SEVENTH EDITION Introduction To Solid State Physics

Transcription

SEVENTH EDITIONIntroduction toSolid State PhysicsCHARLES K IT TEL

14Diamagnetism and ParamagnetismLANGEVIN DIAMAGNETISM EQUATION417QUANTUM THEORY OF DIAMAGNETISM OFMONON UCLEAR SYSTEMS419PARAMAGNETISM420QUANTUM THEORY OF PARAMAGNETISM420Rare earth ions423Hund rules424Iron group ions425Crystal field splitting426Quenching of the orbital angular momentum426Spectroscopie splitting factor429Van Vleck temperature-independent paramagnetism 430COOLING BY ISENTROPIC DEMAGNETIZATIONNuclear demagnetization431432PARAMAGNETIC SUSCEPTffiILITY OFCOND UCTION ELECTRONS433SUMMARY436PROBLEMS4361. Diamagnetic susceptibility of atomic hydrogen2. Hund rules3. Triplet excited states4. Heat capacity from internaI degrees of freedom5. Pauli spin susceptibility6. Conduction electron ferromagnetism7. Two-Ievel system8. Paramagnetism of S 1 system REFERENCES436437437438438438440440440NOTATION : In the problems treated in this chapter the magnetic field B is alwaysclosely equal to the applied field Ba, so that we write B for Ba in most instances.

t -------Or--------- T --------------------------------Pauli paramagnetism (metals)TemperatureDiamagnetismFigure 1 Characteristic magnetic susceptibilities of diamagnetic and paramagnetic substances.416

CHAPT ER14: DIAMAGNE T ISM AND PARAMAGNE TISMMagnetism is inseparable from quantum m echan ics, for a strictly classicalsystem in thermal equilibrium can display no magnetic moment, even in amagnetic field . The magnetic moment of a free atom has three p rincipalsources: the spin with which electrons are endowed; their orbital angular mo mentum about the nucleus; and the change in the orbital moment induced byan applied magne tic field.The first two effects give paramagnetic contributions to the magnetization ,and the third gives a diamagne tic contribution . In the ground Is state of theh ydrogen atpm the orbital moment is zero , and the magnetic moment is that ofthe electron spin along with a small induced diamagne tic moment. In the 1S2state ofhelium the sp in and orbital moments are both zero , and there is only aninduced moment. Atoms with filled electron shells have zero spin and zeroorbital moment: these moments are associated with unfilled shells.The magnetization M is defined as the magnetic moment per unit volume.The magnetic susceptibility p e r unit volume is defined asM(Ce S)x 13 '(SI) X /-LoMB(1)where B is the macroscopic magne tic field intensity. In both systems of units Xis dimensionless. We shall sometimes for convenience refer to MIB as the sus ceptibility without specifying the syste m of units .Quite frequ e ntly a susceptibility is defi ned refe rred to unit mass Or to amole of the substance . The molar susceptibility is written as XM ; the magneticmoment per gram is sometimes writte n as CT. Subs tances with a negative mag netic susceptibility are called diamagnetic. Substances with a positive suscepti bility are called paramagnetic, as in Fig. 1.O rdered arrays of magn etic moments are discussed in Chapter 15; thearrays may be fe rromagne tic, ferrimagnetic, antiferromagne tic, helical, ormore complex in fo rm. N uclear magnetic moments give rise to nuclea rparamagne tism . Magnetic moments of nuclei are of th e order of 10- 3 timessmalle r than the magnetic momen t of th e electron.LANGEVIN DIAMAGNETISM EQUATIONDiamagnetism is associated with the tendency of electrical charges par tially to shield the in terior of a body from an applied magnetic field. In electro magne tism we are fam iliar with Lenz's law: when the fl ux th rough an electricalcircuit is changed, an induced current is set up in such a direction as to opposethe flux change .417

418In a superconductor or in an electron orbit within an atom, the inducedcurrent persists as long as the field is present. The magnetic fie ld of the inducedcurrent is opposite to the applied field, and the magnetic moment associatedwith the current is a diamagnetic moment. Even in a normal metal there is adiamagnetic contribution from the conduction electrons, and this diamag netism is not destroyed by collisions of the electrons.The usual treatment of the diamagnetism of atoms and ions employs theLarmor theorem : in a magnetic field the motion of the electrons around acentral nucleus is , to the first order in B, the same as a possible motion in theabsence of B except for the superposition of a precession of the electrons withangular frequency(ces)(SI)w eB/2mcw eB/2m.(2)If the field is applied slowly, the motion in the rotating reference system will bethe same as the original motion in the rest system before the application of thefield .If the average electron current around the nucleus is zero initially, theapplication of the magnetic field will cause a finite current around the nu cleus. The current is equivalent to a magnetic moment opposite to the appliedfield. It is assumed that the Larmor frequency (2) is mu ch lower than the fre q uency of the original motion in the central field . This condition is not satisfiedin free carrier cyclotron resonance, and the cyclotron frequency is twice thefreq uency (2).The Larmor precession of Z electrons is equivalent to an electric current1 (charge)(revolutions per unit time)(SI) (-eB)Ze) ( - 1 . - .271' 2m(3)The magnetic moment I.L of a current loop is give n by the product(current) X (area of the loop). The are a of the loop of radius p is 7TP'2. We have(S I)JI. -ZtfB4m (p'l) ;(4)Here (p'2) (x'2) (y'2) is the mean square of the perpendicular distance of theelectron from the field axis thro tigh the nucleus. The mean square distance ofthe electrons from the nucleus is (r'2) (x 2) (y2) (Z2). For a sphericallysymmetrical distribution of charge we have (x 2) (y2) (Z2), so that (r 2) i(p2).From (4) the diamagnetic susceptibility per unit volume is, if N is thenumber of atoms per unit volume,2 NI.L NZe (r'2)(ces)(5)XB6mc'2'

142x ILQNIl- (SI)Diamagnetism and ParamagnetismILQNZe (r 2 )B6mThis is the classical Langevin result.The problem of calculating the diamagnetic susceptibili ty of an isolatedatom is reduced to the calculation of (r 2 ) for the electron distribution within theatom . The distribution can be calculated by quantum mechanics .Experimental values for neutral atoms are most easily obtained for theinert gases. Typical experimental values of the molar susceptibilities are thefollowing :XM in CGS in 10- 6 cm3lrnole:HeNeArKrXe-1.9-7 .2-19.4-28.0-43.0In dielectric solids the diamagnetic contribution of the ion cores is de scribed roughly by the Langevin result. The contribution of con duction elec trons is more complicated, as is evident from the de H aas-van Alphen effectdiscussed in Chapter 9.QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTE MSFrom (G . 18) the effect of a magnetic field is to add to the hamiltonian thetermsiehe2J-C -(V' . A A· V') - A22mc2mc2,(6)for an atomic electron these tenns may usually be treated as a small perturba tion . If the magnetic field is uniform and in the z direction , we may writeA x - yB ,hB,Ay Az 0 ,(7)and (6) becomesiehB(dJ-C - - x2mcdyd)2B2e - (x 2 y2)- y- 2dx8mc(8)The first term on the right is proportional to the orbital angular mUlnen tum component Lz if r is measured from th e nucleus. In mononuclear syste msthis term gives rise only to paramagnetism . The second term gives fo r a spheri cally symmetric system a contribution2E' 2e B2 (r 2 )-1n1C 2(9)'419

moment isThenetic:inwithtaisin:lar oxygen and organic """ ' ,","'04. Metals The1l"15" ;U"moment of an atom or ion in free space is givenwhere the total angular momentumangular momenta The constant 1'îs the ratio of thetum; l' is called theagdefined byg 2.the Landé equationFor anfactor isg l IiL andliSmoment to the angular momen asFor a free atom the g ---'---- --'---'-

14s::'If.02ms( ///--',1.00 141 0.75IJ.z0.8. 0 .502ILB".Diamagnetism and Paramagnetism"s::- ILB0251-2ec.!J.o1oI1i i1.00.51.52.0ILBlkBTFigure 2 Energy level splitting for one electronin a magnetic field B directed along the positive zaxis. For an electron the magnetic moment JL isopposite in sign to the spin S, so that JL -gJLBS. In th e low energy state the magneticmoment is paraIJel ta the magnetic field.Figure 3 Fractional populations of a two-levelsystem in thermal equilibrium at temperature Tin a magnetic field B. Th e magnetic moment isproportional ta the difference between the twocurves.The Bohr magneton J-tB is defined as eh/2mc in ces and eh/2m in SI. It isclosely equal to the spin magnetic moment of a free electron .The energy levels of the system in a magnetic field areU- P' B mjgJ-tBB , (14)where mj is the azimuthal quantum number and has the values J, J - l, . ,- J. For a single spin with no orbi tal moment we have mj i and g 2,whence U J-tBB. This splitting is shown in F ig. 2.If a system has only two levels the equilibrium populations are, withT kBT,exp(J-tBIT)NI(15)exp(j.LBiT) exp(- j.LBIT) ,NNzNexp(- J-tB IT)exp(J-tBIT) exp( - j.LBiT) ,(16)here N j , N z are the populations of the lower and upper levels, andN N j N 2 is the total number of atoms. The fractional populations are plot ted in Fig. 3.The projection of the magnetic moment of the upper state along the fielddirection is - J-t and of the lower state is J-t. The resultant magnetization for Natoms per unit volume is , with x J-tB/kBT,M (N I - N 2 )J-t NJ-t ·For x l , tanh x x,eX - e- Xxe , NJ-t tanh x .e(17)and we haveM NJ-t(J-tB/kBT)(18)In a magnetic field an atom with angular momentum quantum number Jhas 2J 1 equally spaced energy levels. The magnetization (Fig. 4) is given byM NgJJ-tB Bj(x) ,(x gJ J-tBB/k BT ) ,(19)421

4227.001TIIInITT':D:o:F:::P:I5F FiTïi'BIT inkG deg- LFigure 4 Plot of magnetic moment versus BIT for sphe rical samples of (1) potassium ch romiumalum, (II) ferric ammonium alum , and (III) gadolinium sulfate octahydrate. Over 99.5% magneticsaturation is achieved at 1.3 K and about 50,000 gauss. (ST). After W. E . Henry.where the Brillouin function BI is defined byB,(x). 2J 1 ctnh ((2J 2J2Jl)x)- - 1 ctnh ( - x )2J2JEquation (17) is a special case of (20) for J t.For x s:; l, we have1xx3ctnh x - - - x345(20)(21)and the susceptibility isM- BNJ(J 1)g2JL C3k B TT(22)H ere p is the effective number of Bohr magnetons, defined asp gU(J 1)F /2.(23)

14 Dianwgnetism and Paranwgnetism40 -------- ----' - ------ siTemperature, JeFigure 5 Plot of l/X vs T for a gadolinium salt, Gd(C zH 5 S0 4hOnnes,)Curie law, (Aftel' L. C. Jackson andRare Earth Ions.straight linethe

Even in theno otheratomstate is charac maximummaximum value of theof S,S allowedmomentumexclusionconsistent withto IL - SI when theshell is more than half fulLruIeL0, soisdifferent

14Table lDiamagnetism and ParamagnetismEffective magneton numbers p for trivalent lanthanide group ions(Near room tempe rature)IonConfigurationc é Pr 3 4P5s2 p64j25s 2 p64P5s 2 p64f 4 5s2 p64f s5s 2p 64f6 5s 2 p64F5s 2 p64j'B5s 2 p64f 9 5s 2 p64po5s2 p64f1l5 s2 p64P 25s 2 p64P 3 5s 2 p6Basic level--- p(calc) gU(] 1)]JJ2p(exp),approximate. :lNd 3 Pm 3 Sm 3 Eu3 Gd 3 Tb 3 D y 3 Ho3 E r3 Tm 3 Yb 3 2Fs I23H441 9125146Hsf27Fo8S 7127F66H1SI2s Is41 1S123H62F7i22. 543. 583.622. 680.8407.949.7210.6310.609.597. 574.542.43. 53.51.53.48.09.510.610.49.57.34.5The second Hund rule is best approached by model calculations. Paulingand Wilson, l for example, give a calculation of the spectral terms that arise fro mthe configuration p2. The third Hund rule is a consequence of the sign of thespin-orbit interaction: For a single electron the energy is lowest when the spinis antiparallel to the orbital angular momentum. But the Iow energy pairs mL,ms are progressively used up as we add electrons to the shell; by the exclusionprinciple when the shell is more th an half full the state of lowest energy neces sarily has the spin parallel ta the orbit.Consider two examples of the Hund fuIes : The ion c é has a single felectron; an f electron has l 3 and s i. Because the f shell is less than halffull, the ] value by the preceding rule is IL - SI L - ! l The ion Pr 3 hastwo f electrons: one of the mIes tells us that the spins add to give S 1. Both felectrons cannot have ml 3 without violating the Pauli exclusion principle, sothat the maximum L consistent with the Pauli principle is not 6, but 5. The]value is IL - si 5 - 1 4.Iron Group IonsTable 2 shows that he experimental magneton numbers for salts of the irontransition group of the p elt'iodic table are in poor agreement with (18). Thevalues often agree quite weil with magneton numbers p 2[S(S 1)]112 calcuIL. Pauling and E . B. Wilson, Introduction to quantum mechanics, McGraw-Hill, 1935,pp. 239-246.425

426Table 2E ffective magneton numbers for iron group ionsIonConfigurationTi3 , y4 y 3 Cr3 , y2 M n 3 , Cr F e 3 , Mn 2 Fe 2 C o 2 Ni 2 Cu 2 3d l3d 23d 33d 43d 53d63d73d 83d 9Basiclevel2D3F3I224F 3/25DO6551 25D44F 9/23F42D5/ 2 p(calc) gU(] 1)]1122[ ( 1)]112p(exp)a1.551.630. 7705.926.706.635.593.551. .44.83.21.9p(calc)"Representative values.lated as if the orbital moment were not there at ail. We say that the orbitalmoments are quenched.Crystal Field SplittingThe difference in behavior of the rare earth and the iron group salts is thatthe 4f shell responsible for paramagnetism in the rare earth ions lies deepinside the ions, within the 5s and 5p sheIls, whereas in the iron group ions the3d shell responsible for paramagnetism is the outermost shell. The 3d shellexperiences the intense inhomogeneous electric field produced by neighboringions. This inhomogeneous electric field is called the crystal field. The interac tion of the paramagnetic ions with the crystal field has two major effects: thecoupling of L and S vectors is largely broken up, so that the states are nO longerspecified by their J values; further, the 2L l sub levels belonging t

Introduction to Solid State Physics . CHARLES KIT TEL . 14 . Diamagnetism and Paramagnetism . LANGEVIN DIAMAGNETISM EQUATION 417 QUANTUM THEORY OF DIAMAGNETISM OF MONONUCLEAR SYSTEMS 419 PARAMAGNETISM 420 QUANTUM THEORY OF PARAMAGNETISM 420 Rare earth ions 423 Hund rules 424 Iron group ions 425 Crystal field splitting 426 Quenching of