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Lecture12

Atomicstructure

Atomicstructure:background

Ourstudiesof hydrogen-likeatom srevealedthat thespectrumof theHamiltonian, ˆ H 0 = ˆp 2 2m - 1

4π?

0 Ze 2 r ischaract erizedbylargen 2 -folddegeneracy. However,althoughthenon-relativisticSchr¨odingerHamiltonian providesausefulplatform, theformulation isalittle toona¨ıve. TheHamiltonianis subjecttoseveral classesof"co rrections",which leadtoimp ortantphysical ramifications(whichreachbeyondthe realmofatomic physics).

Inthislecture, weoutline thesee

ff ects,beforemoving ontodiscuss multi-electronatomsin thenext.

Atomicstructure:hydrogenatomrevisited

Aswithany centrallysymmetricp otential,stationary solutionsof ˆ H 0 indexby quantumnumbersn?m,ψ n?m (r)=R n? (r)Y ?m (θ,φ).

Foratomichydrogen,n

2 -degenerateenergylevels setby E n =-Ry 1 n 2 ,Ry= ? e 2

4π?

0 ? 2 m 2? 2 = e 2

4π?

0 1 2a 0 wheremisreducedmass (ca.electron mass),anda 0 =

4π?0

e 2 ? 2 m .

Forhighersingle-electronions(He

+ ,Li 2+ ,etc.),E n =-Z 2Ry n 2 .

Allowedcombinationsofquantumnumbers:

n?Subshell(s) 101 s

20,12s2p

30,1,23s3p3d

n0···(n-1)ns···

Atomicstructure:hydrogenatomrevisited

However,treatmentofhydrogenatominherentlynon-relativistic : ˆ H 0 = ˆp 2 2m - 1

4π?

0 Ze 2 r isonlythe leadingtermin relativistictreatment( Diractheory ). Suchrelativisticco rrectionsbegin toimpactwhentheelectron becomesrelativistic,i.e.v≂c. Since,for Coulombpotential,2?k.e.?=-?p.e.?(virialtheore m), 1 2 mv 2 =?k.e.?=-E 000 =Z 2

Ry.Usingidentit y,

Z 2 Ry= 1 2 mc 2 (Zα) 2 , α= e 2

4π?

0 1 ?c ? 1 137
whereαdenotesthefinestructureconstant ,we find v c =Zα.

The"real"hyd rogenatom: outline

Termsofhigherorder in

v c =Zαproviderelativisticco rrections whichleadto liftingof thedegeneracy . Thesecorrections (knownasfine-structure)derivefrom three (superficially)di ff erentsources: (a)relativisticcorrections tothekineticenergy; (b)couplingbet weenspinandorbitaldegreesoffreedom; (c)andacontribution knownas theDa rwinterm. (a)Relativisticco rrections tokineticenergy

Fromtherelativisticenergy-momentum invariant,

E= ? p 2 c 2 +m 2 c 4 =mc 2 + p 2 2m - 1 8 (p 2 ) 2 m 3 c 2 +···, wecananticipatetheleading correctionto thenon-relativistic

Hamiltonianisgiven by

ˆ H 1 =- 1 8 (ˆp 2 ) 2 m 3 c 2

Therelativescale ofperturbation

? ˆ H 1 ? ? ˆ H 0 ? ? p 2 m 2 c 2 = v 2 c 2 ?(Zα) 2 whereα= e 2

4π?0

1 ?c ? 1 137
. i.e. ˆ H 1 isonlya smallperturbation for smallatomicnumb er,Z. (a)Relativisticco rrections tokineticenergy ˆ H 1 =- 1 8 (ˆp 2 ) 2 m 3 c 2

Since[

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

Therefore,theo

ff -diagonalmatrixelem entsvanish: ?n?m| ˆ H 1 |n? ? m ? ?=0for??=? ? orm?=m ? andwe canestimateenergyshiftwithout havingtoinvok e degenerateperturbation theory. (a)Relativisticco rrections tokineticenergy

Makinguseof theidentit y,

ˆ H 1 =- 1 8 (ˆp 2 ) 2 m 3 c 2 =- 1 2mc 2 ? ˆ H 0 -V(r) ? 2 ,V(r)=- Ze 2

4π?

0 1 r scaleofresulting energyshift canbe obtainedfromfirst order perturbationtheory, ?n?m| ˆ H 1 |n?m?=- 1 2mc 2 ? E 2 n -2E n ?V(r)? n? +?V 2 (r)? n? ?

Usingtheidentities,

? 1 r ? n? = Z a 0 n 2 , ? 1 r 2 ? n? = Z 2 a 2 0 n 3 (?+1/2) . resultingenergyshift acquiresangula rmomentumdep endence: ? ˆ H 1 ? n?m =- mc 2 2 ? Zα n ? 4 ? n ?+1/2 - 3 4 ? (b)Spin-orbit coupling Spindegreeof freedomofelectron emergesnaturallyfrom relativisticformulation ofquantummechanics.Alongsidethe spin, thisform ulationleadstoafurtherrelativisticcorrection which involvescouplingb etw eenspinandorbitaldegreesoffreedom. ForageneralpotentialV(r),thisspin-o rbitcoupling isgivenby: ˆ H 2 = 1 2m 2 c 2 1 r (∂ r V) ˆ L· ˆ S

Forahydrogen-likeatom,V(r)=-

1

4π?0

Ze 2 r and ˆ H 2 = 1 2m 2 c 2 1

4π?

0 Ze 2 r 3 ˆ L· ˆ S (b)Spin-orbit coupling:physicalorigin Physically,aselectronmoves throughelectric fieldofnucle us,

E=-?V(r)=-ˆe

r (∂ r

V),inits restframeit willexp eriencea

magneticfield,B= 1 c 2 v×E. Inthisfield, thespinmagnetic momentofthe electron,µ s =- e m S, leadstoan additionalinteraction energy, -µ s

·B=-

e (mc) 2

S·(p׈e

r (∂ r V))= e (mc) 2 1 r (∂ r

V)L·S

wherewe haveusedtherelationp׈e r =- 1 r L. Additionalfactor of1/2derivesfrom furtherrelativistice ffect knownasThomasprecession . Thosediscontentwith heuristicderviationneed onlywait forDirac formulation... (b)Spin-orbit coupling ˆ H 2 = 1 2m 2 c 2 1

4π?

0 Ze 2 r 3 ˆ L· ˆ S Withoutspin-orbit interaction,eigenstatesofhydrogen-likeatoms canbe expressedinbasisofmutually commutingoperators, ˆ H 0 , ˆ L 2 , ˆ L z , ˆ S 2 ,and ˆ S z . However,withspin-orbit,totalHamiltoniannolongercommutes with ˆ L z or ˆ S z -usefulto exploitdegeneracy of ˆ H 0 toswitchto new basisinwhich ˆ L· ˆ

Sisdiagonal.

Achievedby turningtobasisofeigenstates oftheop erators, ˆ H 0 , ˆ J 2 , ˆ J z , ˆ L 2 ,and ˆ S 2 ,where ˆ J= ˆ L+ ˆ

S.Since

ˆ J 2 = ˆ L 2 + ˆ S 2 +2 ˆ L· ˆ S,it followsthat, ˆ L· ˆ S= 1 2 ( ˆ J 2 - ˆ L 2 - ˆ S 2 )= 1 2 (j(j+1)-?(?+1)-s(s+1)) (b)Spin-orbit coupling ˆ L· ˆ S= 1 2 ( ˆ J 2 - ˆ L 2 - ˆ S 2 )= 1 2 (j(j+1)-?(?+1)-s(s+1)) ˆ H 2 = 1 2m 2 c 2 1

4π?

0 Ze 2 r 3 ˆ L· ˆ S Combiningspin1 /2withangula rmomentum?,totalangula r momentumcantak evaluesj=?±1/2.Corresp ondingbasisstates |j=?±1/2,m j , ??diagonalizeoperato r, ˆ L· ˆ

S|j=?±1/2,m

j , ??= ? 2 2 ? ? -?-1 ? |?±1/2,m j , ??

Onceagain,o

ff -diagonalmatrixelements of ˆ H 2 vanishallowing correctiontobecomputed infirsto rderperturbationtheory. ?H 2 ? n,j=?±1/2,m j , ? = 1 2m 2 c 2 ? 2 2 ? ? -?-1 ? Ze 2

4π?

0 ? 1 r 3 ? n? (b)Spin-orbit coupling ?H 2 ? n,j=?±1/2,m j , ? = 1 2m 2 c 2 ? 2 2 ? ? -?-1 ? Ze 2

4π?

0 ? 1 r 3 ? n?

Makinguseof identity,

? 1 r 3 ? n? = ? mcαZ ?n ? 3 1 ?(?+1/2)(?+1) , ? >0 ? ˆ H 2 ? n,j=?±1/2,m j , ? = 1 4 mc 2 ? Zα n ? 4 n j+1/2 ? 1 j j=?+1/2 - 1 j+1 j=?-1/2

Rewritingexpression for?

ˆ H 1 ?innewbasis |n,j=?±1/2,m j , ??, ? ˆ H 1 ? n,j=?±1/2,m j , ? =- 1 2 mc 2 ? Zα n ? 4 n ? 1 j j=?+1/2 1 j+1 j=?-1/2 .

Combiningtheseexp ressions,for ?>0,we have

? ˆ H 1 + ˆ H 2 ? n,j=?±1/2,m j , ? = 1 2 mc 2 (Zα) 4 n 4 ? 3 4 - n j+1/2 ? whilefor ?=0,w eretainjust ? ˆ H 1 ? (c)Darwin term Finalrelativisticco rrectiona risesfrom"Zitterbewegung"of electron -giggling- whichsmea rse ff ectivepotential feltbyelectron, ˆ H 3 = ? 2 8m 2 c 2 ? 2 V= ? 2 8m 2 c 2 eQ nuclear (r) ? 0 = π? 2 2m 2 c 2 Ze 2

4π?

0 δ (3) (r)

Sinceperturbation actsonlyatorigin,it e

ff ectsonly?=0states, ? ˆ H 3 ? nj=1/2,m j ?=0 = π? 2 2m 2 c 2 Ze 2

4π?

0 |ψ n00 (0)| 2 = 1 2 mc 2 (Zα) 4 n 3 Thistermis formallyide nticalto thatwhichwouldbe obtained from? ˆ H 2 ?at?=0.As aresult,combining allthree contributions, ΔE n,j=?±1/2,m j , ? = 1 2 mc 2 ? αZ n ? 4 ? 3 4 - n j+1/2 ? independentof?andm j .

Spectroscopicnotation

Todiscussenergyshiftsfo rpa rticularstates, itishelpful to introducesomenomenclature fromatomicphysics. Forastatewithprincipalquantum number n,totalspin s,orbital angularmomentum?,andtotal angularmomentum j,onema y definethestate bythe spectroscopicnotation, n 2s+1 L j Forahydrogen-likeatom,withjust asingleelectron, 2s+1= 2.In thiscase,the factor 2s+1is oftenjustdropp edfo rbrevit y.

Relativisticcorrections

ΔE n,j=?±1/2,m j , ? = 1 2 mc 2 ? αZ n ? 4 ? 3 4 - n j+1/2 ?

Foragivenn,relativisticco rrectionsdep end

onlyonjandn.

Forn=1,?=0and j=1/2:Both1 S

1/2 states,withm j =±1/2,experience negative energyshiftof - 1 4 Z 4 α 2 Ry.

Forn=2,?=0,1:Withj=1/2,both 2S

1/2 and2P 1/2 stateshaveshift, - 5 64
Z 4 α 2 Ry, while2P 3/2 experiencesashift- 1 64
Z 4 α 2 Ry. (Further)relativisticcorrections:Lambshift Perturbativecorrections predictedbyDiractheorypredict that,for hydrogen,the2 S 1/2 and2P 1/2 statesshouldremain degenerate. However,in1951,anexperimentalstudyb yWillisLamb discovered that2P 1/2 stateisslightly lower thanthe 2S 1/2 state-Lamb shift.

Mightseemthat suchatiny e

ff ect wouldbeinsignificant,but shift providedconsiderable insightinto quantumelectrodynamics .

Lambshift

Withinframewo rkofquantumelectrodynamics,Coulomb

interactionismediated byexchange ofphotons- "gaugeparticles". Interactionofelectron withelectromagnetic fieldcaninduce a "self-interaction"?effectivesmearing ofelectronposition, ?(δr) 2 ?? 2α π ? ? mc ? 2 ln 1 αZ ,

δr≂10

-5 a 0 Causeselectronspin g-factortobeslightly differentfrom2. Thereisalso aslightw eakeningof theforce ontheelectron whenit isveryclose tothenucleus, causing2 S 1/2 statetob eslightlyhigher inenergythan the2 P 1/2 state. ΔE Lamb ? 1 2 mc 2 ? αZ n ? 4 n× ? 8 3π

αln

1 αZ ? δ ? ,0

Hyperfinestructure

Finally,weshouldaddress thepotentialinfluenc eofthe nuclear spin,I,whichleads toanuclea rmagnetic moment, M=g N e 2M N I wherenucleushas massM N andgyromagneticratio g N .

Sincenucleushas internalstructure, g

N isnotsimply 2.Fo rp roton, solenuclear constituentofatomichydrogen,g p ≈5.56.Even thoughneutronis chargeneutral, g n ≈-3.83.

Magneticmomentgenerates vectorp otentialA=-

µ0 4π

M×?(1/r)

andmagneticfield

B=?×A=

µ 0 4π ?

3r(r·M)-r

2 M r 5 + 8π 3 Mδ (3) (r) ?

Hyperfineinteraction

B=?×A=

µ 0 4π ?

3r(r·M)-r

2 M r 5 + 8π 3 Mδ (3) (r) ? Asaresult, weobtain hyperfineinteraction withorbitalandspin degreesoffreedom ofelect ron, ˆ H hyp = e 2m ( ˆ L+2 ˆ

S)·B

Energylevelshift oftheground statecanb eestimatedusing perturbationtheory.If weconsider(forsimplicity)justthe ?=0 states,onlylast terminBcontributesatlo westo rder,andleadsto ? ˆ H hyp ? n,1/2,0 = µ 0 4π g N e 2M N e m 8π 3 |ψ n00 (0)| 2 ˆ S· ˆ I/? 2

Hyperfineinteraction

? ˆ H hyp ? n,1/2,0 = µ 0 4π g N e 2M N e m 8π 3 |ψ n00 (0)| 2 ˆ S· ˆ I/? 2

With|ψ

n00 (0)| 2 = 1 πn 3 (

Zαmc

? ) 3 ,we obtain ? ˆ H hyp ? n,1/2,0 = 1 2 mc 2 ? Zα n ? 4 n× 8 3 g N m M N ˆ S· ˆ I/? 2 showingscaleofperturbation suppressed overfines tructureby factorm/M N ≂10 -3 . Finally,aswithspin-orbit interaction,ifw esetF=I+S, 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

Summaryofatomicenergyscales

Grossstructure:Dictatedby

orbitalkineticandpotential energies,ca.1 -10eV.

Finestructure:Relativistic

corrections(spin-orbit,etc.) splitdegeneratemultiplets leadingtosmall shiftin energy, ca.10 -4 -10 -5 eV.

Hyperfinestructure:

Interactionofelectron

magneticmomentwith field generatedby nuclearspinleads tofurthersplitting of multiplets,ca.10 -7 -10 -8 eV

Lecture13-14

Multi-electronatoms

Background

Howcanwedetermine energyleve lsofamulti-electron atom? Wecouldstartwith hydrogenicenergylevels foratomofnuclear chargeZ,andsta rtfillingelectrons fromlowestlevels, accounting forPauliexclusion. Degeneracyfor quantumnumbers(n,?)is2 ×(2?+1).Each energylevel,n,accommodates 2×n 2 electrons: n?Degeneracyinshell Cumulativetotal 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

Expectatomscontaining2,10, 28o r60electrons wouldb e especiallystableandthat, inatomscontaining onemore electron, outermostelectro nwouldbelesstightly bound.

Background:ionizationener giesof elements

Instead,findnoble gases( Z=2,10, 18,36···)are especially stable,andelements containingonemo reelectron(alk alimetals) significantlylesstightly bound.

Background

Failuretopredict stableelectronconfigurationsreflects omissionof electron-electroninte raction(cf.ourdiscussion ofhelium). Infact,first ionizationenergiesof atomsshow onlyaw eak dependenceonZ-outermostelectrons are almostcompletely shieldedfromnuclear charge: E ff ectivenuclea rchargevariesasZ e ff ≂(1+γ) Z whereγ>0 characterizes"ine ff ectivenessofs creening";i.e.ionization energy I Z =-E Z ≂Z 2 e ff ≂(1+2 γZ)(cf.exp eriment).

Multi-electronatoms

Leavingaside(fo rno w)relativistice

ff ects,Hamiltonianfo r multi-electronatomgiven by 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 |. Inadditionto nuclearbinding potential,there isafurtherCoulomb interactionbet weenelectrons. Aswe haveseenwithhelium,thiscontribution canhaveimp ortant consequencesonsp ectraandspin structureofwavefunction. However,electron-electroninteractionmakesproblem"many-bo dy" incharacter andanalyticallyintractable-we must developsome approximationscheme(eventhoughe ff ectsmay notbesmall!).

Multi-electronatoms: outline

Centralfieldapp roximation

Self-consistentfieldmetho d- Hartreeapproximation

Structureofthe perio dictable

Couplingschemes:

1

LScouplingand Hund'srules

2 jjcoupling

Atomicspectra:selectionrules

Zeemane

ff ectrevisited

Centralfieldapp roximation

Electroninteractioncontains larges pherically symmetriccomponent arisingfrom"coreelect rons".Since ? ? m=-? |Y lm (θ,φ)| 2 =const. closedshellhassphericallysymmetric chargedistribution.

Thissuggestsa "partitioning"of theHamiltonian,

ˆ H= ˆ H 0 + ˆ H 1 , with 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 ) wheretheradially-symmetric "single-electronp otentials",U i (r), accommodate"averagee ff ect"ofother electrons,i.e. ˆ H 1 issmall.

Centralfieldapp roximation

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 )

Sincesingle-part icleHamiltonian

ˆ H 0 continuestocommute withthe angularmomentumoperator, [ ˆ H 0 , ˆ

L]=0, itseigenfunctionsre main

indexedby quantumnumbers(n,?,m ? ,m s ).

However,sincee

ff ectivepotential, V(r)+U i (r),isno longer Coulomb-like,?valuesfor agivennneednotb edegenerate.

Buthow dowefixU

i (r);thep otentialenergyexp eriencedbyeach electrondepends onthewavefunctionofall theotherelectrons, whichisonly knownafter theSchr¨ odingerequationhasbeensolved. Thissuggestsan iterativeapproach tosolvingthe problem.

Self-consistentfield method

Beforeembarkingonthis programme,weshould

firstconsiderour ambitions:

Thedevelopmentof computationschemes to

addressquantummechanics ofmany-pa rticle systemsisa specialist (andchallenging)topic commontophysics andchemis try. Ourinteresthere ismerely intheoutcomeofsuchinvestigations, andtheirramifi cationsfo ratomicphysics. Wewillthereforediscuss (general)p rinciplesofthemethodology, butthedetailed technicalaspects ofthe approachneed notbe committedtomemo ry!

Self-consistentfield method

Tounderstandhowthe potentialsU

i (r)canb eestimated,w ewill followavariationalapproach duetoHartree: Ifelectronsa re(for now)considereddistinguishable,w avefunction canbe factorizedinto(normalized) productstate, Ψ ({r i })=ψ i1 (r 1 )ψ i2 (r 2 )···ψ i N (r N ) wherethequantum numbers, i≡n?m ? m s ,indexindividual state occupancies.

Notethat

Ψ ({r i })isnot aprop erlyantisymmetrized Slater determinant-exclusion principletak enintoaccount onlyinsofar thatwe haveassigneddi ff erentquantumnumb ers,n?m ? m s .

Inthisapp roximation,if U

i (r)=0, thegroundstate would involve fillingthelo westshells withelectrons.

Self-consistentfield method

Ψ ({r i })=ψ i1 (r 1 )ψ i2 (r 2 )···ψ i N (r N )

Variationalgroundstateenergy:

E=?Ψ|

ˆ

H|Ψ?=

? i ? d 3 rψ ? i ? - ? 2 ? 2 2m - 1

4π?

0 Ze 2 r ? ψ i + 1

4π?

0 ? iLatterimposed bysetofLagrange multipliers,ε i , δ

δψ

? i ?

E-ε

i ?? d 3 r|ψ i (r)| 2 -1 ?? =0

Self-consistentfield method

δ

δψ

? i ?

E-ε

i ?? d 3 r|ψ i (r)| 2 -1 ?? =0

Followingvariation,obtainHartreeequations,

? - ? 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)

Amongstallp ossibletrialfunctions ψ

i ,setthat minimizesene rgy determinedby e ff ectivepotential, U i (r)= 1

4π?

0 ? j?=i ? d 3 r ? |ψ j (r ? )| 2 e 2 |r-r ? | Tosimplifyprocedure, usefultoengineer radialsymmetryby replacingU i (r)by sphericalaverage,U i (r)= ? dΩ 4π U i (r).

Self-consistentfield method

? - ? 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)

TofixLagrangemultipliers ,ε

i ,we canmultiplyHartreeequations byψ ? i (r)andintegrate, ? i = ? d 3 rψ ? 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

Fromthisresult,we find

E= ? i ? i - 1

4π?

0 ? iSelf-consistentfield method Insummary ,withintheHartreeframework, themulti-electron

Hamiltonianisreplaced by thee

ff ectivesingle-particle Hamiltonian, ˆ H 0 = ? i ? - ? 2 2m ? 2 i - 1

4π?

0 Ze 2 r i +U i (r i ) ? wherethecentral potentials U i dependself-consistentlyonthe single-particlewavefunctions, U i (r)= ? dΩ 4π 1

4π?

0 ? j?=i ? d 3 r ? |ψ j (r ? )| 2 e 2 |r-r ? | OnceU i sare found,perturbationtheory canbe appliedtoresidual

Coulombinteraction,

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

Hartree-Fockmethod

Animprove mentonthisprocedurecanb eachievedb yintro ducting atrialva riationalstatew avefunctioninvolving aSlaterdeterminant, 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 )··· . . . . . . . . . . . . ? ? ? ? ? ? ? ? ? whereψ k (r i ),withi=1,2···N,denotethe single-particle wavefunctionsforelectroni,andk=(n?m ? m s ) Avariational analysisleadstoHartree-Fockequationswith additionalexchangecontribution, ε i ψ i (r)= ? - ? 2 2m ? 2 i - 1

4π?

0 Ze 2 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 ?

Centralfieldapp roximation: conclusions

Althoughstatescha racterizedby quantumnumbersn?m

? m s ,

Hartree-Fockcalculationsshowthatthosewithdi

ff erent?forgiven narenownon-degenerate- large?valuesmore effectivelyscreened andliehighe rin energy. Statescorresp ondingtoparticularnreferredtoas ashell,andthose belongington,?areasubshell.Energylevels ordered as

Subshellname1 s2s2p3s3p4s3d4p5s4d···

n=12 233 43 454 ··· ?=00 101 02 102 ···

Degeneracy22 62 6210 6210 ···

Cumulative24 101218 203036 3848···

Centralfieldapp roximation: conclusions

Subshellname1 s2s2p3s3p4s3d4p5s4d···

Cumulative24 101218 203036 3848···

7s 6s 5s 4s 3s 2s 1s 7p 6p 5p 4p 3p 2p 7d 6d 5d 4d 3d

···

6f 5f 4f

···

5g ? ? ?? ? ? ? ?? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ??

Periodictable

Canuseenergy sequenceto

predictgroundstateelectron configuration-fill levels accountingfor exclusion aufbauprinciple .

Sadly,thereareexceptions to

rule:e.g.Cu (Z=29) expectedtohaveconfiguration (Ar)(4s) 2 (3d) 9 ,actuallyhas (Ar)(4s) 1 (3d) 10 .

1H(1s)13.6

2He(1s)

2 24.6

3LiHe(2s)5.4

4BeHe(2s)

2 9.3

5BHe(2s)

2 (2p)8.3

6CHe(2s)

2 (2p) 2 11.3

7NHe(2s)

2 (2p) 3 14.5

8OHe(2s)

2 (2p) 4 13.6

9FHe(2s)

2 (2p) 5 17.4

10NeHe(2s)

2 (2p) 6 21.6

11NaNe(3s)5.1

12MgNe(3s)

2 7.6

14SiNe(3s)

2 (3p) 2 8.1

16SNe(3s)

2 (3p) 4 10.4

18ArNe(3s)

2 (3p) 6 15.8

19KAr(4s)4.3

Periodictable

Aufbauprinciple formsbasisofPeriodictableofelements: elementswith similarelectronconfigurationsin outermostshells havesimilar chemicalproperties.

Couplingschemes

Theaufbaup rinciplep redictsgroundstateoccupationof subshells- butdoes notspecifyspinand orbitalangula rmomentaofsubshells. Todealwiththisquestion, we mustconsiderspin-o rbitandresidual

Coulombinteractionb etween outerelectrons.

Hamiltonianfor multi-electronatomcanbewritten as,

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 includescentralfield terms, ˆ H 1 isresidualCoulomb interaction,and ˆ H 2 isspin-orbit interaction.

Couplingschemes

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

Forlightatoms,

ˆ H 1 ? ˆ H 2 ,cantreat ˆ H 2 asap erturbationon ˆ H 0 + ˆ H 1 -known asLS(orRussell-Saunders)coupling.

Forheavyatoms(orionizedlight atoms),

ˆ H 2 ? ˆ H 1 ,electrons becomerelativisticandspin-orbit interactiondominates- jj coupling. Bothscenarios areapproximations -realatomsdonotalw aysconformto this"comparatively simple"picture.

Couplingschemes:LS coupling

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 Since ˆ

Hcommuteswith setoftotalangular momenta,

ˆ J 2 , ˆ L 2 ,and ˆ S 2 ,energylevels ofmulti-electron atomsa recharacterized by quantumnumbers L,S,J.

Theirordering inenergysetbyHund'srules.

Asrulesempirical, thereare exceptions.Moreover, asatomicmass increasesandelectrons becomerelativistic, spin-orbitinteractions becomeincreasinglyimportant furtherunderminingrules.

Couplingschemes:LS couplingand Hund'srules

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 1 Combinespinsto obtainp ossiblevaluesof totalspinS.(Remember thatclosedshells contributezerospin.)

Thelargest permittedvalueofSlieslow estinenergy.

Physically:maximisingSmakesspinwavefunctionas symmetricas possible:tendstomake spatialwavefunction antisymmetric,reduces

Coulombrepulsion(cf. helium).

Couplingschemes:LS couplingand Hund'srules

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 2 ForeachvalueofS,findthe possiblevalues oftotal angular momentumL.(Rememb erthatclosedshellscontributezeroorbital angularmomentum.)

Thelargest permittedvalueofLlieslow estinenergy.

Physically:maximisingLalsotendsto keepthe electronsapa rt. Indecidingon permitted valuesofLandS,we alsohavetoensure thatboth quantumstatisticsandtheexclusion principleis respected, i.e.totalelectron wavefunction mustbe antisymmetricunder particleexchange.

Couplingschemes:LS couplingand Hund'srules

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 3 CoupleLandStoobtainvalue sof J(hencenameof scheme). (Rememberthatclosedshells contributezeroangula rmomentum.) Ifsubshellis lessthanhalf full,smallestvalue ofJlieslow est inenergy;otherwise, largest valuelieslo west.

Energyseparation fordi

ff erentJarisesfromtreatingspin-orbit term asap erturbation(finestructure), ?Jm J LS| ? i ξ i (r i ) ˆ L i · ˆ S i |Jm J

LS?=ζ(L,S)?Jm

J LS| ˆ L· ˆ S|Jm J LS? =ζ(L,S)[J(J+1)-L(L+1)-S(S+1)]/2 Sincesignof ζ(L,S)changesacco rdingto thewhetherthesubshell ismore orlessthanhalf-filled, thethirdHund'sruleis established.

LScoupling- Example:helium

Heliumhasground stateelectronconfiguration

(1s) 2 ,i.e.L=S=J=0.

N.B.Fo ranycompletelyfilledsubshell,

L=S=0and henceJ=0.

Forexcitedstate,e.g.(1s)

1 (2p) 1 ,canhave S=1o rS=0,with S=1state lyinglow erin energyaccording toHund'srules. Combiningorbital angularmomentagivesL=1and, withS=0,

J=1,while withS=1,J=0,1,2withJ=0lo westin energy.

Inspectroscopic notation

2S+1 L J ,fourp ossiblestates, 3 P 0 3 P 1 3 P 2 and 1 P 1 wherethree 3

Pstatessepa ratedby spin-orbitinteraction,

andsinglet 1

Pstatelies muchhigherin energydue toCoulomb.

Land´eintervalrule

Sinceseparation ofenergiesforstatesof di

ff erentJarisesfrom spin-orbittermcontribution ˆ H 2 (finestructure), ? |J,m J ,L,S| ? i ξ i (r i ) ˆ L i · ˆ S i |J,m J ,L,S? =

ζ(L,S)

2 [J(J+1)-L(L+1)-S(S+1)] separationbetween pairofadjacentlevelsinafinestructure multipletisp ropo rtionaltolargeroftwoJvalues, Δ J ?J(J+1)-(J-1)J=2J e.g.sepa rationbetween 3 P 2 and 3 P 1 ,and 3 P 1 and 3 P 0 shouldbe in ratio2:1.

LScoupling- Example:ca rbon

Carbonhasgroundstateelectronconfiguration

(1s) 2 (2s) 2 (2p) 2 .

Withtw oidenticalelectronsinsame unfilled

subshell,wavefunc tionmustbeantisymmetric. Totalspincaneither be singletS=0(antisymmetric) orone ofthe tripletS=1states (symmetric).

Toformantisymmetrictotalangular

momentumstate ,twoelectronsmust havedi ff erentvaluesof m ?

Inspectingthevaluesofm

L wecan deducethatL=1. m (1) ? m (2) ? m L 101
1-10 0-1-1

Toformsymmetrictotalangular

momentumstate ,twoelectronsmay haveanyvalues ofm ?

Inspectingthevaluesof m

L weinfer thatL=2o r0. m (1) ? m (2) ? m L 112
101
1-10 000 0-1-1 -1-1-2

LScoupling- Example:ca rbon

Carbonhasgroundstateelectronconfiguration

(1s) 2 (2s) 2 (2p) 2 .

Withtw oidenticalelectronsinsame unfilled

subshell,wavefunc tionmustbeantisymmetric. Toensureantisymmetryofw avefunction,w emusttherefo retake

S=1with L=1and S=0with L=2o r0.

Toaccountforfine structure,state swithS=1and L=1can be combinedintosingle J=0state, threeJ=1states, andfiveJ=2 statesleadingto terms 3 P 0 , 3 P 1 ,and 3 P 2 respectively. SimilarlytheS=0,L=2state canb ecombined togivefiveJ=2 states, 1 D 2 ,whileS=0,L=0state givessingle J=0state, 1 S 0 .

LScoupling- Example:ca rbon

Measuredenergylevels:

E/cm -1 1 S 0 20649
1 D 2 10195
3 P 2 43
3 P 1 16 3 P 0 0

Land´eintervalruleapproximatelyob eyedb y

finestructuretriplet, andseparation betw een

LandSvaluescausedb yCoulomb repulsion

ismuchgreaterthanspin-o rbite ffect.

LScoupling- Example:ca rbon

Forexcitedstatesofcarbon, e.g.(2 p)

1 (3p) 1 ,electronsa reno longerequivalentb ecausethey havedi ff erentradialw avefunctions.

Wecannowcombine anyof S=0,1withany ofL=0,1,2,

yieldingthefollo wingterms (inorderofincreasingenergy ,according toHund'srules): 3 D 1,2,3 3 P 0,1,2 3 S 1 1 D 2 1 P 1 1 S 0

Recap:atomicstructure

Ourstudiesof theenergy spectrum ofatomichydroge nusingthe non-relativisticSchr¨ odingerequationshow edthatstatesare organisedinashellstructure, indexedb yap rinciplequantum numbernandcharacterise dbyann 2 -folddegeneracy. Toaddresstheelectronicstructure ofmultielectronatoms, wehave toaccommodate twoclassesofadditional e ff ects: 1 Evenhydrogenic(i.e. single-electron)atoms aresubjec tto correctionsfromrelativistice ff ects(spin-orbitcoupling,etc.) - finestructure,vacuumfluctuations ofEMfield -Lambshift , andinteractionwith nuclear spin-hyperfinestructurewhich togetherconspireto liftstate degeneracy. 2 Inaddition,in multielectronatoms, thedirectCoulomb interactionbet weenelectronsleadtoscreeningofthenuclear charge,andrearrangethe orderingof theshellstructure.

Recap:atomicstructure

Althoughelectron-electroninteractions make themultielectron systemformally intractable,thesphericalsymmetryoffilled core electronstatesjustifies centralfieldapp roximation inwhichthe principlee ff ectofinteractions iscapturedb yasingle-pa rticle potential, ˆ H 0 = ? i ? - ? 2 ? 2 i 2m - Ze 2

4π?

0 r i +U i (r i ) ? , ˆ H 1 = ? i4π? 0 r ij - ? i U i (r i ) Numericalstudies(based onself-consistent Hartree-Fo ckscheme) provideasimplephenomenologyto describeenergy orderingof core subshells-aufbauprinciple

Influenceofresidual electroninteraction,

ˆ H 1 ,andrelativistic spin-orbitcorrections ˆ H 2 = ? i

ξ(r

i ) ˆ L i · ˆ S i onvalencestates canthen beaddressed withinperturbation theory.

Recap:atomicstructure

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

Forlightatoms,

ˆ H 1 ? ˆ H 2 ,cantreat ˆ H 2 asap erturbationon ˆ H 0 + ˆ H 1 -known asLS(orRussell-Saunders)coupling.

Forheavyatoms(orionizedlight atoms),

ˆ H 2 ? ˆ H 1 ,electrons becomerelativisticandspin-orbit interactiondominates- jj coupling.

Recap:atomicstructure

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 InLScoupling, thegroundstate electronconfigureis specifiedb y anemperical setofrulesknownas Hund'srules.Subjectto Pauli exclusion: 1 Thelarges tpermittedvalueoftotalSlieslow estinenergy. 2 Thelarges tpermittedvalueoftotalLlieslow estinenergy. 3 Ifsubshellis lessthan halffull,smallest valueoftotal Jlies lowestinenergy;otherwiselargestvaluelies lowest.

LScoupling- Example:nitrogen

Nitrogenhasground stateelectron

configuration(1s) 2 (2s) 2 (2p) 3 .

Themaximalvalue ofspinis S=3/2

whileLcantake values3,2,1and 0. Sincespinw avefunction(being maximal)issymmetric,spatial wavefunctionmustbe antisymmetric-allthreestate swith m ? =1,0,-1mustb einvolved.

Wemustthereforehave L=0and J=3/2withthe term,

4 S 3/2 . jjcouplingscheme 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

Whenrelativistice

ff ectsdominatere sidualelectrostaticinteraction, ˆ H 1 ,(i.e.heavy elements)electronsmove independently incentral field,subjectto spin-orbit interaction. Inthislimit,statesare both eigenstatesof ˆ J 2 (asbefo re),andalsoof ˆ J 2 i foreachelectron. Injjcoupling,separate energyshiftsindependentof totalJandM J ,

ΔE=?n

i ? i s i j i Jm J | ? i

ξ(r

i ) ˆ L i · ˆ S i |n i ? i s i j i Jm J ?= ? i ΔE i where ΔE i =ζ(n i , ? i )[j i (j i +1)-? i (? i +1)-s i (s i +1)]/2 Thedegeneracywith respectto Jisthenlifted bythe small electrostaticinteractionb etw eenelectrons, ˆ H 1 . jjcouplingscheme:Example

Considerconfiguration( np)

2 (cf.ca rboninLSscheme):Combining s=1/2with?=1,each electroncan havej=1/2or 3/2. Ifelectronshave samejvalue,theya reequivalent,so wehaveto takecareofsymm etry: (a)j 1 =j 2 =3/2?J=3,2,1,0,ofwhich J=2,0are antisymmetric. (b)j 1 =j 2 =1/2?J=1,0,ofwhich J=0is antisymmetric. (c)j 1 =1/2,j 2 =3/2?J=2,1. TakingintoaccountPauli exclusion,injjcoupling(wherethe term iswritten( j 1 ,j 2 ) J ),we havethefollowingterms: (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 inorder ofincreasingenergy. jjcouplingscheme:Example (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 BothLSand jjcouplinggivesame Jvalues(tw ostateswithJ=0, twowithJ=2and onewithJ=1)and insameo rder.

However,patternoflevelsdi

ff erent:inLS couplingwe foundatriplet( 3 P 0 , 3 P 1 , 3 P 2 )and twosinglets( 1 D 2 and 1 S 0 ),whilein idealjj scenario,wehave twodoubletsanda singlet.

Thesetsof statesint woschemes mustbe

expressibleaslinear combinationsof oneanother, andphysicalstates forreal atomlik elytodi ff er fromeitherapp roximation -e.g.jjcouplingnot seeninPb(6 p) 2 butisseen inCr 18+ whichhas sameconfigurationas carbon, (2p) 2 .

Atomicspectra

Atomicspectraresultfrom transitionsbetween di

ff erentelectronic statesofan atomviaemission or absorptionof photons. Inemissionspectra ,atomis excitedby somem eans(e.g. thermallythroughcollisions), andoneobse rvesdiscrete spectral linesinlight emittedasatoms relax. Inabsorptionspectra,oneilluminates atomsusinga broad wavebandsource,andobserves darkabsorptionlinesin the spectrumoftransmittedlight. Atomsexcitedinthisp rocess subsequentlydeca yby emitting photonsinrandom directions- fluorescence.

Atomicspectra:selectionrules

Basictheory governingemissionandabsorption willbe outlinedindetail whenwe studyradiativetransitions.Herewe anticipatesomeresults: Inelectricdipole approximation,rateof transitionsisp ropo rtional tomatrixelements ofelectric dipoleop erator, ˆ d=-e ? i r i ,

Γ?ω

3 | ?ψ f | ˆ d|ψ i ? | 2 ,

ω=|E

f -E i |

Formofdipoleoperator,

ˆ dmeansthatsome matrixe lementsvanish ?selectionrules.Fo ratransitiontotakeplace: 1

Paritymustchange

2

ΔJ=±1,0(but0 →0isnot allowed) andΔM

J =±1,0

Atomicstatesalways eigenstatesofpa rityand

ˆ J 2 ,soselection rules canbe regardedasabsolutelyvalid inelectricdipoletransitions.

Atomicspectra:selectionrules

Inspecific couplingschemes,furtherselectionrulesapply .Inthe caseofideal LScoupling,w ealsorequire: 1

ΔS=0and ΔM

S =0

Followsfromconservationoftotalspinin transition.

2

ΔL=±1,0(but0 →0isnot allowed) andΔM

L =±1,0

Followsfrom1.andrulesrelatingtoJ.

3 Δ? i =±1ifonly electroniisinvolvedin transition. Followsfromparitychange rulesinc etheparityofatomis productofparitiesofseparate electronwavefunctions, (-1) ? i . However,sinceLScouplingisonlyanappro ximation,theserules shouldthemselvesb eregarded asapproximate.

Atomicspectra:singleelectron atoms

For"singleelectronatoms",e.g.alkalimetalssuchassodium, andalsohy drogen,ground stateis(ns) 1 .

Groundstatehas term

2 S 1/2 whileexcited states alldoubletswith J=L±1/2(exceptfo rsstates whichhaveJ=1/2).

Sinceparit ygivenby(-1)

? ,allow edtransitions involve

Δ?=±1,i.e.s↔p,p↔d,etc.(La rger

changesin?contraveneΔJrule.)

Thes↔ptransitionsare alldoublets.In

sodium,transition3s↔3pgivesriseto familiar yellowsodium"D-lines"at589 nm.

Atomicspectra:singleelectron atoms

p↔dtransitionsinvolvet wo doublets, 2 P

1/2,3/2

and 2 D

3/2,5/2

.How ever,the 2 P 1/2 ↔ 2 D 5/2 transitionforbidden by ΔJ rule,soline isactually atriplet.

Asnincreases,levelsapp roachthose for

hydrogen,asnuc lear chargeisincreasingly screenedby innerelectrons.

Inanabso rptionspectrum, atomsstart

fromgroundstate, soonlyns→n ? plines seen.Inemission, atomsa reexcitedinto essentiallyalltheir excitedlevels, somany morelineswillbe seen inthes pectrum.

Zeemaneffect:revisited

Toconcludesurveyofatomic structure,w enow returntoconsider how atomicspectra areinfluencedbya magneticfield? Beginwithhydrogen-lik eatomsinvolving justasingleelectron.Ina magneticfield, ˆ H= ˆ H 0 + ˆ H rel. + ˆ H

Zeeman

,where ˆ H

Zeeman

=- e 2mc B( ˆ L z +2 ˆ S z )=-µ B B( ˆ L z +2 ˆ S z )/? denotesZeeman term. Sincewe aredealingwithconfinedelec trons,wehaveneglected the diamagneticcontributionto theHamiltonian. Dependingonscaleofmagnetic field,the spin-orbitterm in ˆ H rel. or theZeemanterm maydominate thespectrum oftheatom.

Zeemaneffect:revisited

ˆ H

Zeeman

=- e 2mc B( ˆ L z +2 ˆ S z )=-µ B B( ˆ L z +2 ˆ S z )/? Previouslywe haveseenthat,toleadingo rder,relativistic correctionsleadtofine-structureenergy shift, ΔE rel. n,j = 1 2 mc 2 ? Zα n ? 4 ? 3 4 - n j+1/2 ? forstates|n,j=?±1/2,m j , ??. Forweakmagneticfields,we canalsotreatZeemanenergyin frameworkofperturbationtheory:

Althoughstateswith commonj(e.g.2S

1/2 and2P 1/2 )are degenerate,spatialw avefunctionshavedi ff erentparity,and o ff -diagonalmatrixelements of ˆ H

Zeeman

vanish-avoids needfor degenerateperturbation theory.

Zeemaneffect:revisited

ˆ H

Zeeman

=- e 2mc B( ˆ L z +2 ˆ S z )=-µ B B( ˆ L z +2 ˆ S z )/?=-µ B B( ˆ J z + ˆ S z )/? Makinguseof identity, (exercise-refer backtoadditionofangular momentumandspin) ?n,j=?±1/2,m j , ?| ˆ S z |n,j=?±1/2,m j , ??=± ?m j 2?+1 weobtainthefollowing expressionfo rthefirst orderenergyshift, ΔE

Zeeman

j=?±1,m j , ? =?±1/2,m j , ??=µ B Bm j ? 1± 1 2?+1 ? i.e.alldegenerate levelssplitb yfield.

Incontrastto the"no rmal"Zeeman e

ff ect,themagnitude ofthe splittingnow dependson?.

Zeemaneffect:revisited

ˆ H

Zeeman

=- e 2mc B( ˆ L z +2 ˆ S z )=-µ B B( ˆ L z +2 ˆ S z )/?=-µ B B( ˆ J z + ˆ S z )/? Formulti-electronatominweak field,unperturbedstates givenby |J,M J ,L,S?,whereJ,L,Srefertototal angular momenta.

TodetermineZeemanshift, needtodetermine ?

ˆ S z ?,pre sentingan opportunityutoreviseangularmomenta: 1

Firstwe notethattheoperato r2

ˆ L· ˆ S= ˆ J 2 - ˆ L 2 - ˆ S 2 isdiagonalin thebasisof states,|J,M J ,L,S?. 2

Therefore,recallingthat[

ˆ S i , ˆ S j ]=i?? ijk ˆ S k and[ ˆ L i , ˆ S k ]=0, it followsthatthematrixelement ofthefollo wingoperato rvanishes, ˆ S( ˆ L· ˆ S)-( ˆ L· ˆ S) ˆ S= ˆ L j [ ˆ S i , ˆ S j ]=i?? ijk ˆ L j ˆ S k ≡-i? ˆ S× ˆ L

Zeemaneffect:revisited

-i? ˆ S× ˆ

L≡

ˆ S( ˆ L· ˆ S)-( ˆ L· ˆ S) ˆ S 3

Moreover,since[

ˆ L· ˆ S, ˆ

J]=0, itfollows thatthe matrixelementof

thefollow ingoperatoralsovanishes, -i?( ˆ S× ˆ

L)×

ˆ J= ˆ S× ˆ J( ˆ L· ˆ S)-( ˆ L· ˆ S) ˆ S× ˆ J 4 Ifwe expandlefthandside,we thusfind thatthematrix elementof thefollow ingoperatoralsovanishes, ( 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 5

Therefore,itfollowsthat ?

ˆ S ˆ J 2 ?=? ˆ J( ˆ S· ˆ

J)?.With

ˆ S· ˆ J= 1 2 ( ˆ J 2 + ˆ S 2 - ˆ L 2 ),we have? ˆ S z ?? ˆ J 2 ?=? ˆ J z ?? ˆ S· ˆ

J?,i.e.

? ˆ S z ?=? ˆ J z ?

J(J+1)+ S(S+1)-L(L+1)

2J(J+1)

Zeemaneffect:revisited

? ˆ S z ?=? ˆ J z ?

J(J+1)+ S(S+1)-L(L+1)

2J(J+1)

Asaresult, wecan deducethat,at firstorderinperturbation theory,theenergyshiftarising fromtheZeem antermis givenby ΔE J,M J ,L,S =µ B B?( ˆ J z + ˆ S z )?/?=µ B g J M J B wheree ff ectiveLand´eg-factor g J =1+

J(J+1)+ S(S+1)-L(L+1)

2J(J+1)

N.B.for hydrogen(S=1/2andJ=L±1/2),we recoverprevious result.

Example:atomicsp ectraof sodium

ΔE J,M J ,L,S =µ B g J M J B

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