where p =BS= flux density(in gauss)x area S. The time-changing flux, dap/dt, can happen as a result of 1. A changing magnetic field within a stationary circuit 2. A circuit moving through a 3. A combination of 1 and 2 The electrical circuit may have n turns and ther emf =-N We can write emf=E. dL and in the presence of changing magnetic fields or a moving electrical circuit E.dL no longer required to be equal to 0 as it was for stationary fields and circuits. Maxwell's Equations Because the flux d can be written JB. ds we have emf=E dl=-d/dt B. ds, and by using Stokes'theorer (V×E)·ds=-dB/dtds V×E=-dB/dt That is, a spatially changing electric field produces a time-changing magnetic field. This is one of Maxwells equations linking electric and magnetic fields By a similar argument it can be shown that V×H=J+dDdt This is another of Maxwells equations and shows a spatially changing magnetic field produces a time-changing electric field. The latter dD/dt can be treated as an electric current which flows through a dielectric, e.g., in a capacitor, when an alternating potential is applied across the plates. This current is called the displacement current to distinguish it from the conduction current which flows in conductors. The conduction current involves the movement of electrons from one electrode to the other through the conductor (usually a metal). The displacement current involves no translation of electrons or holes but rather an alternating polarization through out the dielectric material which is between the plates of the capacitor From the last two equations we see a key conclusion of Maxwell: that in electromagnetic fields a time-varying magnetic field produces a spatially varying electric field and a time-varying electric field produces a spatially varying magnetic field. Maxwell's equations in point form, then, are V×E=dB/dt V×H=J+dD/d These equations are supported by the following auxiliary equations: D=EE (displacement permittivity x electric field intensity) B= uh (flux density permeability x magnetic field intensity) e 2000 by CRC Press LLC
© 2000 by CRC Press LLC where F = BS = flux density (in gauss) ¥ area S. The time-changing flux, dF/dt, can happen as a result of 1. A changing magnetic field within a stationary circuit 2. A circuit moving through a steady magnetic field 3. A combination of 1 and 2 The electrical circuit may have N turns and then We can write emf = E · dL and in the presence of changing magnetic fields or a moving electrical circuit E · dL is no longer required to be equal to 0 as it was for stationary fields and circuits. Maxwell’s Equations Because the flux F can be written eB · ds we have emf = E · dL = –d/dt B · ds, and by using Stokes’ theorem (— 2 E) · ds = –dB/dt ds or — 2 E = –dB/dt That is, a spatially changing electric field produces a time-changing magnetic field. This is one of Maxwell’s equations linking electric and magnetic fields. By a similar argument it can be shown that — 2 H = J + dD/dt This is another of Maxwell’s equations and shows a spatially changing magnetic field produces a time-changing electric field. The latter dD/dt can be treated as an electric current which flows through a dielectric, e.g., in a capacitor, when an alternating potential is applied across the plates. This current is called the displacement current to distinguish it from the conduction current which flows in conductors. The conduction current involves the movement of electrons from one electrode to the other through the conductor (usually a metal). The displacement current involves no translation of electrons or holes but rather an alternating polarization throughout the dielectric material which is between the plates of the capacitor. From the last two equations we see a key conclusion of Maxwell: that in electromagnetic fields a time-varying magnetic field produces a spatially varying electric field and a time-varying electric field produces a spatially varying magnetic field. Maxwell’s equations in point form, then, are — 2 E = dB/dt — 2 H = J + dD/dt — · D = rv — · B = 0 These equations are supported by the following auxiliary equations: D = eE (displacement = permittivity 2 electric field intensity) B = mH (flux density = permeability 2 magnetic field intensity) emf d d = -N t F
ALL MATTER Paramagnetic Antiferromagnetic Ferrimagnetic Ferromagnetic e.g. Mno 8.g-1- Fe2 0. FIGURE 36.3 All matter consists of diamagnetic material (atoms having no permanent magnetic dipole moment) aramagnetic material (atoms having magnetic dipole moment). Paramagnetic materials may be further divided into J =oE (current density conductivity X electric field strength) J=P, V (current density volume charge density x carrier velocity) D=EE +P(displacement as function of electric field and polarization) B=H(H M)(magnetic flux density as function of magnetic field strength and magnetization) P=xe E(polarization electric susceptibility x permittivity of free space x electrical field strength) M=x,uH (magnetization magnetic susceptibility x permeability of free space X magnetic field strength) The last two equations relate, respectively, the electric polarization P to the displacement D =E,E and the magnetic moment M to the flux density B=u H. They apply only to "linear"materials, i.e, those for which P is linearly related to E and M to H. For magnetic materials we can say that nonlinear materials are usually of greater practical interest. Dia- and Paramagnetism The phenomenon of magnetism arises ultimately from moving electrical charges(electrons). The movement may be orbital around the nucleus or the other degree of freedom possessed by electrons which, by analogy with the notion of the planets, is referred to as spin. In technologically important materials, i.e., ferromagnetics and ferri magnetics, spin is more important than orbital motion. Each arrow in Fig. 36.3 represents the total spin of an atom. An atom may have a permanent magnetic moment, in which case it is referred to as belonging to a paramagnetic material, or the atom may be magnetized only when in the presence of a magnetic field, in which case it is called diamagnetic. Diamagnetics are magnetized in the opposite direction to that of the applied magnetic field, i.e., they display negative susceptibility(a measure of the induced magnetization per unit of applie magnetic field). Paramagnetics are magnetized in the same direction as the applied magnetic field, i. e, they e 2000 by CRC Press LLC
© 2000 by CRC Press LLC J = sE (current density = conductivity 2 electric field strength) J = rvV (current density = volume charge density 2 carrier velocity) D = eoE + P (displacement as function of electric field and polarization) B = mo(H + M) (magnetic flux density as function of magnetic field strength and magnetization) P = ceeoE (polarization = electric susceptibility 2 permittivity of free space 2 electrical field strength) M = cmmoH (magnetization = magnetic susceptibility 2 permeability of free space 2 magnetic field strength) The last two equations relate, respectively, the electric polarization P to the displacement D = eoE and the magnetic moment M to the flux density B = moH. They apply only to “linear” materials, i.e., those for which P is linearly related to E and M to H. For magnetic materials we can say that nonlinear materials are usually of greater practical interest. Dia- and Paramagnetism The phenomenon of magnetism arises ultimately from moving electrical charges (electrons). The movement may be orbital around the nucleus or the other degree of freedom possessed by electrons which, by analogy with the motion of the planets, is referred to as spin. In technologically important materials, i.e., ferromagnetics and ferrimagnetics, spin is more important than orbital motion. Each arrow in Fig. 36.3 represents the total spin of an atom. An atom may have a permanent magnetic moment, in which case it is referred to as belonging to a paramagnetic material, or the atom may be magnetized only when in the presence of a magnetic field, in which case it is called diamagnetic. Diamagnetics are magnetized in the opposite direction to that of the applied magnetic field, i.e., they display negative susceptibility (a measure of the induced magnetization per unit of applied magnetic field). Paramagnetics are magnetized in the same direction as the applied magnetic field, i.e., they FIGURE 36.3 All matter consists of diamagnetic material (atoms having no permanent magnetic dipole moment) or paramagnetic material (atoms having magnetic dipole moment). Paramagnetic materials may be further divided into ferromagnetics, ferrimagnetics, and antiferromagnetics
TABLE 36.3 The Occurrence of Ferromagnetism omic number 25 262728 64 1.631.821.971.57 Ferromagnetic moment/mass (Am2/kg) 2177516154.390 Curie point, e K Neel temp, e,K have positive susceptibility. All atoms are diamagnetic by virtue of their having electrons. Some atoms are also paramagnetic as well, but in this case they are called paramagnetic since paramagnetism is roughly a hundred times stronger than diamagnetism and overwhelms it. Faraday discovered that paramagnetic are attracted by magnetic field and move toward the region of maximum field, whereas diamagnetic are repelled and move wara a s The total magnetization of both paramagnetic and diamagnetic materials is zero in the absence of an app ld, i. e, they have zero remanence. Atomic paramagnetism is a necessary condition but condition for ferro- or ferrimagnetism, i.e., for materials having useful magnetic properties not a suffici Ferromagnetism and Ferrimagnetism To develop technologically useful materials, we need an additional force that ensures that the spins of the outermost(or almost outermost)electrons are mutually parallel. Slater showed that in iron, cobalt, and nickel this could happen if the distance apart of the atoms(D)was more than 1. 5 times the diameter of the 3delectron shell(d).(These are the electrons, near the outside of atoms of iron, cobalt, and nickel, that are responsible for the strong paramagnetic moment of the atoms. Paramagnetism of the atoms is an essential prerequisite for ferro-or ferrimagnetism in a material. Slater's result suggested that, of these metals, iron, cobalt, nickel, and gadolinium should be ferromagnetic at room temperature, while chromium and manganese should not be ferromagnetic. This is in accordance with experiment. Gadolinium, one of the rare earth elements, is only weakly ferromagnetic in a cool room. Chro mium and manganese in the elemental form narrowly miss being ferromagnetic. However, when manganese is alloyed with copper and aluminum( Cu Mn2 Als)to form what is known as a Heusler alloy [Crangle, 1962 it becomes ferromagnetic. The radius of the 3d electrons has not been changed by alloying, but the atomic e. ing has been increased by a factor of 1.53/1.47. This small change is sufficient to make the difference ween positive exchange, parallel spins, and ferromagnetism and negative exchange, antiparallel spins, and antiferromagnetism. For all ferromagnetic materials there exists a temperature(the Curie temperature)above which the thermal disordering forces are stronger than the exchange forces that cause the atomic spins to be parallel. From Table 36.3 we see that in order of descending Curie temperature we have Co, Fe, Ni, Gd. From Fig. 36.4 we find that this is also the order of descending values of the exchange integral, suggesting that high positive values of the exchange integral are indicative of high Curie temperatures rather than high magnetic intensity in ferromagnetic materials. Negative values of exchange result in an antiparallel arrangement of the spins of adjacent atoms and in antiferromagnetic materials(Fig. 36.3). Until 5 years ago, it was true to say that antiferromagnetism had no practical application. Thin films on antiferromagnetic materials are now used to provide the bias field which is used to linearize the response of some magnetoresistive reading heads in magnetic disk drives. Ferrimag- netism, also illustrated in Fig. 36.3, is much more widely used. It can be produced as soft, i.e., low coercivity ferrites for use in magnetic recording and reading heads or in the core of transformers operating at frequencies up to tens of megahertz. High-coercivity, single-domain particles(which are discussed later )are used in very large quantities to make magnetic recording tapes and flexible disks - O, and cobalt-impregnated iron oxides and to make barium ferrite, the most widely used material for permanent magnets. e 2000 by CRC Press LLC
© 2000 by CRC Press LLC have positive susceptibility. All atoms are diamagnetic by virtue of their having electrons. Some atoms are also paramagnetic as well, but in this case they are called paramagnetics since paramagnetism is roughly a hundred times stronger than diamagnetism and overwhelms it. Faraday discovered that paramagnetics are attracted by a magnetic field and move toward the region of maximum field, whereas diamagnetics are repelled and move toward a field minimum. The total magnetization of both paramagnetic and diamagnetic materials is zero in the absence of an applied field, i.e., they have zero remanence. Atomic paramagnetism is a necessary condition but not a sufficient condition for ferro- or ferrimagnetism, i.e., for materials having useful magnetic properties. Ferromagnetism and Ferrimagnetism To develop technologically useful materials, we need an additional force that ensures that the spins of the outermost (or almost outermost) electrons are mutually parallel. Slater showed that in iron, cobalt, and nickel this could happen if the distance apart of the atoms (D) was more than 1.5 times the diameter of the 3d electron shell (d). (These are the electrons, near the outside of atoms of iron, cobalt, and nickel, that are responsible for the strong paramagnetic moment of the atoms. Paramagnetism of the atoms is an essential prerequisite for ferro- or ferrimagnetism in a material.) Slater’s result suggested that, of these metals, iron, cobalt, nickel, and gadolinium should be ferromagnetic at room temperature, while chromium and manganese should not be ferromagnetic. This is in accordance with experiment. Gadolinium, one of the rare earth elements, is only weakly ferromagnetic in a cool room. Chromium and manganese in the elemental form narrowly miss being ferromagnetic. However, when manganese is alloyed with copper and aluminum (Cu61Mn24Al15) to form what is known as a Heusler alloy [Crangle, 1962], it becomes ferromagnetic. The radius of the 3d electrons has not been changed by alloying, but the atomic spacing has been increased by a factor of 1.53/1.47. This small change is sufficient to make the difference between positive exchange, parallel spins, and ferromagnetism and negative exchange, antiparallel spins, and antiferromagnetism. For all ferromagnetic materials there exists a temperature (the Curie temperature) above which the thermal disordering forces are stronger than the exchange forces that cause the atomic spins to be parallel. From Table 36.3 we see that in order of descending Curie temperature we have Co, Fe, Ni, Gd. From Fig. 36.4 we find that this is also the order of descending values of the exchange integral, suggesting that high positive values of the exchange integral are indicative of high Curie temperatures rather than high magnetic intensity in ferromagnetic materials. Negative values of exchange result in an antiparallel arrangement of the spins of adjacent atoms and in antiferromagnetic materials (Fig. 36.3). Until 5 years ago, it was true to say that antiferromagnetism had no practical application. Thin films on antiferromagnetic materials are now used to provide the bias field which is used to linearize the response of some magnetoresistive reading heads in magnetic disk drives. Ferrimagnetism, also illustrated in Fig. 36.3, is much more widely used. It can be produced as soft, i.e., low coercivity, ferrites for use in magnetic recording and reading heads or in the core of transformers operating at frequencies up to tens of megahertz. High-coercivity, single-domain particles (which are discussed later) are used in very large quantities to make magnetic recording tapes and flexible disks g-Fe2O3 and cobalt-impregnated iron oxides and to make barium ferrite, the most widely used material for permanent magnets. TABLE 36.3 The Occurrence of Ferromagnetism Cr Mn Fe Co Ni Gd Atomic number 24 25 26 27 28 64 Atomic spacing/diameter 1.30 1.47 1.63 1.82 1.97 1.57 Ferromagnetic moment/mass (Am2 /kg) At 293 K — — 217.75 161 54.39 0 At 0 K — — 221.89 162.5 57.50 250 Curie point, Qc K — — 1,043 1,400 631 289 Néel temp., Qn K 475 100 — — — —
FIGURE 36.4 Quantum mechanical exchang es cause a parallel arrar of materials for which the ratio of atomic separation, D, is at least 1.5 x d, the diamter of the 3d orbita Intrinsic Magnetic Properties Intrinsic magnetic properties are those properties that depend on the type of atoms and their composition and rystal structure, but not on the previous history of a particular sample. Examples of intrinsic magneti properties are the saturation magnetization, Curie temperature, magnetocrystallic anisotropy, and magneto- striction Extrinsic magnetic properties depend on type, composition, and structure, but they also depend on the previous history of the sample, e.g., heat treatment. Examples of extrinsic magnetic properties include the technologically important properties of remanent magnetization, coercivity, and permeability. These properties can be substantially altered by heat treatment, quenching, cold-working the sample, or otherwise changing the ze of the magnetic particle A ferromagnetic or ferrimagnetic material, on being heated, suffers a reduction of its magnetization(per unit mass, i. e, O, and per unit volume, M). The slope of the curve of M, vS. T increases with increasing temperature as shown in Fig. 36.5. This figure represents the conflict between the ordering tendency of the exchange interaction and the disordering effect of increasing temperature. At the Curie temperature, the order no longer exists and we have a paramagnetic material. The change from ferromagnetic or ferrimagnetic materials to paramagnetic is completely reversible on reducing the temperature to its initial value. Curie temperatures are always lower than melting points le crystal of iron has the body-centered structure at room temperature. If the magnetization as a function of applied magnetic field is measured, the shape of the curve is found to depend on the direction of ature, and the"easy" directions of magnetization are those directions parallel to the cube edges [100],[010 and [001] or, collectively, <100>. The hard direction of magnetization for iron is the body diagonal [111].At higher temperatures, the anisotropy becomes smaller and disappears above 300C 3. Nickel crystals(face-centered cubic) have an easy direction of [111] and a hard direction of [100]. Cobalt has the hexagonal close-packed (HCP)structure and the hexagonal axis is the easy direction at room temperature Magnetocrystalline anisotropy plays a very important part in determining the coercivity of ferro- or ferri- magnetic materials, i.e., the field value at which the direction of magnetization is reversed. Many magnetic materials change dimensions on becoming magnetized: the phenomenon is known a magnetostriction and can be positive, i. e, length increases, or negative. Magnetostriction plays an important role in determining the preferred direction of magnetization of soft, i. e, low H, films such as those of alloys of nickel and iron, known as Permalloy The origin of both magnetocrystalline anisotropy and magnetostriction is spin-orbit coupling. The magnitude of the magnetization of the film is controlled by the electron spin as usual, but the preferred direction of that e 2000 by CRC Press LLC
© 2000 by CRC Press LLC Intrinsic Magnetic Properties Intrinsic magnetic properties are those properties that depend on the type of atoms and their composition and crystal structure, but not on the previous history of a particular sample. Examples of intrinsic magnetic properties are the saturation magnetization, Curie temperature, magnetocrystallic anisotropy, and magnetostriction. Extrinsic magnetic properties depend on type, composition, and structure, but they also depend on the previous history of the sample, e.g., heat treatment. Examples of extrinsic magnetic properties include the technologically important properties of remanent magnetization, coercivity, and permeability. These properties can be substantially altered by heat treatment, quenching, cold-working the sample, or otherwise changing the size of the magnetic particle. A ferromagnetic or ferrimagnetic material, on being heated, suffers a reduction of its magnetization (per unit mass, i.e., s, and per unit volume, M). The slope of the curve of Ms vs. T increases with increasing temperature as shown in Fig. 36.5. This figure represents the conflict between the ordering tendency of the exchange interaction and the disordering effect of increasing temperature. At the Curie temperature, the order no longer exists and we have a paramagnetic material. The change from ferromagnetic or ferrimagnetic materials to paramagnetic is completely reversible on reducing the temperature to its initial value. Curie temperatures are always lower than melting points. A single crystal of iron has the body-centered structure at room temperature. If the magnetization as a function of applied magnetic field is measured, the shape of the curve is found to depend on the direction of the field. This phenomenon is magnetocrystalline anisotropy. Iron has body-centered structure at room temperature, and the “easy” directions of magnetization are those directions parallel to the cube edges [100], [010], and [001] or, collectively, <100>. The hard direction of magnetization for iron is the body diagonal [111]. At higher temperatures, the anisotropy becomes smaller and disappears above 300°C. Nickel crystals (face-centered cubic) have an easy direction of [111] and a hard direction of [100]. Cobalt has the hexagonal close-packed (HCP) structure and the hexagonal axis is the easy direction at room temperature. Magnetocrystalline anisotropy plays a very important part in determining the coercivity of ferro- or ferrimagnetic materials, i.e., the field value at which the direction of magnetization is reversed. Many magnetic materials change dimensions on becoming magnetized: the phenomenon is known as magnetostriction and can be positive, i.e., length increases, or negative. Magnetostriction plays an important role in determining the preferred direction of magnetization of soft, i.e., low Hc, films such as those of alloys of nickel and iron, known as Permalloy. The origin of both magnetocrystalline anisotropy and magnetostriction is spin-orbit coupling. The magnitude of the magnetization of the film is controlled by the electron spin as usual, but the preferred direction of that FIGURE 36.4 Quantum mechanical exchange forces cause a parallel arrangement of the spins of materials for which the ratio of atomic separation, D, is at least 1.5 ¥ d, the diamter of the 3d orbital