Zeeman effect with a variable magnetic system and a CMOS ...

Student'sSheet

Printed:13.12.201716:15:38|P2511007

Robert-Bosch-Breite10 Tel:+49551604-0 [email protected]

D-37079Göttingen Fax:+49551604-107 www.phywe.com

Difficulty

Difficult

PreparationTime

1Hour

ExecutionTime

2Hours

RecommendedGroupSize

2Students

Zeemaneffectwithavariablemagneticsystemanda

CMOScameraincludingthemeasurementsoftware

(ItemNo.:P2511007)

CurricularRelevance

AdditionalRequirements:

PC

ExperimentVariations:

Keywords:

Bohr’satomicmodel,quantisationofenergylevels,electronspin,Bohr’smagneton,interferenceofelectromagneticwaves,Fabry-Perot

interferometer

Overview

Shortdescription

Principle

The“Zeemaneffect”isthesplittingupofthespectrallinesofatomswithinamagneticfield.Thesimplestisthesplittingupof

onespectrallineintothreecomponentscalledthe“normalZeemaneffect”.ThenormalZeemaneffectisstudiedusinga

cadmiumspectrallampasaspecimen.Thecadmiumlampissubmittedtodifferentmagneticfluxdensitiesandthesplittingup

oftheredcadmiumline(643.8nm)isinvestigatedusingaFabry-Perotinterferometer.Theevaluationoftheresultsleadstoa

fairlyprecisevalueforBohr’smagneton.

Fig.1:ExperimentalsetupwithCMOS-camera.

AreaofExpertise:

Physik

EducationLevel:

Hochschule

Topic:

ModernePhysik

Subtopic:

Quantenphysik

Experiment:

Zeemaneffectwitha

variablemagnetic

systemandaCMOS

cameraincludingthe

measurement

software

Student'sSheet

Printed:13.12.201716:15:38|P2511007



Equipment

Position Material Bestellnr. Menge

1 Fabry-Perotinterferometer 09050-03 1

2 CadmiumlampforZeemaneffect 09050-20 1

3 MagneticSystem,variable 06327-00 1

4 Powersupplyforspectrallamps 13662-97 1

5 Opticalbenchexpert,l=1000mm 08282-00 1

6 Baseforopticalbenchexpert,adjustable 08284-00 2

7 Slidemountforopticalbenchexpert,h=30mm 08286-01 7

8 Slidemountforopticalbenchexpert,h=80mm 08286-02 2

9 Lensholder 08012-00 4

10 Lens,mounted,f+50mm 08020-01 2

11 Lens,mounted,f+300mm 08023-01 1

12 Irisdiaphragm 08045-00 1

13 Polarisingfilter,onstem 08610-00 1

14 Polarizationspecimen,mica 08664-00 1

15 Diaphragmholder 08040-00 1

16 MOTICMoticam1CMOSTechnology 63300-00 1

17 HolderforMoticamonopticalbenchandopticalbaseplate 63300-01 1

Tasks

1. UsingtheFabry-Perotinterferometer,aself-madetelescope,aCMOS-cameraandmeasurementsoftware,thesplittingup

ofthecentrallineintotwoσ-linesismeasuredinwavenumbersasafunctionofthemagneticfluxdensity.Intheclassical

versionwheretheCMOS-Cameraisnotavailable,ascreenwithscaleandaslidingdeviceareusedtomeasurethe

splitting.

2. Fromtheresultsofpoint1.avalueforBohr’smagnetonisevaluated.

3. Thelightemittedwithinthedirectionofthemagneticfieldisqualitativelyinvestigated.

Student'sSheet

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Set-upandprocedure

ThemagneticsystemismountedwiththetwopoleshoesinsuchawaythatagaplargeenoughfortheCd-lamp( )

remainsfortheCd-lamp.Themagneticsystemhastobewelltightenedinsuchawaythatthepoleshoescannotmovelateron

whenthemagneticfluxisestablished.TheCd-lampisinsertedintothegapwithouttouchingthepoleshoesandconnectedto

thepowersupplyforspectrallamps.Theopticalbenchforinvestigationofthelinesplittingcarriesthefollowingelements(their

approximatepositionincmisgiveninbrackets):

(80)CMOS-Camera

(73)

(45)Analyser

(39)

(33)Fabry-PerotEtalon

(25)

(20)Irisdiaphragm

(20)Drilledpole-shoes

(0)Cd-spectrallamponrotatingtable

TheirisdiaphragmiseliminatedforinitialadjustmentandfortheobservationofthelongitudinalZeemaneffect.During

observationofthetransverseZeemaneffecttheirisdiaphragmisilluminatedbytheCdlampandsuchitactsasthelightsource.

Thelens andalensof ,incorporatedintheetalon,createanearlyparallellightbeamwhichtheFabry-Perot

etalonneedsforaproperinterferencepattern.Theetaloncontainsaremovablecolourfilterthatletstheredcadmiumlineat

pass.Thelens producesaninterferencepatternofringswhichcanbeobservedthrough .Theringdiameters

canbemeasuredusingtheCMOS-cameraandthesoftwaresuppliedwithit.Intheclassicalversiontheinterferencepatternis

producedwithintheplaneofthescreenwithascalemountedonaslidemountwhichcanlaterallybedisplacedwithaprecision

of1/100thofamillimeter.Themeasurementherecanbedoneforinstance,bysystematicdisplacementoftheslash

representingthe„0“ofthescale.

Fig.2:Arrangementoftheopticalcomponents.

Theinitialadjustmentisdoneinthefollowingway:

Themagneticsystemisadjustedsothatthecenteroftheholesinthepoleshoesliesabout abovethetable.Theoptical

benchwithallelements(exceptirisdiaphragmandCMOS-camera)mounted,isthenmovedclosertothemagnetinsuchaway

thatoneoftheoutletholesofthepoleshoescoincideswiththepreviouspositionoftheirisdiaphragm. isthenadjustedso

thattheoutletholeiswithinthefocalplaneofit.AllotheropticalelementsofFig.2.aresubsequentlyreadjustedwithrespectto

theirheightcorrespondingly.Themagneticfieldstrengthisadjustedtoapproximately700mTbychangingthedistancebetween

thepoleshoes(turnthespindledrive).Ifnecessary,increaseinlightintensityoftheCd-lamp.Theringinterferencepatternin

axialdirectionisobservedthroughL3bytheeye.Thepatternmustbecenteredandsharpwhichiseventuallyachievedbya

last,slightmovementoftheetalon(totherightortotheleft)andbydisplacementof (verticallyandhorizontally).

FinallytheCMOS-camerawiththe lensattachedismountedtotheopticalbenchandadjustedinhorizontalandvertical

positionaswellasintiltandfocusuntilaclearpictureoftheringpatternisvisibleonthecomputerscreen.Forinstallationand

useofthecameraandsoftwarepleaserefertothemanualsuppliedwiththecamera.

Hint:bestresultsareachievedwhentheexperimentiscarriedoutinadarkenedroom.

Themagnetisnowturnedby ,theirisdiaphragmisinsertedandtheanalyzerturneduntilthe -line(explanationfollows)

disappearscompletelyandthetwo -linesappearclearlyvisible.

Remark:

Forlaterevaluationsthecalibrationcurveofthemagneticfluxdensityversusthepoleshoedistancehastobetracedpreviously.

ThiscanbedoneifaTeslameter,digital(13610-93)withHallprobe,tangential(13610-02)isavailable.Otherwisetheresultsof

Fig.3canbeused.ThecurveofFig.3wastracedbymeasuringthefluxdensityinthecenterofthegapintheabsenceofthe

Cd-lamp.Fortheevaluationsthesecenter-valueswereincreasedby toaccountforthenon-uniformfluxdistributionwithin

thegap.Youwillfindadditionalinformationontheuseofthemagneticsysteminthecorrespondingmanual(06327-00).

Student'sSheet

Printed:13.12.201716:15:38|P2511007



Fig.3:MagneticfluxdensityBinthecentreofthegapwithoutCd-

lampasafunctionofthepoleshoedistance(withoutpoleshoe

tips).

Student'sSheetPrinted:13.12.201716:15:38|P2511007

Robert-Bosch-Breite10 Tel:+49551604-0 [email protected]öttingen Fax:+49551604-107 www.phywe.com

Theoryandevaluation

Theory

Theory

Asearlyas1862,Faradayinvestigatedwhetherthespectrumofcolouredflameschangesundertheinfluenceofamagneticfield,butwithoutsuccess.Itwasnotuntil1885thatFievezfromBelgiumwasabletodemonstrateaneffect,butitwasforgottenandonlyrediscovered11yearslaterbytheDutchphysicistPieterZeeman,andcorrectlyexplainedbyhiscompatriotHendrikAntoonLorentz.Forthisachievement,bothofthemreceivedtheNobelPrizein1902.

HeretheeffectisdemonstratedwiththelightofaCadmiumlampandthehelpofaFabry-Perotinterferometerforresolvingasmallpartofthespectrumpreselectedbyacolorfilteroraninterferencefiltersoonlythelightofasingleatomictransitionlineisobserved.Withoutfieldthemagneticsub-levelshavethesameenergybutwithfieldthedegenerationofthelevelswithdifferent iscancelledandthelineissplit.Cadmiumhastheelectronstructure ,i.e.theoutershelltakingpartinopticaltransitionsiscomposedbythetwo5s^2electronsthatrepresentacompletedelectronshell.(

)ThisissimilartotheouterelectronstructureofHeliumbutalsoofMercury.AschemeoftheenergylevelsofCdisshowninFig.4.Inacompletedshellinit'sgroundstatetheelectronspinsalwayscompensateeachother–theyareanti-parallel.Ifthetotalelectronspiniszero,alsothemagneticmomentconnectedtoelectronspiniszero.Atomicstateswithzerototalspinarecalledsingulettstates.Sointransitionsbetweendifferentsingulettstatesthemagneticmomentofspindoesnotplayarole,asisthecasewiththenormalZeemaneffect.Electricdipoleradiationasincommonopticaltransitionsdoesnotchangetheelectronspinexceptinheavyatomswith -coupling,sotransitionsarenormallybetweendifferentstatesinthesamemultiplicitysystem.ButFig.4showsthereissome -couplinginCadmium.

Fig.4:TheatomicstateofCadmium,wavelengthinÅ=0.1nm.

ThetransitionusedtodemonstratethenormalZeemaneffectis with andthetransitionusedtodemonstratetheanomalousZeemaneffectis with .

Inatermlike thefirstnumber"2"denotesthemainquantumnumberoftheradiatingelectronwithrespecttotheatom'sgroundstate(thatiscountedas"1"),herethisisreallythe6th -shellsince isthegroundstate.(Thisiswhythe –statesarebelowthe –states, denotesthe5th -shellsinceKryptonhas .)Theupper"3"denotesthemultiplicity,thatis with herethespinquantumnumber.Thelower"1"denotesthequantumnumber ofthetotalangularmomentum,i.e.



thetotalangularmomentum,i.e. withlthequantumnumberoftheangularmomentumoftheorbit.'' '','' '','' '','' ''denotetheactualvalueof ,i.e.'' ''means ," "means .

isatransitionwithinthesingulettsystemsothespinmagneticmomentshavenoeffect.Butinthetransitiontriplettstatesareinvolvedandthespinmagneticmomentdoesnotvanishinallsub-states.

Theselectionruleforopticaltransitionsis andtheradiationbelongingtotransitionswith arecalled-linesandtheoneswith arecalled -lines.Withthemagneticfieldturnedonintheabsenceoftheanalyserthree

linescanbeseensimultaneouslyinthenormalZeemaneffectintransversalobservation.InthecaseoftheanomalousZeemaneffectthreegroupsofthreelinesappear.InsertingtheanalyserinthenormalZeemaneffecttwo -linescanbeobservediftheanalyserisintheverticalposition,whileonlythe -lineappearsiftheanalyseristurnedintoitshorizontalposition(transversalZeemaneffect).IntheanomalousZeemaneffecttherearetwogroupsofthree -linesinverticalpolarizationandonegroupofthree -linesinhorizontalpolarization.Turningthemagneticsystemby thelightcomingfromthespectrallampparalleltothedirectionofthefield(longitudinal)canalsobestudiedtroughtheholesinthepolepieces.Itcanbeshownthatthislightiscircularpolarizedlight(longitudinalZeemaneffect).Fig.5summarizesthefacts.A -plateisgenerallyusedtoconvertlinearintoellipticalpolarizedlight.Inthisexperimentthe -plateisusedintheoppositeway.Withthe -plateinsertedbeforetheanalyser,thelightofthelongitudinalZeemaneffectisinvestigated.Iftheopticalaxisofthe -platecoincideswiththevertical,itisobservedthatsomeringsdisappeariftheanalyserisatanangleof withtheverticalwhileotherringsdisappearforapositionof .ThatmeansthatthelightofthelongitudinalZeemaneffectispolarizedinacircular(opposedway).The -linesarelongitudinallynotobservable.

Fig.5:LongitudinalandtransversalZeemaneffect.

InthenormalZeemaneffectwiththetransition with643.847nmtheelectronspinscanceleachotherinboththeinitialandfinalstateandtheenergyofanatomicstateinamagneticfielddependsonlyonthemagneticmomentsoftheelectronorbit.Themagneticmomentoftheorbitalangularmomentumis

withBohr'smagneton

andthegyromagneticfactoroforbitalangularmomentum .

Inthevectormodeloftheatomtheenergyshiftscanbecalculated.Itisassumed,thatangularmomentsandmagneticmomentscanbehandledasvectors.Angularmomentandthemagneticmomentconnectedwithitareantiparallelbecauseofthenegativeelectroncharge.Theamountoftheorbitalmagneticmomentoftheorbitalangularmomentum,withquantumnumber suchthat

,is:

IncaseofLS-coupling(Russel-Saunderscoupling,spin-orbitcoupling)formanyelectronsystemsistheamountofthetotalangularmomentum

with

thesumofthespinsofthesingleelectronsand



thesumoftheorbitangularmomentsofthesingleelectrons.Hereitis

So

Theamountofthecomponentofthecorrespondingmagneticmoment indirectionof is:

with .

Observableisonlytheprojectionofthemagneticmomenton

withit'squantizationwithrespecttoz-axis

withthemagneticquantizationnumberwith Theinteractionenergywiththeoutermagneticfield alongthez-axisisthen

HeretheusedtransitionforthenormalZeemaneffectis .

Sointheinitialstateis , and . mayhavethevalues .Thegyromagneticfactorisandtheenergydifferencebetweentwoneighbouringsubstatesoftheinitialstateisthen .

Inthefinalstateis , and . mayhavethevalues .Thegyromagneticfactoris andtheenergydifferencebetweentwoneighbouringsub-statesofthefinalstateisthen ,too,i.e.fortransitionwiththesame betweeninitialandfinalstatetheenergyshiftisforinitialandfinalstatethesame–sotheyhavealtogetherthesamefrequency.Fig.6showstheresultingtransitiondiagram.Forelectricaldipoletransitionstheselectionrulestates .

Theenergyshiftofatransitionbetweeninitialstatewith and andfinalstatewith and isthen

andherethe( )-valuesaresimplyequalto .SoincaseofLS-couplinginthenormalZeemaneffectthreeequidisantlinesareexpectedinthistransitionwithadistanceinfrequencyorwavenumberproportionaltothemagneticfieldstrength.Thepolarizationofthetransitionswith intransversalobservationisparalleltothemagneticfield(herehorizontal)andoftheothertransitionsthepolarizationisperpendiculartothat.

Fig.6:Energyshiftoftheatomicstates.

TheanomalousZeemaneffectisthemoregeneralcasewheretheelectronspinsdonotcanceleachotherandtheenergyofanatomicstateinamagneticfielddependsonboththemagneticmomentsofelectronorbitandelectronspin.

Themagneticmomentoftheorbitalangularmomentum isasabove(see(*))andthemagneticmomentofthespin is



withthegyromagneticfactoroforbitalangularmomentum .

Additionaltotheorbitalmagneticmomentoftheorbitalangularmomentum theamountofthespinmagneticmomentofthespin ,withquantumnumber suchthat

hastobetakenintoaccount:

IncaseofLS-coupling(Russel-Saunderscoupling,spin-orbitcoupling)formanyelectronsystemstheamountofthetotalangularmomentumis

with

thesumofthespinsofthesingleelectronsand

thesumoftheorbitangularmomentsofthesingleelectrons.Inthevectormodelitisassumed,thatangularmomentsandbothspinandorbitalmagneticmomentscanbehandledasvectors.Sothecosineruleappliesforthesumoftwovectorswithan

anglebetweenthem.Theamountofthecomponentofthecorrespondingmagneticmoment indirectionof iswiththeapproximation :

with

Observableisonlytheprojectionofthemagneticmomenton

withit'squantizationwithrespecttoz-axis

withthemagneticquantizationnumber with Theinteractionenergywiththeoutermagneticfield alongthez-axisisthen

HerefortheanomalousZeemaneffecttheusedtransitionis .Sointheinitialstateis , and . mayhavevalues .Thegyromagneticfactoris

andtheenergydifferencebetweenneighbouringsub-statesoftheinitialstateisthen

Inthefinalstateis , and . mayhavethevalues .Thegyromagneticfactoris

andtheenergydifferencebetweenneighbouredsub-statesofthefinalstateisthen

Fig.7showstheresultingtransitiondiagram.



Fig.7:Energyshiftoftheatomicstates.

Forelectricaldipoletransitionstheselectionrulestates .Theenergyshiftofatransitionbetweeninitialstatewith and andfinalstatewith and isthen

Thefollowingtableshowstheenergyshiftsofthetransitions:

Fig.8:Reflectedandtransmittedraysattheparallelsurfaces(1)and(2)oftheétalon.Theétalonspacingist=3mm.

SoincaseofLS-couplingintheanomalousZeemaneffectnineequidistantlinesareexpectedinthistransitioninsteadofthreewithoutspinmagnetism.Thepolarizationofthetransitionswith intransversalobservationisparalleltothemagneticfield(herehorizontal)andthepolarizationoftheothertransitionsisperpendiculartothemagneticfield.Atobservingthe -linesofthetransversalZeemaneffectitiseasytoseethattheamountofsplittingincreaseswithincreasingmagneticfieldstrength.Foraquantitativemeasurementofthissplittingintermsofnumberofwavelengths,aFabry-Perotinterferometerisused,thefunctioningofwhichhastobeexplained:TheFabry-Perotétalonhasaresolutionofapproximately400000.Thatmeansthatawavelengthchangeoflessthen0.002nmcanstillbedetected.Theétalonconsistsofaquartzglassplateof3mmthicknesscoatedonbothsideswithapartiallyreflectinglayer(90%reflection,10%transmission).Letusconsiderthetwopartiallytransmittingsurfaces(1)and(2)inFig.8seperatedbyadistance .AnincomingrayformingananglewiththeplatenormalwillbesplitintotheraysAB,CD,EF,etc.thepathdifferencebetweenthewavefrontsoftwoadjacentrays(e.g.ABandCD)is

whereBKisdefinednormaltoCDand istherefractiveindexofquartzat509nm, .At644nmis .With

and

weobtain

andforaconstructiveinterferenceitis:

(1)

where isanintegerand thelight'swavelength.Equation(1)isthebasicinterferometerequation.LettheparallelraysB,D,F,etc.bebroughttoafocusbytheuseofalensoffocallength asshowninFig.9.



Fig.9:FocusingofthelightemergingfromaFabry-Perotétalon.Lightenteringtheétalonatanangleθisfocusedontoaringofradiusr=fθwherefisthefocallengthofthelens.

For fulfillingequation(1)brightringswillappearinthefocalplanewiththeradius

(2)

forsmallvalues ,e.g.raysnearlyparalleltotheopticalaxis.Since

with

wefinallyobtain

or

(3)

If correspondstoabrightfringe, isaninteger.Howeveris theinterferenceconditionforthecenter(for )generallynotaninteger.If istheinterferenceorderofthefirstring,itis since .Wethenlet

where istheclosestintegerto (smallerthan ).Ingeneralisforthepthringofthepattern,measuredstartingfromthecenter,thefollowingisvalid:

(4)

Combiningequation(4)withequations(2)and(3),weobtainfortheradiioftherings,writing for ,

(5)

Wenotethatthedifferencebetweenthesquaresoftheradiiofadjacentringsisaconstant:

(6)

canbedeterminedgraphicallyplotting versus andextrapolatingto .Now,iftherearetwocomponentsofaspectralline(splittingofonecentrallineintotwocomponents)withwavelengths and

,whichareveryclosetooneanother,theywillhavefractionalordersatthecenter and :

where and arethecorrespondingwavenumbersand , istheinterferenceorderofthefirstring.Hence,iftheringsdonotoverlapbyawholeorderso andthedifferenceinwavenumbersbetweenthetwocomponentsis

(7)

Usingequations(5)and(6),weget

(8)

Applyingequation(8)tothecomponentsaandb,yields



and

Bysubstitutingthesefractionalordersintoequation(7),wegetforthedifferenceofthewavenumbers:

(9)

Fromequation(6)wegetthedifferencebetweenthesquaresoftheradiiofcomponenta:

thisisequalto(withinaverysmallpart)thesamedifferenceforcomponentb

Henceweassume

forallvaluesofp.Similarly,allvalues

mustbeequal,regardlessof (theorderofinterference)andtheiraveragemaybetakenasmaybedoneforthedifferent -values.With (thedifferenceofsquaresofradiiofdifferentlinesofthesameorderofinterference)and (differenceofsquaresofradiiofdifferentorders)asaveragevalueswegetforthedifferenceofthewavenumbersofthecomponentsaandb:

(10)

Note:Equation(10)showsthatΔkdoesnotdependonthedimensionsusedinmeasuringtheradiioftheringsystem.

Fig.10:NormalZeemaneffect:Interferencepatternwithoutpolarisationfilterfordifferentmagneticfieldstrengths.Ontheleftthereisoneringperorderofinterference,ontherighttherearethreeringsperorderofinterference.

AnomalousZeemaneffect:Forallthesepicturesthecoilcurrentwassetsuchthatthedifferentordersofinterferencewerejuststillseparated.InFig.11alltheringsarevisiblebuthardtocount.InFig.12themiddlethreeringsarevisiblewith .InFig.13theouterandinnerthreeringswith arevisible,that'ssixringsmakingatotalofninerings.Theringsseemtobeequidistantbutnotofthesameintensity.

Fig.11:AnomalousZeemaneffect:Interferencepatternwithoutpolarisationfilterandmagnifiedcut-outofthefirstcompletelyvisibletwoordersofinterference.



Fig.12:AnomalousZeemaneffect:Interferencepatternwithhorizontalpolarisationfilterandmagnifiedcut-outofthefirstcompletetwoorders.

Fig.13:AnomalousZeemaneffect:Interferencepatternwithverticalpolarisationfilterandmagnifiedcut-outofthefirstcompletetwoorders.

Evaluation

Zeeman effect with a variable magnetic system and a CMOS ...

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