12 Magnetic Properties of Materials 12.1·Fundamentals Modern technology would be unthinkable without magnetic ma- terials and magnetic phenomena.Magnetic tapes or disks(for computers,video recorders,etc.)motors,generators,telephones, transformers,permanent magnets,electromagnets,loudspeak- ers,and magnetic strips on credit cards are only a few examples of their applications.To a certain degree,magnetism and elec- tric phenomena can be considered to be siblings since many com- mon mechanisms exist such as dipoles,attraction,repulsion, spontaneous or forced alignment of dipoles,field lines,field strengths,etc.Thus,the governing equations often have the same form.Actually,electrical and magnetic phenomena are linked by the famous Maxwell equations,which were mentioned already in Chapter 10. At least five different kinds of magnetic materials exist.They have been termed para-,dia-,ferro-,ferri-,and antiferromagnet- ics.A qualitative as well as a quantitative distinction between these types can be achieved in a relatively simple way by utiliz- ing a method proposed by Faraday.The magnetic material to be investigated is suspended from one of the arms of a sensitive bal- ance and is allowed to reach into an inhomogeneous magnetic field(Figure 12.1).Diamagnetic materials are expelled from this field,whereas para-,ferro-,antiferro-,and ferrimagnetics are at- tracted in different degrees.It has been found empirically that the apparent loss or gain in mass,that is,the force,F,on the sample exerted by the magnetic field,is: FVxH盟 (12.1)
12 Modern technology would be unthinkable without magnetic materials and magnetic phenomena. Magnetic tapes or disks (for computers, video recorders, etc.) motors, generators, telephones, transformers, permanent magnets, electromagnets, loudspeakers, and magnetic strips on credit cards are only a few examples of their applications. To a certain degree, magnetism and electric phenomena can be considered to be siblings since many common mechanisms exist such as dipoles, attraction, repulsion, spontaneous or forced alignment of dipoles, field lines, field strengths, etc. Thus, the governing equations often have the same form. Actually, electrical and magnetic phenomena are linked by the famous Maxwell equations, which were mentioned already in Chapter 10. At least five different kinds of magnetic materials exist. They have been termed para-, dia-, ferro-, ferri-, and antiferromagnetics. A qualitative as well as a quantitative distinction between these types can be achieved in a relatively simple way by utilizing a method proposed by Faraday. The magnetic material to be investigated is suspended from one of the arms of a sensitive balance and is allowed to reach into an inhomogeneous magnetic field (Figure 12.1). Diamagnetic materials are expelled from this field, whereas para-, ferro-, antiferro-, and ferrimagnetics are attracted in different degrees. It has been found empirically that the apparent loss or gain in mass, that is, the force, F, on the sample exerted by the magnetic field, is: F V $ 0 H d d H x , (12.1) Magnetic Properties of Materials 12.1 • Fundamentals
224 12.Magnetic Properties of Materials FIGURE 12.1.Measure- ment of the magnetic susceptibility in an in- homogeneous mag- netic field.The mag- netic field lines (dashed)follow the iron core. where V is the volume of the sample,uo is a universal constant called the permeability of free space (1.257 X 10-6 H/m or Vs/Am), and X is the susceptibility,which expresses how responsive a ma- terial is to an applied magnetic field.Characteristic values for X are given in Table 12.1.The term dH/dx in Eq.(12.1)is the change of the magnetic field strength H in the x-direction.The field strength H of an electromagnet (consisting of helical windings of a long,in- sulated wire as seen in the lower portion of Figure 12.1)is pro- portional to the current,I,which flows through this coil,and on the number,n,of the windings (called turns)that have been used to make the coil.Further,the magnetic field strength is inversely proportional to the length,L,of the solenoid.Thus,the magnetic field strength is expressed by: H=I L (12.2) The field strength is measured(in SI units)in "Amp-turns per meter"or shortly,in A/m. The magnetic field can be enhanced by inserting,say,iron,into a solenoid,as shown in Figure 12.1.The parameter which ex- presses the amount of enhancement of the magnetic field is called the permeability u.The magnetic field strength within a mate- rial is known by the names magnetic induction!(or magnetic Calling B "magnetic induction"is common practice but should be dis- couraged because it may be confused with electromagnetic induction, as shown in Figure 10.3
where V is the volume of the sample, 0 is a universal constant called the permeability of free space (1.257 106 H/m or Vs/Am), and $ is the susceptibility, which expresses how responsive a material is to an applied magnetic field. Characteristic values for $ are given in Table 12.1. The term dH/dx in Eq. (12.1) is the change of the magnetic field strength H in the x-direction. The field strength H of an electromagnet (consisting of helical windings of a long, insulated wire as seen in the lower portion of Figure 12.1) is proportional to the current, I, which flows through this coil, and on the number, n, of the windings (called turns) that have been used to make the coil. Further, the magnetic field strength is inversely proportional to the length, L, of the solenoid. Thus, the magnetic field strength is expressed by: H I L n . (12.2) The field strength is measured (in SI units) in “Amp-turns per meter” or shortly, in A/m. The magnetic field can be enhanced by inserting, say, iron, into a solenoid, as shown in Figure 12.1. The parameter which expresses the amount of enhancement of the magnetic field is called the permeability . The magnetic field strength within a material is known by the names magnetic induction1 (or magnetic 224 12 • Magnetic Properties of Materials X L N S FX I FIGURE 12.1. Measurement of the magnetic susceptibility in an inhomogeneous magnetic field. The magnetic field lines (dashed) follow the iron core. 1Calling B “magnetic induction” is common practice but should be discouraged because it may be confused with electromagnetic induction, as shown in Figure 10.3.
12.1·Fundamentals 225 TABLE 12.1.Magnetic constants of some materials at room temperature Type of Material X(SI)unitless X(cgs)unitless u unitless magnetism Bi -165×10-6 -13.13×10-6 0.99983 Ge -71.1×10-6 -5.66×10-6 0.99993 Au -34.4×10-6 -2.74×10-6 0.99996 Diamagnetic Ag -25.3×10-6 -2.016×10-6 0.99997 Be -23.2×10-6 -1.85×10-6 0.99998 Cu -9.7×10-6 -0.77×10-6 0.99999 Superconductorsa -1.0 ~-8×10-2 0 B-Sn +2.4×10-6 +0.19×10-6 1 Al +20.7×10-6 +1.65×10-6 1.00002 Paramagnetic W +77.7×10-6 +6.18×10-6 1.00008 Pt +264.4×10-6 +21.04×10-6 1.00026 Low carbon steel 5×103 Fe-3%Si (grain-oriented) Approximately the same as u 4×104 Ferromagnetic Ni-Fe-Mo (supermalloy) because of x=u-1. 106 a See Sections 11.3 and 12.2.1 Note:The table lists the unitless susceptibility,x,in SI and cgs units.(The difference is a factor of 4m,see Appendix Il.)Other sources may provide mass,atomic,molar,volume,or gram equiv- alent susceptibilities in cgs or mks units. Source:Landolt-Bornstein,Zahlenwerte der Physik,Vol.11/9,6th Edition,Springer-Verlag,Berlin (1962). flux density)and is denoted by B.Magnetic field strength and magnetic induction are related by the equation: B=u uoH. (12.3) The SI unit for B is the tesla(T)and that of uo is henries per me- ter (H/m or Vs/Am);see Appendix II.The permeability (some- times called relative permeability,ur)in Eq.(12.3)is unitless and is listed in Table 12.1 for some materials.The relationship be- tween the susceptibility and the permeability is u=1+X. (12.4) For empty space and,for all practical purposes,also for air,one defines X=0 and thus u=1 [See Eq.(12.4)].The susceptibility is small and negative for diamagnetic materials.As a conse- quence,u is slightly less than 1 (see Table 12.1).For para-and antiferromagnetic materials,X is again small,but positive.Thus, u is slightly larger than 1.Finally,X and u are large and positive for ferro-and ferrimagnetic materials.The magnetic constants are temperature-dependent,except for diamagnetic materials,as
flux density) and is denoted by B. Magnetic field strength and magnetic induction are related by the equation: B 0H. (12.3) The SI unit for B is the tesla (T) and that of 0 is henries per meter (H/m or Vs/Am); see Appendix II. The permeability (sometimes called relative permeability, r) in Eq. (12.3) is unitless and is listed in Table 12.1 for some materials. The relationship between the susceptibility and the permeability is 1 $. (12.4) For empty space and, for all practical purposes, also for air, one defines $ 0 and thus 1 [See Eq. (12.4)]. The susceptibility is small and negative for diamagnetic materials. As a consequence, is slightly less than 1 (see Table 12.1). For para- and antiferromagnetic materials, $ is again small, but positive. Thus, is slightly larger than 1. Finally, $ and are large and positive for ferro- and ferrimagnetic materials. The magnetic constants are temperature-dependent, except for diamagnetic materials, as 12.1 • Fundamentals 225 TABLE 12.1. Magnetic constants of some materials at room temperature Type of Material $ (SI) unitless $ (cgs) unitless unitless magnetism Bi 165 106 13.13 106 0.99983 Ge 71.1 106 5.66 106 0.99993 Au 34.4 106 2.74 106 0.99996 Diamagnetic Ag 25.3 106 2.016 106 0.99997 Be 23.2 106 1.85 106 0.99998 Cu 9.7 106 0.77 106 0.99999 Superconductorsa 1.0 8 102 0 -Sn 2.4 106 0.19 106 1 Al 20.7 106 1.65 106 1.00002 Paramagnetic W 77.7 106 6.18 106 1.00008 Pt 264.4 106 21.04 106 1.00026 Low carbon steel 5 103 Fe–3%Si (grain-oriented) 4 104 Ferromagnetic Ni–Fe–Mo (supermalloy) 106 a See Sections 11.3 and 12.2.1 Note: The table lists the unitless susceptibility, $, in SI and cgs units. (The difference is a factor of 4, see Appendix II.) Other sources may provide mass, atomic, molar, volume, or gram equivalent susceptibilities in cgs or mks units. Source: Landolt-Börnstein, Zahlenwerte der Physik, Vol. 11/9, 6th Edition, Springer-Verlag, Berlin (1962). Approximately the same as because of $ 1.
226 12.Magnetic Properties of Materials we will see later.Further,the susceptibility for ferromagnetic ma- terials depends on the field strength,H. The magnetic field parameters at a given point in space are, as explained above,the magnetic field strength H and the mag- netic induction B.In free (empty)space,B and uoHl are identi- cal,as seen in Eq.(12.3).Inside a magnetic material the induc- tion B consists of the free-space component (uoH)plus a contribution to the magnetic field (uoM)which is due to the pres- ence of matter [Figure 12.2(a)],that is, B=uoH uoM, (12.5) where M is called the magnetization of the material.Combining Egs.(12.3)through (12.5)yields: M=XH. (12.6) H,B,and M are actually vectors.Specifically,outside a mater- ial,H(and B)point from the north to the south pole.Inside of a ferro-or paramagnetic material,B and M point from the south N N N HoH H (a) (b) (c) (d) FiGURE 12.2.Schematic representation of magnetic field lines in and around different types of ma- terials.(a)Para-or ferromagnetics.The magnetic induction (B)inside the material consists of the free-space component(uoH)plus a contribution by the material (uoM);see Eq.(12.5).(b)The magnetic field lines outside a material point from the north to the south poles,whereas inside of para-or ferromagnetics,B and poM point from south to north in order to maintain continuity. (c)In diamagnetics,the response of the material counteracts (weakens)the external magnetic field. (d)In a thin surface layer of a superconductor,a supercurrent is created (below its transition tem- perature)which causes a magnetic field that opposes the external field.As a consequence,the magnetic flux lines are expelled from the interior of the material.Compare to Figure 11.27
we will see later. Further, the susceptibility for ferromagnetic materials depends on the field strength, H. The magnetic field parameters at a given point in space are, as explained above, the magnetic field strength H and the magnetic induction B. In free (empty) space, B and 0H are identical, as seen in Eq. (12.3). Inside a magnetic material the induction B consists of the free-space component (0H) plus a contribution to the magnetic field (0M) which is due to the presence of matter [Figure 12.2(a)], that is, B 0H 0M, (12.5) where M is called the magnetization of the material. Combining Eqs. (12.3) through (12.5) yields: M $ H. (12.6) H, B, and M are actually vectors. Specifically, outside a material, H (and B) point from the north to the south pole. Inside of a ferro- or paramagnetic material, B and M point from the south 226 12 • Magnetic Properties of Materials N S N S N S N S 0H 0H B S N 0M 0M 0M 0M (a) (b) (c) (d) FIGURE 12.2. Schematic representation of magnetic field lines in and around different types of materials. (a) Para- or ferromagnetics. The magnetic induction (B) inside the material consists of the free-space component (0H) plus a contribution by the material (0M); see Eq. (12.5). (b) The magnetic field lines outside a material point from the north to the south poles, whereas inside of para- or ferromagnetics, B and 0M point from south to north in order to maintain continuity. (c) In diamagnetics, the response of the material counteracts (weakens) the external magnetic field. (d) In a thin surface layer of a superconductor, a supercurrent is created (below its transition temperature) which causes a magnetic field that opposes the external field. As a consequence, the magnetic flux lines are expelled from the interior of the material. Compare to Figure 11.27
12.2.Magnetic Phenomena and Their Interpretation 227 to the north;see Figures 12.2(a)and (b).However,we will mostly utilize their moduli in the following sections and thus use light- face italic letters. B was called above to be the magnetic flux density in a mate- rial,that is,the magnetic flux per unit area.The magnetic flux is then defined as the product of B and area A,that is,by 中=BA. (12.7) Finally,we need to define the magnetic momentn(also a vector)through the following equation: M=0, (12.8) which means that the magnetization is the magnetic moment per unit volume. A short note on units should be added.This book uses SI units throughout.However,the scientific literature on magnetism(par- ticularly in the United States)is still widely written in electro- magnetic cgs(emu)units.The magnetic field strength in cgs units is measured in Oersted and the magnetic induction in Gauss. Conversion factors from SI into cgs units and for rewriting Eqs. (12.1)-(12.8)in cgs units are given in Appendix II. 12.2.Magnetic Phenomena and Their Interpretation We stated in the last section that different types of magnetism ex- ist which are characterized by the magnitude and the sign of the susceptibility (see Table 12.1).Since various materials respond so differently in a magnetic field,we suspect that several funda- mentally different mechanisms must be responsible for the mag- netic properties.We shall now attempt to unfold the multiplicity of the magnetic behavior of materials by describing some perti- nent experimental findings and giving some brief interpretations. 12.2.1 Ampere postulated more than one hundred years ago that so- Diamagnetism called molecular currents are responsible for the magnetism in solids.He compared these molecular currents to an electric cur- rent in a loop-shaped piece of wire which is known to cause a magnetic moment.Today,we replace Ampere's molecular cur- rents by orbiting valence electrons. To understand diamagnetism,a second aspect needs to be con- sidered.As explained in Chapter 10 a current is induced in a wire loop whenever a bar magnet is moved toward (or from)this loop
to the north; see Figures 12.2(a) and (b). However, we will mostly utilize their moduli in the following sections and thus use lightface italic letters. B was called above to be the magnetic flux density in a material, that is, the magnetic flux per unit area. The magnetic flux is then defined as the product of B and area A, that is, by B A. (12.7) Finally, we need to define the magnetic moment m (also a vector) through the following equation: M V m , (12.8) which means that the magnetization is the magnetic moment per unit volume. A short note on units should be added. This book uses SI units throughout. However, the scientific literature on magnetism (particularly in the United States) is still widely written in electromagnetic cgs (emu) units. The magnetic field strength in cgs units is measured in Oersted and the magnetic induction in Gauss. Conversion factors from SI into cgs units and for rewriting Eqs. (12.1)–(12.8) in cgs units are given in Appendix II. We stated in the last section that different types of magnetism exist which are characterized by the magnitude and the sign of the susceptibility (see Table 12.1). Since various materials respond so differently in a magnetic field, we suspect that several fundamentally different mechanisms must be responsible for the magnetic properties. We shall now attempt to unfold the multiplicity of the magnetic behavior of materials by describing some pertinent experimental findings and giving some brief interpretations. Ampère postulated more than one hundred years ago that socalled molecular currents are responsible for the magnetism in solids. He compared these molecular currents to an electric current in a loop-shaped piece of wire which is known to cause a magnetic moment. Today, we replace Ampère’s molecular currents by orbiting valence electrons. To understand diamagnetism, a second aspect needs to be considered. As explained in Chapter 10 a current is induced in a wire loop whenever a bar magnet is moved toward (or from) this loop. 12.2.1 Diamagnetism 12.2 • Magnetic Phenomena and Their Interpretation 227 12.2 • Magnetic Phenomena and Their Interpretation
228 12.Magnetic Properties of Materials FIGURE 12.3.Induction of a current in a loop-shaped piece of wire by moving a bar magnet toward the wire loop.The current in the loop causes a magnetic field that is directed opposite to the magnetic field of the bar magnet (Lenz law). The current thus induced causes,in turn,a magnetic moment that is opposite to the one of the bar magnet(Figure 12.3).(This has to be so in order for mechanical work to be expended in pro- ducing the current,i.e.,to conserve energy;otherwise,a perpet- ual motion would be created!)Diamagnetism may then be ex- plained by postulating that the external magnetic field induces a change in the magnitude of the atomic currents,i.e.,the external field accelerates or decelerates the orbiting electrons,so that their magnetic moment is in the opposite direction to the external mag- netic field.In other words,the responses of the orbiting electrons counteract the external field [Figure 12.2(c)]. Superconductors have extraordinary diamagnetic properties They completely expel the magnetic flux lines from their interior when in the superconducting state (Meissner effect).In other words,a superconductor behaves in a magnetic field as if B would be zero inside the material [Figure 12.2(d)].Thus,with Eq.(12.5) one obtains: H=-M, (12.9) which means that the magnetization is equal and opposite to the external magnetic field strength.The result is a perfect diamag- net.The susceptibility, 也 (12.6) H in superconductors is therefore-1 compared to about-10-5 in the normal state (see Table 12.1).This strong diamagnetism can be used for frictionless bearings,that is,for support of loads by a repelling magnetic force.The often-demonstrated levitation effect in which a magnet hovers above a superconducting material also can be ex- plained by these strong diamagnetic properties of superconductors
The current thus induced causes, in turn, a magnetic moment that is opposite to the one of the bar magnet (Figure 12.3). (This has to be so in order for mechanical work to be expended in producing the current, i.e., to conserve energy; otherwise, a perpetual motion would be created!) Diamagnetism may then be explained by postulating that the external magnetic field induces a change in the magnitude of the atomic currents, i.e., the external field accelerates or decelerates the orbiting electrons, so that their magnetic moment is in the opposite direction to the external magnetic field. In other words, the responses of the orbiting electrons counteract the external field [Figure 12.2(c)]. Superconductors have extraordinary diamagnetic properties. They completely expel the magnetic flux lines from their interior when in the superconducting state (Meissner effect). In other words, a superconductor behaves in a magnetic field as if B would be zero inside the material [Figure 12.2(d)]. Thus, with Eq. (12.5) one obtains: H M, (12.9) which means that the magnetization is equal and opposite to the external magnetic field strength. The result is a perfect diamagnet. The susceptibility, $ M H , (12.6) in superconductors is therefore 1 compared to about 105 in the normal state (see Table 12.1). This strong diamagnetism can be used for frictionless bearings, that is, for support of loads by a repelling magnetic force. The often-demonstrated levitation effect in which a magnet hovers above a superconducting material also can be explained by these strong diamagnetic properties of superconductors. FIGURE 12.3. Induction of a current in a loop-shaped piece of wire by moving a bar magnet toward the wire loop. The current in the loop causes a magnetic field that is directed opposite to the magnetic field of the bar magnet (Lenz law). 228 12 • Magnetic Properties of Materials m i N S
12.2.Magnetic Phenomena and Their Interpretation 229 FIGURE 12.4.(a)Schematic represen- Nucleus tation of electrons which spin around their own axis.A(para)mag- netic moment um results;its direc- tion depends on the mode of rota- tion.Only two spin directions are shown (called "spin up"and "spin down").(b)An orbiting electron is the source for electron-orbit para- (a) (b) magnetism. 12.2.2Para- Paramagnetism in solids is attributed to a large extent to a mag- magnetism netic moment that results from electrons which spin around their own axis;see Figure 12.4(a).The spin magnetic moments are gen- erally randomly oriented so that no net magnetic moment results An external magnetic field tries to turn the unfavorably oriented spin moments in the direction of the external field,but thermal agitation counteracts the alignment.Thus,spin paramagnetism is slightly temperature-dependent.It is generally weak and is ob- served in some metals and in salts of the transition elements. Free atoms(dilute gases)as well as rare earth elements and their salts and oxides possess an additional source of paramag- netism.It stems from the magnetic moment of the orbiting elec- trons;see Figure 12.4(b).Without an external magnetic field, these magnetic moments are,again,randomly oriented and thus mutually cancel one another.As a result,the net magnetization is zero.However,when an external field is applied,the individ- ual magnetic vectors tend to turn into the field direction which may be counteracted by thermal agitation.Thus,electron-orbit paramagnetism is also temperature-dependent.Specifically,para- magnetics often (not always!)obey the experimentally found Curie-Weiss law: X=。 (12.10) where C and 0 are constants (given in Kelvin),and C is called the Curie Constant.The Curie-Weiss law is observed to be valid for rare earth elements and salts of the transition elements,for ex- ample,the carbonates,chlorides,and sulfates of Fe,Co,Cr,Mn. From the above-said it becomes clear that in paramagnetic ma- terials the magnetic moments of the electrons eventually point in the direction of the external field,that is,the magnetic moments enhance the external field [see Figure 12.2(a)].On the other hand, diamagnetism counteracts an external field [see Figure 12.2(c)]
Paramagnetism in solids is attributed to a large extent to a magnetic moment that results from electrons which spin around their own axis; see Figure 12.4(a). The spin magnetic moments are generally randomly oriented so that no net magnetic moment results. An external magnetic field tries to turn the unfavorably oriented spin moments in the direction of the external field, but thermal agitation counteracts the alignment. Thus, spin paramagnetism is slightly temperature-dependent. It is generally weak and is observed in some metals and in salts of the transition elements. Free atoms (dilute gases) as well as rare earth elements and their salts and oxides possess an additional source of paramagnetism. It stems from the magnetic moment of the orbiting electrons; see Figure 12.4(b). Without an external magnetic field, these magnetic moments are, again, randomly oriented and thus mutually cancel one another. As a result, the net magnetization is zero. However, when an external field is applied, the individual magnetic vectors tend to turn into the field direction which may be counteracted by thermal agitation. Thus, electron-orbit paramagnetism is also temperature-dependent. Specifically, paramagnetics often (not always!) obey the experimentally found Curie–Weiss law: $ T C (12.10) where C and are constants (given in Kelvin), and C is called the Curie Constant. The Curie–Weiss law is observed to be valid for rare earth elements and salts of the transition elements, for example, the carbonates, chlorides, and sulfates of Fe, Co, Cr, Mn. From the above-said it becomes clear that in paramagnetic materials the magnetic moments of the electrons eventually point in the direction of the external field, that is, the magnetic moments enhance the external field [see Figure 12.2(a)]. On the other hand, diamagnetism counteracts an external field [see Figure 12.2(c)]. 12.2 • Magnetic Phenomena and Their Interpretation 229 12.2.2 Paramagnetism m m m e – e – e – Nucleus (a) (b) FIGURE 12.4. (a) Schematic representation of electrons which spin around their own axis. A (para)magnetic moment m results; its direction depends on the mode of rotation. Only two spin directions are shown (called “spin up” and “spin down”). (b) An orbiting electron is the source for electron-orbit paramagnetism.
230 12.Magnetic Properties of Materials FIGURE 12.5.Schematic represen- tation of the spin alignment in a d-band which is partially filled with eight electrons (Hund's rule).See also Appendix I. Thus,para-and diamagnetism oppose each other.Solids that have both orbital as well as spin paramagnetism are consequently para- magnetic(since the sum of both paramagnetic compounds is com- monly larger than the diamagnetism).Rare earth metals are an example of this. In many other solids,however,the electron orbits are essentially coupled to the lattice.This prevents the orbital magnetic moments from turning into the field direction.Thus,electron-orbit para- magnetism does not play a role,and only spin paramagnetism re- mains.The possible presence of a net spin-paramagnetic moment depends,however,on whether or not the magnetic moments of the individual spins cancel each other.Specifically,if a solid has completely filled electron bands,then a quantum mechanical rule, called the Pauli principle,requires the same number of electrons with spins up and with spins down [Figure 12.4(a)].The Pauli prin- ciple stipulates that each electron state can be filled only with two electrons having opposite spins,see Appendix I.The case of com- pletely filled bands thus results in a cancellation of the spin mo- ments and no net paramagnetism is expected.Materials in which this occurs are therefore diamagnetic(no orbital and no spin para- magnetic moments).Examples of filled bands are intrinsic semi- conductors,insulators,and ionic crystals such as NaCl. In materials that have partially filled bands,the electron spins are arranged according to Hund's rule in such a manner that the total spin moment is maximized.For example,for an atom with eight valence d-electrons,five of the spins may point up and three spins point down,which results in a net number of two spins up;Figure 12.5.The atom then has two units of(para-) magnetism or,as it is said,two Bohr magnetons per atom.The Bohr magneton is the smallest unit (or quantum)of the mag- netic moment and has the value: g=品=9274×10-20 =(4·m2). (12.11) (The symbols have the usual meanings as listed in Appendix II.) 12.2.3 Ferro- Figure 12.6 depicts a ring-shaped solenoid consisting of a newly magnetism cast piece of iron and two separate coils which are wound around the iron ring.If the magnetic field strength in the solenoid is tem- porally increased (by increasing the current in the primary wind-
Thus, para- and diamagnetism oppose each other. Solids that have both orbital as well as spin paramagnetism are consequently paramagnetic (since the sum of both paramagnetic compounds is commonly larger than the diamagnetism). Rare earth metals are an example of this. In many other solids, however, the electron orbits are essentially coupled to the lattice. This prevents the orbital magnetic moments from turning into the field direction. Thus, electron-orbit paramagnetism does not play a role, and only spin paramagnetism remains. The possible presence of a net spin-paramagnetic moment depends, however, on whether or not the magnetic moments of the individual spins cancel each other. Specifically, if a solid has completely filled electron bands, then a quantum mechanical rule, called the Pauli principle, requires the same number of electrons with spins up and with spins down [Figure 12.4(a)]. The Pauli principle stipulates that each electron state can be filled only with two electrons having opposite spins, see Appendix I. The case of completely filled bands thus results in a cancellation of the spin moments and no net paramagnetism is expected. Materials in which this occurs are therefore diamagnetic (no orbital and no spin paramagnetic moments). Examples of filled bands are intrinsic semiconductors, insulators, and ionic crystals such as NaCl. In materials that have partially filled bands, the electron spins are arranged according to Hund’s rule in such a manner that the total spin moment is maximized. For example, for an atom with eight valence d-electrons, five of the spins may point up and three spins point down, which results in a net number of two spins up; Figure 12.5. The atom then has two units of (para-) magnetism or, as it is said, two Bohr magnetons per atom. The Bohr magneton is the smallest unit (or quantum) of the magnetic moment and has the value: B 4 e h m 9.274 1024 T J (A m2). (12.11) (The symbols have the usual meanings as listed in Appendix II.) Figure 12.6 depicts a ring-shaped solenoid consisting of a newly cast piece of iron and two separate coils which are wound around the iron ring. If the magnetic field strength in the solenoid is temporally increased (by increasing the current in the primary wind- 12.2.3 Ferromagnetism 230 12 • Magnetic Properties of Materials FIGURE 12.5. Schematic representation of the spin alignment in a d-band which is partially filled with eight electrons (Hund’s rule). See also Appendix I.
12.2.Magnetic Phenomena and Their Interpretation 231 Variable D.C. power supply Primary FIGURE 12.6.A ring-shaped solenoid with primary and secondary windings.The magnetic flux lines are indicated by a dashed circle.Note that a current can flow in the secondary circuit only if the current (and therefore the magnetic flux) Secondary in the primary winding changes with time.An on-off switch in the primary cir- cuit may serve this purpose.A flux meter is an ampmeter without retracting Flux meter springs. ing),then the magnetization (measured in the secondary winding with a flux meter)rises slowly at first and then more rapidly,as shown in Figure 12.7(dashed line).Finally,M levels off and reaches a constant value,called the saturation magnetization,Ms.When H is reduced to zero,the magnetization retains a positive value, called the remanent magnetization,or remanence,M.It is this re- tained magnetization which is utilized in permanent magnets.The remanent magnetization can be removed by reversing the magnetic M H FIGURE 12.7.Schematic representation of a hysteresis loop of a ferromagnetic material. The dashed curve is for a newly cast piece of iron (called virgin iron).Compare to Figure 11.29
ing), then the magnetization (measured in the secondary winding with a flux meter) rises slowly at first and then more rapidly, as shown in Figure 12.7 (dashed line). Finally, M levels off and reaches a constant value, called the saturation magnetization, Ms. When H is reduced to zero, the magnetization retains a positive value, called the remanent magnetization, or remanence, Mr. It is this retained magnetization which is utilized in permanent magnets. The remanent magnetization can be removed by reversing the magnetic 12.2 • Magnetic Phenomena and Their Interpretation 231 Variable D.C. power supply Primary Secondary Flux meter S A FIGURE 12.6. A ring-shaped solenoid with primary and secondary windings. The magnetic flux lines are indicated by a dashed circle. Note that a current can flow in the secondary circuit only if the current (and therefore the magnetic flux) in the primary winding changes with time. An on–off switch in the primary circuit may serve this purpose. A flux meter is an ampmeter without retracting springs. Mr Hc M Ms H FIGURE 12.7. Schematic representation of a hysteresis loop of a ferromagnetic material. The dashed curve is for a newly cast piece of iron (called virgin iron). Compare to Figure 11.29
232 12.Magnetic Properties of Materials field strength to a value Hc,called the coercive field.Solids having a large combination of M,and He are called hard magnetic materi- als(in contrast to soft magnetic materials,for which the area inside the loop of Figure 12.7 is very small).A complete cycle through pos- itive and negative H-values as shown in Figure 12.7 is called a hys- teresis loop.It should be noted that a second type of hysteresis curve is often used in which B(instead of M)is plotted versus H. No saturation value for B can be observed;see Eq.(12.3).Removal of the residual induction requires a field that is called coercivity,but the terms coercive field and coercivity are often used interchange- ably.The area within a hysteresis loop (B times H or M times uoH) is proportional to the energy per unit volume,which is dissipated once a full field cycle has been completed;see also Section 11.9. The saturation magnetization is temperature-dependent. Above the Curie temperature,Tc,ferromagnetics become para- magnetic.Table 12.2 lists Curie temperatures for some elements. In ferromagnetic materials,such as iron,cobalt,and nickel, the spins of unfilled d-bands spontaneously align parallel to each other below Te,that is,they align within small domains(1-100 um in size)without the presence of an external magnetic field; Figure 12.8(a).The individual domains are magnetized to satu- ration.The spin direction in each domain is,however,different, so that the individual magnetic moments for virgin ferromag- netic materials as a whole cancel each other and the net mag- netization is zero.An external magnetic field causes those do- mains whose spins are parallel or nearly parallel to the external field to grow at the expense of the unfavorably aligned domains; Figure 12.8(b).When the entire crystal finally contains only one single domain,having spins aligned parallel to the external field direction then the material is said to have reached technical sat- uration magnetization,Ms [Figure 12.8(c)].An increase in tem- perature progressively destroys the spontaneous alignment,thus reducing the saturation magnetization,Figure 12.8(d). We have not yet answered the question of whether or not the flip from one spin direction into the other occurs in one step, that is,between two adjacent atoms or over an extended range of atoms instead.Indeed,a gradual rotation over several hun- TABLE 12.2.Curie temperature,Te for some ferromagnetic materials Metal Te(K) Fe 1043 Co 1404 Ni 631 Gd 289
field strength to a value Hc, called the coercive field. Solids having a large combination of Mr and Hc are called hard magnetic materials (in contrast to soft magnetic materials, for which the area inside the loop of Figure 12.7 is very small). A complete cycle through positive and negative H-values as shown in Figure 12.7 is called a hysteresis loop. It should be noted that a second type of hysteresis curve is often used in which B (instead of M) is plotted versus H. No saturation value for B can be observed; see Eq. (12.3). Removal of the residual induction requires a field that is called coercivity, but the terms coercive field and coercivity are often used interchangeably. The area within a hysteresis loop (B times H or M times 0H) is proportional to the energy per unit volume, which is dissipated once a full field cycle has been completed; see also Section 11.9. The saturation magnetization is temperature-dependent. Above the Curie temperature, Tc, ferromagnetics become paramagnetic. Table 12.2 lists Curie temperatures for some elements. In ferromagnetic materials, such as iron, cobalt, and nickel, the spins of unfilled d-bands spontaneously align parallel to each other below Tc, that is, they align within small domains (1–100 m in size) without the presence of an external magnetic field; Figure 12.8(a). The individual domains are magnetized to saturation. The spin direction in each domain is, however, different, so that the individual magnetic moments for virgin ferromagnetic materials as a whole cancel each other and the net magnetization is zero. An external magnetic field causes those domains whose spins are parallel or nearly parallel to the external field to grow at the expense of the unfavorably aligned domains; Figure 12.8(b). When the entire crystal finally contains only one single domain, having spins aligned parallel to the external field direction then the material is said to have reached technical saturation magnetization, Ms [Figure 12.8(c)]. An increase in temperature progressively destroys the spontaneous alignment, thus reducing the saturation magnetization, Figure 12.8(d). We have not yet answered the question of whether or not the flip from one spin direction into the other occurs in one step, that is, between two adjacent atoms or over an extended range of atoms instead. Indeed, a gradual rotation over several hun- 232 12 • Magnetic Properties of Materials TABLE 12.2. Curie temperature, Tc, for some ferromagnetic materials Metal Tc (K) Fe 1043 Co 1404 Ni 631 Gd 289
