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 [~Hc/10]), magnetization is well described by Rayleigh's law
(1.22)
where M′ is the magnetization obtained when applying the field H′, χi 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 (Hc), and the remanent magnetization (Mr) 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 Zn0.15Ni0.85Fe2O4 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 (Zn0.15Ni0.85Fe2O4)0.15/(SiO2)0.85 sample.
Source: Reprinted from Caizer et al. (2003), with permission of Elsevier;
(b) Reduced magnetization curve of the (Zn0.15Ni0.85Fe2O4)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):
where (coth α − 1/α) is the Langevin function and
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 (mNP) 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) mp 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.
1.1.7 Magnetic Relaxation in Nanoparticles – Superparamagnetism
When the size of nanoparticles decreases below the critical diameter (Dc), corresponding to the transition from the state of the structure with magnetic domains to the state with the single‐domain structure, there is another nanoparticle‐specific size, the threshold volume (Vth) (or threshold diameter [Dth]) at