The magnetic domains are magnetized uniformly (at saturation), characterized by the spontaneous magnetization of Ms. 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 (Vcr) (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 > Vcr), 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 Dc (or the critical volume Vcr) 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 εP is the energy density of the domain wall, μ0 is the magnetic permeability of the vacuum (μ0 = 4π × 10−7 H m−1), and Ms 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, TC is the Curie temperature of the magnetic material, KV is the constant magnetoscrystalline anisotropy, and KB 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 Dc = 21.6 nm, for the energy of the domain wall εP 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 (Dc) under which the nanoparticle becomes one with a single‐magnetic structure, for which a previous evaluation is needed.
1.1.4 Magnetic Saturation
Another important aspect to consider, when a magnetic material is reduced to the nanoscale, is the saturation magnetization of the material, which is influenced by such reduction.
In the case of bulk magnetic material, the saturation magnetization (Msat) (theoretically obtained in intense magnetic field and at low temperatures) is equal to the spontaneous magnetization (Ms), being a known parameter of material. Example, in the case of Fe, the saturation magnetization at room temperature is 1714 kA m−1 (Cullity and Graham 2009), and in the case of Fe3O4, it is 477.5 kA m−1 (Smit and Wijin 1961). The spontaneous magnetization of the bulk magnetic material decreases with temperature, having the maximum value at 0 K (Ms(0)) and zero at a temperature, generally high (hundreds of degrees), which is Curie temperature in the case of ferromagnetics and Nèel temperature in the case of ferrimagnetic materials. The temperature variation of the spontaneous magnetization of massive ferromagnetic material, such as Fe, Co, and Ni (Figure 1.6a), is a universal function that does not involve indeterminate constants:
(1.11)
Figure 1.6 (a) Relative saturation magnetization of iron, cobalt, and nickel as a function of relative temperature. Calculated curves are shown for three values of J.
Source: Cullity and Graham (2009). Reproduced with permission from John Wiley & Sons;
(b) Curves that represent the dependence of the saturation magnetization on temperature according to Eq. (1.12) (curve α) and Eqs. (1.13) and (1.14) (curve β); experimental curve (□).
Source: Caizer (2005a). Reproduced with permission from Springer Nature.
At low temperatures, this variation is well described by Bloch's law (law in T3/2 (Bloch 1930), deduced from the spin wave model,
which is the exact solution of the Hamiltonian Heisemberg at low temperatures. In Eq. (1.12), B is a constant whose value depends on the exchange integral.
This dependence is well verified experimentally both for bulk ferromagnetic materials (Fe, Ni) (Aldred and Frohle 1972; Aldred 1975) and for some bulk spinel ferrites, such as MnxFe3 − xO4 ferrite for 0.2 < x < 1.0 (Dillon 1962).
However, when the magnetic material has nanometric sizes, such as nanoparticles, some theoretical calculations and some experimental results have shown that the temperature exponent is greater than the value 3/2, corresponding to the bulk material (Hendriksen et al. 1992, 1993; Linderoth et al. 1993). Thus, Morais et al. (2000) showed that, depending on the temperature of the saturation magnetization in the range 4.2–293 K, in the case of a ferrofluid with NiFe2O4 nanoparticles