2. Lecture: Basics of Magnetism: Paramagnetism

1 2. lecture : Basics of Magnetism: ParamagnetismHartmut ZabelRuhr-University BochumGermanyContent2H. Zabel, RUB2. lecture : Properties Temperature of conductionelectronsOrbital momentsof the d-shell3H. Zabel, RUB2. lecture : ParamagnetismHzlm( )1+ll 2 1 012()BBlBlllzmgmm = = =2,1,0,1,2, 5 orthogonal and degenerateorbital wavefunctionsof 3d shellThe upperthreewavefunctionshavemaximain the xy, xz, yzplanes, the lowertwohavemaximiaalongx,yand z coordinate. In a magneticfieldthe degeneratesublevelssplit. 0=lm1+=lm1 =lm2+=lm2 =lm41. Spin-Orbit-couplingCouplingof spinand orbital momentyieldsthe total angular momentumof electrons:The spin-orbit(so) interactionorLS-couplingisdescribedby:Hz L S J ( )( )shell f and d filled half than more for 0shell f and d filled half than less for < > = 01;2drdUrecmrSL r= EeBSO SLJ +=2.

2 lecture : ParamagnetismH. Zabel, RUBSpin-orbitcouplingis due to the Zeeman splittingof the spinmagneticmomentin the magneticfieldthatis producedbythe orbital moment: Bm=ELSSO 5H. Zabel, RUB2. lecture : Paramagnetism In restframeof electron, E and B fieldsacton electrondue topositive chargeof the nucleus: The magneticfieldisproportional toangular momentumof electrons: BL~ L: In magneticfieldBL, S precesseswitha angular velocity Land couplestoL: L S canbeevaluatedvia: Yielding:rEcmpr BL20 =()LrrUrecm BL =1120()SLrrUrecm=EeBSO 12()()2222221;SLJSLSLJ = += ()()()[]1112+ + + sslljj=ESO2. Fine structure6H. Zabel, RUB2. lecture : ParamagnetismTe r m s withsame n and l quantumnumbersareenergetiallysplitaccord ingto whetherthe electronspinis parallel orantiparallel to the orbital moment.

3 Thisis calledthe finestructureof atomicspectra. Examplehydrogen atom:The total splittingof 3/2 increaseswiththe numberof electronsin the atomand becomesin the order of 50 meVfor3d metals. LS-splittinglowersthe energyforL and S antiparallel. Thereforelevelfillingstartswithlowestj-v alues. +1/2 - Whyist changingsign?7H. Zabel, RUB2. lecture : Paramagnetism ( ) 0< SL r= ESO d- electronsLessthanhalf filled: L -SLS( )000> < < SL r= ESLSO if More thanhalf filled: L+SLS( )000<< > SL r= ESLSO if 8LS couplingforlightand heavy atomsSLJ,sS,lLziiZii+=== == 11 ==+=ziiiiijJ,slj1 Russel-Saunders couplingforlightatoms(LS-coupling): Thisapproximationassumesthatthe LS-couplingof individual electronsisweakcomparedtothe momentsof all electronscoupletoa total angular momentumL and spinmomentsof all electronscoupletoS.

4 FinallyL and S coupletoJ:jj-couplingof heavy atoms: In the limitof bigLS coupling, the spinand orbital momentof eachindividual electronscouplestoj, and all j areaddedtototal angular momentJ. 9 Total angular moment and total magnetic momentSLJ+=())SJ()SL()SgLg(mBBSLBSJ+ =+ =+ =+ 2 Total angular momentis : HzL S J SJ +S SJm+ :::JLST otal spinTotal orbital momentTotal angular momentTotal magneticmomentis : Total angular momentJ and the total magneticmomentarenot collinear. However, in an external magenticfield, mJ+Sprecessesfast about J, and Jprecessesmuch slower about Hz. Thus the time average component of the magnetic moment m = m||is parallel to J.||m 2. lecture : ParamagnetismH. Zabel, RUBD ifferent total angular momentsJ10H. Zabel, RUB2. lecture : ParamagnetismL S L S L S Whichoneis the groundstate?

5 Next lecture : Hund sruleSOCoulombEEE+=0< SLJSL +ma xJ J mi nJ 23,3==SL3. Zeeman-splitting In an externalfieldthe quantizationaxisis definedbythe fieldaxisHz A statewithtotal angular momentumJhasa degeneracyof 2J+1 withoutfield. These statesarelabledaccordingto the magneticquantumnumbermJ: J mJ J. In an externalfieldHz the stateswithdifferent mJhavedifferent energyeigenstates, theirdegeneracyis lifted: The energyeigenstateareequidistantand linearlyproportional tothe externalfieldHz.()zJzSOzmJBJSOzmHmEEHmgE EHEJzJ,0000, ++= ++= JmEHzmJ= 3/2mJ= 1/2mJ=-1/2mJ=-3/2zBJHg 0112. lecture : ParamagnetismH. Zabel, RUB 1221+++ =JJJ,..,J,J,JmLS and ZeemannSplitting forL=3, S=3/2, < 012H. Zabel, RUB2. lecture : Paramagnetism < 0 Conversation:1000 cm-1 = eV13 Land factor)1(2)1()1(23)1(2)1()1()1(1++ ++=++ ++++=JJLLSSJJLLSSJJgJ :factor Land Notice: gj=1 for J=L and 2 for J=S BJJjzgmm =,2.

6 lecture : ParamagnetismH. Zabel, RUBZ-componentof the total magneticmoment:Hzjm()1+JJ 212323 21 Evaluatingthe Land -factor14H. Zabel, RUB2. lecture : ParamagnetismFrom1. Lecturewehaveforthe paramagnetic responsein an externalB - field:ConsideringL+2S projectedontoJ and J projectontothe B-axis:Thisyields:Using:Wefind:Whichmust equal:With:()BSLEB + =2()() ()zzBBBJSLSLJJBJJJSLE + + = + =222()zzBBJSSLLJE22223 + + =()()2222223;SLJSLSLJ = +=3 ()zzBBJJSLJE222223 + =zJBJBmgE =( ) ( )()121123++ ++=JJLLSSgJ15Hz=0, T=0gs degenerate, all atoms in the same stateHz>0, T= 0 Lifting of degeneracy, all atoms in the gsHz>0, T>0At high temperature population of higer energy statesmJ10-1mJ10-1 ENEN()JzBJJmHgmE 0-=()JzBJJmHgmE 0-=JBJzmgVNHEVNM == 1-04. Thermal propertiesThermal population of the Zeeman-split levels in the ground state (gs).

7 Example: J=1, mJ=-1,0,+1 Discrete energy levels with mJ=-J,..JAverage thermal energy:Magnetization:2. lecture : ParamagnetismH. Zabel, RUB16()() =jjmjmjjjmmmmexpexpTkHgBzBj = jBjmgVNM =Thermal average of the magnetizationThermal average of the magnetic moment follows from the partition function:with:)~(),( =jBjzjBgVNHTMBiis the Brillouin function. The Brillouin function replaces the Langevin function in case of discrete energy levels. 2. lecture : ParamagnetismH. Zabel, RUB17 ++= jajjjjjBj2~coth21~212coth212)~(Brillouin FunctionmitTkjHgjBzBj = = ~2. lecture : ParamagnetismH. Zabel, RUB18 JJJE xamplesforthe Brillouin-FunctionBJ:2. lecture : ParamagnetismH. Zabel, RUB191>>=TkjHg~BzBj )(0 TMMjgVNMSBj==== 1B )~( 4. Low and high temperature approximationsLow temperature approximation (LTA) for: the Brillouinfunction approaches 1 The thermally averaged magnetization then becomes:This corresponds to the saturation magnetization MS.

8 The saturation magnetization can not become bigger than given by j. It corresponds to a state in which all atoms occupy the ground state. 2. lecture : ParamagnetismH. Zabel, RUB2031)coth(xxx+ zBzBeffHTCTkHpVNTM= =3)(22)1(+=JJgpjeffBBeffkpVNC322 =High temperature approximationIn HTA for the Brillouin function can be approximated by )~(Bj 1<< ~Then follows for the magnetizationWith the effective moment:And the Curie constant:2. lecture : ParamagnetismH. Zabel, RUB21 TCHMz= = With C we can calculate peffand j. From j the valence of a chemical bond can be determined. Thuspeffis important for chemistry. Curie law of the magnetic susceptibility01002003004005006000510152 0253035404550 T010020030040050060001 T2. lecture : ParamagnetismH. Zabel, RUB5. Van Vleckparamagnetism22 ForJ = 0 the paramagnetic susceptibilitybecomeszero.

9 J = 0 occursforshells, whicharelessthanhalf filledbyoneelectron In thiscasehigherorder termscontributeto the susceptibility, in particulara diamagnetic termof secondorder, whichis positiv. The higherorder termsaredue to excitedstateswhichmayhavea J 0, evenif forthe groundstateJ = 0. Calculatingin secondorder perturbationtheorycontributionstoonlythe groundstate, oneobtains: Van Vleckcontributiontothe susceptibilityisweak, positive and temperatureindependent. But itplaysa decisiveroleforthe Paramagnetism of Smand Eu 3. lecture . () 020001,2,..m--=+ = EEHSgLmEmzzSzBVleckVan2. lecture : ParamagnetismH. Zabel, RUB6. Paramagnetism of conductionelectrons232. lecture : ParamagnetismForFermi particleswithspinS=1/2 weexpect:Expectedmagnetization:whichhas1 /T dependence. Experiment shows.

10 Is independentof T value1/100 of the calculatedvalueat 300 K. Contradiction!TkHVNHMBzBzz2 ==()TkVNTkVNTkSSgVNTkpVNBBBBBBSBeffB2222 222323212313 = =+==H. Zabel, RUB24 Magnetization of a free electron gas( )( )VEDHVHEDVNVNNMFzBzBFBBB2212 = = = = ( )ED ( )ED H2B ( )FED21 With:( )FFENED23=Follows:zH( ) ===FBz2 BFBz2 BFz2 BTTTkHVN 23 TkHVN 23 VEDH ME()FED2. lecture : ParamagnetismH. Zabel, RUBSpin splitDOS forfreeelectronsin an externalfield25 Pauli Spin Suszeptibility of a free electron gasFBBFBBP auliTkVNTkHHVNHM12323 22 = == Pauli spinsusceptibilityhasthe correctform. It is independentof temperature It is reducedbythe factorT/TF ~ 100. Closedshellshavenodensityof statesat the Fermi level, thusclosedshellsdo not contributeto Pauli. Onlys,pand d-electronsof unfilledshellscontribute.