上海交通大学：《材料与文明》课程教学资源（参考资料）Understanding Mater_Chapter 12 - Magnetic Properties of Materials

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 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, $
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 ex￾plained 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 mag￾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: $
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 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)]. 12.2 • Magnetic Phenomena and Their Interpretation 229 12.2.2 Para￾magnetism m m m e – e – e – Nucleus (a) (b) FIGURE 12.4. (a) Schematic represen￾tation of electrons which spin around their own axis. A (para)mag￾netic moment m 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￾magnetism.

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 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: 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 tem￾porally increased (by increasing the current in the primary wind- 12.2.3 Ferro￾magnetism 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.