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The magnetization of the bulk magnetic material in the external magnetic field (Cullity and Graham 2009), between two maximum values corresponding to the magnetic saturation, is generally with hysteresis (Figure 1.14a and b), due to the existence of a phase shift between the magnetization of a material and the applied magnetic field. Magnetization of the magnetic material, represented by the type of magnetization curve in Figure 1.14c, takes place both by processes of displacement of the walls of the magnetic domain (in low fields) and by processes of rotations of spontaneous magnetization (in high fields and near saturation), processes that are both reversible and irreversible. The basic macroscopic magnetic quantities characteristic of the hysteresis cycle, which are determined experimentally, are the saturation magnetization (M_sat), the remanent magnetization (M_r), the rectangular ratio, r = M_r/M_sat, and the coercive field (H_c) (Figure 1.14a). Depending on their applications, certain values and different shapes of the hysteresis curve are targeted. For example in the case of use of magnetic materials in high‐frequency fields, those materials are used that have the hysteresis cycle as narrow as possible with H_c and r as small as possible, close to 0 (preferably r < 0.1) (Figure 1.14b curve (2)); and in the case of applications for memory information, there magnetic materials are used that have the hysteresis cycle as rectangular as possible, with H_c and r as large as possible, theoretically close to the value 1 (preferably r > 0.9) (Figure 1.14b curve (3)), compared to the general case (curve (1)).

Figure 1.14 (a) Typical hysteresis loop for ferromagnetic materials.

Source: Reprinted from Sung and Rudowicz (2003), with permission of Elsevier.

(b) A typical hysteresis loop such as that obtained for soft and hard ferromagnetic materials.

Source: Mody et al. (2013). Reproduced with permission from Walter de Gruyter GmbH;

(c) Magnetization curves of iron, cobalt, and nickel at room temperature (H‐axis schematic). The SI values for saturation magnetization in A m⁻¹ are 10³ times the cgs values in emu cm⁻³.

Source: Cullity and Graham (2009). Reproduced with permission from John Wiley & Sons.

For magnetization of the bulk magnetic material in the external field (Figure 1.14c), there is no universal function, the magnetization curve being specific to each material time. Only in low and high fields, there are mathematical functions that describe well the magnetization obtained experimentally. Thus, in low magnetic fields (lower than the coercive field of the material [~H_c/10]), magnetization is well described by Rayleigh's law

(1.22)

where M′ is the magnetization obtained when applying the field H′, χ_i the initial magnetic susceptibility, and alpha a coefficient. The ± sign corresponds to the case when H > H′(+) and H < H′ (−), respectively.

In intense magnetic fields, close to saturation, magnetization is well described by the experimentally established Weiss–Forer law:

(1.23)

where a, b, and c are some coefficients. The last term (χ₀H) is determined by the contribution of χ₀, independent of the magnetic field H. This term becomes more important when the material is brought to a temperature close to the Curie temperature.

However, in the case of magnetic nanoparticles, the magnetization can significantly change, depending on the nanoparticle system considered. Typically, hysteresis is absent in the case of small nanoparticles, the coercive field (H_c), and the remanent magnetization (M_r) becoming zero. In Figure 1.15. shows (□) the experimental and (‐) theoretical curves calculated with the Langevin function (Jacobs and Bean 1963) for ferrimagnetic nanoparticles of Zn_0.15Ni_0.85Fe₂O₄ having mean magnetic diameter of 8.9 nm dispersed in amorphous silica matrix (SiO₂) with volume fraction of 0.15 (magnetic nanocomposites) (Caizer et al. 2003; Caizer 2008).

Figure 1.15 (a) M versus H for (Zn_0.15Ni_0.85Fe₂O₄)_0.15/(SiO₂)_0.85 sample.

Source: Reprinted from Caizer et al. (2003), with permission of Elsevier;

(b) Reduced magnetization curve of the (Zn_0.15Ni_0.85Fe₂O₄)_0.15/(SiO₂)_0.85 nanocomposite registered at room temperature and 50 Hz frequency of the magnetization field (H).

Source: Reprinted from Caizer (2008), with permission of Elsevier.

Magnetization (Figure 1.15.b) in this case follows a Langevin type function as in the case of paramagnetic atoms (Caizer 2004a):

(1.24)

where (coth α − 1/α) is the Langevin function and

(1.25)

In Eqs. (1.24) an (1.25), the observables are the following: N is the number of atoms and μ is magnetic moment of atom.

In low fields, the magnetization varies linearly with the magnetic field (Figure 1.16a):

(1.26)

whereas in high fields close to saturation, the magnetization is described by the following relationship:

(1.27)

where M_∞ is the saturation magnetization (theoretically, in the infinite magnetic field).

Figure 1.16 (a) M versus H in low fields and (b) M versus 1/H in high fields.

Source:Caizer (2003a). Reprinted by permission of IOP Publishing;

(c) magnetic structure in single‐domain nanoparticles with uniaxial anisotropy.

Source: Caizer (2017). Reprinted by permission of Springer Nature.

In Figure 1.16a and b, these cases are given for Fe₃O₄ nanoparticles covered with oleic acid and dispersed in kerosene (magnetic ferrofluid with a magnetic packing fraction of 0.024) having an average magnetic diameter of 11.8 nm. Figure 1.16b shows the dependence M = f(1/H), which is a linear function with a negative slope near the magnetic saturation (M_∞).

This magnetic behavior in the external field, totally different from the bulk magnetic material (ferro‐ or ferrimagnetic), results from the existence of the superparamagnetism phenomenon, evidenced by Nèel and introduced by Bean in the case of nanoparticles. Nèel shows that the magnetic moment of the nanoparticle (m_NP) can be reversed at 180° along the easy magnetization axis due to thermal activation (at a temperature), in the absence of the external magnetic field (Figure 1.16c). This behavior is similar to paramagnetic atoms, whose magnetic moments are oriented at a temperature in all directions.

However, in the case of nanoparticles, there is an essential difference, namely that they are not dealing with individual atoms but with nanoparticles containing a multitude of magnetically aligned atoms, characterized by the magnetic moments (resulting) m_p instead of the atomic magnetic moment (see Section 1.1.2). Formally, these systems behave the same; quantitatively the aspects being different, instead of atoms, there are nanoparticles.

In the case of larger nanoparticles, tens of nm, the magnetic behavior in the external magnetic field is similar to that of bulk magnetic material, namely creating a narrow hysteresis, determined mainly by the presence of a structure of incipient magnetic domains, or increased magnetic anisotropy in the case of nanoparticles that are still unidominal, as is discussed in the next paragraph.