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process of abstaining from nanoparticles, as a result of physico‐chemical methods of preparation, or when the nanoparticles are surfacted or embedded in different solid matrices, elastic stresses can occur which induces an additional magnetic anisotropy (stress anisotropy) compared to those above. And this anisotropy, in the case of nanoparticles, can become large or high compared to magnetocrystalline anisotropy, and it must be taken into account when it appears. Example Coey (Coey and Khalafalla 1972) obtains a value of 1.2 × 105 J m−3 for nanoparticles of 6.5 nm in diameter and Vassiliou et al. (1993) obtains the value of the anisotropy constant of 4.4 × 105 J m−3, values that are approximately twice higher in magnitude than the magnetocrystalline anisotropy constant of the α‐Fe2O3 massive ferrite (K1 = 4.6 × 103 J m−3).

      To conclude, in the case of magnetic nanoparticles, a magnetic anisotropy determined by magnetocrystalline anisotropy, shape anisotropy, surface anisotropy, and induced anisotropy must be considered:

      (1.20)equation

      and an effective magnetic anisotropy constant

      (1.21)equation

      respectively.

      Typically, in the case of magnetic nanoparticles, this effective anisotropy constant increases when the magnetic nanoparticles become smaller, generally below 15–20 nm, depending on the nature of the material (Figure 1.12).

Schematic illustrations of (a) the general spin canting geometry. (b) Theoretical Ms and Keff versus magnetite nanoparticle diameter D at 300 K.

      Source: Wu et al. (2017). Reprinted by permission of IOP Publishing.

      The Figure 1.12b also shows the variation of the saturation magnetization (see Section 1.1.4) when the diameter of the nanoparticles decreases, this becoming smaller when the size of the nanoparticles decreases.

Schematic illustrations of (a) a core-shell structure and (b) transmission electron microscopy image of an oxidized Co particle.

      Source: Reprinted from Nogues et al. (2005), with permission from Elsevier.

      Such situations may occur frequently in the case of different more complex magnetic nanostructures which are currently developed in nanotechnology and bionanotechnology for various applications.

      1.1.6 Magnetic Behavior in External Magnetic Field

Schematic illustrations of (a) typical hysteresis loop for ferromagnetic materials. (b) A typical hysteresis loop such as that obtained for soft and hard ferromagnetic materials. (c) Magnetization curves of iron, cobalt, and nickel at room temperature.

      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,

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