Physcs 97 Secton 15: Magnetc propertes of materals Defnton of fundamental quanttes When a materal medum s placed n a magnetc feld, the medum s magnetzed. Ths magnetzaton s descrbed by the magnetzaton vector M, the dpole moment per unt volume. Snce the magnetzaton s nduced by the feld, we may assume that M s proportonal to H. That s, M = χ. (1) The proportonalty constant χ s known as the magnetc susceptblty of the medum. Note that the magnetc susceptblty χ bears no physcal relatonshp to the electrc susceptblty, although the same symbol s used for both. Note also that our dscusson assumes that the medum s magnetcally sotropc. ut real crystals are ansotropc, and the susceptblty s represented by a second-rank tensor. In order to avod mathematcal complcatons, however, we shall gnore ansotropc effects n our treatment. Note, that n Eq. (1) we assumed that M s proportonal to, the external feld, and n dong so we gnored such thngs as demagnetzaton feld, whch were ncluded n the electrc case. The neglect of these factors s justfable n the case of paramagnetc and damagnetc materals because M s 5 very small compared to (typcally χ = / M ~ 10 ), unlke the electrc case, n whch χ ~ 1. ut when we deal wth ferromagnetc materals, where M s qute large, ths omsson s no longer tenable, and the above effects must be ncluded. ecause of small value of the magnetc susceptblty we wll not make dstncton between magnetc feld and magnetc nducton. Note also that χ n Eq.(1) can be dependent on the appled magnetc feld. In ths case, we can defne the magnetc susceptblty as follows M χ =. () The magnetzaton can be defned as E M =, (3) where E s the total energy of the system. Defntons () and (3) are more general and can be used n calculatons. Classfcaton of materals All magnetc materals may be grouped nto three magnetc classes, dependng on the magnetc orderng and the sgn, magntude and temperature dependence of the magnetc susceptblty. We wll dscuss propertes of fve classes of materals: damagnetc, paramagnetc, ferromagnetc, antferromagnetc and ferrmagnetc. There s no magnetc order at any temperature n damagnetc and paramagnetc materals, whereas there s a magnetc order at low temperatures n ferromagnetc, antferromagnetc and ferrmagnetc materals. 1
Physcs 97 In damagnetc materals the magnetc susceptblty s negatve. Usually ts magntude s of the order of -10-6 to -10-5. The negatve value of the susceptblty means that n an appled magnetc feld damagnetc materals acqure the magnetzaton, whch s ponted opposte to the appled feld. In damagnetc materals the susceptblty nearly has a constant value ndependent of temperature. Ionc crystals and nert gas atoms are damagnetc. These substances have atoms or ons wth complete shells, and ther damagnetc behavor s due to the fact that a magnetc feld acts to dstort the orbtal moton. Another class of damagnetc materals s noble metals. All the other classes of materals have postve susceptblty. Wthn these classes the magntude of the susceptblty vares over a very wde range. However, at suffcently hgh temperatures the susceptblty decreases wth ncreasng temperature for all materals n these classes. It was found expermentally that all these materals follow the relatonshp χ = C T ± T C more or less exactly for suffcently hgh T. Here C and T C are postve constants ndependent of temperature and dfferent for each materal. It was found that n some materals T C =0 and ths equaton s obeyed down to the lowest temperatures at whch measurements have been made. Ths class of materals s called paramagnetc. In paramagnetc materals χ s postve - that s, for whch M s parallel to. The susceptblty s however s also very small: 10-4 to 10-5. The best-known examples of paramagnetc materals are the ons of transton and rare-earth ons. The fact that these ons have ncomplete atomc shells s what s responsble for ther paramagnetc behavor. In all other materals equaton (4) breaks down as temperature decreases. They all have a crtcal temperature below whch the varaton of susceptblty wth temperature s very dfferent from ts varaton above ths temperature. In ferromagnetc materals the crtcal temperature s called the Cure temperature. Above the Cure temperature the susceptblty follow relatonshp (4) wth a negatve sgn. When temperature approaches T C the magnetc susceptblty tends to be nfnte. An nfnte susceptblty means that a fnte magnetzaton can exst even n zero appled feld, whch s the case n permanent magnets. The problem s that the magnetzaton of ferromagnetc materals n zero feld can have a range of dfferent values and consequently cannot be regarded as a property of the materal. However, t s found that f a relatvely small magnetc feld s appled to these materals, the magnetzaton tends to a constant value, whch s called the saturaton magnetzaton M S or spontaneous magnetzaton. elow Cure temperature M S (T) aganst T follows a unversal curve: t tends to a constant value as T=0; as T ncreases, the spontaneous magnetzaton decreases more and more rapdly. At the Cure temperature the magnetzaton dsappeared. Ferrmagnetc materals have non-zero magnetzaton below the Cure temperature whch s smlar to ferromagnetc materals. However, sgnfcant departures from (4) occur over a range of temperatures. Ths behavour s only followed at temperatures large compared wth the Cure temperature. Another dfference between ferrmagnets and ferromagnets s that n ferrmagnetc materals the saturaton magnetzaton aganst temperature behave n a more complcated way. For (4)
Physcs 97 example, for some ferrmagnets the magnetzaton can ncrease wth ncreasng temperature and then drops down. Antferromagnetc materals have small postve susceptbltes at all temperatures. At hgh temperatures they follow eq. (4) wth T C usually havng a postve sgn. A crtcal temperature n ths case s called Neel temperature. elow the Neel temperature the susceptblty generally decreases wth decreasng temperature. There s no spontaneous magnetzaton n antferromagnetc materals. Calculaton of atomc susceptbltes In the presence of a unform magnetc feld the Hamltonan of an on (atom) s modfed n the two major ways: (1) In the total knetc energy term the momentum of each electron s replaced by e p p + A, (5) c where A s the vector potental assocated wth the magnetc feld such that = A. (6) We assume that the appled feld s unform so that 1 = A r. (7) () The nteracton energy of the feld wth each electron spn must be added to the Hamltonan: H = µ S, (8) spn where µ s the ohr magneton e µ = = mc 8 0.58 10 / ev G. (9) As the result the total energy of electrons wll have a form 1 e H = + µ m p c r S. (10) We denote by T 0 the knetc energy n the absence of the appled feld,.e. T 0 1 = p. (11) m The cross term s the brackets can be rewrtten takng nto account that ( ) ( ) p r = r p. (1) We note that also r and p are quantum-mechancal operators, here we can work wth these quanttes as wth classcal varables because only non-dagonal components enter ths product (.e. 3
Physcs 97 there no terms whch contan, e.g., x components of both r and p whch do not commute). Note: r µ, pν = δ. µν Assumng that the feld s along z drecton, we can rewrte ( ) ( = x + y ) r. (13) Fnally we fnd for the feld-dependent correcton to the total Hamltonan: e H = H T = + + x + y, (14) 0 µ ( L S) 8mc where L s the total orbtal momentum: = ( ) ( ) L r p. (15) The energy correcton due to the appled electrc feld s small compared to electron energes. For 4 example, 1T= µ 1Tesla = 0.58 10 ev. Therefore one can compute the changes n the energy levels nduced by the feld wth ordnary perturbaton theory. Equaton (14) s the bass for theores of the magnetc susceptblty of ndvdual atoms, ons, or molecules. Langevn damagnetsm Let us now apply these results to a sold composed of ons or atoms wth all electronc shells flled. Such atoms have zero spn and orbtal angular momentum n ts ground state,.e. 0 S 0 = 0 L 0 = 0. (16) Consequently only last term n eq.(14) contrbutes to the feld-nduced shft n the ground state energy: e e E = 0 H 0 = 0 x + y 0 = 0 r 0 8mc 1 mc ( ), (17) where the last form follows from the sphercal symmetry of the closed-shell on, 0 x 0 = 0 y 0 = 0 z 0 = 0 r 0 3 1. (18) It s conventonal to defne a mean square onc radus by r = 1 0 r 0, (19) Z where Z s the total number of electrons n an on. We obtan then for the magnetzaton nduced by the appled magnetc feld, accordng to (3): 4
Physcs 97 E e NZ r M = = 6mc whch mples a negatve magnetc susceptblty:, (0) e NZ r χ =, (1) 6mc where N s the number of atoms per unt volume. Damagnetsm s assocated wth the tendency of electrcal charges partally to sheld the nteror of a body from an appled magnetc feld. In electro-magnetsm we are famlar wth Lenz's law: when the magnetc energy flux through an electrcal crcut s changed, an nduced current s set up n such a drecton as to oppose the flux change. Formula (1) can be derved classcally. Consder an electron rotatng about the nucleus n a crcular orbt, and let a magnetc feld be appled perpendcular to the plane of the paper, as shown n Fg. 1. efore ths feld s appled, we have, accordng to Newton's second law, F = mω r () 0 0 where F 0 s the attractve Coulomb force between the nucleus and the electron, and ω 0 s the angular velocty. Fg. 1 Atomc orgn of damagnetsm. The Lorentz force F L opposes the Coulomb force F 0 ; v s the electron velocty. When the feld s appled, an addtonal force starts to act on the electron: the Lorentz force e / c v. For the geometry of Fg.1, the effect s to produce a radally outward force gven by ( ) eω 0 r/c, and Eq. () should therefore be amended to e c ω = ω. (3) F0 0r m r Assumng that s small we can look for a soluton s a form ω = ω0 + ω. (4) 5
Physcs 97 Substtutng (4) n the rght-hand part of the Eq.(3) we fnd e ω =, (5) mc whch shows that the rotaton of the electron has been slowed down. Ths reducton n frequency produces a correspondng change n the magnetc moment. Ths can be calculated as follows. The change n the frequency of rotaton s equvalent to the change n the current around the nucleus, whch s I = (charge) x (revolutons per unt tme) = ( Ze) 1 e π mc. The magnetc moment µ of a current loop s gven by the product (current) x (area of the loop)/c, where c appears due to CGS unts. The area of the loop of radus r s πr. We have then 1 e π r e Z r µ = ( Ze) = π mc c 4mc, (6) Here <r > = <x > + <y > s the mean square of the perpendcular dstance of the electron from the feld axs through the nucleus. The mean square dstance of the electrons from the nucleus s <r > = <x > + <y > + <z >. For a sphercally symmetrcal dstrbuton of charge we have <x > = <y > =<z >, so that s <r > n eq.(6) should be replaced by 3/<r >, whch gves dentcal result to eq.(0). Damagnetsm can be found n onc crystals and crystals composed of nert gas atoms, because these substances have atoms or ons wth complete electronc shells. Another class of damagnetc materals s noble metals whch wll be dscussed later. Paramagnetsm of nsulators If atoms n a sold have non-flled electronc shells than we have to take nto account the frst term n the Hamltonan (14). Its contrbuton s then much larger than the contrbuton from the second term so that we can gnore t. We consder the effect of ths term on an on n a ground state whch can be descrbed by quantum numbers L, S, J and J z, where J s the total angular momentum and J z s the projecton of ths momentum nto a quantzaton axs. It can be shown that LSJJ L + S LSJJ = g LSJJ J LSJJ, (7) z z z z where g s the g-factor, whch s gven by J ( J + 1) + S( S + 1) L( L + 1) g = 1+. (8) J ( J + 1) We stress that ths relaton s vald only wthn the (J + 1) dmensonal set of states that make up the degenerate atomc ground state n zero feld;.e., (8) s obeyed only for matrx elements taken between states that are dagonal n J, L, and S. If the splttng between the zero-feld atomc groundstate multplet and the frst excted multplet s large compared wth k T (as s frequently the case), then only the (J + 1) states n the ground-state multplet wll contrbute apprecably to the free 6
Physcs 97 energy. In that case (and only n that case) Eq. (8) permts one to nterpret the frst term n the Hamltonan (14) as expressng the nteracton E = (9) of the feld wth a magnetc moment that s proportonal to the total angular momentum of the on, so that = gµ J. (30) The appled magnetc feld lfts degeneracy of the manfold of states and splts t nto J+1 equdstant levels, whch hs known as Zeeman splttng. The energes of these levels are gven by E = gµ J, (31) J z z Where J z s an nteger and has values from J to J. If thermal energy s less or comparable wth the Zeeman splttng, these levels wll be populated dfferently and gve a dfferent contrbuton to the magnetc moment of the on. The magnetzaton of the sold s determned by the average value of the magnetc moment, so that M = N, where N s the concentraton of ons f the sold and s the value of magnetc moment averaged over the oltzmann dstrbuton: J gµ J z kt J ze J z = J J gµ J z kt M = N = Ngµ. (3) J z = J e The summaton can be easly performed due to geometrc progresson and the result for the magnetzaton s M = NgJ µ ( x), (33) where x J gµ J kt = and J ( ) x s the rlloun functon defned by J + 1 J + 1 1 x J ( x) = coth x coth J J J J. (34) Fgure shows the dependence of the magnetzaton for three dfferent ons as a functon of appled magnetc feld. Note that n order to reach the saturaton, very low temperatures and very hgh magnetc felds are requred. At relatvely low felds and not too low temperatures we can expand the coth n (34) assumng that x << 1, so that coth ( x) 1 x +. (35) x 3 We can then fnd for the susceptblty: 7
Physcs 97 M NJ ( J + 1) g µ Np µ C χ = = = =, (36) 3k T 3k T T where p s the effectve number of ohr magnetons, defned as [ ( 1) ] 1/ p = g J J +. (37) C s the Cure constant and the form (37) s known as the Cure low. Fg. Magnetc moment versus /T for samples of (I) potassum chromum alum, (II) ferrc ammonum alum, and (III) gadolnum sulfate octahydrate. Over 99.5% magnetc saturaton s acheved at 1.3 K and about 5T. Hund Rules The Hund rules as appled to atoms and ons affrm that electrons wll occupy orbtals n such a way that the ground state s characterzed by the followng: 1. The maxmum value of the total spn S allowed by the excluson prncple;. The maxmum value of the orbtal angular momentum L consstent wth ths value of S; 3. The value of the total angular momentum J s equal to L-S when the shell s less than half full and to L+S when the shell s more than half full. Ths due to the spn-orbt nteracton the constant of whch has opposte sgn dependng on whether s less than half full or more than half full. When the shell s just half full, the applcaton of the frst rule gves L = 0, so that J = S. 8
Physcs 97 Table 1. Ground states of ons wth partally flled d- or f-shells, as constructed from Hund's rules Rare-earth ons Experments on rare-earth ons n crystals show that they obey the Cure law, wth an effectve number of magnetons n agreement wth the theory of spn-orbt nteracton. Table confrms ths. In these ons, therefore, the angular momenta L and S are strongly coupled, and the moment of the on can respond freely to the external feld. Table. Effectve Number of Magnetons for Rare-Earth Ions 9
Physcs 97 Ths result s not surprsng. In these ons - from La to Lu n the perodc table - the 4f shell s ncompletely flled. The outer 5p shell s completely flled, whle the 5d and 6s shells whch are stll further out are strpped of ther electrons to form the onc crystal. Thus the only ncomplete shell s the 4f shell, and ths s the one n whch the magnetc behavor occurs. Snce electrons n ths shell le deep wthn the on, screened by the outer 5p and 5d shells, they are not apprecably affected by other ons n the crystal. Magnetcally ther behavor s much lke that of a free on. Another reason why the free-on treatment apples to the rare-earth ons s that the spn-orbt nteracton s strong n these substances, because ths nteracton s proportonal to Z, the atomc number of the element concerned, and all the rare-earth ons have large Z's. Typcal values for the spn-orbt and the crystal-feld nteractons n these materals are 10-1 ev and 10 - ev, respectvely. Iron-group ons Table 3 shows that ron-group ons behave magnetcally as f J = S, that s, only the spn moment can contrbute to magnetzaton. We can see ths by means of the followng argument. The magnetc propertes of ths group of elements are due to the electron n the ncomplete 3d shell. Snce electrons n ths outermost shell nteract strongly wth neghborng ons, the orbtal moton s essentally destroyed, or quenched, leavng only the spn moment to contrbute to the magnetzaton. In other words, n these ons, the strength of the crystal feld s much greater than the strength of the spn-orbt nteracton, just the reverse of the stuaton n rare-earth ons. Typcal strengths of the crystal feld and spn-orbt nteractons n the ron group are 1 ev and 10 - ev, respectvely. Table 3. Iron-Group Ions In the case of the transton metal ons from the ron group (partally flled 3d shells) the crystal feld s very much larger than the spn-orbt couplng. Ths perturbaton wll not lft the spn degeneracy, snce t depends only on spatal varables and therefore commutes wth S, but t can completely lft the degeneracy of the orbtal L-multplet, f t s suffcently asymmetrc. The result wll then be a ground-state multplet n whch the mean value of every component of L vanshes (even though L stll has the mean value L(L + 1)). One can nterpret ths classcally as arsng from a precesson of the orbtal angular momentum n the crystal feld, so that although ts magntude s unchanged, all ts components average to zero. 10
Physcs 97 Paramagnetsm of conducton electrons Spn paramagnetsm arses from the fact that each conducton electron carres a spn magnetc moment whch tends to algn wth the feld. An electron has spn ½. One mght expect that the conducton would make a Cure-type paramagnetc contrbuton (36) to the magnetzaton of the metal wth J=S =1/: Nµ χ =, (38) k T Ths shows that the susceptblty s nversely proportonal to temperature. Experments show, however, that spn susceptbltes n metals are essentally ndependent of temperature. The observed values are also consderably smaller than those predcted by (38). These facts clearly cast strong doubts on the applcablty of (36) to the conducton electrons. The source of the dffculty les n the fact that Eq. (36) was derved on the bass of localzed electrons obeyng the oltzmann dstrbuton, whereas the conducton electrons are delocalzed and satsfy the Ferm dstrbuton. The proper treatment, takng ths nto account, s llustrated n Fg.4. In the absence of the feld, half the electrons have spns pontng n the postve z-drecton, and the other half n the negatve drecton (Fg. 4a), resultng n a vanshng net magnetzaton. When a feld s appled along the z-drecton, the energy of the spns parallel to s lowered by the amount µ, whle the energy of spns opposte to s rased by the same amount (Fg.4b). The stuaton whch ensues s energetcally unstable, and hence some electrons near the Ferm level begn to transfer from the opposte-spn half to the parallel-spn one, leadng to a net magnetzaton. Note that only relatvely few electrons near the Ferm level are able to flp ther spns and algn wth the feld. The other electrons, lyng deep wthn the Ferm dstrbuton, are prevented from dong so by the excluson prncple. Fg. 4. (a) When = 0, the two halves of the Ferm-Drac dstrbuton are equal, and thus M = 0; (b) When a feld s appled, spns n the antparallel half flp nto the parallel half, resultng n a net parallel magnetzaton. We can now estmate the magnetc susceptblty. The electrons partcpatng n the spn flp occupy an energy nterval of µ (Fg.4). Thus ther concentraton s gven by N eff = ½D(E F )µ, where D(E F ) s the densty of states at the Ferm energy level [the factor ½ s nserted because D(E F ) as defned us earler ncludes both spn drectons, whle n the present crcumstances only one spn drecton s nvolved n the flppng]. Snce each spn flp ncreases the magnetzaton by µ (from -µ to + µ ), t follows that the net magnetzaton s gven by 11
Physcs 97 M = N µ = µ D E, (39) eff ( F ) leadng to a paramagnetc susceptblty χ = µ D( E ) (40) F The susceptblty s thus determned by the densty of states at the Ferm level, D(E F ). Accordng to eq. (40), χ s essentally ndependent of temperature. Ths s seen from the fact that temperature has only a small effect on the Ferm-Drac dstrbuton of the electrons, and consequently the dervaton leadng to (40) remans vald. If we apply the results for free electrons for whch D(E F ) = 3N/E F =3N/k TF, eq.(40) then leads to 3Nµ χ =, (41) k T F where T F the Ferm temperature (E F = k T F ). Snce T F s very large, often 30,000 K or hgher, we can see that (41) s smaller than (36) by factor of 10 - n agreement wth experment. In transton metals, the paramagnetc susceptblty s exceptonally large, because D(E F ) s large, by vrtue of the narrow and hgh 3d band. Damagnetsm Conducton electrons also exhbt damagnetsm on account of the cyclotron moton they execute n the presence of the magnetc feld. Each electron loop s equvalent to a dpole moment whose drecton s opposte to that of the appled feld. Classcal treatment shows that the total damagnetc contrbuton of all electrons s zero. Quantum treatment however shows that for free electrons ths causes a damagnetc moment equal to 1/3 of the paramagnetc moment. Therefore the total susceptblty of a free electrons gas s Nµ χ = (4) k T F The net response s therefore paramagnetc. In comparng theoretcal results wth experment, one must also nclude the damagnetc effect of the on cores. Table 4 gves the results for some metals. Table 4. Susceptbltes of Some Monovalent and Dvalent Metals x 10 6 (Room Temperature) 1