**3.1. Magnetic nanoparticles**

There are four assumptions which need to be taken into account to study the dependence of

**1.** The electrons which carry the magnetization in a metal are assumed to lie at the top of the Fermi distribution of the conduction electrons and to move with constant velocity.

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

**3.** The spin of each particle is a quantum independent variable. It is unaffected by collisions

**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

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

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

2 1/2

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

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

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

1. The electrons which carry the magnetization in a metal are assumed to lie at the top of the Fermi distribution of the

3. The spin of each particle is a quantum independent variable. It is unaffected by collisions and only the local

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

7

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‐

*A +*

*H*

*H*

= ( /2 ) *c* (5)

is the radio frequency. If the skin depth is larger than

) (5)

 *A‐*

*R* = *A*

‐ /*A* +

 psw

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

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

and only the local magnetic field has an effect on them.

will be limited by the skin depth, which is given by [11]

magnetic field has an effect on them.

d

*P*

/d

*H*

spectrum (metallic Na) is shown in Figure 5.

where *c* is the speed of light,

d

electron will not be randomly distributed by collisions during this interval.

(*c* / 2

is the conductivity of the metal, and

the magnetization because of the skin depth. A typical spectrum (metallic Na) is shown in Figure 5.

conduction electrons and to move with constant velocity.

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

the line width [8,11]:

218 Advanced Electromagnetic Waves

direction.

interval.

er and Kip [8].

Iron-based oxide nanoparticles materials have been widely studied because of their technological importance. γ-Fe2O3 is a very popular material for magnetic tape purposes. Another popular material in high density recording media is BaFe12O19 and its substitut‐ ed derivatives since it can be doped with other cations in order to reduce their magnetocrys‐ talline anisotropy [10].

Bulk Fe2O3 exists as a ferrimagneticγ -Fe2O3 (maghemite) or antiferromagnetic α-Fe2O3 (hematite). Maghemite nanoparticles are a representative model for the experimental study of nanoparticles and for testing theoretical concepts. In contrast, bulk hematite is antiferromag‐ netic below the Morin temperature and weakly ferromagnetic above this temperature [10].

At room temperature, the EPR of γ-Fe2O3 and BaFe2O4 and nanoparticles show a tow line pattern as shown in Figure 6 [9], which is typical of superparamagnetic resonance in contrast with the BaFe12O19 where the narrow line is more pronounced (as shown in Figure 6c). In these sample, the line width Δ*H* ∼120 Oe and *g* = 2.00; this value of the *g* factor is typical paramagnetic.

**Figure 6.** Room temperature EPR spectra of nanoparticles: (a) Fe2O3, (b) BaFe2O4, and (c) BaFe12O19. Adapted from Kok‐ sharob et al. [10].

It is shown in Figure 7a that below 100 K, the broad line 1 shows a typical superparamagnetic (SPM) behavior of single-domain particles in the absence of transitions to a magnetic ordered state. SPM behavior will be discussed in the following section. Below 50 K, new resonances appear such as 2 and 3. Point 2 is a typical paramagnetic resonance signal. On the other hand, point 3 is characteristic of a phase transition α-Fe2O3 of the hematite to antiferromagnetic phase.

**Figure 7.** (a) EPR spectra of Fe2O3 nanoparticles; (b) EPR spectra of BaFe2O4 nanoparticles. Adapted from Koksharob et al. [10].

The bulk thermodynamics phase diagram [14] shows several interchangeable iron oxide phases. For nanoparticles, the mutual phase transition should be easier. For this reason, it is accurate to expect a multiphase composition in Fe2O3 nanoparticles. In point 3, the value *g* is 4.03, which is characteristic of a high spin state Fe3+ in the rhombic crystal field. The antifer‐ romagnetic transition is associated with the α-Fe2O3 phase. In rhombohedralα -Fe2O3 as well as in γ-Fe2O3, there is a departure from axial symmetry. The paramagnetic signal in point 2 is attributed to the γ-Fe2O3 phase; it has a *g* value of 2.02, and it is due to the octahedral symmetry sites of Fe3+ in the spinel structure.

Bulk BaFe2O4 is nonferrimagnetic. On the contrary, BaFe2O4 demonstrate an EPR anomaly near 125 K that could indicate the presence of a Fe2O3 phase. On the other hand, bulk hexaferrite BaFe12O19 is a representative case of FMR because of its very high uniaxial anisotropy. However, in the nanoparticles, the contribution of the particle's surface is appreciable and can reduce the total anisotropy energy as well as crystalline defects due to stress and strain. The EPR spectra reveal the effect of superparamagnetic fluctuations narrowing the resonance [10] and will be discussed in the next section.
