Materials for Biomedical Engineering. Mohamed N. Rahaman

electrons of its constituent atoms or ions.

There are two contributions to the magnetic dipole moment of the electron (Figure 4.18). One is due to the intrinsic spin of the electron analogous to a charged body spinning about its own axis. The other is due to its orbital motion around the nucleus. Each contribution is quantized, that is, it depends on

       The spin quantum number ms of the electron, which can have values of +1/2 (sometimes referred to as spin up) and −1/2 (spin down)

       The magnetic orbital quantum orbital quantum number ml of the electron, which can have values from −l to +l, where l is the angular momentum (azimuthal) quantum number.

Figure 4.18 Schematic diagram illustrating the two contributions to the magnetic dipole moment of an electron in an atom due to its intrinsic spin and its orbital motion around the nucleus.

      The important quantity is the Bohr magneton μ_B, the magnetic moment of an unpaired electron, equal to 9.27 × 10⁻²⁷ J/T. For each electron in an atom, for example, the intrinsic spin magnetic moment is +μ_B (spin up) or −μ_B (spin down), whereas the orbital magnetic moment is m_lμ_B.

      4.6.2 Meaning and Definition of Relevant Magnetic Properties

      If a magnetic material is placed in an applied magnetic field of strength H, the individual atomic magnetic moments contribute to its overall response such that the internal field strength B in the material, also called the magnetic induction or magnetic flux density, is given by

      (4.40)

      where, μ_o is a physical constant called the permeability of free space (equal to 1.257 × 10⁻⁶ H/m) and M is the magnetic moment per unit volume of the material, commonly referred to as its magnetization. B, H, and M are vectors but to simplify the discussion, we will not emphasize this point and just consider their magnitude. As the magnetic flux density in a vacuum B_o is equal to μ_o H, the term μ_o M represents the change in the flux density due to the presence of the material.

      A few different parameters are used to classify the magnetic properties of magnetic materials. The magnetic susceptibility χ_m, defined by

      (4.41)

      relates the magnetization induced in a material by an applied field H. As both M and H have the same units (A m), χ_m is unitless, that is, a number without units. The relative permeability μ_r is defined as

      (4.42)

      where, μ is the permeability of the material. The susceptibility or the relative permeability provides a measure of the degree to which a material can be magnetized by an applied field. They are related by

      (4.43)

      4.6.3 Diamagnetic and Paramagnetic Materials

      For most atoms (or ions), the magnetic effects of the electrons, including their spin and orbital motions, cancel out and, consequently, the atom is not magnetic. This is true for atoms of the inert gasses and atoms of elements such as zinc that have completely filled electron shells. The Pauli exclusion principle requires that two electrons in any energy level of an atom must have opposite spins and, thus, the resultant spin magnetic moment of paired electrons is zero. Materials composed of atoms in which all the electrons are paired have no intrinsic magnetic moment and are described as diamagnetic. A weak magnetic dipole may be induced in these materials when they are placed in a magnetic field but the induced dipole points in a direction opposite to that of the field.

      On the other hand, for other atoms or ions such as those of some transition elements, the magnetic effects of the electrons do not cancel, since one or more unpaired electrons are present, and the atom as a whole has a magnetic moment. In the absence of an applied magnetic field, the orientations of the atomic dipoles are random and, thus, the material as a whole has no net magnetic moment. However, when placed in an applied magnetic field, the atomic dipoles line up in the direction of the applied field. This type of magnetic behavior is referred to as paramagnetism.

Diamagnetic materials have χ_m values in the range approximately −10⁻⁶ to −10⁻³, whereas paramagnetic materials have χ_m values in the range ~10⁻⁶ to 10⁻¹. The susceptibility of diamagnetic materials is negative whereas it is positive for paramagnetic materials. This is because in an applied magnetic field, the induced atomic dipoles in a diamagnetic material point in a direction opposite to the magnetic field, making the field within the material lower than that of a vacuum. On the other hand, the atomic dipoles present in a paramagnetic material line up in the direction of the applied field, in much the same way as a compass needle lines up in the earth’s magnetic field. This has the effect of increasing the magnetization of the material.

      4.6.4 Ferromagnetic Materials

      Diamagnetism and paramagnetism are weak forms of magnetism and, consequently, materials that show these two types of behavior are often considered nonmagnetic. Certain materials that contain paramagnetic atoms or ions, however, do show a strong permanent magnetization even in the absence of a magnetic field. This type of response, called ferromagnetism, is shown by materials composed of atoms of certain transition elements such as iron and cobalt, and certain rare earth elements such as gadolinium. In ferromagnetic materials, a special type of coupling occurs between adjacent atoms, resulting in cooperative alignment of their magnetic moment throughout the crystal. This coupling is purely a quantum mechanical effect that cannot be adequately explained in terms of classical physics. It leads to a high net magnetic moment of the material even in the absence of a magnetic field. Ferromagnetism is a property not just of the individual atoms but is a result of the interaction of each atom with its neighbors in the crystal lattice of the solid. Although the magnetization can vary with the magnetic field, magnetic susceptibility χ_m values as high as 10⁶ are possible for ferromagnetic materials.

Another property of ferromagnetism is that each grain in a polycrystalline material (or the whole crystal of a single crystalline material) is made up of a number of magnetic domains, tiny regions in which the atomic dipoles are essentially perfectly aligned. These domains are originally oriented in such a way that they cancel each other as far as their external magnetic effects are concerned (Figure 4.19a). However, when a ferromagnetic material is subjected to an applied magnetic field of gradually increasing strength, the domains that are favorably oriented grow at the expense of those that are not, and the material becomes highly magnetized (Figure 4.19b).