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rel="nofollow" href="#ulink_00806266-7d44-5dfa-bb58-824ff9380222">Figure 4.19 Schematic illustration of magnetic domains in a ferromagnetic material: (a) Randomly oriented domains in an unmagnetized material. (b) The domains become oriented upon application of a magnetic field, resulting in a highly magnetized material. Each arrow represents a huge number of atoms.

The maximum possible magnetization of a ferroelectric material, called the saturation magnetization M_max, corresponds to the magnetization that would result if all the atomic dipoles of the material were completely aligned with the magnetic field. There is also a saturation magnetic flux density, B_max (Eq. (4.40)). For the transition metal atoms (those with a partially filled d electron subshell), the contribution from the orbital angular momentum is negligible and, thus, the magnetic moment of the atom is determined by the number of unpaired electrons. Consequently, in the transition metals where the 3d shells of the atoms are partially filled, we can determine the magnetic moment of the atoms by filling the 3d shell with aligned spins up to a maximum of five beyond which the spins must have an opposite or antiparallel alignment to the first five. This is required by the Pauli exclusion principle. In the iron (Fe) atom, for example, the net number of unpaired electrons is four and, thus, the net magnetic moment of the atom is 4 μ_B, where μ_B is the Bohr magneton. Experimental magnetization curves of M (or B) versus H show a saturation magnetization that is lower than the ideal value due to microstructural factors and pinning of the domain walls by defects such as grain boundaries.

      4.6.5 Ferrimagnetic Materials

      Another type of magnetism, called ferrimagnetism, more common in ionic‐bonded ceramics, refers to a type of ferromagnetism in which the magnetic moment of ions at one type of atomic sites is partly cancelled by antiparallel interactions with ions of another site. However, there remains a net magnetic moment of the material in the absence of a magnetic field. Ferrimagnetic ceramics have a lower saturation magnetization than ferromagnetic metals but their electrically insulating properties provide an advantage in some engineering applications where a low electrical conductivity is required.

The ferrimagnetic iron oxides Fe₃O₄, commonly referred to as magnetite, and ferric oxide, γ‐Fe₂O₃, referred to as maghemite, are the most widely studied nanoparticles for hyperthermia treatment of tumors because of their favorable biocompatibility. Magnetite has a crystal structure that belongs to a class of compounds called spinels. The formula of normal spinels has the general formula AB₂O₄ (or AO∙B₂O₃), where A²⁺ and B³⁺ are divalent and trivalent ions, respectively. These normal spinels have a characteristic crystal structure based on a close packed face‐centered cubic (FCC) lattice of O²⁻ ions, with the A²⁺ ions and B³⁺ ions occupying tetrahedral and octahedral interstitial sites within the lattice. Magnetite has an inverse spinel structure B(AB)O₄ in which the occupancy of the interstitial sites is slightly different. One‐half of the B³⁺ ions in magnetite occupy tetrahedral sites (called a sites) whereas the A²⁺ ions and remaining half of B³⁺ ions occupy octahedral sites (called b sites) (Figure 4.20a). In magnetite, the O²⁻ ions are nonmagnetic due to all their electrons occupying filled shells. Consequently, we need to consider only the interactions between one‐half of the Fe³⁺ ions (with five unpaired electrons in their 3d shell) on the a sites and the Fe²⁺ ions (four unpaired electrons in their 3d shell) and the other half of the Fe³⁺ ions on the b sites (Figure 4.20b). The antiparallel a–b interaction between the ions in the a and b sites dominates and, thus, the saturation magnetization per formula unit (Fe₃O₄) is 4 μ_B.

Figure 4.20 (a) Part of Fe₃O₄ crystal structure showing the tetrahedral a sites and octahedral b sites. (b) Illustration showing the arrangement of the electron magnetic moments of the Fe ions at the a and b sites in the Fe₃O₄ crystal structure.

      Maghemite (γ‐Fe₂O₃) has the same crystal structure as magnetite but it is often considered a ferrous ion (Fe²⁺) deficient magnetite because there are no Fe²⁺ ions in the crystal structure when compared to magnetite. Although there is no clear agreement about the distribution of the cations in the maghemite crystal structure, it is often stated that the Fe³⁺ ions occupy both the tetrahedral a sites and octahedral b sites, with five Fe³⁺ ions and one vacancy in the b sites for every three Fe³⁺ ions in the a sites. This can be expressed by the formula Fe[Fe_5/3◻_1/3]O₄ where the symbol ◻ represents a vacant site.

      4.6.6 Magnetization Curves and Hysteresis

Measured curves of magnetization M versus applied magnetic field H (or magnetic flux density B vs H) for diamagnetic and paramagnetic materials are straight lines and they retrace themselves as H is increased or decreased. In comparison, magnetization curves for multi‐domain ferromagnetic and ferrimagnetic materials are not straight lines and they do not retrace themselves as H is increased or decreased (Figure 4.21). This lack of reversibility is a well‐known phenomenon called hysteresis. Another feature of the magnetization curves for ferromagnetic and ferrimagnetic materials is that they show a remanent magnetization (point C or point D) even though there is no applied field. This is the phenomenon of permanent magnetism. Hysteresis is due to the presence of multiple magnetic domains within the material. The motion of the domain boundaries and the reorientation of the domain directions are not fully reversible. The domains do not return completely to their original arrangement but, instead, some alignment of the magnetic dipoles remains even when the magnetic field is completely removed.

Figure 4.21 Magnetization curve for a ferromagnetic or ferrimagnetic material showing a hysteresis loop caused by domain motion.

      4.6.7 Hyperthermia Treatment of Tumors using Magnetic Nanoparticles

Hysteresis in a time varying or alternating magnetic field leads to the generation of heat in ferromagnetic and ferrimagnetic materials. The amount of heat P generated per unit volume is given by

      (4.44)

      where, μ_o is the permeability of free space, f is the frequency of the alternating applied field, and the circular integral sign represents integration over the closed hysteresis loop, yielding the area of the loop.

      The generation of heat in ferromagnetic and ferrimagnetic nanoparticles when they are subjected to an alternating magnetic field forms the basis of hyperthermia treatment of tumors. Nanoparticles of the ferrimagnetic oxide Fe₃O₄ or γ‐Fe₂O₃ are most commonly used because of their favorable biocompatibility. The procedure involves dispersing nanoparticles

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