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In the case of a bulk paramagnetic, ferro‐, or ferrimagnetic material, the magnetism is due to the existence of the magnetic moment (total) at the atomic (or ionic/molecular) level (Kneller 1962; Jacobs and Bean 1963; Vonsovskii 1974; Caizer 2004a):

(1.1)

as a result of the spin–orbit coupling (vector summation of the spin magnetic moments (total) () and the orbital magnetic moments (total) (): the vector model of the atom (= + ). In Eq. (1.1), g_J is the spectroscopic splitting factor (Lande factor) at the atomic level,

(1.2)

m_J is the internal magnetic quantum number (total), which can take (2J + 1) values (according to quantum physics, respectively –J, …, 0, …, +J), and μ_B is the Bohr magnetone:

(1.3)

with the observables: e is the electron charge (e = 1.6 × 10⁻¹⁹ C), m₀ is the resting electron mass (m₀ = 9.1 × 10⁻³¹ kg), and h is the Planck constant (h = 6.63 × 10⁻³⁴ Js). In Eq. (1.2), L is the internal orbital quantum number (total), and S is the internal spin quantum number.

Macroscopically, the quantity that characterizes the bulk magnetic material, from a magnetic point of view, is the magnetization (), defined as a numerical quantity equal to the resulting magnetic moment (, being the total magnetic moment of the atom/ion (Eq. 1.1), and i the number of atoms/ions in volume V) of the volume unit (Caizer 2004a),

(1.4)

respectively, in the hypothesis of a continuous environment. According to formula (1.4), the magnetization vector has the same direction and sense as the elementary magnetic moment 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)

(1.7)

where m_NP is the magnetic moment of the nanoparticle, V_NP the volume of the nanoparticle, and M_s the spontaneous magnetization of the magnetic material (the magnetization of a magnetic domain [M] corresponds to the spontaneous magnetization [or saturation]) (M_s) (M ≡ M_s). When the nanoparticle is spherically approximated, formula (1.7) is written as

(1.8)

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 () and elementary magnet moment () for an elementary volume dV of the bulk magnetic material of finite volume V, and an example of multidomain magnetic structures (in magnified image).

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 M_s is the spontaneous magnetization of the nanoparticle material which is a known observable (M_s is a material parameter), and V_NP is the effective volume of the nanoparticle. V_NP 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.