum for the atom *μ* !, it is proportional to angular momentum *L* ! given by *μ* ! *<sup>L</sup>* ¼ *e* <sup>2</sup>*mc <sup>L</sup>* ! , where *L* ! is the orbital angular momentum, *e* is the electron charge, *m* is the electron mass, and *c* is the speed of light. The gyromagnetic ratio is defined by *<sup>γ</sup> <sup>e</sup>* <sup>2</sup>*mc* <sup>¼</sup> <sup>1</sup>*:*<sup>7</sup> 107 *rad gauss*-1 , so the atomic momentum is given by *μ* ! *<sup>L</sup>* ¼ *γL* ! [9].

Otherwise, the intrinsic magnetic momentum for electron is *μ* ! *<sup>S</sup>* ¼ 2 *e* <sup>2</sup>*mc <sup>S</sup>* ! , where *S* ! is the spin vector associated to electron spin. The total magnetic moment is given by the spin and orbital momentum addition *μ* ! ¼ *μ* ! *<sup>L</sup>* þ *μ* ! *<sup>S</sup>*. If the total magnetic moment is placed into magnetic field *H* ! , the total magnetic energy is given *Paramagnetic Transitions Ions as Structural Modifiers in Ferroelectrics DOI: http://dx.doi.org/10.5772/intechopen.95983*

by *W* ¼ *μ* ! ∙ *H* ! ¼ *μH* cos *θ*, where *θ* is the angle between vectors *μ* ! and *H* ! , the angle could varies continually in the classical description, however, in the quantum mechanics description the variation is quantized with 2*J* þ 1 orientations, where *J* is the quantum number for the total angular momentum given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *J J*ð Þ <sup>þ</sup> <sup>1</sup> <sup>p</sup> <sup>ℏ</sup> [9, 12]. The allowed projections *J*, when the system is quantized along the magnetic field direction, are given by *m <sup>j</sup>*ℏ, where *m <sup>j</sup>* is the quantum magnetic number with values from *J*, *J* - 1, … , -*J* [15].

The simplest case is just only the spin electronic momentum (atoms in base state 2 *S*1*<sup>=</sup>*2) with *mS* ¼ *S*, *S* - 1, … , -*S*, with *S* the total electronic spin, and the projections are *μ* ! *S* ¼ 2 *eh* <sup>4</sup>*πmc mS* where for *<sup>S</sup>* ! operator the eigenvalue *mS*ℏ was substituted. The quantity *eh* <sup>4</sup>*πmc β<sup>e</sup>* ¼ 9*:*2741 10-<sup>21</sup>*erg=gauss* is called Bohr's magneton, then *μ* ! *S* ¼ *βemS* and the energy values allowed for the atom placed into a magnetic field *H* ! , are *EmS* ¼ *μSH* ¼ 2*βemSH* (Zeeman's energy). For an electron isolate quantum electrodynamic correction is necessary [9–15] replacing the number 2 for the *ge* ¼ 2*:*00023. In the case for isolate spin *S* ¼ 1*=*2, the 2*S* þ 1 energy levels are *geβeH* equally spaced, the **Figure 2** shows the case for *S* ¼ 1*=*2 [9–15].

For degenerated orbital state (*L* ¼6 0) where exist and important L and S coupling (Russell-Saunders coupled) [9–15] there are *J* ¼ *L* þ *S*, *L* þ *S* - 1,…, ∣*L* - *S*∣, and for each *J* there are *mJ* ¼ *J*, *J* - 1,…, -*J* values, so the energy for each degenerated state is *EmJ* ¼ *gJ <sup>β</sup>emJH*, where *gJ* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *J J*ð Þþ <sup>þ</sup><sup>1</sup> *S S*ð Þ þ1 *L L*ð Þ þ1 <sup>2</sup>*J J*ð Þ <sup>þ</sup><sup>1</sup> is the splitting Landé factor. For example, the state <sup>2</sup> *P*1*=*<sup>2</sup> have *S* ¼ 1*=*2, *L* ¼ 1, *J* ¼ 1*=*2 and then *g* ¼ 3*=*2. Of course, when *L* ¼ 0 the *gJ* ¼ *ge* [9].

#### **Figure 2.**

*EPR scheme for system with S = 1/2. The Zeeman's effect for S* ¼ 1*=*2 *split the degenerate state energy into W*<sup>1</sup> *and W*<sup>2</sup> *and the microwave photon hν provide the energy for the transition between them [14].*
