3 Theoretical basis. Spectrum of a free electron

An electron is a charged particle that has an angular momentum of spin with a quantum number of spin, $s=1/2$, and an associated magnetic moment. In the presence of an external magnetic field, $H$, the electronic energy is:

$$E(m_s) = E_0 + g_e \cdot \beta\cdot H\cdot m_s ,$$ (1)

where $E_0$ is the energy in absence of the magnetic field; $g_e$, an dimensionless constant, is the Landé g factor for a free electron whose value is 2.002319; $\beta$ is the Bohr magneton, 9.274$\cdot$10$^{-24}$ J$\cdot$T$^{-1}$; $H$ is the external magnetic field in Tesla and $m_s$ is the magnetic quantum number, $\pm 1/2$.

In an EPR experiment, the transition between two Zeeman states ($m_s = +1/2$ and $m_s = -1/2$) generated by a strong magnetic field is induced by means of a radiation of microwave frequency $\nu_0$ which determines the ''resonance condition'':

$$\Delta E = E(m_s = +1/2) - E(m_s = -1/2) = h \nu_0$$ (2)

that replacing in Eq. (1) yield:

$$h \nu_0 = g_e \cdot {\beta} \cdot H$$ (3)

The energy diagram and the EPR spectrum of a hypothetical free electrons are shown in the Fig. 1. In the diagram, the absorption is represented against the magnetic field, because the conventional equipment usually works with a fixed frequency and variable magnetic field. The resonance condition or the absorption will be obtained increasing the value of $H$ until the Eq. (3) is fulfilled. The magnetic field corresponding to the absorption is denominated resonance field, $H_r$.

Figure 1: Electronic Zeeman splitting and EPR absorption spectrum of a free electron.

The separation between the energy levels is a linear function of the applied magnetic field (see Eq. (3) and Fig. 1).

The spectrometers of Electron Paramagnetic Resonance denominated of ''X-band'' have a frequency of 9.5 GHz that correspond to $H_r \approx$ 340 mT (3400 G)¹. The spectrometers called of ''Q-band'' have a frequency of 34 GHz ($H_r \approx$ 1250 mT).