**4. Spin resonance in superparamagnetic phases**

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

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

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‐

sharob et al. [10].

220 Advanced Electromagnetic Waves

al. [10].

The superparamagnetic (SPM) state is characterized by random fluctuations of the magneti‐ zation due to thermal excitation. It appears in ferro- and ferrimagnetic particles, which are sufficiently small and thus single domain, typically in the nanometric range (1–100 nm). It is important to note that in contrast with the paramagnetic state, characterized by random fluctuations of *individual, noninteracting* spins in the SPM phase, these fluctuations involve the *whole* magnetization vector, which is the sum of all individual moments of the particle. As SPM occurs at temperatures below the Curie transition, exchange interaction is effective, and spins are therefore coupled. It is by this reason that some authors describe SPM as "macro-spin" fluctuations.

A simple approach to SPM can be obtained by considering the magnetocrystalline anisotropy energy, *E*K = *K V*, for a nanoparticle of volume *V* and anisotropy constant *K*. When this product becomes small and comparable to the thermal energy *E*T = *k*B*T* (where *k*<sup>B</sup> is the Boltzmann constant), the magnetization oscillates by thermal excitations and easy directions vanish. A simplified graphical model is shown in Figure 8. A uniaxial nanoparticle has two easy magnetization directions, i.e., two magnetization orientations that lead to a minimum in anisotropy energy. These orientations are separated by an energy barrier *E*K = *K V*. Superpar‐ amagnetism appears when the thermal energy becomes comparable to anisotropy energy, and therefore the energy barrier is dominated by thermal oscillations.

A nanoparticle of a given composition and constant volume typically shows a ferro/ferrimag‐ netic behavior at low temperatures and can become superparamagnetic as *T* increases. At low temperatures, *K* is large and *T* is small, and the opposite is true as *T* increases. The temperature at which SPM occurs is known as the blocking temperature, *T*B.

The probability of reversal of magnetization is given by the Néel–Arrhenius or Néel–Brown relation [15,16]:

"macro-spin" fluctuations.

$$
\pi\_m = \pi\_0 \exp\left(KV / k\_0 T\right) \tag{6}
$$

where *V* is the volume of the nanoparticle, *K* is the anisotropy constant, *T* is the temperature, *τ*<sup>m</sup> is the average length of time for a magnetization reversal to occur, and *τ*<sup>0</sup> is a characteristic time constant related with the gyromagnetic precession, known also as the *attempt time* for a given material. Typical values are in the 10–9 to 10–10 s range. Under an applied magnetic field, the blocking temperature variations can be expressed (for uniaxial materials) as [17] The probability of reversal of magnetization is given by the Néel–-Arrhenius, or Néel-–Brown relation [15,16]: m = 0 exp (*KV*/*k*BkB*T*) (6) where *V* is the volume of the nanoparticle, *K* is the anisotropy constant, *T* is the temperature,<sup>m</sup>m is the average length of time for a magnetization reversal to occur, and <sup>0</sup>0 is a characteristic time constant related with the gyromagnetic

opposite is true as *T* increases. The temperature at which SPM occurs is known as the blocking temperature, *T*B.

precession, known also as the *attempt time* for a given material. Typical values are in the 10-9

The superparamagnetic (SPM) state is characterized by random fluctuations of the magnetization, due to thermal excitation. It appears in ferro- and ferrimagnetic particles, which are sufficiently small and thus single domain; , typically, in the nanometric range (1–-100 nm). It is important to note that in contrast with the paramagnetic state, characterized by random fluctuations of *individual, non-interacting* spins, in the SPM phase, these fluctuations involve the *whole* magnetization vector, which is the sum of all individual moments of the particle. As SPM occurs at temperatures below the Curie transition, exchange interaction is effective, and spins are therefore coupled. It is by this reason that some authors describe SPM as

A simple approach to SPM can be obtained by considering the magnetocrystalline anisotropy energy, *E*K = *K·V*, for a nanoparticle of volume *V* and anisotropy constant *K*. When this product becomes small and comparable to the thermal energy *E*T = *k*B*T* (where *k*BkB is the Boltzmann constant), the magnetization oscillates by thermal excitations and easy

separated by an energy barrier *E*K = *K·V*. Superparamagnetism appears when the thermal energy becomes comparable to

temperatures, and can become superparamagnetic as *T* increases. At low temperatures, *K* is large and *T* is small, and the

anisotropy energy, and therefore the energy barrier is dominated by thermal oscillations.

$$T\_{\rm B}(H) = \frac{KV}{k\_{\rm B} \ln(\tau\_{\rm m} / \tau\_{\rm o})} \left[ I - \left(\frac{H}{H\_{\rm k}}\right) \right]^2. \tag{7}$$

(7)

to 10-10 s range. Under an

**Formatted:** Font: Italic

**Formatted:** Font: Italic **Formatted:** Font: Italic

**Formatted:** Font: Italic

where *H* is the applied field, *H*K is the anisotropy field, and *a* has a typical value of 1.5 [18]. where *H* is the applied field, *H*K is the anisotropy field, and *a* has a typical value of 1.5 [18].

Figure 8. Schematics of the energy in a single-domain particle (with uniaxial anisotropy) as a function of the angle between easy axes and the magnetization direction. For SPM nanoparticles, the energy barrier is comparable to the thermal energy, thus resulting in magnetization random oscillations. **Figure 8.** Schematics of the energy in a single-domain particle (with uniaxial anisotropy) as a function of the angle be‐ tween easy axes and the magnetization direction. For SPM nanoparticles, the energy barrier is comparable to the ther‐ mal energy, thus resulting in magnetization random oscillations.

10 It is interesting to note that the blocking temperature, *T*B, depends on the time window, *t*w, of the particular experimental technique. *t*w is the effective length of time during the measure‐ ment. If *t*w >> *τ*m, the magnetization will show several reversals and the state of the particle will we taken as SPM. On the contrary, if *t*w << *τ*m, magnetization will exhibit a stable state, and therefore the material will be considered as in the ordered phase. The time window for *T*<sup>B</sup> determination from magnetization measurements in the ZFC-FC technique (in a SQUID machine) is about 10–1 s, while for Mössbauer spectroscopy or spin resonance techniques *t*w is about 10–6 s. A model to describe the crossover from the SPM state to the blocked phase in magnetic nanoparticles has been recently proposed [19]. It is based on the Stoner–Wohlfarth model and also assumes noninteracting nanoparticles with uniaxial anisotropy and uniform random reversal of magnetization.

**Formatted:** Font: Italic **Formatted:** Font: Italic SPM materials show no hysteresis in *M* vs. *H* plots, i.e., absence of both coercive field and remanent magnetization, as illustrated in Figure 9. For an ensemble of noninteracting particles at temperatures low enough (*T*B < T < *KV*/10*k*B), a simple model for the magnetization behavior, based on the paramagnetic theory, is

$$M(H) \approx n\mu \tanh\left(\frac{\mu\_0 H \mu}{k\_B T}\right) \tag{8}$$

At higher temperatures (*T* > *KV*/*k*B),

**Formatted:** Font: Italic

**Formatted:** Font: Italic

t t

m = 

where *V* is the volume of the nanoparticle, *K* is the anisotropy constant, *T* is the temperature,

precession, known also as the *attempt time* for a given material. Typical values are in the 10-9

where *H* is the applied field, *H*K is the anisotropy field, and *a* has a typical value of 1.5 [18].

kB*T*

anisotropy energy, and therefore the energy barrier is dominated by thermal oscillations.

"macro-spin" fluctuations.

222 Advanced Electromagnetic Waves

B

Energy

mal energy, thus resulting in magnetization random oscillations.

time for a magnetization reversal to occur, and

thus resulting in magnetization random oscillations.

where *V* is the volume of the nanoparticle, *K* is the anisotropy constant, *T* is the temperature, *τ*<sup>m</sup> is the average length of time for a magnetization reversal to occur, and *τ*<sup>0</sup> is a characteristic time constant related with the gyromagnetic precession, known also as the *attempt time* for a given material. Typical values are in the 10–9 to 10–10 s range. Under an applied magnetic field,

A nanoparticle of a given composition and constant volume typically shows a ferro/ferrimagnetic behavior at low temperatures, and can become superparamagnetic as *T* increases. At low temperatures, *K* is large and *T* is small, and the

The superparamagnetic (SPM) state is characterized by random fluctuations of the magnetization, due to thermal excitation. It appears in ferro- and ferrimagnetic particles, which are sufficiently small and thus single domain; , typically, in the nanometric range (1–-100 nm). It is important to note that in contrast with the paramagnetic state, characterized by random fluctuations of *individual, non-interacting* spins, in the SPM phase, these fluctuations involve the *whole* magnetization vector, which is the sum of all individual moments of the particle. As SPM occurs at temperatures below the Curie transition, exchange interaction is effective, and spins are therefore coupled. It is by this reason that some authors describe SPM as

A simple approach to SPM can be obtained by considering the magnetocrystalline anisotropy energy, *E*K = *K·V*, for a nanoparticle of volume *V* and anisotropy constant *K*. When this product becomes small and comparable to the thermal energy *E*T = *k*B*T* (where *k*BkB is the Boltzmann constant), the magnetization oscillates by thermal excitations and easy directions vanish. A simplified graphical model is shown in Figure 8. A uniaxial nanoparticle has two easy magnetization directions, i.e., two magnetization orientationswhich that lead to a minimum in anisotropy energy. These orientations are separated by an energy barrier *E*K = *K·V*. Superparamagnetism appears when the thermal energy becomes comparable to

the blocking temperature variations can be expressed (for uniaxial materials) as [17]

applied magnetic field, the blocking temperature variations can be expressed (for uniaxial materials) as [17]:

opposite is true as *T* increases. The temperature at which SPM occurs is known as the blocking temperature, *T*B. The probability of reversal of magnetization is given by the Néel–-Arrhenius, or Néel-–Brown relation [15,16]:

é ù æ ö <sup>=</sup> ê ú - ç ÷

where *H* is the applied field, *H*K is the anisotropy field, and *a* has a typical value of 1.5 [18].

10

Figure 8. Schematics of the energy in a single-domain particle (with uniaxial anisotropy) as a function of the angle between easy axes and the magnetization direction. For SPM nanoparticles, the energy barrier is comparable to the thermal energy,

**Figure 8.** Schematics of the energy in a single-domain particle (with uniaxial anisotropy) as a function of the angle be‐ tween easy axes and the magnetization direction. For SPM nanoparticles, the energy barrier is comparable to the ther‐

It is interesting to note that the blocking temperature, *T*B, depends on the time window, *t*w, of the particular experimental technique. *t*w is the effective length of time during the measure‐ ment. If *t*w >> *τ*m, the magnetization will show several reversals and the state of the particle will we taken as SPM. On the contrary, if *t*w << *τ*m, magnetization will exhibit a stable state, and therefore the material will be considered as in the ordered phase. The time window for *T*<sup>B</sup> determination from magnetization measurements in the ZFC-FC technique (in a SQUID machine) is about 10–1 s, while for Mössbauer spectroscopy or spin resonance techniques *t*w is about 10–6 s. A model to describe the crossover from the SPM state to the blocked phase in magnetic nanoparticles has been recently proposed [19]. It is based on the Stoner–Wohlfarth


state A state B

Angle

B m0 ( ) . ln( / ) *<sup>k</sup> KV H T H <sup>I</sup> k H* t t

m 0 = exp / (*KV k T*<sup>B</sup> ) (6)

0 exp (*KV*/*k*BkB*T*) (6)

2

M

(7)

<sup>m</sup>m is the average length of

to 10-10 s range. Under an

<sup>0</sup>0 is a characteristic time constant related with the gyromagnetic

(7)

ç ÷ ê ú ë û è ø

$$M(H) \approx n\mu L \left(\frac{\mu\_0 H \mu}{k\_B T}\right) \tag{9}$$

The kinetic approach [21,22] is known with this name because it makes use of the Fokker–Planck-type equation used by Brown [16] to analyze viscosity problems. The main criticism for this approach is that it predicts an increase of the line width with temperature, which is opposite to the large majority of experimental results. On some cases, a tendency to show a constant line width has been observed, but to our knowledge, no significant increase in ∆*H* has been reported in

100 200 300 400 500

T(K)

The SPM state has elicited the proposal of a number of theoretical models to account for its magnetization, and more specifically to understand the features of the resonance spectrum. Some models [24, 25] discuss the longitudinal modes, which are more directly associated with magnetization reversal. For spin resonance, however, it is the transverse modes, which are more directly related with magnetization precession and therefore with magnetization resonance. There are

In the statistical approach [26,27], which has a phenomenological character, the resonance conditions are evaluated by averaging over the equilibrium statistical distribution of all possible directions of particle magnetization moment. Typically, this model assumes that the magnetocrystalline anisotropy is smaller than the interaction energy between the magnetization and the applied field. While modeling can effectively represent some of the experimental results, the

Another approach is also based on the Landau–Lifshitz equation of magnetization dynamics. In this model [26], *H is*  changed by *H*loc to take into account both the external field and the effective field of the intrinsic anisotropy of the NPs.

 ∆*H* = ∆*H*T *L*(*X*) (10) where ∆*H*T is the limiting value of the line width for bulk particles and *L*(*X*) is a Langevin type (*L*(*X*) = cot h *X*– 1/*X*) of a

Experimentally, the behavior of the line width and the resonance field of SPM phases of metallic and oxide NPs is quite clear; as temperature increases, the line width decreases and the resonance field increases, as illustrated by means of

240

two main approaches, the so-called "statistical" approach and the "kinetic" approach.

260

280

H

averaging of the magnetic anisotropy is not fully clear.

some recent results, see Figure 11 [23].

The dependence of the line width with temperature is expressed by

phenomenological temperature factor *X* (see Berger et al. [26] for more details).

res (mT)

300

320

340

Zn0.7Ni0.3Fe2

Figure 10. Resonance field as a function of temperature for Zn-Ni ferrite NPs, at *f* = 9.45 GHz [23].

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where *n* is the density (atoms/volume), *μ* is the magnetic moment (the "macrospin"), *M* is the magnetization, and *L* is the Langevin function, *L* (*x*) = 1/tanh (*x*) – 1/*x*, with *x* = *μ*0*H μ* / *T k*B. This behavior is very convenient for many applications, especially in biomedicine, where magnetic nanoparticles manifest a magnetization only when subjected to a magnetic field, thus avoiding interparticle interactions (which could lead to aggregation) at *H* = 0. In contrast, SPM is useless for magnetic recording since the absence of a remanent state eliminates any possi‐ bility of memory.

Figure 9. Hysteresis loop of ZnNi nanoparticles in the superparamagnetic phase [20]. **Figure 9.** Hysteresis loop of ZnNi nanoparticles in the superparamagnetic phase [20].

SPM phases as *T* increases.

The kinetic approach [21,22] is known with this name because it makes use of the Fokker– Planck-type equation used by Brown [16] to analyze viscosity problems. The main criticism for this approach is that it predicts an increase of the line width with temperature, which is opposite to the large majority of experimental results. On some cases, a tendency to show a constant line width has been observed, but to our knowledge, no significant increase in ∆*H* has been reported in SPM phases as *T* increases. The kinetic approach [21,22] is known with this name because it makes use of the Fokker–Planck‐type equation used by Brown [16] to analyze viscosity problems. The main criticism for this approach is that it predicts an increase of the line width with temperature, which is opposite to the large majority of experimental results. On some cases, a tendency to show a constant line width has been observed, but to our knowledge, no significant increase in ∆*H* has been reported in SPM phases as *T* increases.

**Figure 10.** Resonance field as a function of temperature for Zn-Ni ferrite NPs, at *f* = 9.45 GHz [23].

The SPM state has elicited the proposal of a number of theoretical models to account for its magnetization, and more specifically to understand the features of the resonance spectrum. Some models [24, 25] discuss the longitudinal modes, which are more directly associated with magnetization reversal. For spin resonance, however, it is the transverse modes, which are more directly related with magnetization precession and therefore with magnetization resonance. There are two main approaches, the so-called "statistical" approach and the "kinetic" approach. In the statistical approach [26,27], which has a phenomenological character, the resonance conditions are evaluated by averaging over the equilibrium statistical distribution of all possible directions of particle magnetization moment. Typically, The SPM state has elicited the proposal of a number of theoretical models to account for its magnetization, and more specifically to understand the features of the resonance spectrum. Some models [24, 25] discuss the longitudinal modes, which are more directly associated with magnetization reversal. For spin resonance, however, it is the transverse modes, which are more directly related with magnetization precession and therefore with magnetization resonance. There are two main approaches, the so-called "statistical" approach and the "kinetic" approach.

Figure 10. Resonance field as a function of temperature for Zn-Ni ferrite NPs, at *f* = 9.45 GHz [23].

As temperature increases, the resonance field increases from 243 mT for 102 K to 333 mT for 473 K. This is due to the fact that for temperatures below the Curie transition, there is an internal field formed by all the possible contribution to the

where *H*exch is the exchange interaction, *H*anis is the anisotropy field, and *H*dem is the demagnetization field. As a result,

the magnetization vector is subjected to a total field composed by the internal field and the applied field,

*H*int = *H*exch + *H*anis + *H*dem + ... (11)

*H*tot = *H*appl + *H*int (12)

this model assumes that the magnetocrystalline anisotropy is smaller than the interaction energy between the magnetization and the applied field. While modeling can effectively represent some of the experimental results, the averaging of the magnetic anisotropy is not fully clear. Another approach is also based on the Landau–Lifshitz equation of magnetization dynamics. In this model [26], *H is*  changed by *H*loc to take into account both the external field and the effective field of the intrinsic anisotropy of the NPs. The dependence of the line width with temperature is expressed by ∆*H* = ∆*H*T *L*(*X*) (10) In the statistical approach [26,27], which has a phenomenological character, the resonance conditions are evaluated by averaging over the equilibrium statistical distribution of all possible directions of particle magnetization moment. Typically, this model assumes that the magnetocrystalline anisotropy is smaller than the interaction energy between the magnetiza‐ tion and the applied field. While modeling can effectively represent some of the experimental results, the averaging of the magnetic anisotropy is not fully clear.

where ∆*H*T is the limiting value of the line width for bulk particles and *L*(*X*) is a Langevin type (*L*(*X*) = cot h *X*– 1/*X*) of a phenomenological temperature factor *X* (see Berger et al. [26] for more details). Experimentally, the behavior of the line width and the resonance field of SPM phases of metallic and oxide NPs is quite clear; as temperature increases, the line width decreases and the resonance field increases, as illustrated by means of some recent results, see Figure 11 [23]. Another approach is also based on the Landau–Lifshitz equation of magnetization dynamics. In this model [26], *H is* changed by *H*loc to take into account both the external field and the effective field of the intrinsic anisotropy of the NPs. The dependence of the line width with temperature is expressed by

ordered state:

Characterization of Magnetic Phases in Nanostructured Ferrites by Electron Spin Resonance http://dx.doi.org/10.5772/61508 225

$$
\Delta H = \Delta H\_T L \text{(X)}\tag{10}
$$

show a constant line width has been observed, but to our knowledge, no significant increase in ∆*H* has been reported in where ∆*H*T is the limiting value of the line width for bulk particles and *L*(*X*) is a Langevin type (*L*(*X*) = cot h *X*– 1/*X*) of a phenomenological temperature factor *X* (see Berger et al. [26] for more details).

The kinetic approach [21,22] is known with this name because it makes use of the Fokker– Planck-type equation used by Brown [16] to analyze viscosity problems. The main criticism for this approach is that it predicts an increase of the line width with temperature, which is opposite to the large majority of experimental results. On some cases, a tendency to show a constant line width has been observed, but to our knowledge, no significant increase in ∆*H*

100 200 300 400 500

two main approaches, the so-called "statistical" approach and the "kinetic" approach.

T(K)

The SPM state has elicited the proposal of a number of theoretical models to account for its magnetization, and more specifically to understand the features of the resonance spectrum. Some models [24, 25] discuss the longitudinal modes, which are more directly associated with magnetization reversal. For spin resonance, however, it is the transverse modes, which are more directly related with magnetization precession and therefore with magnetization resonance. There are two main approaches, the so-called "statistical" approach and the

averaging of the magnetic anisotropy is not fully clear.

In the statistical approach [26,27], which has a phenomenological character, the resonance conditions are evaluated by averaging over the equilibrium statistical distribution of all possible directions of particle magnetization moment. Typically, this model assumes that the magnetocrystalline anisotropy is smaller than the interaction energy between the magnetiza‐ tion and the applied field. While modeling can effectively represent some of the experimental

Another approach is also based on the Landau–Lifshitz equation of magnetization dynamics. In this model [26], *H is* changed by *H*loc to take into account both the external field and the effective field of the intrinsic anisotropy of the NPs. The dependence of the line width with

some recent results, see Figure 11 [23].

results, the averaging of the magnetic anisotropy is not fully clear.

ordered state:

temperature is expressed by

The dependence of the line width with temperature is expressed by

phenomenological temperature factor *X* (see Berger et al. [26] for more details).

The kinetic approach [21,22] is known with this name because it makes use of the Fokker–Planck‐type equation used by Brown [16] to analyze viscosity problems. The main criticism for this approach is that it predicts an increase of the line width with temperature, which is opposite to the large majority of experimental results. On some cases, a tendency to

changed by *H*loc to take into account both the external field and the effective field of the intrinsic anisotropy of the NPs.

As temperature increases, the resonance field increases from 243 mT for 102 K to 333 mT for 473 K. This is due to the fact that for temperatures below the Curie transition, there is an internal field formed by all the possible contribution to the

where *H*exch is the exchange interaction, *H*anis is the anisotropy field, and *H*dem is the demagnetization field. As a result,

the magnetization vector is subjected to a total field composed by the internal field and the applied field,

*H*int = *H*exch + *H*anis + *H*dem + ... (11)

*H*tot = *H*appl + *H*int (12)

has been reported in SPM phases as *T* increases.

224 Advanced Electromagnetic Waves

240

260

280

H

"kinetic" approach.

res (mT)

300

320

340

SPM phases as *T* increases.

Zn0.7Ni0.3Fe2

**Figure 10.** Resonance field as a function of temperature for Zn-Ni ferrite NPs, at *f* = 9.45 GHz [23].

O4

Experimentally, the behavior of the line width and the resonance field of SPM phases of metallic and oxide NPs is quite clear; as temperature increases, the line width decreases and the resonance field increases, as illustrated by means of some recent results, see Figure 11 [23].

As temperature increases, the resonance field increases from 243 mT for 102 K to 333 mT for 473 K. This is due to the fact that for temperatures below the Curie transition, there is an internal field formed by all the possible contribution to the ordered state:

$$H\_{\rm int} = H\_{\rm each} + H\_{\rm axis} + H\_{\rm dem} + \dots \tag{11}$$

where *H*exch is the exchange interaction, *H*anis is the anisotropy field, and *H*dem is the demagnet‐ ization field. As a result, the magnetization vector is subjected to a total field composed by the internal field and the applied field,

$$H\_{\text{tot}} = H\_{\text{appl}} + H\_{\text{int}} \tag{12}$$

 ∆*H* = ∆*H*T *L*(*X*) (10) where ∆*H*T is the limiting value of the line width for bulk particles and *L*(*X*) is a Langevin type (*L*(*X*) = cot h *X*– 1/*X*) of a Figure 11. (a) ESR results on Zn0.7Ni0.3Fe2O4 ferrite NPs at a microwave frequenyfrequency of 9.45 GHz, in the 102–-473 temperature range [22]. Signals were normalized to have the same value for the maximum (section of the resonance signal for *H*<*H*res) to facilitate comparisons; (b) temperature behavior of the line width (peak to peak value) [23]. **Figure 11.** (a) ESR results on Zn0.7Ni0.3Fe2O4 ferrite NPs at a microwave frequency of 9.45 GHz, in the 102–473 tempera‐ ture range [22]. Signals were normalized to have the same value for the maximum (section of the resonance signal for *H* < *H*res) to facilitate comparisons; (b) temperature behavior of the line width (peak to peak value) [23].

Experimentally, the behavior of the line width and the resonance field of SPM phases of metallic and oxide NPs is quite clear; as temperature increases, the line width decreases and the resonance field increases, as illustrated by means of In order to fulfill the resonance conditions (Larmor equation), a smaller external field is required. In the paramagnetic state, thermal energy has overwhelmed the exchange coupling, and the resonance field is identical to the applied field. On the other hand, it is interesting to note again that in the SPM state, the exchange interaction is fully effective, and it remains active up to the Curie transition. In order to fulfill the resonance conditions (Larmor equation), a smaller external field is required. In the paramagnetic state, thermal energy has overwhelmed the exchange coupling, and the resonance field is identical to the applied field. On the other hand, it is interesting to

> In SPM phases, *H*anis should be small and decreasing as *T* increases and thermal energy progressively overwhelms it, but it certainly retains some influence, especially at temperatures close to *T*B. The transition from the ordered (ferrimagnetic) to the SPM phase is essentially continuous. This is more evident in the *H*>*H*res section of the resonance signal, as this section exhibits a larger broadening and becomes asymmetric. As we will see, this section of the signal is also associated with the

magnetocrystalline anisotropy in the case of ferrimagnetic phases.

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(b) behavior of *R* for the measured temperature range [23].

473 K

Zn0.7Ni0.3Fe2

*R*= *A*‐ /*A*<sup>+</sup>

the section in the negative part of the spectrum, *A-*

paramagnetic phase.

dP /d

H (a.u.)

13

Figure 12. (a) Definition of the *R* symmetry parameter, as the ratio of resonance signal amplitude *L*A (*H*>*H*res) to *L*B (*H*<*H*res);

0.5

0.6

0.7

R

0.8

0.9

Zn0.7Ni0.3Fe2

O4

1.0

As an attempt to get more insight into the changes associated with the SPM phase, we have proposed a parameter *R*, which

many ferrite NPs, *R* has been observed to tend to unity as temperature increases and the ferrite progresses to the

, to and the amplitude of the positive part, *A+*

/*A*<sup>+</sup>

100 200 300 400 500

T(K)

, between the amplitude of

, as shown in Figure 12. In

measures the symmetry of the resonance signal. This parameter is defined as the ratio *R* = *A-*

100 200 300 400 500 600

H(mT) LB

*A*+

R = LA /L *R* = *A* B - /*A*<sup>+</sup>

LA

*A*‐

dP /d

f= 9.45 GHz

Zn0.7Ni0.3Fe2

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H (a.u.)

100 200 300 400 500

H (mT)

note again that in the SPM state, the exchange interaction is fully effective, and it remains active up to the Curie transition. In order to fulfill the resonance conditions (Larmor equation), a smaller external field is required. In the paramagnetic state, thermal energy has overwhelmed the exchange coupling, and the resonance field is identical to the applied field. On the

for *H*<*H*res) to facilitate comparisons; (b) temperature behavior of the line width (peak to peak value) [23].

a

temperature range [22]. Signals were normalized to have the same value for the maximum (section of the resonance signal

10

20

30

H (mT)

40

50

60

100 200 300 400 500

Zn0.7Ni0.3Fe2

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b

T(K)

In SPM phases, *H*anis should be small and decreasing as *T* increases and thermal energy progressively overwhelms it, but it certainly retains some influence, especially at temperatures close to *T*B. The transition from the ordered (ferrimagnetic) to the SPM phase is essentially continuous. This is more evident in the *H* > *H*res section of the resonance signal, as this section exhibits a larger broadening and becomes asymmetric. As we will see, this section of the signal is also associated with the magnetocrystalline anisotropy in the case of ferrimagnetic phases. other hand, it is interesting to note again that in the SPM state, the exchange interaction is fully effective, and it remains active up to the Curie transition. In SPM phases, *H*anis should be small and decreasing as *T* increases and thermal energy progressively overwhelms it, but it certainly retains some influence, especially at temperatures close to *T*B. The transition from the ordered (ferrimagnetic) to the SPM phase is essentially continuous. This is more evident in the *H*>*H*res section of the resonance signal, as this section exhibits a larger broadening and becomes asymmetric. As we will see, this section of the signal is also associated with the magnetocrystalline anisotropy in the case of ferrimagnetic phases.

Figure 12. (a) Definition of the *R* symmetry parameter, as the ratio of resonance signal amplitude *L*A (*H*>*H*res) to *L*B (*H*<*H*res); (b) behavior of *R* for the measured temperature range [23]. **Figure 12.** (a) Definition of the *R* symmetry parameter, as the ratio of resonance signal amplitude *L*A (*H* > *H*res) to *L*B (*H* < *H*res); (b) behavior of *R* for the measured temperature range [23].

As an attempt to get more insight into the changes associated with the SPM phase, we have proposed a parameter *R*, which measures the symmetry of the resonance signal. This parameter is defined as the ratio *R* = *A-* /*A*<sup>+</sup> , between the amplitude of the section in the negative part of the spectrum, *A-* , to and the amplitude of the positive part, *A+* , as shown in Figure 12. In many ferrite NPs, *R* has been observed to tend to unity as temperature increases and the ferrite progresses to the paramagnetic phase. As an attempt to get more insight into the changes associated with the SPM phase, we have proposed a parameter *R*, which measures the symmetry of the resonance signal. This param‐ eter is defined as the ratio *R* = *A*– /*A*<sup>+</sup> , between the amplitude of the section in the negative part of the spectrum, *A*– , and the amplitude of the positive part, *A+* , as shown in Figure 12. In many ferrite NPs, *R* has been observed to tend to unity as temperature increases and the ferrite progresses to the paramagnetic phase.

13

### **5. Ferromagnetic resonance**

While the physical principals behind EPR experiments are due to the Zeeman effect, ferro‐ magnetic resonance has a different description because FMR arises from the precessional motion of the whole magnetization **M** of a ferromagnetic material in an external magnetic field **H** as shown in Figure 13 [27].

**Figure 13.** Magnetization vector *M* precessing around an external field *H*.

note again that in the SPM state, the exchange interaction is fully effective, and it remains active

In order to fulfill the resonance conditions (Larmor equation), a smaller external field is required. In the paramagnetic state, thermal energy has overwhelmed the exchange coupling, and the resonance field is identical to the applied field. On the other hand, it is interesting to note again that in the SPM state, the exchange interaction is fully effective, and it remains

for *H*<*H*res) to facilitate comparisons; (b) temperature behavior of the line width (peak to peak value) [23].

a

Figure 11. (a) ESR results on Zn0.7Ni0.3Fe2O4 ferrite NPs at a microwave frequenyfrequency of 9.45 GHz, in the 102–-473 temperature range [22]. Signals were normalized to have the same value for the maximum (section of the resonance signal

10

20

30

H (mT)

40

50

60

100 200 300 400 500

Zn0.7Ni0.3Fe2

O4

b

T(K)

In SPM phases, *H*anis should be small and decreasing as *T* increases and thermal energy progressively overwhelms it, but it certainly retains some influence, especially at temperatures close to *T*B. The transition from the ordered (ferrimagnetic) to the SPM phase is essentially continuous. This is more evident in the *H* > *H*res section of the resonance signal, as this section exhibits a larger broadening and becomes asymmetric. As we will see, this section of the signal is also associated with the magnetocrystalline anisotropy in the case of ferrimagnetic phases.

In SPM phases, *H*anis should be small and decreasing as *T* increases and thermal energy progressively overwhelms it, but it certainly retains some influence, especially at temperatures close to *T*B. The transition from the ordered (ferrimagnetic) to the SPM phase is essentially continuous. This is more evident in the *H*>*H*res section of the resonance signal, as this section exhibits a larger broadening and becomes asymmetric. As we will see, this section of the signal is also associated with the

13

While the physical principals behind EPR experiments are due to the Zeeman effect, ferro‐ magnetic resonance has a different description because FMR arises from the precessional motion of the whole magnetization **M** of a ferromagnetic material in an external magnetic field

Figure 12. (a) Definition of the *R* symmetry parameter, as the ratio of resonance signal amplitude *L*A (*H*>*H*res) to *L*B (*H*<*H*res);

**Figure 12.** (a) Definition of the *R* symmetry parameter, as the ratio of resonance signal amplitude *L*A (*H* > *H*res) to *L*B (*H*

0.5

0.6

0.7

R

0.8

0.9

Zn0.7Ni0.3Fe2

O4

1.0

As an attempt to get more insight into the changes associated with the SPM phase, we have proposed a parameter *R*, which

As an attempt to get more insight into the changes associated with the SPM phase, we have proposed a parameter *R*, which measures the symmetry of the resonance signal. This param‐

many ferrite NPs, *R* has been observed to tend to unity as temperature increases and the ferrite progresses to the

ferrite NPs, *R* has been observed to tend to unity as temperature increases and the ferrite

, to and the amplitude of the positive part, *A+*

, between the amplitude of the section in the negative part

/*A*<sup>+</sup>

, as shown in Figure 12. In many

100 200 300 400 500

T(K)

, between the amplitude of

, as shown in Figure 12. In

measures the symmetry of the resonance signal. This parameter is defined as the ratio *R* = *A-*

/*A*<sup>+</sup>

, and the amplitude of the positive part, *A+*

100 200 300 400 500 600

< *H*res); (b) behavior of *R* for the measured temperature range [23].

H(mT) LB

*A*+

R = LA /L *R* = *A* B - /*A*<sup>+</sup>

LA

*A*‐

up to the Curie transition.

226 Advanced Electromagnetic Waves

magnetocrystalline anisotropy in the case of ferrimagnetic phases.

/*A*<sup>+</sup>

O4

100 200 300 400 500

H (mT)

(b) behavior of *R* for the measured temperature range [23].

eter is defined as the ratio *R* = *A*–

progresses to the paramagnetic phase.

**5. Ferromagnetic resonance**

**H** as shown in Figure 13 [27].

473 K

Zn0.7Ni0.3Fe2

*R*= *A*‐

the section in the negative part of the spectrum, *A-*

of the spectrum, *A*–

paramagnetic phase.

active up to the Curie transition.

dP /d

H (a.u.)

dP /d

f= 9.45 GHz

Zn0.7Ni0.3Fe2

O4

H (a.u.)

Actually ferromagnetic resonance at microwave frequencies is similar in principle to nuclear spin resonance. The total electron magnetic moment of the sample precesses around the direction of the external field and the absorption is due to the energy absorbed from the *rf* transverse field when its frequency is equal to the precessional frequency [27,28].

The ferromagnetic resonance was unknowingly discovered by V. K. Arkad'yev when he observed the absorption of ultrahigh frequency radiation by ferromagnetic materials. How‐ ever, it was officially observed by Griffiths in the Clarendon Laboratory in Oxford in 1946 and then confirmed by Yager and Bozorth who found a sharp peak in a supermalloy (Ni 75%–Fe 20%–Mo 5%) sample for a field strength near 5 kOe with a 24,000-Mc/s frequency [28].

Ferromagnetic resonance was theoretically discussed before it was knowingly observed, particularly in the paper written by Landau and Lifshitz in Kharkov in 1935. C. Kittel provided a more complete theoretical formulation in 1951 [27,28]. The resonant absorp‐ tion in ferromagnetic metals for a given frequency is controlled by an effective field *H*eff, which is the sum of an external applied field and the contributions of internal magnetiza‐ tion, while in EPR the effective field in which the absorption takes place is the same as the external applied field [27–29].

From the theoretical point of view, the ferromagnetic resonance is described by the FMR equations. These are differential equations that connect time derivatives of the magnetization components with the components of the magnetization, the external field, the static magnetic susceptibility, and the time relaxation [9,27]. The FMR equations are different depending on the temperature regime which one is working in. Below the Curie temperature, the FMR equations are called the Landau–Lifshitz equations. Over the Curie temperature, the FMR equations lead to the Bloch-type ferromagnetic-resonance equations [9].

These equations were obtained empirically; however, for a better understanding of the microscopic phenomena, a quantum statistical derivation was developed by O. A. Olkhov and B. N. Provotorov, taking into account a system with spin magnetism placed in a magnetic field composed by a constant field *H*0 (which causes the state of full saturation), and the alternating field *H*1 [9].

The Hamiltonian for this derivation contemplates the potential energy *μH.S*, the exchange interaction term, the dipolar interaction term, the lattice energy, and the spin–lattice interac‐ tion. For the quantum statistical derivation, the relationship between the relaxation time and the correlation time is very important. The correlation time *ω*C is much less than the relaxation time *ω*R. This means that the equilibrium in the spin system is established by the strong exchange interaction, while the relaxation is conditioned by the weak dipolar interaction, so the equilibrium of the system is established by the strong exchange interaction much more quickly than the relaxation process. This assumption is confirmed by experimental data on the inelastic magnetic scattering of neutrons [9].

The principal result of the theory is that the resonance condition is given by [9,28]

$$\rho o\_0 = g \frac{e}{2mc} \{ [H\_z + (N\_y - N\_z)M\_z] \times [H\_z + (N\_x - N\_z)M\_z] \}^{1/2} \tag{13}$$

Instead of the Larmor condition, where *ω*0 is the frequency at resonance, *γ* = *ge*/2*mc*, is the magnetomechanical ratio for an electron spin, *Hz* is the static magnetic field, *Mz* is the compo‐ nent of the magnetization along the *z* axis, and *Nx*,*y*,*<sup>z</sup>* are the components of the demagnetization.

It is important to take into account that the resonance condition is closely related to the demagnetization field, and this depends on the shape of the sample. For this reason, there are some special cases for the resonance frequency, for instance, the plane, the sphere, and a long circular cylinder [27,28].

Another important consideration is that the energy of ferromagnetic crystals depends in part on the anisotropy energy. The anisotropy energy has an effect in the resonance condition. In the case of a single crystal, the value of the magnetic field required for resonance at fixed frequency depends on the direction of the crystal axes relative to the shape axes of the sample.

In a polycrystalline sample, the absorption line will, in general, look broader than in a single crystal sample because the distribution in direction of the crystalline axes causes a distribution in the field strengths for resonance [27,28].

It is convenient to consider the effect of the anisotropy energy as an equivalent magnetic field *H*s ; this field is defined such that the torque exerted on the sample by this field is equal to the torque exerted by the anisotropy energy,

$$\frac{\partial f}{\partial \theta} = \mathbf{M}\_{\text{g}} \times \mathbf{H}^{\text{s}} \tag{14}$$

However, this equivalent magnetic field is not completely determined by equation 2 because the magnitude and the direction are arbitrary. One can express the components of **H**<sup>S</sup> in terms of an effective demagnetization factor, which will be added to the usual demagnetization factor in the equation for the resonance frequency. The resonance condition depends on the shape of the sample as well as on the orientation of the crystal.

**Figure 14.** Comparison of the theoretical resonance condition for [100] and [110] directions in Fe-Si single crystal with a (001) plane surface. Adapted from Kittel [28].

A classic example of this is the difference of the magnetic field required for resonance with the (100) crystal face in a crystal with cubic anisotropy when the static field *Hz* is in the [110] direction and when *Hz* is in the [100], see Figure 14. In the first case, *Hz* is greater than in the second. The difference is of the order of 4K1/MS.[6].

### **5.1. Ferrimagnetic nanoparticles**

tion. For the quantum statistical derivation, the relationship between the relaxation time and the correlation time is very important. The correlation time *ω*C is much less than the relaxation time *ω*R. This means that the equilibrium in the spin system is established by the strong exchange interaction, while the relaxation is conditioned by the weak dipolar interaction, so the equilibrium of the system is established by the strong exchange interaction much more quickly than the relaxation process. This assumption is confirmed by experimental data on the

1/2

**M H** (14)

in terms

The principal result of the theory is that the resonance condition is given by [9,28]

<sup>0</sup> {[ ( ) ] [ ( ) ]} <sup>2</sup> *z y zz z x zz <sup>e</sup> g H N NM H N NM*

Instead of the Larmor condition, where *ω*0 is the frequency at resonance, *γ* = *ge*/2*mc*, is the magnetomechanical ratio for an electron spin, *Hz* is the static magnetic field, *Mz* is the compo‐ nent of the magnetization along the *z* axis, and *Nx*,*y*,*<sup>z</sup>* are the components of the demagnetization.

It is important to take into account that the resonance condition is closely related to the demagnetization field, and this depends on the shape of the sample. For this reason, there are some special cases for the resonance frequency, for instance, the plane, the sphere, and a long

Another important consideration is that the energy of ferromagnetic crystals depends in part on the anisotropy energy. The anisotropy energy has an effect in the resonance condition. In the case of a single crystal, the value of the magnetic field required for resonance at fixed frequency depends on the direction of the crystal axes relative to the shape axes of the sample.

In a polycrystalline sample, the absorption line will, in general, look broader than in a single crystal sample because the distribution in direction of the crystalline axes causes a distribution

It is convenient to consider the effect of the anisotropy energy as an equivalent magnetic field

; this field is defined such that the torque exerted on the sample by this field is equal to the

*S*

*S*

However, this equivalent magnetic field is not completely determined by equation 2 because

of an effective demagnetization factor, which will be added to the usual demagnetization factor in the equation for the resonance frequency. The resonance condition depends on the shape

¶ = ´ ¶

the magnitude and the direction are arbitrary. One can express the components of **H**<sup>S</sup>

*f* q

= + - ´+ - (13)

inelastic magnetic scattering of neutrons [9].

w

228 Advanced Electromagnetic Waves

circular cylinder [27,28].

*H*s

in the field strengths for resonance [27,28].

torque exerted by the anisotropy energy,

of the sample as well as on the orientation of the crystal.

*mc*

Nanoparticles of magnetic systems are of particular interest since the reduction to nanosized dimensions of the magnetic lattice gives rise to many interesting and different properties with respect to bulk materials. For instance, the surface spins, which constitute an important fraction of the total spins, undergo decrease in the coordination number and, therefore, a deficiency in exchange interactions. This situation can lead to severe changes in magnetization and aniso‐ tropy behavior. The relaxation processes of magnetization are also strongly temperature and size dependent.

In order to make a comparison between the three phases in ferrite NPs, Figure 15, we have selected the spectra obtained at 103 K (ordered, ferrimagnetic phase), 323 K (SPM phase), and 448 K (paramagnetic phases) [23]. The main difference exhibited by the ordered phase (in

addition to the decrease in the resonance field) is the broadening in the section at *H* > *H*res, i.e., *A*– . The origin of this feature should be found in the effect of the internal field, which is the main difference with the other phases, and in particular, in the magnetocrystalline anisotropy field. anisotropy behavior. The relaxation processes of magnetization are also strongly temperature and size dependent. In order to make a comparison between the three phases in ferrite NPs, Figure 15, we have selected the spectra obtained at 103 K (ordered, ferrimagnetic phase), 323 K (SPM phase), and 448 K (paramagnetic phases) [23]. The main difference exhibited by the ordered phase (in addition to the decrease in the resonance field) is the broadening in the section at *H* >

Nanoparticles of magnetic systems are of particular interest since the reduction to nanosized dimensions of the magnetic

therefore, a deficiency in exchange interactions. This situation can lead to severe changes in magnetization and

We present here two examples to show the complexity of changes driven by the reduction to nanosize dimensions, as illustrated by FMR. The first example involves cobalt-doped zinc ferrite nanoparticles (Co0.73*y*Zn0.73(1-*<sup>y</sup>*) Fe2.18□0.09O4). After synthesis, they were solubilized in aqueous solution containing 10% of polyvinyl alcohol [29]. Upon evaporation and during the polymerization process, the samples were subjected to a magnetic field in order to obtain an alignment of anisotropy axes. *H*res, i.e., *A*–. The origin of this feature should be found in the effect of the internal field, which is the main difference with the other phases, and in particular, in the magnetocrystalline anisotropy field. We present here two examples to show the complexity of changes driven by the reduction to nanosize dimensions, as illustrated by FMR. The first example involves cobalt‐doped zinc ferrite nanoparticles (Co0.73*y*Zn0.73(1‐*<sup>y</sup>*)Fe2.18□0.09O4). After synthesis, they were solubilized in aqueous solution containing 10% of polyvinyl alcohol [29]. Upon evaporation and during the polymerization process, the samples were subjected to a magnetic field in order to obtain an alignment of

The magnetization curves for these samples showed that the coercitivity increases as the percentage of Co increased, as well as the ratio of the remanence magnetization to saturated magnetization *M*R/*M*S. This latter result was interpreted as a change from uniaxial anisotropy for the Zn ferrite to a cubic anisotropy for the ferrites containing Co. This result was confirmed by the FMR spectrums as shown in Figure 15, associated with a broadening of the line width. This is attributed to the strong cubic magnetocrystalline anisotropy of cobalt ions in octahedral sites. This is somewhat contradictory as in bulk materials, it is well known that Co2+ on octahedral sites leads to change in the anisotropy sign (negative in most ferrites) from negative to positive (hence, uniaxial) [30]. anisotropy axes. The magnetization curves for these samples showed that the coercitivity increases as the percentage of Co increased, as well as the ratio of the remanence magnetization to saturated magnetization *M*R/*M*S. This latter result was interpreted as a change from uniaxial anisotropy for the Zn ferrite to a cubic anisotropy for the ferrites containing Co. This result was confirmed by the FMR spectrums as shown in Figure 15, associated with a broadening of the line width. This is attributed to the strong cubic magnetocrystalline anisotropy of cobalt ions in octahedral sites. This is somewhat contradictory as in bulk materials, it is well known that Co2+ on octahedral sites leads to change in the anisotropy sign (negative in most ferrites) from negative to positive (hence, uniaxial) [30].

and in the middle, the SPM phase which shows a progressive behavior from ferrimagnetic to paramagnetic as *T* increases [23]. The FMR spectra of 3.7-nm nanoparticles show a drastic change when 10% of cobalt ions were added. A line width broadening and a shift in resonance toward lower fields are observed as the temperature decreases, and this behavior is **Figure 15.** Typical signals associated with the three magnetic phases; paramagnetic at 448 K, ferrimagnetic at 103 K, and in the middle, the SPM phase which shows a progressive behavior from ferrimagnetic to paramagnetic as *T* in‐ creases [23].

Figure 15. Typical signals associated with the three magnetic phases; paramagnetic at 448 K, ferrimagnetic at 103 K,

more evident for the nanoparticles containing Co (Figure 16). The FMR spectra of 3.7-nm nanoparticles show a drastic change when 10% of cobalt ions were added. A line width broadening and a shift in resonance toward lower fields are observed as the temperature decreases, and this behavior is more evident for the nanoparticles containing Co (Figure 16).

addition to the decrease in the resonance field) is the broadening in the section at *H* > *H*res, i.e.,

We present here two examples to show the complexity of changes driven by the reduction to nanosize dimensions, as illustrated by FMR. The first example involves cobalt-doped zinc

aqueous solution containing 10% of polyvinyl alcohol [29]. Upon evaporation and during the polymerization process, the samples were subjected to a magnetic field in order to obtain an

We present here two examples to show the complexity of changes driven by the reduction to nanosize dimensions, as illustrated by FMR. The first example involves cobalt‐doped zinc ferrite nanoparticles (Co0.73*y*Zn0.73(1‐*<sup>y</sup>*)Fe2.18□0.09O4). After synthesis, they were solubilized in aqueous solution containing 10% of polyvinyl alcohol [29]. Upon evaporation and during the polymerization process, the samples were subjected to a magnetic field in order to obtain an alignment of

The magnetization curves for these samples showed that the coercitivity increases as the percentage of Co increased, as well as the ratio of the remanence magnetization to saturated magnetization *M*R/*M*S. This latter result was interpreted as a change from uniaxial anisotropy for the Zn ferrite to a cubic anisotropy for the ferrites containing Co. This result was confirmed by the FMR spectrums as shown in Figure 15, associated with a broadening of the line width. This is attributed to the strong cubic magnetocrystalline anisotropy of cobalt ions in octahedral sites. This is somewhat contradictory as in bulk materials, it is well known that Co2+ on octahedral sites leads to change in the anisotropy sign (negative in most ferrites) from negative

The magnetization curves for these samples showed that the coercitivity increases as the percentage of Co increased, as well as the ratio of the remanence magnetization to saturated magnetization *M*R/*M*S. This latter result was interpreted as a change from uniaxial anisotropy for the Zn ferrite to a cubic anisotropy for the ferrites containing Co. This result was confirmed by the FMR spectrums as shown in Figure 15, associated with a broadening of the line width. This is attributed to the strong cubic magnetocrystalline anisotropy of cobalt ions in octahedral sites. This is somewhat contradictory as in bulk materials, it is well known that Co2+ on octahedral sites leads to change in the anisotropy sign

323 K

448 K

100 200 300 400 500

H(mT)

Figure 15. Typical signals associated with the three magnetic phases; paramagnetic at 448 K, ferrimagnetic at 103 K, and in the middle, the SPM phase which shows a progressive behavior from ferrimagnetic to paramagnetic as *T*

**Figure 15.** Typical signals associated with the three magnetic phases; paramagnetic at 448 K, ferrimagnetic at 103 K, and in the middle, the SPM phase which shows a progressive behavior from ferrimagnetic to paramagnetic as *T* in‐

The FMR spectra of 3.7-nm nanoparticles show a drastic change when 10% of cobalt ions were added. A line width broadening and a shift in resonance toward lower fields are observed as

The FMR spectra of 3.7-nm nanoparticles show a drastic change when 10% of cobalt ions were added. A line width broadening and a shift in resonance toward lower fields are observed as the temperature decreases, and this behavior is

Fe2.18□0.09O4). After synthesis, they were solubilized in

. The origin of this feature should be found in the effect of the internal field, which is the main difference with the other phases, and in particular, in the magnetocrystalline anisotropy

In order to make a comparison between the three phases in ferrite NPs, Figure 15, we have selected the spectra obtained at 103 K (ordered, ferrimagnetic phase), 323 K (SPM phase), and 448 K (paramagnetic phases) [23]. The main difference exhibited by the ordered phase (in addition to the decrease in the resonance field) is the broadening in the section at *H* > *H*res, i.e., *A*–. The origin of this feature should be found in the effect of the internal field, which is the main difference with

Nanoparticles of magnetic systems are of particular interest since the reduction to nanosized dimensions of the magnetic lattice gives rise to many interesting and different properties with respect to bulk materials. For instance, the surface spins, which constitute an important fraction of the total spins, undergo decrease in the coordination number and, therefore, a deficiency in exchange interactions. This situation can lead to severe changes in magnetization and anisotropy behavior. The relaxation processes of magnetization are also strongly temperature and size dependent.

*A*–

anisotropy axes.

increases [23].

creases [23].

field.

ferrite nanoparticles (Co0.73*y*Zn0.73(1-*<sup>y</sup>*)

the other phases, and in particular, in the magnetocrystalline anisotropy field.

alignment of anisotropy axes.

230 Advanced Electromagnetic Waves

to positive (hence, uniaxial) [30].

(negative in most ferrites) from negative to positive (hence, uniaxial) [30].

Zn0.7Ni0.3Fe2

103 K

103 61.3 0.60 323 22.2 0.59 448 13.9 0.99

O4

dP /d

more evident for the nanoparticles containing Co (Figure 16).

T H R

H (a.u.)

0.0

**Figure 16.** FMR spectra of 3.7-nm diameter ferrite nanoparticles dispersed in PVA, at various temperatures. (a) Zinc ferrite nanoparticles with no cobalt (*y* = 0). (b) Zinc ferrite with y = 10 of cobalt. Adapted from Gazeau et al. [31].

The next example is a ferrite nanoparticle system of maghemite (γ-Fe3O2) [31]. The oxide γ-Fe3O2 is an inverse spinel structure with all iron in the trivalent state and ion vacancies in the octahedral sublattice. Anisotropy properties studies have been done for γ-Fe3O2 nanoparticles of diameter from 4.8 to 10 nm dispersed in glycerol forming a magnetic fluid. This study is done by measuring the samples in a temperature range from 3.5 K to 300 K with (i) zero field cooled (ZFC) and thus randomly oriented anisotropy axes and (ii) field cooled (FC) with *H*fr = 10 kOe. The FMR spectra showed a decrease in the resonance field, and a broadening of the line width, Figure 17. The parameters are shown in Figure 5 [31].

The resonance fields and the line widths of the γ-Fe3O2 samples are represented as a function of temperature, see Figure 18. The ZF experiments are done for *θ* = 0° and for *θ* = 90°, where *θ* is the angle between the directions of the freezing field and the magnetizing field.

In the field cooled samples, the orientation distribution of the anisotropy axes results from the competition between the magnetic energy, the anisotropy energy and the thermal energy. The distribution in orientation of the anisotropy axes affects the FMR spectrum in angular variation and line width. On the other hand, there is also an effect of the particle size in the anisotropy. This is observed in Figure 19, where *H*res(90°)–*H*res(0°) and Δ*H* are plotted as a function of temperature for different particle sizes. The observed reduction of *H*res(90°)–*H*res(0°) is related

**Figure 17.** FMR spectrum of maghemite (γ-Fe3O2) nanoparticles of 7 nm of average diameter at room temperature and mi‐ crowave frequency *f* = 9.26 GHz. The inset shows the spinel structure of maghemite. Adapted from Gazeau et al. [31].

**Figure 18.** (a) Temperature dependence of the resonance field for 10-nm diameter γ-Fe3O2 nanoparticles cooled under a 10-kOe magnetic field (FC) and cooled without field (ZFC). The arrow indicates the melting temperature of the ferro‐ fluid matrix. (b) Temperature dependence of the peak-to-peak line width for the same samples. Adapted from Gazeau et al. [31].

to the particle size decrease because of the orientation distribution of the anisotropy axes. The FMR study indicates that nanosized particles possess uniaxial anisotropy, even though bulk maghemite has a cubic anisotropy.

By using a theoretical model [32] based on the Landau–Lifshitz–Gilbert dynamics, it is possible to show that for a single domain assembly of magnetic NPs with randomly distributed anisotropy axes, a large broadening of the absorption line is obtained, see Figure 19. In addition to a shift of the resonance frequency, the distribution of anisotropy axes gives rise to a significant change in the shape and symmetry of the line. Figure 20 was calculated at zero temperature; as temperature increases, the shift in resonance field decreases, and due to the typical decrease in anisotropy energy with *T*, the line broadening is also reduced.

Characterization of Magnetic Phases in Nanostructured Ferrites by Electron Spin Resonance http://dx.doi.org/10.5772/61508 233

**Figure 19.** (a) Difference of the resonance fields for anisotropy axis orientated at 90° and 0° form the magnetic field. (b) Peak-to-peak line width as a function of microwave field for two different size of γ-Fe3O2 nanoparticles. (c) Anisotropy field deduced for angular variations at 3.5 K as a function a particle size. Adapted from Sukhov et al. [32].

to the particle size decrease because of the orientation distribution of the anisotropy axes. The FMR study indicates that nanosized particles possess uniaxial anisotropy, even though bulk

**Figure 18.** (a) Temperature dependence of the resonance field for 10-nm diameter γ-Fe3O2 nanoparticles cooled under a 10-kOe magnetic field (FC) and cooled without field (ZFC). The arrow indicates the melting temperature of the ferro‐ fluid matrix. (b) Temperature dependence of the peak-to-peak line width for the same samples. Adapted from Gazeau

**Figure 17.** FMR spectrum of maghemite (γ-Fe3O2) nanoparticles of 7 nm of average diameter at room temperature and mi‐ crowave frequency *f* = 9.26 GHz. The inset shows the spinel structure of maghemite. Adapted from Gazeau et al. [31].

By using a theoretical model [32] based on the Landau–Lifshitz–Gilbert dynamics, it is possible to show that for a single domain assembly of magnetic NPs with randomly distributed anisotropy axes, a large broadening of the absorption line is obtained, see Figure 19. In addition to a shift of the resonance frequency, the distribution of anisotropy axes gives rise to a significant change in the shape and symmetry of the line. Figure 20 was calculated at zero temperature; as temperature increases, the shift in resonance field decreases, and due to the

typical decrease in anisotropy energy with *T*, the line broadening is also reduced.

maghemite has a cubic anisotropy.

et al. [31].

232 Advanced Electromagnetic Waves

**Figure 20.** Absorbed FMR power (arbitrary units) vs reduced magnetic field at zero temperature and for different val‐ ues of the damping parameter in the Landau–Lifshitz–Gilbert dynamics [32].

**Figure 21.** Interpretation of the line broadening in ferrites with cubic anisotropy. On a [110] crystal plane, containing two easy axis of the <111> family, which have an influence on the magnetization dynamics, leading to an increase in the precession amplitude.

These results were obtained for the case of uniaxial materials; in the case of most ferrites with cubic anisotropy axes, it would be expected a larger broadening. In a very cartoon-like model, it is possible to imagine that the magnetization is first oriented along the applied field, which has a value closer to the anisotropy field. Once close to the saturation and once satisfied the resonance conditions, the magnetization is driven by the microwave radiation to precess uniformly, that is in phase. However, magnetocrystaline anisotropy is strong at these temper‐ atures, and it is possible to consider that the magnetization vector can still feel its influence and precesses with a tendency toward the cubic axes. This dynamics leads to a broadening of the line, as far as the main result will be the combination of precession along several anisotropy axes, and has to be averaged over a large NPs population. Figure 21 illustrates this interpre‐ tation.
