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Chapter9

Atomicstructure

Previously,weha veseenthatthequan tummechanicsofatomichydrogen,and hydrogen-likeatomsischaracterizedb yalarge degeneracywitheigen values separatinginto multipletsofn 2 -folddegeneracy, wherendenotestheprinc iple quantumnumb er.However,althoughtheidealizedSc hr¨odingerHamiltonian, ˆ H 0 = ˆp 2 2m +V(r),V(r)=- 1

4π?

0 Ze 2 r ,(9.1) providesausefulplatformfrom whichdev elopourin tuition,thereare seve ral importante ff ectswhich meanthattheformulation isalittle toona ¨ıve. These "corrections",which derivefromseveralsources, areimportantasthey leadto physicalramificationswhich extendbey ondtherealmofatomic physics.Here weoutlinesomeofthe e ff ectswhich needtobetaken into accountev enfor atomichydrogen, beforemovingon todiscussthequantumph ysicsofmulti- electronatoms.In broadterms, thee ff ectstob econs ideredcanbe grouped intothosecausedby theinternal propertiesof thenucleus,andthosewhic h derivefromrelativisticcorrections. Toorientourdiscussion, itwillbehelpfulto summarizesomeof key aspects ofthesolutions ofthe non-relativisiticSchr¨ odingerequation, ˆ H 0

ψ=Eψon

whichwewilldra w:

Hydrogenatomrevisited:

?Aswithan ycen trallysymmetricpotential,the solutionsoftheSchr¨odinger equationtake theformψ ? m (r)=R(r)Y ? m (θ,φ),wherethe spheric alhar- monicfunctionsY ? m (θ,φ)depend onlyonsphericalpolar coordinates, andR(r)represents theradialcomponent ofthew avefunction. Solving theradialw av eequationintroducesaradialquantumnumb er,n r ≥0. Inthecase ofa Coulombp otential,the energydep endsontheprincipal quantumnumb ern=n r +?+1≥1,andnot onn r and?separately. ?Foratomichydrogen( Z=1),the energylev elsofthe Hamiltonian(9.1) aregiven byThefinestructure constantis knowntogreataccuracyand is givenby,

α=7.297352570(5)×10

-3 = 1

137.035999070(9)

. E n =- Ry n 2 ,Ry= ? e 2

4π?

0 ? m 2? 2 = e 2

4π?

0 1 2a 0 = 1 2 mc 2 α 2 , wherea 0 =

4π?

0 e 2 ? 2 m istheBohr radius,α= e 2

4π?

0 1 ?c denotesthefine structureconstant ,andstrictly speakingmrepresentsthereduced massofthe electronand proton.Appliedto singleelectronions with higheratomicw eight, suchasHe + ,Li 2+ ,etc.,the Bohrradiusis reduced byafactor1/Z,whereZdenotesthen uclearc harge,andtheenergyis givenbyE n =- Z 2 n 2 Ry=- 1 2n 2 mc 2 (Zα) 2 .

AdvancedQuantumPh ysics

9.1.THE"REAL" HYDROGENA TOM90

?Sincen≥1andn r ≥0,theallo wedcom binationsofquantumnum- bersareshownon therigh t,wherewehav eintro ducedthecon ventional n ?Subshell(s) 101 s

20,12s2p

30,1,23s3p3d

40,1,2,34s4p4d4f

n0···(n-1)ns··· notationwhereby valuesof?=0,1,2,3,4···arerepresented byletters s,p,d,f,g···respectively. ?SinceE n dependsonlyonn,thisimplies, forexample, anexactdegen- eracyofthe 2sand2p,andof the3s,3pand3dlevels. Theseresultsemerge fromatreatmen tofthe hydrogen atomwhic his inherentlynon-relativistic.Infact, aswe willseelater inourdiscussion ofthe Diracequationin chapter 15,theHamiltonian (9.1)representsonlytheleading terminan expansioninv 2 /c 2 ?(Zα) 2 ofthefull relativisticHamiltonian Toseethatv 2 /c 2 ?(Zα) 2 ,we mayinvoke thevirialtheorem.

Thelattersho wsthat theaver-

agekineticenergy isrelatedto thepote ntialenergyas?T?= - 1 2 ?V?.Therefore,the av er- ageenergyis given by?E?= ?T?+?V?=-?T?≡- 1 2 mv 2 .

Wethereforehave that

1 2 mv 2 =

Ry≡

1 2 mc 2 (Zα) 2 fromwhich followstherelationv 2 /c 2 ? (Zα) 2 . (seebelo w).Higherordertermsproviderelativisticcorrections, whichimpact significantlyinatomicandcondensed matterphysics, andleadto aliftingof thedegeneracy. Inthefollowingw ewilldiscuss andobtainthe heirarchyof leadingrelativisticcorrections. 1

Thisdiscussionwill provide aplatformto

describemulti-electronatoms.

9.1The"r eal"hydrogen atom

Therelativisticcorrections (sometimesknown asthefine-structurecor- rections)tothe spectrum ofhydrogen-lik eatomsderivefrom threedifferent sources: ?relativisticcorrectionsto thekinetic energy; ?couplingbet weenspinandorbitaldegreesoffreedom; ?andacon tributionkno wnastheDarwinterm. Inthefollo wing,we willdiscusseachofthese correctionsinturn.

9.1.1Relativisticcor rectionto thekineticenergy

Previously,wehav etakenthekineticenergy tohavethefamiliarnon-relativistic form, ˆp 2 2m .How ever,fromtheexpressionfortherelativisticenergy-momen tum invariant,wecanalreadyanticipatethatthe leadingcorrectiontothenon- relativisticHamiltonianapp earsat orderp 4 , E=(p µ p µ ) 1/2 = ? p 2 c 2 +m 2 c 4 =mc 2 + p 2 2m - 1 8 (p 2 ) 2 m 3 c 2 +··· Asaresult, wecan inferthefollo wingperturbationtothekinetic energyof theelectron, ˆ H 1 =- 1 8 (ˆp 2 ) 2 m 3 c 2 . Whencomparedwith thenon-relativistic kineticenergy, p 2 /2m,onecan see thatthep erturbationissmaller byafactorof p 2 /m 2 c 2 =v 2 /c 2 ?(Zα) 2 ,i.e. ˆ H 1 isonlya smallp erturbationforsmall atomicnum ber,Z?1/α?137. 1 Itma yseemoddtodiscussrelativistic correctionsinadv anceoftheDirac equation andtherel ativi sticformulationofquantummechanics.However,suc hadiscussionwould presentalengthy andunnecessarily complexdigressionwhichw ouldnot leadtofurther illumination.Wewillthereforefollowthenormalpracticeof discussingrelativisticcorrections asperturbati onstothefamiliarnon-relativistictheory .

AdvancedQuantumPhysics

9.1.THE"REAL" HYDROGENA TOM91

Wecanthereforemak eus eofaperturbative analysistoestimatethescaleof thecorrection. Inprinciple,the large-scaledegeneracyof thehydrogen atomwould de- mandananalysis basedon thedegeneratep erturbationtheory. Howev er, fortunately,sincetheo ff -diagonalmatrixelemen tsvanish, 2 ?n?m| ˆ H 1 |n? ? m ? ?=0for ??=? ? orm?=m ? , degeneratestatesare uncoupledand suchan approachis unnecessary.Then makinguseof theidentit y, ˆ H 1 =- 1 2mc 2 [ ˆ H 0 -V(r)] 2 ,thescale oftheresulting energyshiftcan be obtainedfromfirst orderperturbationtheory,Bymakinguse ofthe formofthe radialwa vefunctionforthehy- drogenatom, onemayobtainthe identities, ? 1 r ? n ? = Z a 0 n 2 ? 1 r 2 ? n ? = Z 2 a 2 0 n 3 (?+1/2) . ? ˆ H 1 ? n ? m ≡?n?m| ˆ H 1 |n?m?=- 1 2mc 2 ? E 2 n -2E n ?V(r)? n ? +?V 2 (r)? n ? ? 2 . Sincethecalculation ofthe resultingexp ectationvalues isnotparticularly illuminating,we refertotheliteraturefora detailedexposition 3 andpresen t hereonlythe requiredidentities (right).F romtheseconsiderations, weobtain thefollow ingexpressionforthefirstorderenergyshift, ? ˆ H 1 ? n ? m =- mc 2 2 ? Z α n ? 4 ? n ?+1/2 - 3 4 ? .(9.2) Fromthistermalone,w eexpect thedegerenacy betw eenstateswithdi ff erent valuesoftotalangularmomen tum?tobe lifted.However, aswell willsee, thisconclusionis alittle hasty. Weneed togatherall termsofthesame orderofp erturbationtheory beforewecanreac hadefinite conclusion.We can,how ever,confirmthat(asexpected)thescaleof thecorrectionis oforder ? ˆ H 1 ? n ? m En ≂( Z α n ) 2 .We nowturntothe secondimportantclassof corrections.

9.1.2Spin-orbit coupling

Aswell asrevealingtheexistenceof aninternal spindegreeoffreedom,the relativisticformulation ofquantummechanics shows thatthereisafurther relativisticcorrectionto theSchr¨ odingerop eratorwhich involvesacoupling betweenthespinandorbitaldegreesof freedom.For ageneralp otentialV(r), thisspin-orbitcoupling takes theform, ˆ H 2 = 1 2m 2 c 2 1 r (∂ r V(r)) ˆ S· ˆ L.

Forahydrogen-like atom,V(r)=-

1

4π?

0 Ze 2 r ,and ˆ H 2 = 1 2m 2 c 2 Ze 2

4π?

0 1 r 3 ˆ S· ˆ L. ?Info.Physically,theoriginofthe spin-orbitin teractioncanbe understoon fromthefollo wingconsiderations. Astheelectronismoving throughtheelectric fieldofthe nucleusthen, initsres tframe,itwillexperience thisas amagnetic field.Therewill bean additionalenergyterm intheHamiltonianassociatedwith theorientation ofthespinmagneticmomen twithresp ecttothis field.We canmake anestimateof thespin-orbit interactionenergy asfollows: Ifwe haveacentral field determinedby anelectrostaticpotential V(r),thecorresp ondingelectric fieldisgiven 2

Theproof runsasfollows: Since[

ˆ H1, ˆ L 2 ]=0, ? 2 [? ? (? ? +1)-?(?+1)]?n?m| ˆ H1|n? ? m ? ?=

0.Simi larly,since[

ˆ H1, ˆ

Lz]=0, ?(m

? -m)?n?m| ˆ H1|n? ? m ? ?=0. 3 see,e.g., Ref[1].

AdvancedQuantumPh ysics

9.1.THE"REAL" HYDROGENA TOM92

byE=-?V(r)=-e r (∂ r V).For anelectronmoving atvelo cityv,thistranslates toane ff ectivemagneticfieldB= 1 c 2 v×E.Themagnetic momentof theelectron associatedwithitsspinis equaltoµ s =g s q 2m

S≡-

e m

S,andth usthein teraction

energyisgiv enb y -µ s

·B=

e mc 2

S·(v×E)=-

e (mc) 2

S·(p×e

r (∂ r V))= e (mc) 2 1 r (∂ r

V)L·S,

wherewe haveused therelationp×e r =p× r r =- L r .Infact thisisn'tquite correct; thereisa relativistice ff ectconnected withtheprecession ofaxesunder rotation,called Thomasprecessionwhichmultipliestheform ulabyafurther factorof 1 2 .

Onceagain,w ecan estimatethee

ff ectofspin-orbit couplingby treating ˆ H 2 asap erturbation.Inthe absenceofspin-orbitinteraction, onemay express theeigenstatesof hydrogen-like atomsinthe basisstatesofthemutually commutingoperators, ˆ H 0 , ˆ L 2 , ˆ L z , ˆ S 2 ,and ˆ S z .How ever,inthepresence ofspin-orbitc oupling,the totalHamiltoniannolongercommutes with ˆ L z or ˆ S z (exercise).It isthereforehelpful tomake useofthe degeneracyof theunperturb edHamiltoniantoswitchtoanewbasis inwhic htheangular momentumcomponents oftheperturbedsystemarediagonal.This canbe achievedbyturningtothebasisof eigenstatesoftheoperators , ˆ H 0 , ˆ J 2 , ˆ J z , ˆ L 2 ,and ˆ S 2 ,where ˆ J= ˆ L+ ˆ

Sdenotesthetotal angularmomen tum.(For a

discussionofthe formofthese basisstates,w ereferbac ktosection 6.4.2.)

Makinguseof therelation,

ˆ J 2 = ˆ L 2 + ˆ S 2 +2 ˆ L· ˆ

S,inthis basis,itfollo ws

that, ˆ L· ˆ S= 1 2 ( ˆ J 2 - ˆ L 2 - ˆ S 2 ). Combiningthespinand angularmomentum, thetotalangular momentum takesvaluesj=?±1/2.Thecorresp ondingbasis states|j=?±1/2,m j , ?? (withs=1/2implicit)therefore diagonalizethe operator, ˆ S· ˆ

L|j=?±1/2,m

j , ??= ? 2 2 ? ? -?-1 ? |?±1/2,m j , ??, wherethebrac kets indexj=?+1/2(top)and j=?-1/2(bottom). Asfor theradialdep endenceof theperturbation,onceagain,the o ff -diagonalmatrix elementsvanishcircum ventingtheneedto invokedegenerateperturbation theory.Asaresult,at firstorder inperturbation theory,one obtains ?H 2 ? n,j=?±1/2,m j , ? = 1 2m 2 c 2 ? 2 2 ? ? -?-1 ? Ze 2

4π?

0 ? 1 r 3 ? n ? .

Thenmakinguse ofthe identity (right),

4 oneobtainsFor?>0, ? 1 r 3 ? n ? = ? mc α Z ?n ? 3 1 ?(?+ 1 2 )(?+1) .? ˆ H 2 ? n,j=?±1/2,m j , ? = 1 4 mc 2 ? Z α n ? 4 n j+1/2 ? 1 j - 1 j+1 ? . Notethat,for ?=0,the reisno orbitalangularmomentumwith whichto couple!Then,if we rewritetheexpression for? ˆ H 1 ?(9.2)inthe newbasis, ? ˆ H 1 ? n,j=?±1/2,m j , ? =- 1 2 mc 2 ? Z α n ? 4 n ? 1 j 1 j+1 ? , andcombining bothofthesee xpressions,for?>0,we obtain ? ˆ H 1 + ˆ H 2 ? n,j=?±1/2,m j , ? = 1 2 mc 2 ? Z α n ? 4 ? 3 4 - n j+1/2 ? , whilefor?=0,w eretainjust thekineticenergyterm (9.2). 4

Fordetailssee,e. g.,Ref[1].

AdvancedQuantumPhysics

9.1.THE"REAL" HYDROGENA TOM93

9.1.3Darwinter m

Thefinalcon tributiontothe Hamiltonianfromrelativistice ff ectsiskno wnas theDarwinterm andarises fromthe"Zitterbewegung"oftheelectron - tremblingmotion-whic hsmearsthe e ff ectivepotentialfelt bytheelectron. Suche ff ectslead toap erturbationofthe form, ˆ H 3 = ? 2 8m 2 c 2 ? 2 V= ? 2 8m 2 c 2 ? e ? 0 Q nuclear (r) ? = Ze 2

4π?

0 ? 2 8(mc) 2

4πδ

(3) (r), whereQ nuclear (r)=Zeδ (3) (r)denotesthe nuclear chargedensit y.Sincethe perturbationactsonlyat theorigin,it e ff ectsonlystates with?=0.As a result,onefinds that ? ˆ H 3 ? njm j ? = Ze 2

4π?

0 ? 2 8(mc) 2

4π|ψ

? n (0)| 2 = 1 2 mc 2 ? Z α n ? 4 n δ ? ,0 . Intriguingly,thistermisformallyiden ticaltothat whichw ouldbe obtained from? ˆ H 2 ?at?=0.As aresult,com biningall threecontributions, thetotal energyshiftis given simplyby ΔE n,j=?±1/2,m j , ? = 1 2 mc 2 ? Z α n ? 4 ? 3 4 - n j+1/2 ? ,(9.3) aresultthat isindependen tof?andm j . Todiscussthepredictedenergy shiftsfor particularstates,it ishelpfulto introducesomenomenclaturefromatomicphysics. Fora statewithprinciple quantumnumb ern,totalspin s,orbitalangular momentum?,andtotal an- gularmomentum j,onema yusespectroscopicnotationn 2s+1 L j todefine thestate. Forahydrogen-like atom,withjustasingle electron,2s+1= 2. Inthiscase, thefactor2 s+1is oftenjustdropp edfor brevity. Ifwe applyourperturbative expressionforthe relativisticcorrections(9.3), howdoweexp ectthelev elstoshiftforhydrogen-lik eatoms?Aswe have seen, forthenon-relativistic Hamiltonian,eac hstateof givennexhibitsa2 n 2 -fold degeneracy.Foragiv enmultipletspecifiedb yn,therelativistic corrections dependonlyonjandn.For n=1,w ehav e?=0and j=1/2:Both1S 1/2 states,withm j =1/2and-1/2,experience anegativeenergyshift byan amount ΔE

1,1/2,m

j ,0 =- 1 4 Z 4 α 2

Ry.Forn=2,?cantake thevaluesof0 or1.

Withj=1/2,both theformer2S

1/2 state,andthe latter2 P 1/2 statesshare thesamenegativ eshiftin energy, ΔE

2,1/2,m

j ,0 = ΔE

2,1/2,m

j ,1 =- 5 64
Z 4 α 2 Ry, whilethe2 P 3/2 experiencesashiftof ΔE

2,3/2,m

j ,1 =- 1 64
Z 4 α 2

Ry.Finally,for

n=3,?cantake valuesof0,1 or2.Here,thepairsof states3S 1/2 and3P 1/2 , and3P 3/2 and2D 3/2 eachremaindegeneratewhilethe state3 D 5/2 isunique.

Thesepredictedshifts aresummarizedin Figure9.1.

Thiscompletesour discussionof therelativisticcorrections whichdev elop fromthetreatmen tof theDiractheoryfortheh ydrogenatom.Ho wever, thisdoes notcompleteourdiscriptionofthe "real"hydrogen atom.Indeed, therearefurther correctionsw hichderiv efromquan tumelectrodynamicsand nucleare ff ectswhich wenowturnto address.

WillisEugeneLamb, 1913-2008

Aphysicist who

wontheNobel

PrizeinPhysics

in1955"forhis discoveriescon- cerningthefine structureofthe hydrogenspectrum". Lamband

PolykarpKuschwereabletopre-

ciselydeterminecertain electromag- neticprop ertiesoftheelectron.

9.1.4Lamb shift

Accordingtothe perturbationtheory abo ve,therelativisticcorrections which followfromtheDiractheory forhydrogen leave the2S 1/2 and2P 1/2 states degenerate.How ever,in1947,acarefulexperimentalstudyb yWillis Lamb

AdvancedQuantumPh ysics

9.1.THE"REAL" HYDROGENA TOM94

Figure9.1:Figureshowing the

heirarchyofenergyshiftsofthe spec- traofh ydrogen-likeatoms asaresult ofrelativisticcorrections. Thefirst columnshows theenergyspectrum predictedby the(non-relativistic)

Bohrtheory. Thesecondcolumn

showsthepredictedenergyshifts fromrelativisticcorrections arising fromtheDirac theory. Thethirdcol- umnincludescorrections duequan- tumelectrodynamics andthefourth columnincludesterms forcoupling tothen uclearspindegrees offree- dom.TheH- αline,particularly importantintheastronomy,corre- spondstothetransition bet weenthe levelswithn=2and n=3. andRobert Retherforddiscoveredthatthis wasnot infactthecase: 5 2P 1/2 stateissligh tlylo werinenergythanthe 2S 1/2 stateresultingin asmall shift ofthecorresp ondingsp ectralline-theLamb shift.Itmightseemthat such a tinye ff ectwould bedeemedinsignificant,but inthiscase,theobserv edshift (whichwasexplainedb yHansBetheinthesame year)pro videdconsiderable insightintoquan tumelectrodynamics.

HansAlbrecht Bethe1906-2005

AGerman-

American

physicist,and

Nobellaureate

inphysics"for hiswo rkonthe theoryofstellar nucleosynthesis."

Aversatile theo-

reticalphysicist,

Bethealsomade important contribu-

tionstoquantumelectrodyna mics, nuclearphysics,solid-state physics andparticle astrophysics.During

WorldWarII,hew asheadofthe

TheoreticalDivisionatthesecret

LosAlamoslab oratory developing

thefirstatomic bombs.There he playedakeyrolein calculatingthe criticalmassofthew eapons,and didtheoretical workonthe implosion methodusedinboth theTrinit ytest andthe"F atMan" weapondropped onNagasaki. Inquantum electrodynamics,aquantized radiationfieldhasazero-poin t energyequivalen ttothemean-squareelectricfieldsothatev enina vacuum therearefluctuations. Thesefluctuations causeanelectron toexecutean oscillatorymotionand itsc hargeistherefore smeared.Ifthe electronisbound byanon-uniformelectric field(asin hydrogen),it experiencesadi ff erent potentialfromthatappropriatetoitsm eanposition. Hencetheatomic levels areshifted.In hydrogen-lik eatoms,the smearingoccursoveralength scale, ?(δr) 2 ?? 2α π ? ? mc ? 2 ln 1 α Z , somefive ordersofmagnitudesmallerthan theBohrradius. Thiscauses the electronsping-factortob esligh tlydifferentfrom2, g s =2 ? 1+ α 2π -0.328 α 2 π 2 +··· ? . Thereisalso as lightw eakening oftheforceontheelectronwhenitis very closetothe nucleus, causingthe2 S 1/2 electron(which haspenetratedallthe waytothenucleus)to bes lightlyhigher inenergythanthe2P 1/2 electron. Takingintoac countthesecorrections,oneobtains apositiveenergyshift ΔE Lamb ? ? Z n ? 4 n α 2

Ry×

? 8 3π

αln

1 α Z ? δ ? ,0 , forstateswith ?=0.

Hydrogenfinestructure andh y-

perfinestructureforthen=3to n=2transition (seeFig.9.1). 5 W.E.Lam bandR. C.Retherfod,FineStructure oftheHydrogenA tombya Microwave

Method,Phys. Rev.72,241(1947).

AdvancedQuantumPhysics

9.1.THE"REAL" HYDROGENA TOM95

9.1.5Hyper finestructure

Sofar,w eha veconsideredthenucleus assimplyamassivepointc hargerespon- sibleforthe largeelectrostaticin teractionwiththe chargedelectrons which surroundit.Ho wev er,thenucleushasaspinangularmomentumwhich is associatedwithafurtherset ofhyperfinecorrectionstotheatomic spec- traofatoms. Aswith electrons,theprotons andneutronsthat makeup a nucleusarefermions,each withintrinsic spin1/2.Thismeans thata nucleus willhav esometotalnuclearspinwhich islabe lledb ythequantumnum ber,

I.Thelatter leadstoa nuclear magneticmoment,

µ N =g N Ze 2M N I. whereM N denotesthemass ofthe nucleus,and g N denotesthegyromagnetic ratio.Sincethe nucleus hasinternal structure,thenucleargyromagneticratio isnotsimply 2asit (nearly)isfor theelectron.F ortheproton, thesole nuclearconstituen tofatomichydrogen,g P ≈5.56.Even thoughtheneutron ischarge neutral,itsgyromagneticratiois about-3.83.(Theconsitituen t quarkshav egyromagneticratiosof2(pluscorrections)lik etheelectron but theproblemis complicatedby thestrongin teractionswhich makeithardto defineaquark's mass.)W ecancompute (tosomeacc uracy)thegyromagnetic ratioofn ucleifromthat ofprotonsandneutronsasw ecan computethe proton'sgyromagneticratio fromits quarkconstituents. Sincethen uclear massissev eralorders ofmagnitudehigherthanthatof theelectron,the nuclear magneticmomen tprovidesonlyasmallperturbation. Accordingtoclassical electromagnetism,the magneticmomen tgenerates amagneticfield B= µ 0

4πr

3 (3(µ N ·e r )e r -µ N )+ 2µ 0 3 µ N δ (3) (r).

Toexplorethee

ff ectofthis field,let usconsiderjust thes-electrons,i.e.?=0, forsimplicity . 6 Inthiscase, thein teractionofthe magneticmoment ofthe electronswiththe fieldgenerated bythe nucleus,giv esriseto thehyperfine interaction, ˆ H hyp =-µ e

·B=

e mc ˆ

S·B.

Forthe?=0state, thefirstcon tributionto Bvanisheswhilesecondleads to thefirstorder correction, ?H hyp ? n,1/2,0 = ? Z n ? 4 n α 2

Ry×

8 3 g N m M N 1 ? 2

S·I.

Onceagain,to evaluate theexpectation valuesonthespindegrees offree- dom,itis convenien ttodefine thetotalspinF=I+S.We thenhave 1 ? 2

S·I=

1 2? 2 (F 2 -S 2 -I 2 )= 1 2 (F(F+1)-3/4-I(I+1)) = 1 2 ?

IF=I+1/2

-I-1F=I-1/2 Therefore,the1 sstateofHydrogen issplit intot wo,corresp ondingtothe twopossiblevaluesF=0and 1.Thetransition bet weenthese two levelshas frequency1420Hz, orwa velength21 cm,solies intheradiowaveband.It 6 Forafulldi scussion oftheinfluenceoftheorbital angularmomentum,werefer toRef. [6].

AdvancedQuantumPhysics

9.2.MULTI-ELECTR ONATOMS96

isanimp ortanttrans itionforradioastronomy.Afurther contributiontothe hyperfinestructurearisesifthenuclearshapeisnotspherical thus distorting theCoulomb potential;thiso ccursfordeuteriumandformanyothern uclei. Finally,beforelea vingthissection,weshould notethatthenucleusis not point-likebuthasasmallsize.Thee ff ectoffinitenuclear sizecanbe Amuonisaparticle somewhat likeanelectron,butab out200 timesheavier. Ifamuonis capturedby anatom,thecor- respondingBohrradiusis 200 timessmaller,th usenhancing thenucle arsizee ff ect. estimatedperturbativ ely.Indoingso,onefindsthatthes(?=0)lev elsare thosemoste ff ected,because thesehave thelargestprobabilityoffinding the electroncloseto then ucleus;butthe e ff ectisstill verysmall inhydrogen. It canbe significant,however, inatomsofhighnuclear chargeZ,orfor muonic atoms. Thiscompletesour discussionofthe "one-electron"theory .We now turn toconsiderthe properties ofmulti-electron atoms.

9.2Multi-electronatoms

Toaddresstheelectronicstructure ofam ulti-electronatom,w emight begin withtheh ydrogenicenergy levelsforanatomof nuclearc hargeZ,andstart fillingthelo westlev elswithelectrons,accountingfortheexclusionprinciple. Thedegeneracyfor quantum numb ers(n,?)is2 ×(2?+1),where (2?+1)is thenum berofavailablem ? values,andthefactor of2accoun tsforthe spin degeneracy.Hence,then umber ofelectronsaccommo datedinshell,n,would be2×n 2 , n ?Degeneracyinshell Cumulativ etotal 102 2

20,1(1+ 3)×2=8 10

30,1,2(1+ 3+ 5)×2=18 28

40,1,2,3(1+ 3+ 5+7) ×2=32 60

Wewouldthereforeexp ectthatatomscontaining 2,10,28 or60electrons wouldbeespecially stable,andthatinatomscon tainingonemoreelectron thanthis,the outermostelectron wouldb elesstigh tlybound. Infact,ifwe lookatdata(Fig.9.2) recordingthe firstionizationenergy ofatoms,i.e. the minimumenergyneededtoremo veone electron,w efindthat thenoblegases, havingZ=2,10, 18,36···areespecially tightlybound,andthe elements containingonemoreelectron, thealkali metals,aresignific antlyless tightly bound. Thereasonfor thefailure ofthissimple-minded approachis fairlyobvious - wehaveneglected therepulsionbetweenelectrons.Infact, thefirstionization energiesofatoms show arelatively weakdependenceon Z;thistells usthat the outermostelectronsare almostcom pletelyshielded fromthen uclearcharge. 7 Indeed,whenw etreated theHeliumatomasanexample ofthev ariational methodinchapter7, wefound thatthee ff ectofelectron-electron repulsion wassizeable,andreallyto olargeto betreated accuratelyby perturbation theory. 7 Infact,the shieldi ngis notcompletelyperfect.Foragivenenergyshell, thee ff ective nuclearchargevaries foranatomicnum berZasZ e ff ≂(1+α) Z whereα>0characteri zes theine ff ectivenessofscreening.Thisimpl iesthatthe ionizationenergy IZ=-EZ≂Z 2 e ff ≂ (1+2 αZ).Thenear-l ineardep endenceofIZonZisreflectedinFi g.9. 2.

AdvancedQuantumPh ysics

9.2.MULTI-ELECTR ONATOMS97

Figure9.2:Ionizationenergiesof theelemen ts.

9.2.1Central fieldapproximation

Leavingasideforno wtheinfluence ofspinor relativistice ff ects,theHamilto- nianfora multi-electron atomcanb ewrittenas ˆ H= ? i ? - ? 2 2m ? 2 i - 1

4π?

0 Ze 2 r i ? + ? i4π? 0 e 2 r ij , wherer ij ≡ |r i -r j |.Thefirst termrepresents the"single-particle" contri- butiontothe Hamiltonianarising frominteraction ofeach electronwiththe nucleus,whilethelastterm representsthe mutualCoulom binteraction be- tweentheconstituentelectrons.Itisthis lattertermthat makesthegeneric problem"many-b ody"incharacterandthereforeverycomplicated.Y et,asw e havealreadyseeninthepe rturbative analysisofthe excitedstatesof atomic Helium,thisterm canha veimp ortantph ysicalconsequencesbothontheov er- allenergyof theproblem andonthe associatedspin structureofthe states. Thecentralfieldapproximationisbasedup onthe observationthat theelectronin teractionterm containsalargecentral (sphericallysymmetric) componentarisingfromthe"coreelectrons". Fromthe followingrelation, ? ? m=-? |Y lm (θ,φ)| 2 =const. itisapparen tthata closedshellhasanelectron densitydistribution which isisotropic(indep enden tofθandφ).We canthereforedevelopa perturbative schemebysetting ˆ H= ˆ H 0 + ˆ H 1 ,where ˆ H 0 = ? i ? - ? 2 2m ? 2 i - 1

4π?

0 Ze 2 r i +U i (r i ) ? , ˆ H 1 = ? i4π? 0 e 2 r ij - ? i U i (r i ).

Heretheone-e lectronp otential,U

i (r),which isassumedcentral(se ebelo w), incorporatesthe"av erage"e ff ectofthe otherelectrons.Before discussinghow tocho osethepotentialsU i (r),letus notethat ˆ H 0 isseparablein toa sumof termsforeac helectron, sothatthetotalwa vefunctioncan befactorized into componentsforeachelectron.The basicideais firsttosolvetheSc hr¨odinger equationusing ˆ H 0 ,andthe nto treat ˆ H 1 asasmall perturbation.

Ongeneralgrounds, sincethe Hamiltonian

ˆ H 0 continuestocommutewith theangularmomen tumop erator,[ ˆ H 0 , ˆ

L]=0, wecan seethat theeigenfunc-

tionsof ˆ H 0 willbe characterizedby quantumnumbers(n,?,m ? ,m s ).How ever,

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9.2.MULTI-ELECTR ONATOMS98

sincethee ff ectivepotentialis nolongerCoulomb-like,the ?valuesforagiven nneednotb edegenerate. Ofcourse,thedifficultpartof thisprocedure is toestimateU i (r);thep otential energyexperiencedbyeachelectrondep ends onthew avefunction ofalltheotherelectrons,whichisonlyknownafter the Schr¨odingerequationhasbeensolv ed.Thissuggests thataniterativ eapproach tosolvingthe problemwillb erequired.

DouglasRayner HartreeFRS

1897-1958

AnEnglish

mathematician andphysicist mostfamousfo r thedevelopment ofnumerical analysisandits applicationto atomicphysics.

Heentered

StJohn'sCollege Cambridge in

1915butW orl dWarIinterrupted

hisstudies andhejoinedateam studyinganti-aircraft gunnery.He returnedtoCamb ridgea fterthewar andgraduatedin 1921but,p erhaps becauseofhisinterruptedstudies, heonlyobtained asecond class degreeinNatural Sciences.In 1921, avisitb yNiels BohrtoCambridge inspiredhimtoapplyhis knowledge ofnumericalanalysis tothe solution ofdi ff erentialequationsfo rthe calculationofatomicwavefunctions.

Tounderstandhowthe poten tialsU

i (r)canb eestimated -theself- consistentfieldmethod-itis instructiveto considerav ariationalap- proachdueoriginallyto Hartree.Ifelectrons areconsideredindep endent,the wavefunctioncanbefactorizedintothepro ductstate , Ψ ({r i })=ψ i 1 (r 1 )ψ i 2 (r 2 )···ψ i N (r N ), wherethequan tumn umbers,i k ≡(n?m ? m s ) k ,indicatethe individuals tate occupancies.Notethatthis productstate isnota properlyan tisymmetrized Slaterdeterminant -theexclusionprincipleistak eninto accountonly inso farasthe energyof thegroundstate istaken tobe thelow estthat isconsistent withtheassignmen tofdi ff erentquantumnum bers,n?m ? m s toeach electron. Nevertheless,usingthisw avefunction asatrial state,thevariationalenergyis thengive nby

E=?Ψ|

ˆ

H|Ψ?=

? i ? d 3 r ψ ? i ? - ? 2 ? 2 2m - 1

4π?

0 Ze 2 r ? ψ i + 1

4π?

0 ? iδψ ? i ?

E-ε

i ?? d 3 r|ψ i (r)| 2 -1 ?? =0.

Followingthevariation,

8 oneobtainsthe Hartreeequations, ? - ? 2 ? 2 2m - 1

4π?

0 Ze 2 r ? ψ i + 1

4π?

0 ? j?=i ? d 3 r ? |ψ j (r ? )| 2 e 2 |r-r ? | ψ i (r) =ε i ψ i (r).(9.4) Thenaccordingto thevariational principle,amongstall possibletrial functions ψ i ,theset thatminimizes theenergyare determinedby thee ff ectivepotential, U i (r)= 1

4π?

0 ? j?=i ? d 3 r ? |ψ j (r ? )| 2 e 2 |r-r ? | . Equation(9.4)has asimple interpretation:The firsttw otermsrelate tothe nuclearpotentialexp eriencedbytheindividualelectrons,whilethethird term representstheelectrostaticp otentialdue totheother electrons.However,to simplifythepro cedure,itis usefultoengineertheradialsymmetry ofthe potentialbyreplacingU i (r)by itssphericalaverage, U i (r)?→U i (r)= ? dΩ 4π U i (r). 8 Notethat,i napplyi ngthevariation, thewavefunctionψ ? i canbe consideredindependent ofψi-you mightli ketothinkwhy.

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9.2.MULTI-ELECTR ONATOMS99

Finally,torelatethe Lagrangemultipliers, ε

i (whichhave theappearance ofone-electronenergies ),to thetotalenergy,we canmultiply Eq.(9.4)b y ψ ? i (r)andin tegrate, ? i = ? d 3 r ψ ? i ? - ? 2 ? 2 2m - 1

4π?

0 Ze 2 r ? ψ i + 1

4π?

0 ? j?=i ? d 3 r ? d 3 r|ψ j (r ? )| 2 e 2 |r-r ? | |ψ i (r)| 2 . Ifwe comparethisexpressionwiththe variationalstate energy,w efindthat E= ? i ? i - 1

4π?

0 ? i1.Firstly, onemakesaninitial"guess" fora(common) centralpotential, U(r).Asr→0,screeningb ecomesincreasingly ineffectiveandweex- pectU(r)→0.Asr→∞,we anticipatethatU(r)→ 1

4π?

0 (Z-1)e 2 r , correspondingtoperfect screening.So,as astartingpoint,w emake takesomesmooth functionU(r)interp olatingbetweentheselimits.F or thistrialp otential, wecansolve(numerically)forthe eigenstatesofthe single-particleHamiltonian.W ecan thenusethesestatesasa plat- formtobuild thepro ductwa vefunctionand inturndeterminetheself- consistentpotentials,U i (r).

2.Withthese poten tials,U

i (r),we candetermineanewset ofeigenstates fortheset ofSchr¨ odingerequations, ? - ? 2 2m ? 2 - 1

4π?

0 Ze 2 r +U i (r) ? ψ i =ε i ψ i .

3.Anestimate forthe groundstateenergy ofanatom canbe foundby

fillingupthe energylev els,startingfrom thelow est,andtakingaccount oftheexclusion principle.

4.Usingthese wa vefunctions, onecanthenmakeanimprovedestimateof

thepoten tialsU i (r i )andreturn tostep 2iteratingun tilcon vergence. Sincethepractical implemention ofsuch analgorithmdemandsalarge degree ofcomputationalflair, ifyou remaincurious,y oumay finditusefultorefer to theMathematicaco depreparedb yRef.[4]whereboth theHartree andthe Hartree-Fockprocedures(describedbelow) areillustrated. ?Info.Animprov ementtothisprocedure,knowntheHartree-Fockmethod, takesaccountofexc hangeinteractions.Inorder todothis, itisnecessarytoensure thatthew avefunc tion,includingspin,isantisymmetricunderinterchangeofanypair ofelectrons.This isachiev edby introducing theSlaterdeterminant .Writing the individualelectronw avefunction forthei th electronasψ k (r i ),wherei=1,2···Nand kisshorthandfor theset ofquantum numb ers(n?m ? m s ),theo verall wavefunction isgiven by Ψ = 1 ⎷ N! ? ? ? ? ? ? ? ? ? ψ 1 (r 1 )ψ 1 (r 2 )ψ 1 (r 3 )··· ψ 2 (r 1 )ψ 2 (r 2 )ψ 2 (r 3 )··· ψ 3 (r 1 )ψ 3 (r 2 )ψ 3 (r 3 )··· . . . . . . . . . . . . ? ? ? ? ? ? ? ? ? .

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9.2.MULTI-ELECTR ONATOMS100

Notethateac hof theN!termsin Ψisapro ductofw avefunctionsforeac hindividual electron.The1 / ⎷ N!factorensures thew avefunction isnormalized.A determinant changessignifany tw ocolumnsare exchanged,correspondingtor i ↔r j (say);this ensuresthatthe wav efunctionisan tisymmetricunderexchangeofelectronsiandj. Likewise,adeterminant iszeroif anytworo wsareiden tical;henceall theψ k smust bedi ff erentandtheP auliexclusionprinciple issatisfied. 9

Inthisappro ximation,a

variationalanalysisleadsto theHartree-Fo ckequations (exercise),

VladimirAleksandrovichF ock

1898-1974

ASovietphysi-

cist,whodid foundational workonquan- tummechanics andquantum electrodynamics.

Hisprimary

scientificcon- tributionlies inthedevelopmentofquantum physics,althoughhealsocontributed significantlytothefieldsofme- chanics,theoretical optics,theoryof gravitation,physics ofcontinuous medium.In1926 hederived the

Klein-Gordonequation.Hegave

hisnameto Fo ckspace,the Fock representationandFockstate,and developedtheHartreeF ockmetho d in1930.Fock madesignificant contributionstogeneral relativity theory,specifically forthemany bodyproblems. ε i ψ i (r)= ? - ? 2 2m ? 2 i - Ze 2

4π?r

i ? ψ i (r) + ? j?=i ? d 3 r j 1

4π?

0 e 2 |r-r ? | ψ ? j (r ? ) ? ψ j (r ? )ψ i (r)-ψ j (r)ψ i (r ? )δ ms i ,ms j ? . Thefirstterm inthe lastsetof brackets translatestothe ordinaryHartreecon tribution aboveanddescribestheinfluenceofthe chargedensit yoftheotherelectrons,while thesecondterm describes thenon-lo calexchangecontribution,amanifestatio nof particlestatistics. Theoutcomeof suchcalculations isthatthe eigenfunctionsare,asfor hydrogen,characterizedby quantumnumbers n,?,m ? ,with?Subshellname1 s2s2p3s3p4s3d4p5s4d··· n=12 23 343 454 ··· ?=00 10 102 102 ···

Degeneracy22 62 6210 6210 ···

Cumulative2410121820 303638 48···

Notethatthe value sofZcorrespondingtothenoble gases,2,10, 18,36,at whichtheionizationenergy isunusually high,now emergenaturallyfrom this fillingorder,corresp ondingtothe numbersofelectrons justb eforeanew shell (n)isen tered.There isahandymnemonictoremem berthis fillingorder.By writingthesubshells downas shownrigh t,theorderofstatescan be reado ff 7s 6s 5s 4s 3s 2s 1s 7p 6p 5p 4p 3p 2p 7d 6d 5d 4d 3d

···

6f 5f 4f

···

5g ? ? ??? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ??? alongdiagonalsfrom lower rightto upperleft,startingatthebottom. Wecanusethiss equenceof energylevels topredictthe groundstate electronconfiguration ofatoms.Wesimplyfill upthelev elsstarting from thelow est,accountingfortheexclusionprinciple,un tiltheelectronsareall accommodated(theaufbauprinciple).Hereare afewexamples:

ZElementConfiguration

2S+1 L J

Ioniz.Pot. (eV)

1H(1s)

2 S 1/2 13.6

2He(1s)

21
S 0 24.6

3LiHe(2s)

2 S 1/2 5.4

4BeHe(2s)

21
S 0 9.3

5BHe(2s)

2 (2p) 2 P 1/2 8.3

6CHe(2s)

2 (2p) 23
P 0 11.3

7NHe(2s)

2 (2p) 34
S 3/2 14.5

8OHe(2s)

2 (2p) 43
P 2 13.6

9FHe(2s)

2 (2p) 52
P 3/2 17.4

10NeHe(2s)

2 (2p) 61
S 0 21.6

11NaNe(3s)

2 S 1/2 5.1 9 Notethatfor N=2,the determinant reducestothe familiarantisymmetricwav efunc- tion, 1 ⎷ 2 [ψ1(r1)ψ2(r2)-ψ2(r1)ψ1(r2)].

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9.2.MULTI-ELECTR ONATOMS101

Figure9.3:Periodictableofelements.

Sinceitis generallythe outermostelectronswhic hareof mostinterest, con- tributingtoc hemicalactivity oropticalspectra,oneoften omitstheinner closedshells,and justwrites Oas(2 p) 4 ,forexample. Howev er,the configura- tionisnot always correctlypredicted,esp eciallyintheheavierelements,where levelsmayb eclosetogether.Itmay befavourableto promoteoneor even twoelectronsonelevelab ove thatexpecte dinthissimplepicture,in orderto achieveafilledshell.For example,Cu( Z=29)w ouldb eexpectedtohav econ- figuration···(4s) 2 (3d) 9 ,andactually hasconfiguration···(4s) 1 (3d) 10 .There areseveral similarexamplesinthe transitionelements wherethedsubshells arebeing filled,andmanyamong thelanthanides (rareearths)and actinides wherefsubshellsareb eingfilled. ?Info.Sincetheassignmen tofan electronconfigurationrequiresonlythe enumerationofthevalues ofnand?forallelectrons, butnot thoseofm ? andm s , eachconfigurationwillbe accompaniedb yadegeneracyg.Ifν n ? denotesthen umber ofelectronso ccupying agivenlevelE n, ? ,andδ ? =2×(2?+1)is thedegeneracyof thatlevel, thereare d n ? = δ ? ! ν n ? !(δ ? -ν n ? )! (9.6) waysofdistributingtheν n ? electronsamongthe δ ? individualstates.The total degeneracy,g,isthen obtainedfromthe product. Thisscheme providesabasistounderstand theperiodictableofel- ements(seeFig.9.3). Wew ouldexpect thatelementswhich havesimilar electronconfigurationsin theiroutermost shells(such asLi,Na, K,Rb,Cs, Fr whichallhave (ns) 1 orF,Cl, Br,I,whic hall have (np) 5 )would havesimilar chemicalproperties,suc hasvalency,since itistheunpairedouterelectrons whichespeciallyparticipatein chemicalbonding.Therefore, ifonearranges theatomsin orderofincreasing atomicn umber Z(whichequalsthen umber ofelectronsin theatom), periodic behaviour isseen wheneveranewsubshell ofagiv en?isfilled.

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9.3.COUPLING SCHEMES102

9.3Couplingsc hemes

Theprocedure outlinedabove allowsus topredicttheoccupationofsubshells inanatomic groundstate. Thisis notingeneral su ffi cienttospecify theground statefully. Ifthereareseveralelectrons inapartially filledsubshell,then their spinsandorbital angularmomen tacancom bineinsev eraldi ff erentways,to givedi ff erentvaluesoftotal angularmomentum,withdi ff erentenergies.In ordertodeal withthisproblem, itisnecessary toconsiderthe spin-orbit interactionaswell astheresidual Coulombinteractionbet weenthe outer electrons. Schematicallywecan writetheHamiltonianforthis systemasfollo ws: ˆ

H≈

ˆ H 0 + ? i4π? 0 e 2 r ij - ? i U i (r) ???? ˆ H 1 + ? i ξ i (r i ) ˆ L i · ˆ S i ???? ˆ H 2 , where ˆ H 0 includesthekinetic energyand centralfield terms, ˆ H 1 istheresidual

Coulombinteraction,and(with ξ

i (r i )= 1 2m 2 c 2 1 r (∂ r

V(r)))

ˆ H 2 isthespin-orbit interaction.Wecan thenconsidertwop ossiblescenarios : ˆ H 1 ? ˆ H 2 :Thistendsto applyin thecaseof lightatoms. Inthissituation, onecons idersfirsttheeigenstatesof ˆ H 0 + ˆ H 1 ,andthen treats ˆ H 2 asa perturbation.Thisleadstoa scheme calledLS(orRussell-Saunders) coupling. ˆ H 2 ? ˆ H 1 :Thiscanapply inv eryheavy atoms,orin heavilyionizedlight atoms,inwhic htheelectrons aremovingathigherv elocitiesand rela- tivistice ff ectssuch asthespin-orbitinteractionare moreimp ortant.In thiscase,a scheme calledjjcouplingapplies. Itisimp ortantto emphasisethatbothofthesescenariosrepresen tapproxima- tions;realatoms donot always conformtothe comparativelysimple picture whichemergesfromthesesc hemes,which weno wdiscussin detail.

9.3.1LScoupling scheme

Inthisappro ximation,w estartbyconsideringthe eigenstatesof ˆ H 0 + ˆ H 1 .We notethatthis Hamiltonianmus tcommute withthetotal angularmomentum ˆ J 2 (becauseofinv arianceunderrotations inspace),andalsoclearlycommutes withthetotal spin ˆ S 2 .Italso commuteswith thetotal orbitalangularmomen- tum ˆ L 2 ,since ˆ H 1 onlyinv olvesinternalinteractions,andmustthereforeb e invariantunderglobalrotationofalltheelectrons. Thereforetheenergy levels canbe characterisedbythe correspondingtotalangularmomentum quantum numbersL,S, J.Theirordering inenergyis givenb yHund'srules:

1.Combine thespinsoftheelectrons toobtainp ossiblevalues oftotalspin

S.Thelargest permittedv alueof Slieslow estinenergy.

2.For eachvalueof S,findthe possible valuesof totalangularmomentum

L.Thelargest valueof Llieslow estinenergy.

3.Couplethe values ofLandStoobtainthe valuesof J(hencethename

ofthesc heme).Ifthe subshellislessthanhalf full,thesmallest value ofJlieslow est;otherwise,thelargestvalueofJlieslow est.

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9.3.COUPLING SCHEMES103

Indecidingon thep ermittedvalues ofLandS,inaddition toapplyingthe usualrulesfor addingangular momenta, onealsohas toensurethat theexclu- sionprincipleis respected,as wewill seelaterwhenconsideringsome examples. Theserulesare empirical;there areexceptions,esp eciallyto theLandJ rules(2and 3).Nev ertheless,Hund'srules areaus efulguide,andweshould trytounderstand theirph ysicalorigin.

1.MaximisingSmakesthespinwa vefunction assymmetricas possible.

Thistendsto make thespatialw avefunctionantisymmetric,and hence reducestheCoulom brepulsion,as wesawwhendiscussing theexc hange interactionsinHelium.

2.MaximisingLalsotendsto keepthe electronsapart. Thisislessobvious,

thoughasimple classicalpicture ofelectronsrotating roundthen ucleus inthesame ordi ff erentsensesmakes itatleast plausible.

3.Theseparation ofenergiesfor statesofdi

ff erentJarisesfromtreating thespin-orbitterm ˆ H 2 asap erturbation(finestructure). Itcanbe shown(usingtheWigner-Eck arttheorem -bey ondthescopeofthese lectures)that ? |J,m J ,L,S| ? i ξ i (r i ) ˆ L i · ˆ S i |J,m J ,L,S? =ζ(L,S)?J,m J ,L,S| ˆ L· ˆ S|J,m J ,L,S? =

ζ(L,S)

2 [J(J+1)-L(L+1)-S(S+1)],(9.7) wherethematrix element ζ(L,S)depends onthetotalLandSvalues. Sinceonema yshow thatthesignofζ(L,S)changes accordingtothe whetherthesubshell ismoreor lessthanhalf-filled, thethirdHund's ruleisestablished. Tounderstandtheapplicationof LScoupling, itisb esttow orkthrough someexamples.Starting withthesimplest multi-electronatom ,helium,the groundstatehas anelectronconfiguration (1s) 2 ,andm ustthereforeha ve L=S=J=0.In fact,foran ycompletely filledsubshell,w ehav eL=S=0 andhenceJ=0,since thetotalm L andm S mustequalzeroifall substates areoccupied. Considernowanexcited stateofhelium, e.g.(1s) 1 (2p) 1 ,in whichoneelectronhasb eenexcited tothe2 plevel.Wecanno whaveS=1 orS=0,with theS=1state lyinglow erin energyaccordingto Hund's rules.Combining theorbitalangularmomenta oftheelectrons yieldsL=1 andthus, withS=0,J=1,while withS=1,J=0,1,2withJ=0lying lowestinenergy. Onceagain,as withthe hydrogen-like states,wemayindex thestates of multi-electronatomsby spectroscopicterm notation, 2S+1 L J .Thesup erscript

2S+1giv esthem ultiplicityofJvaluesintowhich thelevelissplit bythe

spin-orbitinteraction; theLvalueisrepresentedb ya capitalletter,S,P,D, etc.,andJisrepresented byitsnumerical value.Thus,forthe (1s) 1 (2p) 1 stateofhelium, thereare fourpossible states,withterms: 3 P 0 3 P 1 3 P 2 1 P 1 , wherethethree 3 Pstatesare separated by thespin-orbitinteraction,andthe singlet 1 Pstatelies much higherin energyowingtotheCoulombinteraction.

Theseparationsb etween the

3 P 2 and 3 P 1 andthe 3 P 1 and 3 P 0 shouldbe inthe

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9.3.COUPLING SCHEMES104

ratio2:1.This isan exampleofthe Land´eintervalrule,which statesthat theseparationb etween apairofadjacentlevelsinafine structuremultiplet is proportionaltothelargerof thetw oJvaluesinvolved. Thisiseasilyshown usingEq.(9.7) -theseparation inenergy betw eenstatesJandJ-1is ?J(J+1)-(J-1)J=2J. Actuallyinthe caseof heliumthesituation isabit morecomplicated,b ecause itturnsout thatthespin-orbit interactionb etween differentelectronsmakes anon-negligibleadditional contribution tothefine structure.Otherexcited statesofhelium, ofthe form(1s) 1 (n?) 1 ,canb ehandled similarly,andagain separateinto singletandtripletstates. ?Exercise.Forthecaseofboron,withthe electronconfiguration(1 s) 2 (2s) 2 (2p), useHund'srules tosho wthatthe groundstateis 2 P 1/2 . Wenextconsiderthecase ofcarbon,which hasgroundstateelectron configuration(1s) 2 (2s) 2 (2p) 2 .Thisin troduces afurthercomplication;wenow havetwoidenticalelectronsinthe sameunfilledsubshell,andw eneedto ensurethattheir wav efunctionisan tisymmetricwithrespecttoelectronex- change.Thetotalspin caneitherb ethesinglet S=0state, whichhas an antisymmetricwav efunction 1 ⎷ 2 [|↑ 1 ?? | ↓ 2 ?- | ↓ 1 ?? | ↑ 2 ?],orone ofthetriple t

S=1states, whichare symmetric,

1 ⎷ 2 [|↑ 1 ?? | ↓ 2 ?+|↓ 1 ?? | ↑ 2 ?],|↑ 1 ?? | ↑ 2 ? or|↓ 1 ?? | ↓ 2 ?.We mustthereforecho osevaluesofLwiththeappropriate symmetrytopartner each valueof S.To formanantisymmetricstate,the tw o m (1) ? m (2) ? m L 101
1-10 0-1-1 electronsmust havedi ff erentvaluesofm ? ,sothe possibilitiesare assho wn right.Inspectingthev aluesofm L wecandeducethatL=1. 10

Bycontrast,

m (1) ? m (2) ? m L 112
101
1-10 000 0-1-1 -1-1-2 toforma symmetric totalangularmomen tumstate,thetwo electronsmay haveanyvaluesofm ? ,leadingto thep ossibilitiesshown right.Insp ectingthe valuesofm L weinferthatL=2or 0. Wemustthereforetak eS=1with L=1and S=0with L=2or 0. Finally,toaccountfor thefinestructure, wenotethatthestates withS=1 andL=1can becom binedin toasingleJ=0state, threeJ=1states, and fiveJ=2states leadingtothe terms 3 P 0 , 3 P 1 ,and 3 P 2 respectively.Similarly theS=0,L=2state canbe combined togive fiveJ=2states, 1 D 2 ,whilethe

S=0,L=0state givesthe singleJ=0state,

1 S 0 .Altogtherw erec overthe

1+3+5+5+1=15possiblestates(cf.Eq.(9.6)withtheorderinginenergy

E/cm -1 1 S 0 20649
1 D 2 10195
3 P 2 43
3 P 1 16 3 P 0 0 givenbyHund'srules (showntotherigh t).Theexp erimentalenergy values aregiven usingtheconven tionalspec troscopicunits ofinversewavelength. Notethatthe Land´ einterv alruleisapproximatelyobeyedb ythefinestructure triplet,andthat theseparation betw eenLandSvaluescausedbythe electron- electronrepulsionis muchgreaterthanthe spin-orbiteffect.

Inanexcited stateof carbon,e.g. (2p)

1 (3p) 1 ,theelectrons arenolonger equivalent,becausetheyha vedi ff erentradialwav efunctions.So nowonecan combineanyofS=0,1withan yofL=0,1,2,yieldingthe following terms (inorderof increasingenergy ,accordingto Hund'srules): 3 D 1,2,3 3 P 0,1,2 3 S 1 1 D 2 1 P 1 1 S 0 . Fornitrogen,theelectron configurationisgiv enb y(1s) 2 (2s) 2 (2p) 3 .The maximalvalue ofspinisS=3/2whileLcantake values3,2,1 and0.Since 10 Thisresultwoul dalsobe apparentifwerecallthethatangular momentumstatesare eigenstatesoftheparity operatorwi theigen value(-1) L .Since therearejusttw oelectrons, thisresultshowsthat boththe L=0and L=2w avefuncti onmustbesymmetricunder exchange.

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9.3.COUPLING SCHEMES105

thespinw av efunction(beingmaximal)issymmetric,thespatialwavefunction mustbecompletelyan tisymmetric.Thisdemandsthatall threestateswith m ? =1,0,-1must beinvolv ed.Wemusttherefore haveL=0,leading to

J=3/2andthe term,

4 S 3/2 .

Levelschemeofthe carbon

atom(1s) 2 (2s) 2 (2p) 2 .Draw- ingisnot toscale.On the lefttheenergy isshown with- outany two-particleinteraction.

Theelectron-electronin teraction

leadstoa three-foldenergy split- tingwithLandSremaining goodquantumnumbers. Spin- orbitcouplingleads toa further splittingofthe stateswith Jre- mainingago od quantumnum- ber.Finallyontherigh t,thelev- elsshow Zeemansplittingsinan externalmagneticfie ld.In this case,thefull setof 15levels be- comenon-degenerate. ?Exercise.ConstructtheL=0state involving theaddition ofthree?=1 angularmomentum states.Hint:makeuse ofthetotal antisymmetrycondition. Asafinal example,let usconsider thegroundstate ofoxygen,which has electronconfiguration(2 p) 4 .Althoughthere arefour electronsinthe (2p) subshell,themaxim umv alueofS=1.This isbecause thereare onlythree availablevaluesofm ? =±1,0,andtherefore oneof thesemust containt wo electronswithopp ositespins. Therefore,themaximumvalue ofm S =1, achievedbyhavingelectronswithm s =+ 1 2 inboth theotherm ? states.By pursuingthisargumen t,itis quiteeasytoseethatthe allowed values ofL,S andJarethesame asfor carbon(2 p) 2 .Thisis infact ageneralresult -the allowedquantumnumbersfor asubshellwithnelectronsarethe sameas for thatofa subshellwith n"holes".Therefore,the energylev elsforthe oxygen groundstateconfiguration arethe sameasfor carbon,except thatthefine structuremultiplet isinverted,inaccordance withHund's thirdrule.

9.3.2jjcouplingscheme

Whenrelativitice

ff ectstake precedenceoverelectronin teractione ff ects,we muststartbyconsidering theeigenstates of ˆ H 0 + ˆ H 2 = ˆ H 0 + ? i ξ i (r i ) ˆ L i · ˆ S i .

Thesemust beeigenstatesof

ˆ J 2 asbefore, becauseoftheo verallrotational invariance,andalsoof ˆ J 2 i foreach electron.Therefore,in thiscase,the cou- plingprocedure istofindtheallo wedjvaluesofindividualelectrons ,whose energieswillb eseparated bythespin-orbitin teraction.Thenthese individual jvaluesarecombinedto findthe allowedvaluesof totalJ.Thee ffectofthe residualCoulomb interactionwillbeto splittheJvaluesforagiv ensetof js. Sadly,inthiscas e,thereare nosimplerules toparallelthoseofHund.

Asanexample, consideraconfiguration (np)

2 inthejjcouplingsche me,to becomparedwiththeexample ofcarb onwhich westudied intheLS scheme. Combinings=1/2with?=1,eac helectroncan havej=1/2or3 /2.If theelectronsha vethe samejvalue,theyareequivalen t,so weha vetotake careofthe symmetryof thewa vefunction.W ethereforeha vethe following possibilities: ? j 1 =j 2 =3/2?J=3,2,1,0,ofwhic hJ=2,0arean tisymmetric. ? j 1 =j 2 =1/2?J=1,0,ofwhic hJ=0is antisymmetric. ? j 1 =1/2,j 2 =3/2?J=2,1.

Injjcoupling,theterm iswritten( j

1 ,j 2 ) J ,sow ehav ethefollowingterms in ourexample: (1/2,1/2) 0 (3/2,1/2) 1 (3/2,1/2) 2 (3/2,3/2) 2 (3/2,3/2) 0 inorderof increasingenergy .Notethat bothLS andjjcouplinggive thesame valuesofJ(inthiscase, tw ostateswith J=0,t wowith J=2and onewith J=1)and inthe sameorder.Ho wever, thepatternof levelsis different;in LScouplingw efounda tripletandtwosinglets, whileinthis idealjjscenario, wehavet wodoubletsandasinglet.Thesets ofstatesinthetwocoupling

AdvancedQuantumPh ysics

9.4.ATOMIC SPECTRA106

schemesmustbe expressibleaslinearcombinations ofoneanother,andthe physicalstatesforareal atomarelik elytodi ff erfromeither approximation. Infact,this idealizedformof jjcouplingisnot seenin theheaviest such atominthe periodic table,lead(6 p) 2 .How ever,itisseeninsomehighly ionizedstates,for examplein Cr 18+ ,which hasthesameelectronconfigu- ration(2p) 2 ascarbon, butwhere,becauseof thelargeruns creenedcharge onthen ucleus,the electronsaremovingmorere lativistically,enhancing the spin-orbite ff ect.Ho wever,aclassicexampleofthetransitionfromLSto jj couplingisseen inthe seriesC-Si-Ge-Sn-Pb intheexcited states(2p)(3s), (3p)(4s),···(6p)(7s)(seefigure right).Here, theelectrons arenotinthesame subshell,sotheir wa vefunctionso verlapless,andtheCoulombrepulsionis reducedcomparedto thespin-orbitin teraction.Analysingthis situationin theLScoupling approximation,one expectsa tripletandasinglet: 3 P 0,1,2 1 P 1 , whileinthe jjschemeoneexpectst wodoublets: (1/2,1/2) 0,1 (1/2,3/2) 2,1 . Experimentally,CandSiconformtotheLS expectationand Pbtothe jj scheme,whileGeandSn areintermediate.

9.4Atomic spectra

Atomicspectraresult fromtransitionsbetw eendi

ff erentelectronicstatesof anatomvia emissionorabsorption ofphotons.In emissionspectra ,an atomisexcited bysome means(e.g.thermally throughcollisions,orbyan electricdischarge), andoneobservesdiscrete spectrallines intheligh temitted astheatoms relax.Inabsorptionspectra ,oneilluminates atomsusinga broadwa vebandsource,andobservesdarkabsorptionlinesinthe spectrum oftransmittedligh t.Ofcourse theatomsexcitedinthispro cesssubsequently decaybyemitting photonsinrandomdirections;by observingindirections awayfromtheincidentlightthisfluorescenceradiationmay bestudied.The observationofthesespectral linesisan important wayofprobing theatomic energylevels experimentally.In thecaseofopticalspectraandthenearby wavebands,theexcitedstatesresponsiblegenerallyin volve theexcitation of asingleelectron fromtheground statetosome higherleve l.Insome cases, itmay simplyinvolve adi ff erentcouplingoftheangular momentawithin the sameelectron configuration.Thesearethekindsof excitationswhich we are abouttodiscuss. Howev er,othertypesof excitationsarealsopossible.For example,X-ray emissionoccurswhenan electronhasb eenremovedfromone oftheinnermost shellsofa heavyatom; aselectronscascade downto fillthe hole,highe nergy photonsmaybeemitted. Thebasictheory governing stimulated emissionandabsorption,andspon- taneousemissionof photonswillb eoutlinedin detailwhen we studyradiative transitionsin chapter13. Herewemustan ticipatesomeof thebasiccon- clusionsofthat study. Intheelectricdip oleapproximation,therate of transitionsis proportionalto thesquareofthematrixelemen tofthe electric dipoleoperatorbet weentheinitialandfinal states,|?ψ f | ˆ d|ψ i ? | 2 .Inaddition, therateof spontaneous transitionsisprop ortionaltoω 3 ,whereω=|E f -E i | denotestheenergy separationb etwe enthestates.

Theformof thedip oleoperator,

ˆ dmeansthatthe matrixelemen tsmay vanishidentically.This leadstoasetofselectionrules definingwhic htran- sitionsareallo wed. Hereweconsiderthesimplestcase ofasingle electron,but

AdvancedQuantumPh ysics

9.4.ATOMIC SPECTRA107

theprinciplescan be generalized.Referringto chapter13fora moredetailed discussion,onefinds that,fora transitiontotak eplace: ?Paritymustchange; ?ΔJ=±1,0(but0 →0isnot allowed) andΔM J =±1,0. Atomicstatesarealwa yseigenstatesof parityand oftotalangularmomentum, J,sothese selectionrules canb eregardedasabsolutelyvalid inelectricdip ole transitions.It shouldbe emphasizedagain,though, thattheelectricdipole approximationisanapproxim ation,andhigher orderprocesses mayoccur, albeitataslo werrate, andhav etheirownselectionrules. Inspecific couplingschemes,furtherselectionrules mayapply .Inthecase ofidealLS coupling,w ealsorequire: ?ΔS=0and ΔM S =0; ?ΔL=±1,0(but0 →0isnot allowed) andΔM L =±1,0; ?andΔ? i =±1ifonly electroniisinv olvedinthetransition. InLScoupling, thestatesare eigenstatesoftotal spin;sincethe dipoleop erator doesnotoperateon thes pinpartofthewa vefunction,the ruleson

ΔSand

ΔM S followstraightforwardly .This,andtheabsoluterulesrelatingtoJ, implytherules forLandM L .Therule for Δ? i followsfromtheparit ychange rule,sincethe parityof theatomis theproductoftheparities ofthe separate electronwa vefunctions,givenby(-1) ? i .How ever,sinceLScouplingisonly anapproximation, theserulesshouldthemselvesb eregardedas approximate. Withthispreparation, we nowturn totheconsequencesoftheselection rules ontheatomic spectraof atoms.

9.4.1Singleelectr onatoms

Inthiscon text,"single electronatoms"refertoatomswhose groundstate consistsofa singleelectronin anslevel,outsideclosedshells; itisthis electron whichisactivein opticalsp ectroscopy.Ourdiscussion thereforeencompasses thealkalimetals,such assodium,andalso hydrogen.W etakesodium, whosegroundstate configurationis (3s) 1 ,asour example: ?Thegroundstate hasterm 2 S 1/2 .Theexcited state sareall doublets withJ=L±1/2,exceptfor thesstates,which areobviouslyrestricted toJ=1/2. ?Theparity isgivenby (-1) ? i ,sothe allow edtransitionsin volvechanges in?by±1unit,i.e. s↔p,p↔d,d↔f,etc.Changes ofmore than oneunitin ?wouldfallfoulofthe ΔJrule. ?Thes↔ptransitionsare alldoublets.All thedoubletsstarting or endingona given pstatehav ethesamespacinginenergy.The transition

3s↔3pgivesrisetothe familiaryello wsodium "D-lines"at589 nm

(seeright). ?Thep↔dtransitionsinv olvetwodoublets, 2 P

1/2,3/2

and 2 D

3/2,5/2

.

However,the

2 P 1/2 ↔ 2 D 5/2 transitionisforbidden by the

ΔJrule,so

thelineis actuallya triplet.In practice,thespin-orbit interactionfalls quiterapidlywith increasing?(andwithincreasing n)asthe effectof screeningincreases, sothatthee ff ectofthe 2 D

3/2,5/2

splittingmay not beresolvedexpe rimentally.

AdvancedQuantumPhysics

9.4.ATOMIC SPECTRA108

?Asnincreases,thee nergy levelsapproach(frombelow)those forhy- drogen,be causethenuclearchargeisincreasingly e ff ectivelyscreened bytheinnerelectrons.This happensso onerfor thehigher?values,for whichtheelectrontendsto liefurtherout fromthen ucleus. ?Inanabsorption spectrum,the atomswill startfromthegroundstate, soonlythe 3s→nplineswillb eseen. Inemission,theatomsareexcited intoessentiallyallthe irexcitedlevels,so manymore lineswillbeseen inthesp ectrum. Thecomments abovefor sodiumalsoapplyforhydrogen,exceptthat,inthis case,(2s,2p),(3s,3p,3d),etc.are degenerate.One consequenceisthat the2s stateinh ydrogenis metastable-itcannotdecay totheonly lower lyinglevel (1s)by anelectricdipoletransition. Infactits favoured spontaneousdecay isby emissionoftwo photons;apro cesswhich isdescribedbysecond-order perturbationtheory.In practice,hydrogenatomsina 2sstatearemore likely todeexcitethrough collisionpro cesses.Duringan atomiccollision,the atoms aresubject tostrongelectricfields,and wekno wfromour discussionofthe

Starke

ff ectthatthiswillmix the2sand2pstates,anddec ayfrom 2pto1s isreadilyp ossible.

9.4.2Heliumand alkali earths

Wenextdiscussatomswhose groundstate consistsoft woelectrons inans level.Ourdiscussionthereforeco vers helium(1s) 2 ,andthe alkaliearths: beryllium(2s) 2 ,magnesium(3 s) 2 ,calcium(4 s) 2 ,etc.W estartwith helium. ?Thegroundstate hasterm 1 S 0 .Theexcited statesareof theform (1s)(n?)(theenergy requiredto exciteboth ofthe1 selectronstohigher statesisgreater thanthefirst ionizationenergy ,andtherefore these formdiscretestates withina continuum ofionizedHe + +e - states).The excitedstates canhaveS=0or S=1,with S=1lying lower inenergy (Hund). ?TheLScoupling approximationis pretty goodforhelium,so theΔS=0 selectionruleimplies thattheS=0and S=1states formcompletely independentsystemsasfarassp ectroscopy isconc erned. ?Thelinesin theS=0system areallsinglets. Theycan beobserv ed inemission,and those startingfromthe groundstatecanbeseen in absorption. ?Thelinesin theS=1system areallm ultiplets.They canbe observed inemissiononly .T ransitionsoftheform 3 S 1 ↔ 3 P 2,1,0 areobserved as triplets,spacedaccording tothe Land´e interval rule.Transitions ofthe form 3 P 2,1,0 ↔ 3 D 3,2,1 areobserved assextuplets,asiseas ilyseenb y applicationof the ΔJ=±1,0rule.Actually ,as mentionedabov e,the finestructureis alittle moresubtlein thecaseof helium. Thealkali earthsfollowthesame principles.Inthe caseofcalcium,the triplet4pstateisthe low estlying tripletstate,andthereforemetastable.In factafain temissionline correspondingtothe 3 P 1 → 1 S 0 decaytotheground statemay beobserved; thisviolatesthe

ΔS=0rule, andindicatesthat theLS

couplingapproximation isnotsogood inthiscase. Amore extremeexampleis seeninMercury ,ground state(6s) 2 (5d) 10 .Excitedstates inv olvingpromotion ofoneof the6 selectronstoa higherle velcan betreated justlikethealkali

AdvancedQuantumPhysics

9.5.ZEEMANEFFECT 109

earths.Inthis case the"forbidden" 3 P 1 → 1 S 0 isactuallya prominent feature oftheem ission spectruminthevisible,implyingasignificant breakdownof theLSappro ximation.

9.4.3Multi-electronatoms

Similarprinciplescan be usedtomak esenseofthespectra ofmorecomplicated atoms,thoughunsurprisingly theirstructure ismorecomplex. Forexample, carbon,withgroundstate (2s) 2 (2p) 2 ,corresponds toterms 3 P 0,1,2 , 1 D 2 and 1 S 0 asdiscussedab ov e.Theexcitedstatesareoftheform(2s) 2 (2p) 1 (n?) 1 , andcanb eseparated intosingletsandtriplets,and inadditionexcitations oftheform (2s) 1 (2p) 3 canarise.Nitrogen, withground state(2s) 2 (2p) 3 ,has threeunpairedelectrons, sotheground stateand excitedstatesform doublets (S=1/2)andquartets (S=3/2)withc orrespondingly complexfinestructure tothesp ectrallines.

9.5Zeemane

ff ect

9.5.1Single-electron atoms

Beforeleaving thischapteronatomic structure,we arenowina positionto revisitthequestion ofhow atomicspectra areinfluencedb yamagneticfield. Toorientourdiscuss ion,letusbegin withthestudy ofhydrogen-likeatoms involvingjustasingleelectron.In amagneticfield, theHamiltonianof such asystemis describedb y ˆ H= ˆ H 0 + ˆ H rel. + ˆ H

Zeeman

,where ˆ H 0 denotesthe non-relativisticHamiltonianfor theatom, ˆ H rel. incorporatestherelativistic correctionsconsideredearlier inthe chapter,and ˆ H

Zeeman

=- eB 2mc ( ˆ L z +2 ˆ S z ), denotestheZeeman energyasso ciatedwiththe couplingofthe spinandorbital angularmomentum degreesoffreedomtothemagnetic field.Here,since wearedealingwithconfined electrons,we have neglectedthe diamagnetic contributiontotheHamiltonian. Dependingon thescaleof themagneticfield, thespin-orbitterm in ˆ H rel. ortheZeeman termma ydominatethe spectrum oftheatom. Previously,wehav eseenthat,toleadingorder, therelativisticcorrections leadtoa fine-structureenergy shiftof ΔE rel. n,j = 1 2 mc 2 ? Z α n ? 4 ? 3 4 - n j+1/2 ? , forstates|n,j=?±1/2,m j , ??.For weakmagneticfields,we canalsotreat theZeemanenergy inthe framework ofperturbation theory.Here, although stateswithcommon jvalues(suchas2 S 1/2 and2P 1/2 )aredege nerate ,the twospatialwavefunctions have di ff erentparity(?=0and 1inthis case), andtheo ff -diagonalmatrixelemen tof ˆ H

Zeeman

couplingthesestates vanishes. Wemaythereforea voidusingdegeneratep erturbationtheory. Makinguseof therelation(exercise -referbac ktothe discussionofthe additionof angular momentaandspinin section6.4.2), ?n,j=?±1/2,m j , ?|S z |n,j=?±1/2,m j , ??=± ?m j 2?+1 , weobtainthefollowing expressionfor thefirst orderenergyshift, ΔE

Zeeman

j=?±1,m j , ? =?±1/2,m j , ??=µ B Bm j ? 1± 1 2?+1 ? ,

AdvancedQuantumPh ysics

9.5.ZEEMANEFFECT 110

Figure9.4:Thewell knowndoubletwhich isresponsibleforthebrigh tyello wlight fromaso diumlampma ybeusedto demonstrate severaloftheinfluenceswhic h causesplittingof theemission linesofatomic spectra.The transitionwhich gives risetothe doubletisfrom the3 ptothe3 slevel.Thefactthatthe 3sstateislo wer thanthe3 pstateisa good exampleof thedependenceofatomicenergylevelson orbitalangularmomen tum.The3 selectronpenetrates the1sshellmoreand isless e ff ectivelyshieldedthanthe3 pelectron,sothe 3slevelislower. Thefact thatthere isadoublet showsthe smallerdependence oftheatomicenergylevels onthe total angularmomentum. The3plevelissplitinto stateswith totalangularmomen tum J=3/2andJ=1/2by thespin-orbitinteraction.Inthe presenceof anexternal magneticfield, theselevelsarefurther splitby themagneticdipoleenergy, showing dependenceoftheenergies onthez-componentofthetotalangularmomen tum. whereµ B denotestheBohr magne ton.Therefore,w eseethatalldegenerate levelsaresplitdue tothemagnetic field.Incon trasttothe "normal"Zeeman e ff ect,themagnitude ofthesplitting dependson ?. ?Info.Ifthefieldisstrong ,theZeeman energybecom eslargein comparison withthespin-orbit contribution. Inthiscase, wemustwork withthebasis states |n,?,m ? ,m s ?=|n,?,m ? ?? |m s ?inwhich both ˆ H 0 and ˆ H

Zeeman

arediagonal.Within firstorderof perturbation theory,one thenfindsthat(exercise) ΔE n, ? ,m ? ,ms =µ B (m ? +m s )+ 1 2 mc 2 ? Z α n ? 4 ? 3 4 - n ?+1/2 - nm ? m s ?(?+1/2)(?+1) ? , thefirstterm arisingfrom theZeemanenergy andtheremaining termsfrom ˆ H rel. .At intermediatevaluesofthe field,wehave toapplydegenerate perturbation theoryto thestatesin volvingthe linearcombinationof|n,j=?±1/2,m j , ??.Such acalculation reachesbeyond thescopeoftheselecturesand,fordetails,we refertothe literature (see,e.g., Ref.[6].Letusinstead considerwhathapp ensinm ulti-electronatoms.

9.5.2Multi-electronatoms

Foramulti-electronatom inaw eakmagneticfield,theappropriateunp er- turbedstatesaregiv enby |J,M J ,L,S?,whereJ,L,Srefertothe total angularmomenta. Todetermine theZeemanenergyshift,we needtodeter- minethematrix elementof ˆ S z .To doso,we canmake useofthefollowing

AdvancedQuantumPh ysics

9.5.ZEEMANEFFECT 111

argument.Since2 ˆ L· ˆ S= ˆ J 2 - ˆ L 2 - ˆ S 2 ,thisop eratorisdiagonal inthebasisof states,|J,M J ,L,S?.Therefore,the matrixelemen tofthe operator(exercise, hint:recallthat[ ˆ S i , ˆ S j ]=i?? ijk ˆ S k and[ ˆ L i , ˆ S k ]=0),

Figure9.5:Inthew eakfield

case,thev ectormo del(top)im- pliesthatthe couplingofthe or- bitalangularmomen tumLto thespin angularmomentumS isstrongerthan theircoupling totheexternal field.Inthis casewherespin-orbit coupling is dominant,theycanb evisual- izedascom biningto formato- talangularmomen tumJwhich thenprecesses aboutthemag- neticfielddirection. Inthe strongfieldcase, SandLcou- plemorestrongly tothe exter- nalmagneticfield thanto each other,andcan be visualizedas independentlyprecessingabout theexternalfield direction. -i? ˆ S× ˆ

L≡

ˆ S( ˆ L· ˆ S)-( ˆ L· ˆ S) ˆ S mustvanish.Moreov er,fromtheiden tity[ ˆ L· ˆ S, ˆ

J]=0, itfollows thatthe

matrixelement ofthevectorpro duct, -i?( ˆ S× ˆ

L)×

ˆ J= ˆ S× ˆ J( ˆ L· ˆ S)-( ˆ L· ˆ S) ˆ S× ˆ J, mustalsovanish.If we expandthelefthandside,w ethus findthatthematrix elementof ( ˆ S× ˆ

L)×

ˆ J= ˆ L( ˆ S· ˆ J)- ˆ S( ˆ L· ˆ J) ˆ L= ˆ J- ˆ S = ˆ J( ˆ S· ˆ J)- ˆ S ˆ J 2 , alsovanishes. Therefore,itfollowsthat ? ˆ S ˆ J 2 ?=? ˆ J( ˆ S· ˆ

J)?,wherethe expec-

tationvalue istakenov erthebasis states.Then,with ˆ S· ˆ J= 1 2 ( ˆ J 2 + ˆ S 2 - ˆ L 2 ), wehave

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