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respectively, in the hypothesis of a continuous environment. According to formula (1.4), the magnetization vector
In accordance with Eq. (1.4), the magnetic moment of a volume of magnetic material will be
(1.5)
In the case of reducing the volume of ferrous‐ or ferrimagnetic material in the nanometer range (nm – tens of nm), as in the case of magnetic nanoparticles, when there is a single magnetic domain (Weiss domain) (or in the case of a nanoparticle volume even smaller than the one corresponding to a magnetic domain), the magnetization (M) is uniform in the finite volume of material. Thus, in this case, of the single‐domain magnetic nanoparticle, the resulting magnetic moment can be written as (Caizer 2016)
(1.6)
or by using the common notations (Caizer 2019)
where mNP is the magnetic moment of the nanoparticle, VNP the volume of the nanoparticle, and Ms the spontaneous magnetization of the magnetic material (the magnetization of a magnetic domain [M] corresponds to the spontaneous magnetization [or saturation]) (Ms) (M ≡ Ms). When the nanoparticle is spherically approximated, formula (1.7) is written as
where D is the diameter of the nanoparticle, an approximation widely used both in theoretical calculations and in practical applications. From a magnetic point of view, it is important if the nanoparticle is spherical or has another shape, e.g. ellipsoidal, as the magnetic behavior in the external magnetic field may change a lot, especially due to soft magnetic materials case (see Section 1.1.5).
To conclude, it can be said that, from a magnetic point of view, in the case of bulk magnetic material, the base observable for the magnetic characterization is the magnetization given by relationship (1.4) or the elementary magnetic moment du, where the magnetization is nonuniform (Figure 1.3a), whose field and space dependence must be known for the calculation of the integral.
Figure 1.3 (a) Representation of the magnetization vectors (
Source: Caizer (2016). Reprinted by permission from Springer Nature;
(b) Spherical nanoparticle for uniaxial crystalline symmetry; e.m.a. is the easy magnetization axis.
Source: Caizer et al. (2020). Reprinted by permission from Springer Nature.
In the case of magnetic nanoparticles (Figure 1.3b), the aspects are simplified, these being characterized by the magnetic moment of the nanoparticles given by Eq. (1.7) (or Eq. (1.8) for spherical nanoparticles), where Ms is the spontaneous magnetization of the nanoparticle material which is a known observable (Ms is a material parameter), and VNP is the effective volume of the nanoparticle. VNP and in most theoretical or practical cases can be easily approximated by the volume of a sphere, ellipsoid of revolute or flattened, cylinder, etc., which radically simplifies the calculations. However, for this reason, the exact given situation will have to be taken into account, in order not to introduce errors.
1.1.3 Magnetic Structures
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.