**3. Electron paramagnetic resonance**

cation oxygen cation

The magnetic interactions in spinel ferrites can be easily understood by considering two B sites and one A site about an oxygen, see Figure 3. The center and right side of this figure, where the large circle represents the oxygen ion, the small circle above it stands for the cation on the A site, and the two small circles below represent B sites. Note that this arrangement is 3D, but for this schematic illustration, it has been simplified on a plane. Nickel ferrite is an inverse spinel, Ni2+ goes to B sites, one ferric ion occupies the other B site, and the other Fe3+ is located on the A site. All magnetic interactions between cations are antiparallel (superexchange).

A

A

B B

B B

**Figure 3.** Schematic spin arrangements in spinel ferrites. Left: a section of the crystal structure showing the cation occu‐ pancy about an oxygen. Center: the ferrimagnetic spin orientation of the crystal sites for nickel ferrite. Right: the anti‐

There are A–O–B interactions between the cation on A site and the cations on B sites, and an B–O–B interaction between the cations on the two B sites. However, due to the axial symmetry of oxygen *p*-orbitals, superexchange interaction becomes more efficient when cations and oxygen are in an axial arrangement. This makes a strong difference between interactions; A– O–B interaction is far more efficient than B–O–B interactions since the latter has a 90° angle, while the former is closer to 180°. As a result, both spins on B sites are antiparallel to the A spin. The magnetic moment of Fe3+ on A site cancels with the magnetic moment of Fe3+ on B site, and the resulting magnetic moment is the spin value for nickel Ni2+ (2.3 Bohr magnetons), as shown in Figure 3 center. This is a strong interaction; the Curie transition is about 858 K.

**Figure 2.** Superexchange interactions in ferrites.

214 Advanced Electromagnetic Waves

ferromagnetic structure for the zinc ferrite.

Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) is a technique of observing resonance absorption of microwave power by unpaired electron spins aligned with a magnetic field [4]. The electron paramagnetic resonance (EPR) spectroscopy is similar to the nuclear magnetic resonance (NMR) except for the fact that EPR arises from the interaction between unpaired electrons and electromagnetic radiation.

The history of EPR can be told beginning with the Stern–Gerlach experiment in 1921. They observed that a beam of silver atoms split into two lines when it was subjected to a magnetic field. In contrast with the line splitting in optics found by Zeeman in 1896, these phenomena could not be explained by the angular momentum of the electrons. At the time quantum mechanics was still developing and was still an emerging field. Three more years passed before Uhlenbeck and Goudsmit found an explanation for these anomalous Zeeman effect when they postulated the spin [5,6].

The development of the EPR was furthered by the World War II since after the war was over, there was available a great amount of microwave instruments from the radar equipment used in the war. It was in 1945 when a Russian physicist, Zavoisky, observed the first EPR spectrum. He observed a radiofrequency absorption line from a CuCl2⋅2H2O sample. One year later, the first NMR spectrum was observed and through the first decade both techniques developed simultaneously. Nevertheless, EPR spectroscopy had a few challenges from lack of microwave components and limited microwave power to expensive instrumentation and consequently was left behind by NMR for a couple of decades [4–6].

It was until the 1980s that the instrumentation became cheaper and more manageable and the first pulse EPR was released to the market. A decade later the high field spectrometer was released and since then the interest in EPR as a characterization technique has considerably increased.

The spin angular momentum *S* gives rise to a magnetic moment *μ* =– g*μ*B*S*, where *g* is the g factor; its value for free electrons is *g* = 2.0023193043617 and *μ*<sup>B</sup> is the Bohr magneton. When the electrons are subjected to an external field *H* (it is customary to place the field along the *z* axis), the energy levels of the degenerate spin states split depending on their quantum magnetic moment *m*<sup>s</sup> = ± 1/2 and the strength of the magnetic field as shown in Figure 4 [4–6].

**Figure 4.** Splitting of the energy levels of an electron spin in a magnetic field at the resonance frequencies.

When the irradiation at a given frequency *ω*<sup>0</sup> is the same as the energy difference ∆*E* between both states, a resonant absorption takes place. This frequency is named the Larmor frequency *ω*0 = –γ *H*0 after J. Larmor [6]:

$$
\Delta W = \hbar \alpha\_0 = \text{g}\,\mu\_\text{g}H \tag{1}
$$

The detection of that absorption is the most important principle of this spectroscopy.

For temperatures above the Curie temperature, the magnetic moments in an external field obey the Bloch equations. These equations follow the Larmor theorem where the motion of the magnetic moments in a magnetic field originates a torque [6–9]:

$$
\hbar \frac{d\mathbf{M}}{dt} = \mathbf{M} \times \boldsymbol{\gamma} \mathbf{H} \tag{2}
$$

To detect the absorption, there is an oscillating microwave field **H**1 = (*Hx*,*Hy*,0) = (*H*1cos(*ω*mw*t*), *H*1sin(*ω*mw*t*),0) aside from the external magnetic field along the *z* axis. This magnetic field deviates the magnetization away from its equilibrium position. To describe accurately the motion of the magnetization vector, it is important to take into account the relaxation effects. The relaxation time has two components, the longitudinal relaxation time, which is the average time in which the magnetization vector returns to its thermal equilibrium state, and the transverse relaxation time, which characterizes the loss of coherence in the transverse plane due to spin interactions [5–8].

The evolution of the magnetization as a function of time and its dependence of the magnetic field is given by the equation motion derived by F. Bloch [5–6]:

Characterization of Magnetic Phases in Nanostructured Ferrites by Electron Spin Resonance http://dx.doi.org/10.5772/61508 217

$$i\hbar \frac{d\mathbf{M}}{dt} = \mathbf{M}(t) \times \gamma \mathbf{H}(t) - \mathbf{R}(\mathbf{M}(t) - \mathbf{M}\_0) \tag{3}$$

whereγ is the gyromagnetic ratio, **H** = (*H*1cos(*ω*mw*t*), *H*1sin(*ω*mw*t*),*Hz*), and **R** is the relaxation tensor given by [6]:

$$\mathbf{R} = \begin{pmatrix} 1/T\_2 & 0 & 0 \\ 0 & 1/T\_2 & 0 \\ 0 & 0 & 1/T\_1 \end{pmatrix} \tag{4}$$

Relaxation phenomena need to be taken into account to describe the motion of the magneti‐ zation vector. *T*1 is the longitudinal relaxation time; this parameter characterizes the process that makes the magnetization vector return to its thermal equilibrium. *T*2 is the transverse relaxation time which describes the loss of coherence in the transverse plane due to spin-spin interactions.

**Figure 4.** Splitting of the energy levels of an electron spin in a magnetic field at the resonance frequencies.

*ω*0 = –γ *H*0 after J. Larmor [6]:

216 Advanced Electromagnetic Waves

due to spin interactions [5–8].

When the irradiation at a given frequency *ω*<sup>0</sup> is the same as the energy difference ∆*E* between both states, a resonant absorption takes place. This frequency is named the Larmor frequency

> m

For temperatures above the Curie temperature, the magnetic moments in an external field obey the Bloch equations. These equations follow the Larmor theorem where the motion of the

g

To detect the absorption, there is an oscillating microwave field **H**1 = (*Hx*,*Hy*,0) = (*H*1cos(*ω*mw*t*), *H*1sin(*ω*mw*t*),0) aside from the external magnetic field along the *z* axis. This magnetic field deviates the magnetization away from its equilibrium position. To describe accurately the motion of the magnetization vector, it is important to take into account the relaxation effects. The relaxation time has two components, the longitudinal relaxation time, which is the average time in which the magnetization vector returns to its thermal equilibrium state, and the transverse relaxation time, which characterizes the loss of coherence in the transverse plane

The evolution of the magnetization as a function of time and its dependence of the magnetic

<sup>0</sup> *<sup>B</sup>* D= = h (1)

**<sup>M</sup>** <sup>h</sup> **M H** (2)

*W gH* w

The detection of that absorption is the most important principle of this spectroscopy.

*dt* = ´

magnetic moments in a magnetic field originates a torque [6–9]:

field is given by the equation motion derived by F. Bloch [5–6]:

*d*

Electron paramagnetic resonance is a very useful technique to study the properties of bulk paramagnetic compounds including their transitions to the magnetic ordered state. Below this temperature, ferromagnetic resonance (FMR) or antiferromagnetic resonance (AFMR) are detected. EPR in paramagnetic samples give information such as the resonance active ion valence and the symmetry of the ligand environment. In the case of nanoparticles, the theory of EPR is not quite the same as in bulk; however, a study of the thermal variations of EPR spectra can be very informative because of the high sensibility of the EPR technique [10].

Let's recall that magnetic resonance is observable only in materials that contain a sufficient number of permanent magnetic dipoles; if the origin of these dipoles is electronic, then the resonance is detectable with a population even fewer than 1011 dipoles [8].

The first paramagnetic resonance absorption in metals due to the conduction electrons was observed by Feher and Kip [8]. The conduction electrons have an effect on the shape and intensity of the resonance lines. However, until the theoretical study of Dyson, there were not many studies in conduction electron paramagnetic resonance in metals [11].

In a metal, the electrons are assumed to diffuse as free particles and the magnetic moments of each of them can be seen as free-particle moments. When the metal is placed in a radiofre‐ quency electromagnetic field and at the same time in a perpendicular uniform magnetic field, a certain macroscopic magnetization is created as a result of the magnetic moments of the conduction electrons. The penetration of the radio frequency field into the metal is modified by the magnetization. Actually, only the layers near the surface contribute, since the excitation field *H*1 penetrates only a small depth into the metal. This is called the skin effect [11]. The magnetization *M* shows a resonant behavior and becomes large when the frequency of the field is nearly equal to the resonant frequency. The absorption field observed is a measure of the total energy absorbed in the metal both by eddy currents and by the resistive out of phase component of the magnetization.

There are four assumptions which need to be taken into account to study the dependence of the line width [8,11]:


These assumptions give as a result a finite line width proportional to 1/*U* in the resonance signal from nondiffusing electrons. The penetration of the radio frequency field into the metal will be limited by the skin depth, which is given by [11] In a metal, the electrons are assumed to diffuse as free particles and the magnetic moments of each of them can be seen as free-particle moments. When the metal is placed in a radiofrequency electromagnetic field and at the same time in a perpendicular uniform magnetic field, a certain macroscopic magnetization is created as a result of the magnetic moments of the conduction electrons. The penetration of the radio frequency field into the metal is modified by the magnetization. Actually, only the layers near the surface contribute, since the excitation field *H*1 penetrates only a small depth into the metal. This is called the skin effect [11]. The magnetization *M* shows a resonant behavior and becomes large when the frequency of the field is nearly equal to the resonant frequency. The absorption field observed is a measure of the total energy absorbed in

the metal both by eddy currents and by the resistive out of phase component of the magnetization.

2. Each electron moves as an independent classical particle with random changes in their direction.

conduction electrons and to move with constant velocity.

$$\mathcal{S} = \left(\mathbf{c}^2 \mid 2\pi\sigma\alpha\phi\right)^{1/2} \tag{5}$$

where *c* is the speed of light, *σ* is the conductivity of the metal, and *ω* is the radio frequency. If the skin depth is larger than the main free path, we are in the domain of the classical skin effects; if it is equal to the main free path, then we are in the anomalous case. In fact, unlike paramagnetic insulators, in which the varying part of the magnetization depends on the magnitude of the applied oscillating field, in metals, there is also a dependence of the local values of the oscillating field on the magnetization because of the skin depth. A typical spectrum (metallic Na) is shown in Figure 5. 3. The spin of each particle is a quantum independent variable. It is unaffected by collisions and only the local magnetic field has an effect on them. 4. Given any time interval *t* and *U,* a volume relaxation time there is a probability ~exp(-*t*/*U*) that the spin state of an electron will not be randomly distributed by collisions during this interval. These assumptions give as a result a finite line width proportional to 1/*U* in the resonance signal from nondiffusing electrons. The penetration of the radio frequency field into the metal will be limited by the skin depth, which is given by [11] 2 1/ 2 (*c* / 2) (5) where *c* is the speed of light, is the conductivity of the metal, and is the radio frequency. If the skin depth is larger than the main free path, we are in the domain of the classical skin effects; if it is equal to the main free path, then we are in the anomalous case. In fact, unlike paramagnetic insulators, in which the varying part of the magnetization depends on the magnitude of the applied oscillating field, in metals, there is also a dependence of the local values of the oscillating field on the magnetization because of the skin depth. A typical spectrum (metallic Na) is shown in Figure 5.

Figure 5. Typical first derivative EPR spectrum observed in colloidal samples of Na, with mean diameter particle small compared to the skin depth. The basic parameters and the definition of *R* are indicated. Adapted from Feher and Kip [8]. **Figure 5.** Typical first derivative EPR spectrum observed in colloidal samples of Na, with mean diameter particle small compared to the skin depth. The basic parameters and the definition of *R* are indicated. Adapted from Feh‐ er and Kip [8].

7

The magnetic susceptibility in metals has a diamagnetic component due to the circulation of electrons in the field *H*. This is opposed by the normal paramagnetic component due to unpaired electrons. This is related with another EPR parameter that has been studied in metals: the *g* factor [8,12]. The relation between the resonance values of the constant magnetic field *H*<sup>0</sup> and the frequency is determined by the magnitude of the *g* factor of the conduction electrons. The existence of internal interactions in a metal leads to a shift in the value of the *g* factor.

Experimentally, the *g* factor has been measured in different paramagnetic metals giving rise to values very close to the *g* free electron value *g* = 2.0023 as shown in Table 3. This is a measure of the spin-orbit coupling. It has been possible to study *S*-state ions in metals in order to understand the interaction between the conduction electrons and the inserted ions.


**Table 3.** Experimental values for the *g* shift. Adapted from Feher and Kip [8].

EPR spectroscopy has made contributions in understanding the bonding and the electronic structure of molecular species with metal–metal bonds. Some of the information obtained through this technique includes determining whether unpaired electrons reside in metal-based or ligand-based molecular orbitals, giving information of the metal center's total electronic spin, which also provides information of its oxidation state, broadly have information of the distribution of the unpaired electrons between metals and organic ligands [13].
