**3. The basic principle of the electron paramagnetic resonance (EPR) spectroscopy**

The splitting of the degenerate energy levels applying magnetic field is the principle of the paramagnetic resonance spectroscopy useful for study the paramagnetic materials with total electronic spin *S* ! 6¼ 0. The splitting of the electronic energy levels occurs when a magnetic field *H* ! is applied, this phenomenon is called Zeeman's effect. In the experiment an oscillating microwave field is applied to give energy to the electrons so the electronic transition of the electrons can occur; these electrons are arranged in the electronic levels according to Hund's rule. From quan-

tum mechanics point of view, when the microwave energy photon *hν* is equal to the energy difference between Zeeman levels there are electrons transitions from one low electronic energy level (*W*1) to other with high energy (*W*2); the energy difference between this electronic energy levels is Δ*W* ¼ *W*<sup>2</sup> - *W*<sup>1</sup> ¼ *hν*, according to quantum mechanics selection rules [10, 12]. When Δ*W* ¼ *hν* occurs, there are peaks of microwave energy absorptions observed in the EPR spectrum. Experimentally a high resolution for the EPR spectrum is obtained taken the first derivative of absorption [9–15].

In general, the splitting electronic levels effect is written tensorial form by *μβ S* ! ∙ *g* \$ ∙ *H* ! . The photon energy *hv* of the microwaves is provided to the system applying an oscillating microwave electromagnetic field *H* ! <sup>1</sup> at frequency *ν*0; the *H* ! <sup>1</sup>ð Þ*t* is applied perpendicularly to *H* ! <sup>0</sup> magnetic field, and the magnitude of *H*<sup>0</sup> is varied until the resonance condition given in Eq. (1) be satisfied.

$$
\hbar\nu = \mathbf{g}\beta\mathbf{H} \tag{1}
$$

For the simplest case for "free ion" and spin *S* ¼ 1*=*2 system the Eq. (1) is reduced to Δ*W* ¼ *hν*<sup>0</sup> ¼ *geβeH*0, where *ge* is the gyromagnetic ratio and *β<sup>e</sup>* is the Bohr's magneton for free electron *e*; the EPR scheme is shown in **Figure 2**. The constants value *ge* ¼ 2*:*0023 and *β<sup>e</sup>* ¼ 9*:*2741 10-<sup>21</sup>*erg=gauss*, the Planck's constant is *h* ¼ 6*:*6262 10-<sup>27</sup>*ergs* [9, 15]. In general, these energy levels are described by tensorial treatment.
