Magnetic Nanoparticles in Human Health and Medicine. Группа авторов

of 11.1 nm, it deviates from the law corresponding to the bulk material. In the case of magnetite (Fe₃O₄) nanoparticles coated with oleic acid, Caizer (2003b) shows that the law is well verified for T² instead of T^3/2, and the constant of M_s(0) in Eq. (1.12) becomes in this case a function of temperature:

      (1.13)

      However, in the case of Mn_0.6Fe_2.4O₄ nanoparticles coated with oleic acid (Caizer 2005a), it is found that is verified in the law of Eq. (1.12), but M_s(0) is no longer a constant but is temperature‐dependent according to the formula:

(1.14)

      In this equation, n is the concentration of the nanoparticles, and c_i are constants whose value is known, resulting from the fitting of the experimental data. In Figure 1.6b is shown the dependence on M_sat − T in this case, where the deviation is highlighted: (α) is the curve for bulk and (β) is the curve for nanoparticles. These results as well as others (Caizer 2002) show that T^3/2 valid in the case of bulk magnetic material is no longer verified in the case of magnetic nanoparticles, and there are different interpretations for this.

Another important aspect in the case of nanoparticles is the fact that the saturation magnetization measured experimentally at room temperature is lower than that corresponding to the same bulk magnetic material (Berkowitz et al. 1975; Zhang et al. 1997; Kodama 1999; Caizer and Stefanescu 2002; Caizer 2003b). It has also been found experimentally that the decrease in saturation magnetization is all the more pronounced the smaller the nanoparticles, a few nm. In Figure 1.7 is shown the dependency of saturation magnetization as a function of the average nanoparticle diameter in the case of Ni_0.35Zn_0.65Fe₂O₄ nanoparticles (Caizer and Stefanescu 2002). Also, a similar dependency is shown in Figure 1.12b for the magnetic moment of the nanoparticles (Wu et al. 2017).

Figure 1.7 (a) Specific saturation magnetization as a function of the mean diameter of nanocrystallites.

      Source: Caizer and Stefanescu (2002). Reprinted by permission from IOP Publishing.

      (b) Core‐shell pattern of the spherical nanoparticle.

      Source: Caizer (2016). Springer Nature.

In Ref. (Caizer 2016) is presented in more detail the aspects that lead to the decrease of the saturation magnetization of the nanoparticles, considering the core‐shell model in which there is a layer on the surface of the nanoparticles where the magnetic moments are no longer aligned as in the ordered magnetic core. As a result, in the case of nanoparticles, a magnetic diameter (D_m) determined by the nanoparticle core in which the spins are magnetically aligned must be considered, which is generally smaller than the physical diameter (D) of the nanoparticles: D_m < D. Difference (D − D_m) becomes even more pronounced in the case of ferrimagnetic nanoparticles (Berkowitz et al. 1975, 1980; Kodama et al. 1996), surfactated (Caizer 2002) or in SiO₂ matrix, this reaching even up to 2–3 nm (Caizer et al. 2003; Caizer 2008).

      To conclude, if in the case of bulk magnetic material, the saturation magnetization is a well‐defined value, being a material parameter, characteristic of the substance type, in the case of nanoparticles it is generally smaller, and decreases with the decrease in diameter to nanometers size. This is a very important aspect that must be taken into account in biomedical applications. Thus, in order not to introduce errors in the application of magnetic nanoparticles, it is recommended, before conducting any experiment, to determine/measure the saturation magnetization of the nanoparticles, and also the variation of saturation magnetization with temperature, if there is such a dependency in the targeted application.

      1.1.5 Magnetic Anisotropy

      Experimentally, it was found that in the case of ferro‐ or ferrimagnetic crystalline materials, there is a dependence of the magnetization of the single crystal on the crystallographic directions. The dependence of the magnetization of the crystalline magnetic material on the crystallographic axes determines the magnetocrystalline anisotropy (Kneller 1962; Caizer 2004a, 2019). Thus, the magnetization curves that are obtained in the same external magnetic field depend on the direction in which the crystalline material is magnetized. This type of magnetic anisotropy is characteristic of all bulk single crystalline (ferro‐ or ferromagnetic) magnetic materials (Fe, Co, Ni, Cd, their alloys, Fe oxides [Fe₃O₄, γ‐Fe₂O₃], etc.).

For example, in the case of the Ni bulk single crystal ferromagnetic material, which crystallizes in the cube system with centered volume (vcc), the magnetization curves have the shape shown in Figure 1.8 (Baberschke 2001). Thus, the magnetization is made easiest following the crystallographic direction [1,1,1], which represents the large diagonal of the cube, and its magnetization is made the hardest following the crystallographic direction [1,0,0], which is the edge of the cube. The direction [1,1,1] in this case is called the direction or axis of easy magnetization (e.m.a.), and the direction [1,0,0] is called the axis of hard magnetization (h.m.a).

Figure 1.8 (a) The crystallographic systems for Ni‐single crystal.

      Source: Caizer (2016). Reprinted by permission from Springer Nature;

      (b) Room temperature magnetization curves for Ni along the easy ([111]) and hard ([100]) direction.

      Source: Based on Wijn (1986).

      In the case of bulk ferromagnetic monocrystalline material with cubic symmetry, the energy of magnetocrystalline anisotropy can be determined with the following formula:

(1.15)

written as a series development of powers using the model proposed by Beker, Doring, Akulov, Mason (Kneller 1962; Herpin 1968), based on the symmetry properties of the crystal. In general, it was found that it is sufficient to use only the first two terms of development, in K₁ and K₂. In Eq. (1.15), K₁ and K₂ are the magnetocystalline anisotopy constants, and α₁, α₂ and α₃ are the cosine directors of the vector of spontaneous magnetization (M_s) in relation to the main crystallographic axes of the cube. In some cases, even the first term of development is