5.3ism •ismoriginatesfromtheangularmomentaofunpairedelectrons...i.e.,randomlyoriented,rapidlyreorientingmoments. •Inaicmaterial,thereisnopermanent,spontaneousicmoment.i.e,ifH=0,thenM=
0 •Theelectronspins(ormorecorrectly,theicmoments)arenon-cooperative. •Allformsofism,otherthanismandism,aregovernedbyacriticaltemperature(TC,TN)belowwhichthereissomecooperativityoftheicmoments.Abovethesecriticaltemperatures→PARAMAGNET!
•χparavarieswithT....let’slookatthismorecloselynow... (NOTE:inthetextbelow,χreferstomolaricsusceptibility) VII.Themolaricsusceptibilityχisabulkmeasurablequantitythatcanberelatedtotheaverageizationpermole(m),i.e.,averagemomentpermolecule: !
χ=mN!

A H whereNAisAvogadro’snumber VIII.Foracollectionofidenticalmolecules,eachwillhaveitsownicmoment(µ)andwillbefreetoorientitselfinanappliedicfieldH.Somealignmentwillur,butthethermalmotionofthemoleculeswillinterferepreventing“perfect”alignment…sotheaveragemolarizationmisrelatedtotheindividualmolecularicmomentvectorsµbybothappliedfieldHandbytemperatureT(provenbyLangevin): !
!
µ2Hm= 3kT wherekistheBoltzmannconstantk=0.69503877cm-1K-
1 Section5.3-
1 IX.Ifbinethelasttwoequations,wegettheCurielaw–“icsusceptibilityisinverselyproportionaltotheabsolutetemperature”. χ=NAµ2=C3kTT whereCistheCurieconstantC=NAµ2/3k Thisequationdirectlyrelatesthemolaricsusceptibilitytotheicmomentofeachmolecule!
!
ItpredictsalinearrelationshipbetweenicsusceptibilityandT-1,withχpara=0asT→∞ TheCurieLawistheicanalogtotheidealgaslaw.(Justasintermolecularinteractionscausedeviationsfromtheidealgaslaw,interactionsbetweenunpairedelectronscausedeviationsfromtheCurieLaw,especiallyatlowT.) Itisnotsurprising,then,thattrueCurieismisnotoftenobserved,butmanyicmaterialshavenearCuriebehaviour...sotheCurieLawprovidesagoodmodelformanysystems.
X.ItisconvenienttoexpressµinunitsofBohrons(β),whereoneBohronis4.66864374x10-5cm-1G-1andisthefundamentalquantumof icmoment…sowewriteµ=µeffβ.
1 ⎛3kTχ⎞
2 µeff = ⎜⎜⎝
N Aβ
2 ⎟⎟⎠ ≈ 8χ
T XI.Now,allthatremainsistorelateµefftothetotalspinangularmomentumquantumnumberS,ormorecorrectly,tothetotalangularmomentumquantumnumberJ. µeff=gS(S+1) spinonlycase µeff=gJ(J+1) spin-orbitcouplingcase Notethatgcanbeestimatedby:g=1+S(S+1)−L(L+1)+J(J+1)2J(J+1) Section5.3-
2 HOMEWORK: •ProvetoyourselfthatwhenL=0,g=
2.•Findthespin-onlyvalueofµeffinunitsofBohronsforaNi2+freeion. •DeriveanequationtocalculateχfromSinthespin-onlycase.•FindthevalueofχTinunitsofcm3Kmol-1fortheNi2+freeion(spin-onlycase). •Usinganapproximationthathasalreadybeengivenabove,showthatthis“quick”calculationisvalid:χT≈[S(S+1)]/2 Valuesof(4/g2)χTandµeffasafunctionoftotalspinS (4/g2)χ
T µeff(spinonly,forg=2)
S (cm3Kmol-1) (unitsofBohrons) 1/2 0.375 1.73
1 1.000 2.83 3/2 1.876 3.87
2 3.001 4.90 5/2 4.377 5.92
3 6.002 6.93 7/2 7.878 7.94 The
aboveounthasgivenusaneasystep-by-stepmethodforrelatingthemeasuredicsusceptibilitytothevalueofS(orthevalueofJ)foramolecularicmaterialthathasnearCuriebehaviour. Inordertounderstandplicatedicbehaviour,wehavetodelvealittledeeperbyderivingtheVanVleckequation,andshowingthattheCurieLawisaspecialcasethatcanbeobtainedfromthisequation. ...butfirstthingsfirst... Whatistheoriginofg?
•Consideranatomwithoneunpairedelectronthatissphericallydistributedaboutthenucleussuchthatthereisnoorbitalangularmomentum(L=0). •Theicmomentoftheelectroncanalignwithanappliedexternalfieldineithera“parallel”or“anti-parallel”orientation,dependingonthetwopossiblequantumstates(ms=+1/2,-1/2) Landeconstantg=2.0023192778forafreeelectron "↑ µ!
n=−mSgβ ↓ icmomentofanelectroninquantumstaten Section5.3-
3 •gisacoefficientthatmustbeusedinordertoountforthefactthattheratiooftheicmomentofanelectronandthespinangularmomentumdifferfromtheclassicalvalue(i.e.,g=1)...inotherwordstooffsetthefactthatmsisnotaunitvector!
µn=−(!
±21)gβ ↓ The“unit”vectorofmsisNOTinfactunity!
•Foralongtimeitwasthoughtthatgforelectronspinwasexactly2(aswouldfollowfromtheaboveequation),butithasnowbeenfound,bothexperimentallyandtheoretically,thatthevalueforafreeelectronis2(1.001145±0.000013) •gcandeviatedramaticallyfrom≈2,dependingonhow“free”theunpairedelectronisinmaterial.ForthoseofyoufamiliarwithEPRspectroscopy,youwillknowthattheg-valueisakintochemicalshiftinNMRspectroscopy.Themoredelocalizedtheelectronisinamolecularorbital,theclosertheg-valueisto≈
2. •Furthermore,gmaynothavethesamevalueinallspatialdirections.Inotherwords,theatomormoleculeinquestionmaybeicallyanisotropic.SolidstateorfrozensolutionEPRspectroscopycanbeusedtodeterminethevaluesofgx,gyandgz.Inaroomtemperaturesolution(particularlyofanicradical)thesoluteistumblinginsolutionrapidlyenoughthatanaverageg-valueisusuallyobserved: gavg2=(gx2+gy2+gz2)/3 Inmostcases,icsusceptibilitydataarerecordedonpolycrystallinesamples,soanaverageg-valueisobservedforthesemeasurementsaswell. DerivingtheVanVleckequation •Molaricsusceptibility(χ)characterizesthewayinwhichanappliedicfieldHinteractswiththeangularmomentumassociatedwiththethermallypopulatedstatesofamolecule. •InCLASSICALMECHANICS,whenasampleisperturbedbyanexternalicfield,itsizationisrelatedtoitsenergyvariationwithappliedfield: M=−∂E∂
H •InQUANTUMMECHANICS,wecandefineamicroscopicizationµnforeachenergylevelEn(n=1,2,3,...)as: µn=−∂∂HEn Section5.3-
4 •ThemacroscopicmolarizationMisthenobtainedbysummingthemicroscopicizationsweightedordingtotheBoltzmanndistributionlaw: ⎛ ∑⎜ − ∂En ⎞⎟ exp(− En kT) M=Nn⎝∂H⎠ ∑
A exp(−EnkT) n Thisexpressionmaybeconsideredasthefundamentalexpressioninmolecularism.Itdoesnotleanonanyapproximations!
•However,itisdifficulttoapplythisequationbecauseitrequirestheknowledgeofEn=f(H)variationsforallthermallypopulatedstatesinordertocalculatethederivatives∂En/∂
H. •In1932,VanVleckproposedasimplificationoftheabovegeneralequationbasedonafewapproximations:
1.ItislegitimatetoexpandtheenergiesEnordingtoanincreasingpowerseriesinH. Energyoflevelninzerofield(H=0) !
↓ En = E(0)n + "En
(1)
H + "En
(2)H2 +# ↑ ↑ FirstandSecondZeemancoefficients(i.e.,energiesdependentuponthe magnitudeoftheappliedicfield) Notethatusingµn=-∂En/∂H,wegettheVanVleckexpressionformicroscopicization: µn=−En
(1)−2En
(2)H+!

2.ThesecondapproximationisthatH/kT<<1sothat exp⎜⎛ − En ⎞⎟ = ⎛exp⎜ − E(0)n ⎞⎛⎟⎜1− En
(1)
H ⎞⎟ ⎝kT⎠ ⎜⎝ kT ⎟⎜⎠⎝ kT ⎟⎠
Q.WhenistheV.V.eq’nnotvalid?
Section5.3-
5 Sowheredothesetwoapproximationsgetus?
Wearestartingwithourfundamentalexpression: ∑µnexp(−EnkT)∑M=NAnexp(−EnkT) n whereµnistheicmomentofstatenµn=-∂En/∂
H •Let’slookatthetermexp(-En/kT)... ()exp⎜⎛−EnkT⎟⎞ = exp ⎡⎢ − En
(0) +HEn
(1) +H2En
(2) +!
⎤⎥ ⎝ ⎠⎣ kT ⎦ = ⎛exp⎜ − En
(0) ⎞⎟ ⎛exp⎜ − HE(1)n ⎞⎟ ⎛exp⎜ −
H E2(2)n ⎟⎞!
⎜⎝ kT ⎟⎠ ⎜⎝ kT ⎟⎠ ⎜⎝ kT ⎟⎠ Forthesakeofsimplicity,wewillignoretheH2andhigherorderpotentials...Whycanwedothis?
•Recall:forsmallx,exp(-x)≈1–x Then,ignoringH2andhigherorderexponentials... ⎛exp⎜ − E(0)n ⎞⎛⎟exp⎜ − HEn
(1) ⎞⎟ ≈ ⎛exp⎜ − E(0)n ⎞⎛⎟⎜1− HEn
(1) ⎞⎟ ⎜⎝ kT ⎟⎠ ⎜⎝ kT ⎟⎠ ⎜⎝ kT ⎟⎜⎠⎝ kT ⎟⎠ NOTE:ThisassumesthatthefirstorderZeemansplittingismuchsmallerthanexchangecoupling(wewilllookatthissoon...) •Theicmomentofstaten,ignoringH2andhigherordertermsis...µn=−∂∂HEn≅−En
(1)−2HEn
(2) Section5.3-
6 Substitutingtheseapproximationsintoourfundamentalexpression,weget... ∑()NA ∑M=n −
E
(1) −2HE
(2) ⎛⎜1− HEn
(1) ⎞⎛⎟exp⎜ − E(0)n ⎞⎟ n n⎜⎝ kT ⎟⎠ ⎜⎝ kT ⎟⎠ ⎛⎜1− HEn
(1) ⎞⎟ ⎛exp⎜ − E(0)n ⎞⎟ ⎜n⎝ kT ⎟⎠ ⎜⎝ kT ⎟⎠ •Inzerofield(H=0),theizationvanishes(M=0) ∑i.e.
E
(1) ⎛exp⎜ − En
(0) ⎞⎟ =
0 provethistoyourself!
nn ⎜⎝ kT ⎟⎠ NOTE:bydefinition,thisexcludesmaterialsexhibitingspontaneousization...weareonlyconsideringicspecies. •Ifbinethesetwoequations,retainingonlylineartermsinH,andwedividebothsidesbyH,weget...
N ⎛⎜ E(1)2n − 2E
(2)⎞⎛⎟exp⎜ − En
(0) ⎞⎟ ∑M=χ= An ⎜⎝ kT n⎟⎠ ⎜⎝ kT ⎟⎠
H ⎛exp⎜ − En
(0) ⎞⎟ ∑n ⎜⎝ kT ⎟⎠ VanVleckEquation•Now,allweneedtoknowarethequantities:En
(0),En(1)andEn
(2) •Fromatheoreticalstandpoint,theVanVleckequationcanbeusedwhentheeigenvaluesEn(0)andeigenfunctions|n>oftheHamiltonianinzero-fieldareknown. •ThenEn(1)andEn(2)canbecalculatedthroughperturbationtheoryas: En
(1)=nHˆZEn ∑En
(2)= 'nHˆZEm2
(0)
(0) m(En−Em) whereΣ'meansthesummationrunsoverthelevelsmwithEm
(0)≠En
(0). ĤZEistheZeemanoperator,whichountsfortheinteractionbetweentheicfieldandtheelectronicangularmomenta: ĤZE=βΣi(Ii+gesi)·
H whereIiandsiaretheorbitalandspinmomentaoperatorsofelectroni,respectively. Section5.3-
7 Okay,sothisisstillnotveryhelpfulintermsofapplyingtheVanVleckequation,butatleastwecanstarttoseeafewusefulthings... •WhenallenergiesEnarelinearinH,thesecond-orderZeemancoefficientsEn(2)vanishandwegetasimplifiedversionoftheVanVleckequation: ∑()χ=NAnEn(1)2exp−En(0)kT∑()kTexp−En(0)kT n •WecannowhaveacloserlookatTemperatureIndependentism(TIP): Iftheonlythermallypopulatedstateofamoleculeisaspinsinglet(S=0)withoutfirst-orderangularmomentum,thentheicsusceptibilityisintuitivelyexpectedtobezero,andthemeasuredsusceptibilitywillbenegative. i.e.χpara=
0 thereforeχexp=χdia However,insomecasesthisisnotthefullstory...andthiscanbeountedforbylookingattheVanVleckequation. LettheenergyE0(0)ofthegroundstatebetheenergyorigin(i.e.equaltozero).Sincethisstatehasnoangularmomentum–becauseitisic–... E0
(1)=
0 andχpara=-2NAE0
(2) fromtheV.V.equation Or ∑χ=−2NA 0HˆZEm2
(0)
(0) m≠0(E0−Em) So...theicgroundstatemaycouplewithexcitedstatesthroughtheZeemanperturbation–providedthattheyarecloseenoughinenergy. χispositivesincealldenominatorsarenegative,andtemperatureindependent.Thiscontributioniscalledtemperatureindependentism. TIPisusuallysmall,oftenonthesameorderofmagnitudeastheiccontributiontotheicsusceptibility,butofoppositesign. Forexample,itisestimatedtobeabout60x10-6cm3mol-1forcopper(II)mononuclearspecies100x10-6cm3mol-1fornickel(II)mononuclearspecies200x10-6cm3mol-1foroctahedral[Co(NH3)6]3+possessinga1A1ggroundstate Section5.3-
8