**2.3.3 Inversred spinel ferrites**

Inversed spinel structure, where all Me2+ are in B-positions and Fe3+ ions are equally distributed betweenAand B-sites: structural formula of these ferrites are Fe3+[Me2+Fe3+] O4 <sup>2</sup>−. Magnetite Fe3O4, ferrites NiFe2O4 and CoFe2O4 have inversed spinel structure [12].In the inversed ferrites one half of Fe3+ is placed in A-sites and another half in B-sites. Their magnetic moments are mutually compensated and the resulting moment of the ferrite is due to the magnetic moments of bivalent cations Me2+ in the B-positions. This type spinel ferrite are schematically illustrated in Figure5.

#### **2.4 Domains**

Domains, which are groups of spins all pointing in the same direction and acting cooperatively are separated by domain walls, which have a characteristic width and energy associated with their formation and existence. The motion of domain walls is a primary means of reversing magnetization. Experimental investigation of the dependence of coercivity on particle size showed a behavior similar to that schematically illustrated in Figure 6 [13].

It was found that the coercivity Hc increases with decreasing grain size D down to values of about 40 nm, independent of the kind of material. The increase of Hc is proportional to 1/D. The reason for this is that in small particles the formation of a closed magnetic flux becomes energetically less favorable so that the magnetic domain size with a uniform magnetization becomes more and more identical with the grain size. This grain size is defined as the first critical size (Dc, which is characteristic of each material) where the multidomain materials

Crystalization in Spinel Ferrite Nanoparticles 355

Ek=K1sin2θ+K2sin4θ… (Hexagonal structre)

<sup>2</sup> α<sup>2</sup>

where K is the anisotropy constant, *θ* is the angle between the easy axis and the direction of magnetization, and *α* ; are the direction cosines, which are the ratios of the individual components of the magnetization projected on each axis divided by the magnitude of the magnetization. A crystal is higher in anisotropy energy when the magnetization points in the hard direction rather than along the easy direction. The formation of domains permits the magnetization to point along the easy axis, resulting in a decrease in the net anisotropy

c. *Magnetostrictive energy:* In a magnetic field, the material may change its dimensions on the order of several parts per million. This change in dimension results in what is called magnetostrictive energy, which is lowered by a reduction in the size of the domains,

The magnetization curve describes the change in magnetization or magnetic flux of the material with the applied field. When a field is applied to a material with randomly oriented magnetic moments, it will be progressively magnetized due to movement of domain boundaries. Initially, when no field is applied, the magnetic dipoles are randomly oriented in domains, thus the net magnetization is zero. When a field is applied, the domains begin to rotate, increasing their size in the case of the domains with direction favorable with respect to the field, and decreasing for the domains with unfavorable direction. As the field increases, the domains continue to grow until the material becomes a single domain, which is oriented in the field direction. At this point, the material has reached saturation (Figure7) [3].As the magnetic field is increased or decreased continuously, the magnetization of the material increases or decreases but in a discontinuous fashion. This phenomenon is called the Barkhausen effect and is attributed to discontinuous domain boundary motion and the discontinuous rotation of the magnetization direction within a domain [14]. The typical

a. *Reversible region:* The material can be reversibly magnetized or demagnetized. Charges

If the field is reduced from saturation, with eventual reversal of field direction, the magnetization curve does not retrace its original path, resulting in what is called a hysteresis loop. This effect is due to a decrease of the magnetization at a lower rate. The area inside the hysteresis loop is indicative of the magnetic energy losses during the magnetization process. When the field reaches zero, the material may remain magnetized (i.e., some domains are

b. *Irreversible region:* Domain wall motion is irreversible and the slope increases greatly. c. *Saturation region:* Irreversible domain rotation. It is characterized by a required large

in magnetization occur due to rotation of the domains with the field.

amount of energy to rotate the domains in the direction of the field [5].

<sup>2</sup> α32+…) (Cubic and Garnet structure) (1)

2) + K2 (α<sup>1</sup>

magnetization rotates.

energy.

Ek=K1 (α12 α22+ α22 α32+ α32 α<sup>1</sup>

requiring the formation of more domains.

**2.5 Magnetization curve and hysteresis loops** 

magnetization curve can be divided into three regions:

a hard direction. For materials with cubic crystalline structure (such as ferrites), the energy is expressed in terms of anisotropy constants and the direction to which the

change to a monodomain material. This leads to a strong increase of the coercivity (or high remanence) because a change of magnetization in this case cannot happen only by shifting the domain walls which normally requires only a weak magnetic field. As the size of magnetic element scales below20 nm, the transformation from ferromagnetic to superparamagnetic behavior occurs. In the superparamagnetic state of the material, the room temperature thermal energy overcomes the magnetostatic energy well of the domain or the particle, resulting in zero hysteresis. In other words, although the particle itself is a single-domain ferromagnet, the ability of an individual magnetic "dot" to store magnetization orientation information is lost when its dimension is below a threshold. Consequently, the magnetic moments within a particle rotate rapidly in unison, exhibiting the superparamagnetic relation phenomenon.

Fig. 6. Qualitative illustration of the behavior of the coercivity in ultrafine systems as the particle size changes, where H is the magnetic field amplitude (Oe) and *D* is the particle diameter (nm).


change to a monodomain material. This leads to a strong increase of the coercivity (or high remanence) because a change of magnetization in this case cannot happen only by shifting the domain walls which normally requires only a weak magnetic field. As the size of magnetic element scales below20 nm, the transformation from ferromagnetic to superparamagnetic behavior occurs. In the superparamagnetic state of the material, the room temperature thermal energy overcomes the magnetostatic energy well of the domain or the particle, resulting in zero hysteresis. In other words, although the particle itself is a single-domain ferromagnet, the ability of an individual magnetic "dot" to store magnetization orientation information is lost when its dimension is below a threshold. Consequently, the magnetic moments within a particle rotate rapidly in unison, exhibiting

Fig. 6. Qualitative illustration of the behavior of the coercivity in ultrafine systems as the particle size changes, where H is the magnetic field amplitude (Oe) and *D* is the particle

a. *Magnetostatic or demagnetization energy:* The magnetized material behaves like a magnet, with a surrounding magnetic field. This field acts to magnetize the material in the direction opposite from its own magnetization, causing a magnetostatic energy which depends on the shape of the material. This magnetostatic energy can be reduced by reducing the net external field through the formation of domains inside the material. b. *Magnetocrystalline anisotropy energy:* In some materials the domain magnetization tends to align in a particular crystal direction (the so-called easy axis). The material is easiest to magnetize to saturation or demagnetize from saturation if the field is applied along an easy axis. The energy difference between aligning the domain in the easy and another direction (hard direction) is called magnetocrystalline anisotropy energy. Anisotropy energy is the energy needed to rotate the moment from the easy direction to

the superparamagnetic relation phenomenon.

diameter (nm).

a hard direction. For materials with cubic crystalline structure (such as ferrites), the energy is expressed in terms of anisotropy constants and the direction to which the magnetization rotates.

Ek=K1sin2θ+K2sin4θ… (Hexagonal structre)

 Ek=K1 (α12 α22+ α22 α32+ α32 α12) + K2 (α12 α22 α<sup>3</sup> 2+…) (Cubic and Garnet structure) (1)

where K is the anisotropy constant, *θ* is the angle between the easy axis and the direction of magnetization, and *α* ; are the direction cosines, which are the ratios of the individual components of the magnetization projected on each axis divided by the magnitude of the magnetization. A crystal is higher in anisotropy energy when the magnetization points in the hard direction rather than along the easy direction. The formation of domains permits the magnetization to point along the easy axis, resulting in a decrease in the net anisotropy energy.

c. *Magnetostrictive energy:* In a magnetic field, the material may change its dimensions on the order of several parts per million. This change in dimension results in what is called magnetostrictive energy, which is lowered by a reduction in the size of the domains, requiring the formation of more domains.

### **2.5 Magnetization curve and hysteresis loops**

The magnetization curve describes the change in magnetization or magnetic flux of the material with the applied field. When a field is applied to a material with randomly oriented magnetic moments, it will be progressively magnetized due to movement of domain boundaries. Initially, when no field is applied, the magnetic dipoles are randomly oriented in domains, thus the net magnetization is zero. When a field is applied, the domains begin to rotate, increasing their size in the case of the domains with direction favorable with respect to the field, and decreasing for the domains with unfavorable direction. As the field increases, the domains continue to grow until the material becomes a single domain, which is oriented in the field direction. At this point, the material has reached saturation (Figure7) [3].As the magnetic field is increased or decreased continuously, the magnetization of the material increases or decreases but in a discontinuous fashion. This phenomenon is called the Barkhausen effect and is attributed to discontinuous domain boundary motion and the discontinuous rotation of the magnetization direction within a domain [14]. The typical magnetization curve can be divided into three regions:


If the field is reduced from saturation, with eventual reversal of field direction, the magnetization curve does not retrace its original path, resulting in what is called a hysteresis loop. This effect is due to a decrease of the magnetization at a lower rate. The area inside the hysteresis loop is indicative of the magnetic energy losses during the magnetization process. When the field reaches zero, the material may remain magnetized (i.e., some domains are

Crystalization in Spinel Ferrite Nanoparticles 357

Ferrites materials can be classified based on differences between their internal and external flux and the variation of the magnetization M or magnetic induction B when a magnetic field is applied (Figure 9) [3,4]. There are two quantities that relate M and B to H: the

(2)

(3)

Fig. 9. Representation of the behavior of the flux density with respect to the magnetic field

In SI the permeability μ has units of Henry/m. The susceptibility is a measure of the increase in magnetic moment caused by an applied field, whereas permeability represents

Diamagnetism is an inherent result of the orbital motion of the electrons in a magnetic field. It is present when the atom has zero net magnetic moment. In this case the orbital motion generates a field opposite to the applied field (magnetization is directed oppositely to the

the relative increase in flux caused by the presence of the magnetic material [15].

**2.6 Magnetic behaviors** 

susceptibility χ and the permeability μ .

for different classes of magnetic materials.

**2.6.1 Diamagnetism** 

oriented in the former direction). This residual magnetization is commonly called remanence Mr. To reduces this remanent magnetization to zero; a field with opposite direction must be applied. The magnitute of field required to lower the sample magnetization to zero is called the coercivity Hc (Figure 8). A material can present different hysteresis loops depending on the degree of magnetization. If the maximum magnetization is less than the saturation magnetization, the loop is called a minor loop [3,4].

Fig. 7. Magnetization curve with domain configurations at different stages of magnetization.

Fig. 8. Hysteresis cycle of a multidomain magnetic material, where*H*is the magnetic field amplitude (Oe) and M is the magnetization of the material (emu/g).

#### **2.6 Magnetic behaviors**

356 Advances in Crystallization Processes

oriented in the former direction). This residual magnetization is commonly called remanence Mr. To reduces this remanent magnetization to zero; a field with opposite direction must be applied. The magnitute of field required to lower the sample magnetization to zero is called the coercivity Hc (Figure 8). A material can present different hysteresis loops depending on the degree of magnetization. If the maximum magnetization

Fig. 7. Magnetization curve with domain configurations at different stages of magnetization.

Fig. 8. Hysteresis cycle of a multidomain magnetic material, where*H*is the magnetic field

amplitude (Oe) and M is the magnetization of the material (emu/g).

is less than the saturation magnetization, the loop is called a minor loop [3,4].

Ferrites materials can be classified based on differences between their internal and external flux and the variation of the magnetization M or magnetic induction B when a magnetic field is applied (Figure 9) [3,4]. There are two quantities that relate M and B to H: the susceptibility χ and the permeability μ .

$$\chi = \frac{\mathbf{M}}{H},\tag{2}$$

$$
\mu = \frac{B}{H}.\tag{3}
$$

Fig. 9. Representation of the behavior of the flux density with respect to the magnetic field for different classes of magnetic materials.

In SI the permeability μ has units of Henry/m. The susceptibility is a measure of the increase in magnetic moment caused by an applied field, whereas permeability represents the relative increase in flux caused by the presence of the magnetic material [15].

#### **2.6.1 Diamagnetism**

Diamagnetism is an inherent result of the orbital motion of the electrons in a magnetic field. It is present when the atom has zero net magnetic moment. In this case the orbital motion generates a field opposite to the applied field (magnetization is directed oppositely to the

Crystalization in Spinel Ferrite Nanoparticles 359

Fig. 12. Ordering of the atomic dipoles in a) ferromagnetic and b) ferrimagnetic material.

In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism. Generally, antiferromagnetic order may exist at sufficiently low temperatures, vanishing at and above a certain temperature, the Néel temperature (Neel temperature is the temperature at which an antiferromagnetic material becomes paramagnetic; hence losing its magnetic properties) [16].Above the Néel temperature, the material is typically paramagnetic. Figure13 shows ordering of the atomic

Fig. 13. Ordering of the atomic dipoles in an antiferromagnetic material.

Superparamagnetism is a phenomena by which magnetic materials may exhibit a behavior similar to paramagnetism at temperatures below the Neel or the Curie temperature (The Curie temperature is the temperature at which a ferromagnetic or a ferromagnetic material becomes paramagnetic; hence losing its magnetic properties).Normally, coupling forces in magnetic materials cause the magnetic moments of neighboring atoms to align, resulting in very large internal magnetic fields. At temperatures above the Curie temperature (or the Neel temperature for antiferromagnetic materials), the thermal energy is sufficient to overcome the coupling forces, causing the atomic magnetic moments to fluctuate randomly.

**2.6.4 Antiferromagnetism** 

**2.6.5 Superparamagnetic** 

dipoles in an antiferromagnetic material

field, as illustrated in Figure 10), described by a negative susceptibility. These materials tend to move toward regions of weaker field [5, 15].

Fig. 10. Atomic dipole configuration for a diamagnetic material.
