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The bulk ferromagnetic magnetic material consists of magnetic domains (Kneller 1962) spontaneously magnetized to saturation, resulting from the balance of exchange forces, which tend to align the atomic (ionic) magnetic moments in the network, and magnetostatic forces, which, through the created magnetic poles, tend to disorient the magnetic moments from their parallel alignment. The magnetic structure is stable when there is a balance between the exchange and magnetostatic forces, respectively, in the condition of minimum magnetocrystalline energy. Experimentally, different structures of magnetic domains were observed, the most common being those with free magnetic poles (Figure 1.4a) and magnetic structures without free magnetic poles (with magnetic flux closing domains) (Figure 1.4b). The first magnetic structure is characteristic of uniaxial crystals and the second magnetic structure is characteristic of the magnetic crystals with cubic symmetry.

Figure 1.4 Magnetic structures of nanoparticles: multidomain nanoparticles with (a) uniaxial and (b) cubic symmetry.

Source: Caizer et al. (2017). Reprinted by permission from Springer Nature.

The magnetic domains are separated from each other by narrow regions in the crystal (transition) called walls of magnetic domains. Within the walls is a continuous change in orientation of spins, from the direction of magnetization in one domain to the direction of magnetization in the neighboring domain. The most common walls found in magnetic structures are the Bloch‐type walls (Bloch 1930) or 180 walls, which separate 2 neighboring domains with opposite magnetizations. They are also the most stable in magnetic structures. But there are also Nèel or 90's walls, which separate adjacent domains, where the magnetizations in the domains are oriented at 90°. Nèel‐type walls are generally unstable.

The magnetic domains are magnetized uniformly (at saturation), characterized by the spontaneous magnetization of M_s. In the closing domains, the spontaneous magnetization is oriented at 45 in relation to the direction of separation of the domains (Figure 1.4b) so that the normal component of the magnetization is continuous along the boundaries separating the domains, and, thus, no magnetostatic energy will occur.

The thickness of the domain walls is generally less than 10⁵ A, and that of the walls in the range 10–10³ A, strongly depending on the anisotropy of the material and the exchange forces.

When the volume of the magnetic material is reduced in the nanometer range, the magnetic structure changes radically, reaching a unidominal structure, under a certain critical volume (V_cr) (Kittel 1946). Schematically, this aspect is shown in Figure 1.5, in the case of the spherical nanoparticle (Caizer 2004a). Above the critical volume (V > V_cr), the nanoparticle has an incipient structure of magnetic domains, which depending on the crystalline symmetry of the material, can be of the form: (a) case of uniaxial symmetry or (b) case of cubic symmetry.

Figure 1.5 Multidomain magnetic nanoparticles with (a) uniform magnetization (uniaxial symmetry) and (b) nonuniform magnetization (cubic symmetry), and (c) single‐domain nanoparticle.

Source: Adapted from Caizer and Stefanescu (2003).

Using the classic model of the single‐domain particle, it can estimate the critical diameter D_c (or the critical volume V_cr) at which the transition from the state with the structure of magnetic domains (multidomains) to the one with the single‐domain structure takes place. Thus, for the critical diameter, the following formula is obtained:

(1.9)

where ε_P is the energy density of the domain wall, μ₀ is the magnetic permeability of the vacuum (μ₀ = 4π × 10⁻⁷ H m⁻¹), and M_s the spontaneous magnetization of the material.

The energy density of the wall was determined by Landau–Lifschtz, finding the following formula:

(1.10)

where D is the constant in the crystalline network, T_C is the Curie temperature of the magnetic material, K_V is the constant magnetoscrystalline anisotropy, and K_B is the Boltzmann constant.

The critical diameter at which the transition from the multidomains structure to the single‐domain structure takes place depending a lot on the magnetic anisotropy of the nanoparticle. For Co, the value of ~60 nm was found (…). However, in the case of Ni–Zn ferrite nanoparticles, Caizer finds a value D_c = 21.6 nm, for the energy of the domain wall ε_P of 0.145 erg cm⁻² (Caizer 2003a).

In conclusion, when conducting theoretical and practical studies on the use of nanoparticles, it is very important to know the critical diameter (D_c) under which the nanoparticle becomes one with a single‐magnetic structure, for which a previous evaluation is needed.