Magnetic Nanoparticles in Human Health and Medicine. Группа авторов

of the ferromagnetic Fe single crystal, K₁ = 4.8 × 10⁴ J m⁻³ and K₂ = 5 × 10³ J m⁻³ were found, where K₁ is in this case with approximately one order of magnitude larger than K₂.

In the case of uniaxial, hexagonal symmetry, as in the case of the ferromagnetic Co monocrystalline (Figure 1.9), the magnetocrystalline anisotropy energy is expressed as a function of the angle fi between the spontaneous magnetization vector M_s and the main axis of symmetry:

(1.16)

Figure 1.9 The crystallographic systems for Co‐single crystal.

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

      Also, in this case, the first two terms are used (in K₁ and K₂) in the energy expression of uniaxial magnetocrystalline anisotropy.

      In this case, the main axis of symmetry is the easy magnetization axis (e.m.a), and the direction perpendicular to it is the hard magnetization axis (h.m.a).

      In the case of bulk magnetic material, there is another important form of magnetic anisotropy, which should not be neglected as it can become dominant in some cases. This is the anisotropy of the shape (Kneller 1962; Caizer 2004a), which shows that the magnetization of a sample depends on its shape.

As a general case, approximating the shape of the sample by the ellipsoid of revolution: a > b = c, where a, b, and c are the semiaxes of the ellipsoid (Figure 1.10), the anisotropy energy due to the shape of the sample is expressed by the following formula:

(1.17)

      where N_a and N_b are the demagnetization factors along the a and b directions of the ellipsoid, and θ is the angle that the spontaneous magnetization vector M_s makes with the main axis (a) of the ellipsoid.

Figure 1.10 The crystal approximated by an ellipsoid (general case).

      Source: Caizer (2019). Reprinted by permission of Taylor & Francis Ltd.

When the magnetization of the ellipsoid in the external magnetic field is done along the direction of the a‐axis, the shape anisotropy energy is minimum (or zero). In contrast, when magnetization is done along a direction perpendicular to the a‐axis (e.g. the b, c, or other directions), the shape anisotropy energy is maximum. In the latter case, taking into account the Eqs (1.17) and (1.16), the shape anisotropy constant can be expressed by the following formula:

      (1.18)

      in the approximation of the first order.

      When the magnetic material is reduced to the nanoscale, these forms of magnetic anisotropy remain valid. In addition, in the case of magnetic nanoparticles, the shape anisotropy becomes very important, reaching in some cases even larger, or much larger than the magnetocrystalline anisotropy. Thus, the neglect of this first aspect leads to important errors from a magnetic point of view, incompatible with the physical reality. For example, if the nanoparticle is spherical in shape (Figure 1.5), the semiaxes a, b, c become equal (Figure 1.10), and equal to the radius of the sphere. According to Eqs. the shape anisotropy constant in this case is zero, as is the energy. So there is no shape anisotropy in the case of spherical nanoparticles. In contrast, in the case of elongated nanoparticles, when a ≫ b = c, and when they are soft magnetic (magnetocrystalline anisotropy is reduced), the shape anisotropy exceeds the magnetocrystalline anisotropy, or even becomes dominant. This is an important aspect in the case of magnetic nanoparticles that must be taken into account not only in practical applications, including biomedical ones, but also in theoretical calculations and models/experiments.

Moreover, in the case of nanoparticles, generally in the field of nanometers, when the volume‐to‐surface ratio of spherical nanoparticles increases: S/V ~ 1/D, D is the diameter of the nanoparticle, a new form of anisotopy appears which must be taken into account, namely: surface anisotropy (Caizer 2019). This is because in the case of small and very small nanoparticles, and for soft magnetic materials such as ferrites (e.g. Fe₃O₄, γ‐Fe₂O₃, Ni–ZnFe₂O₄, MnFe₂O₄), this form of anisotropy becomes dominant, sometimes much more larger than the magnetocrystalline (Caizer 2004b). This form of anisotropy results from the surface effects that occur in the case of small nanoparticles where the surface spines have an important contribution, the symmetry of the bonds with the nearest neighbors being different from that inside the core of nanoparticle. These aspects were first highlighted by Néel (1954) who showed that in the case of crystals with cubic symmetries and for surfaces of type (111) or (100) the surface anisotropy energy can be written as follows:

      (1.19)

where B is the angle that the spontaneous magnetization vector M_s makes with the direction of the external normal to the considered surface (Figure 1.11), and K_s is the surface anisotropy constant (expressed in J m⁻²).

Figure 1.11 The orientation of spontaneous magnetization

relative to normal on the surface (a) (100) for the monocrystal with cubic symmetry.

      Source: Caizer (2019). Reprinted by permission of Taylor & Francis Ltd.

      For example (Caizer 2004a), in the case of spherical nanoparticles with a diameter D of 10 nm, the value of ~6 × 10³ expressed in J m⁻³ is obtained for the surface anisotropy constant. This value is five times higher than the magnetocrystalline anisotropy of the Ni–Zn ferrite, which is 1.5 × 10³ J m⁻³ (Broese Van Groenou et al. 1967). Therefore, this form of magnetic anisotropy must be considered in the case of magnetic nanoparticles.

      Moreover,