英文原子物理

ShanxiUniversityAtomicPhysics

MarkProportion:

FinalExam:45%Midsemester:30%Homework:15%Attendingclass:10%第1页/共33页第一页，共34页。ShanxiUniversityAtomicPhysics

MainContents1Earlyatomicphysics2Thehydrogenatom3Helium4Thealkalis5TheLS-couplingscheme6Hyperfinestructureandisotopeshift7Theinteractionofatomswithradiation8Doppler-freelaserspectroscopy9Lasercoolingandtrapping10(1)Magnetictrapping10(2)EvaporativecoolingandBose-Einsteincondensation11Atominterferometry12Iontraps13Quantumcomputing第2页/共33页第二页，共34页。ShanxiUniversityAtomicPhysics

Chapter1Earlyatomicphysics1.1Introduction1.2Spectrumofatomichydrogen1.3Bohr'stheory1.4Relativisticeffects1.5Moseleyandtheatomicnumber1.6Radiativedecay1.7EinsteinAandBcoeffecients1.8TheZeemaneffect

1.8.1ExperimentalobservationoftheZeemaneffect

1.9SummaryofatomicunitsExercises第3页/共33页第三页，共34页。ShanxiUniversityAtomicPhysics

1.1IntroductionTheoriginsofatomicphysics:quantummechanicsBohrmodeloftheHThisintroductorychaptersurveyssomeoftheearlyideas:SpectrumofatomicHandBohrTheoryEinstein'streatmentofinteractionofatomwithlighttheZeemaneffectRutherfordscatteringAndsoon第4页/共33页第四页，共34页。ShanxiUniversityAtomicPhysics

1.1IntroductionBeforedescribingthetheoryofanatomwithoneelectron,someexperimentalfactsarepresented.Thisorderingofexperimentfollowedbyexplanationreflectstheauthor'sopinionthatatomicphysicsshouldnotbepresentedasappliedquantummechanics,butitshouldbemotivatedbythedesiretounderstandexperiments.Thisrepresentswhatreallyhappensinresearchwheremostadvancescomeaboutthroughtheinterplayoftheoryandexperiment.第5页/共33页第五页，共34页。ShanxiUniversityAtomicPhysics

1.2Spectrumofatomichydrogen_2nandn’:wholenumbers;R:Rydbergconstant.Balmerseries:spectrallinesforwhichn=2andn’=3;4;…Thefirstlineat656nmiscalledtheBalmer-(orH)line

thecharacteristicspectrumforatomsiscomposedofdiscretelinesthatarethe‘fingerprint'oftheelement.

In1888,theSwedishprofessorJ.Rydbergfoundthatthespectrallinesinhydrogenobeythefollowingmathematicalformula:第6页/共33页第六页，共34页。ShanxiUniversityAtomicPhysics

1.2Spectrumofatomichydrogen_3Wavenumbersmayseemratherold-fashionedbuttheyareveryusefulinatomicphysicsWavenumberInpractice,theunitsusedforagivenquantityarerelatedtothemethodusedtomeasureit,e.g.spectroscopesandspectrographsarecalibratedintermsofwavelength.Ritzcombinationprinciple:thewavenumbersofcertainlinesinthespectrumcanbeexpressedassums.第7页/共33页第七页，共34页。ShanxiUniversityAtomicPhysics

1.2Spectrumofatomichydrogen_4Examinationofthesumsanddifferencesofthewavenumbersoftransitionsgivescluesthatenabletheunderlyingstructuretobededuced,ratherlikeacrosswordPuzzle--someexamplesofthisaregiveninlaterchapters.Theobservedspectrallinesinhydrogencanallbeexpressedasdifferencesbetweenenergylevels.WhichasshowninFig.1.1,wheretheenergiesareproportionalto1/n2.Thefollowingsectionlooksathowthesespectracanbeexplainedtheoretically.第8页/共33页第八页，共34页。ShanxiUniversityAtomicPhysics

Lymanseries:n’=2;3;4;…n=1.

Balmer(n=2),

Paschenseries:(n=3),Brackett(n=4)andPfund(n=5)第9页/共33页第九页，共34页。ShanxiUniversityAtomicPhysics

Bohrassumedthateachelectronorbitsthenucleusinacircle,whoseradiusrisdeterminedbythebalancebetweencentripetalaccelerationandtheCoulombattractiontowardstheproton.Forelectronsofmassmeandspeedvthisgives

(1.3)InSIunitsthestrengthoftheelectrostaticinteractionbetweentwochargesofmagnitudeeischaracterisedbythecombinationofconstantse2/40.Thisleadstothefollowingrelationbetweentheangularfrequency=v/randtheradius:

(1.4)ThisisequivalenttoKepler'slawsforplanetaryorbitsrelatingthesquareoftheperiod2=!tothecubeoftheradius(asexpectedsinceallstepshavebeenpurelyclassicalmechanics).Thetotalenergyofanelectroninsuchanorbitisthesumofitskineticandpotentialenergies:

(1.5)1.3Bohr’stheory-2第10页/共33页第十页，共34页。ShanxiUniversityAtomicPhysics

1.3Bohr’stheory-3Usingeqn1.3wendthatthekineticenergyhasamagnitudeequaltohalfthepotentialenergy(anexampleofthevirialtheorem).Takingintoaccounttheoppositesignsofkineticandpotentialenergy,wefind(1.6)Thistotalenergyisnegativebecausetheelectronisboundtotheprotonandenergymustbesuppliedtoremoveit.TogofurtherBohrmadethefollowingassumption.AssumptionI:Therearecertainallowedorbitsforwhichtheelectronhasaxedenergy.Theelectronlosesenergyonlywhenitjumpsbetweentheallowedorbitsandtheatomemitsthisenergyaslightofagivenwavelength.第11页/共33页第十一页，共34页。ShanxiUniversityAtomicPhysics

Thatelectronsintheallowedorbitsdonotradiateenergyiscontrarytoclassicalelectrodynamics|achargedparticleincircularmotionundergoesaccelerationandhenceradiateselectromagneticwaves.Bohr'smodeldoesnotexplainwhytheelectrondoesnotradiatebutsimplytakesthisasanassumptionthatturnsouttoagreewiththeexperimentaldata.Wenowneedtodeterminewhichoutofallthepossibleclassicalorbitsaretheallowedones.Therearevariouswaysofdoingthisandwefollowthestandardmethod,usedinmanyelementarytexts,thatassumesquantisationoftheangularmomentuminintegralmultiplesofħ(Planck'sconstantover2):Mevr=nħ(1.7)wherenisaninteger.Combiningthiswitheqn1.3givestheradiioftheallowedorbitsasr=a0n2

(1.8)wheretheBohrradiusa0isgivenby

(1.9)1.3Bohr’stheory-4第12页/共33页第十二页，共34页。ShanxiUniversityAtomicPhysics

1.3Bohr’stheory-5Thisisthenaturalunitoflengthinatomicphysics.Equations1.6and1.8combinetogivethefamousBohrformula:(1.10)Thepositiveintegerniscalledtheprincipalquantumnumber.Bohr'sformulapredictsthatinthetransitionsbetweentheseenergylevelstheatomsemitlightwithawavenumbergivenby(1.11)Thisequationfitsverycloselytotheobservedspectrumofatomichydrogendescribedbyeqn1.1.TheRydbergconstantR∞ineqn1.11isdefinedby(1.12)第13页/共33页第十三页，共34页。ShanxiUniversityAtomicPhysics

1.3Bohr’stheory-6ThefactorofhcmultiplyingtheRydbergconstantistheconversionfactorbetweenenergyandwavenumberssincethevalueofR∞isgiveninunitsofm(orcmincommonly-usedunits).ThemeasurementofthespectrumofatomichydrogenusinglasertechniqueshasgivenanextremelyaccuratevaluefortheRydbergconstantR∞=10973731.568525m-1.However,thereisasubtledifferencebetweentheRydbergconstantcalculatedforanelectronorbitingaxednucleusR∞andtheconstantforrealhydrogenatomsineqn1.1(weoriginallywroteRwithoutasubscriptbutmorestrictlyweshouldspecifythatitistheconstantforhydrogenRH).Thetheoreticaltreatmentabovehasassumedaninfinitelymassivenucleus,hencethesubscript∞.Inrealityboththeelectronandprotonmovearoundthecentreofmassofthesystem.ForanucleusoffinitemassMtheequationsaremodiedbyreplacingtheelectronmassmebyitsreducedmass(1.13)第14页/共33页第十四页，共34页。ShanxiUniversityAtomicPhysics

1.3Bohr’stheory-7Forhydrogen

(1.14)wheretheelectron-to-protonmassratioisme/Mp'1≈1/1836.Thisreduced-masscorrectionisnotthesamefordifferentisotopesofanelement,e.g.hydrogenanddeuterium.Thisleadstoasmallbutreadilyobservabledifferenceinthefrequencyofthelightemittedbytheatomsofdifferentisotopes;thisiscalledtheisotopeshift(seeExercises1.1and1.2).

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Complementarity:第16页/共33页第十六页，共34页。ShanxiUniversityAtomicPhysics

Largesize:

2n2a0

~1μm(n=100)Largedipolemoments:

n2ea0

~10000D(H2O–2.6D)Longlifetime:n3

~1msEasytoionizeEion

n-4~10V/cmRydbergatoms:“Semi-classicalatoms”Rydbergsystem(atom,molecule,…)=BignHydrogenatom

Largesize:

2n2a0

~1μm(n=100)Largedipolemoments:

n2ea0

~10000D(H2O–2.6D)Longlifetime:n3

~1msEasytoionizeEion

n-4~10V/cmDipole-dipoleinteraction第17页/共33页第十七页，共34页。ShanxiUniversityAtomicPhysics

1.4Relativisticeffects-1Applyingthisquantisationruletomomentumaroundacircularorbitgivestheequivalentofeqn1.7:mev2r=nh(1.15)Bohr'stheorywasagreatbreakthrough.Howeverassumptionofcircularorbitsistoomuchofanover-simplication.Sroducedquantisationthroughageneralrulethatstated'theintegralofthemomentumassociatedwithacoordinatearoundoneperiodofthemotionassociatedwiththatcoordinateisanintegralmultipleofPlanck'sconstant'.第18页/共33页第十八页，共34页。ShanxiUniversityAtomicPhysics

1.4Relativisticeffects-2Sommerfeldalsoconsideredquantisationoftheradialdegreeoffreedomr.ANDfoundthatsomeoftheellipticalorbitsexpectedforapotentialproportionalto1=rarealsostationarystates.InspecialrelativityaparticleofrestmassmmovingatspeedvhasanenergyE(v)=mc2(1.16)Mucheortwasputintocomplicatedschemesbasedonclassicalorbitswithquantisation,andbyincorporatingspecialrelativitythis'oldquantumtheory'couldexplainaccuratelythenestructureofspectrallines第19页/共33页第十九页，共34页。ShanxiUniversityAtomicPhysics

1.4Relativisticeffects-3wherethegammafactoris.ThekineticenergyofthemovingparticleisE=E(v)-E(0)=(-1)mec2.Thusrelativisticeffectproduceafractionalchangeinenergy:(1.17)Thisleadstoenergydierencesbetweenthevariousellipticalorbitsofthesamegrossenergybecausethespeedvariesindierentwaysaroundtheellipticalorbits,e.g.foracircularorbitandahighlyellipticalorbitofthesamegrossenergy.Fromeqns1.3and1.7wendthattheratioofthespeedintheorbittothespeedoflightis(1.18)Wherethefine-structureconstantisgivenby

(1.19)第20页/共33页第二十页，共34页。ShanxiUniversityAtomicPhysics

1.4Relativisticeffects-4Theideaofellipticalorbitsprovidesaconnectionwithourintuitionbasedonclassicalmechanicsandweoftenretainsometracesofthissimplepictureofelectronorbitsinourminds.However,foratomswithmorethanoneelectron,e.g.helium,classicalmodelsdonotworkandwemustthinkintermsofwavefunctions.Thisfundamentalconstantplaysanimportantrolethroughoutatomicphysics.Numericallyitsvalueisapproximately=137ItisnotnecessarytogointoalltherenementsofSommerfeld'srelativistictheorythatgavetheenergylevelsinhydrogenveryprecisely,byimposingquantisationrulesonclassicalorbits,sinceultimatelyaparadigmshiftwasnecessary.第21页/共33页第二十一页，共34页。ShanxiUniversityAtomicPhysics

1.5Moseleyandtheatomicnumber_1AtthesametimeasBohrwasworkingonhismodelofthehydrogenatom,H.G.J.MoseleymeasuredtheX-rayspectraofmanyelements.MoseleyestablishedthatthesquarerootofthefrequencyoftheemittedlinesisproportionaltotheatomicnumberZ(thathedenedasthepositionoftheatomintheperiodictable,startingcountingatZ=1forhydrogen),i.e.(1.20)Moseley'soriginalplotisshowninFig.1.2.Asweshallsee,thisequationisaconsiderablesimplicationoftheactualsituationbutitwasremarkablypowerfulatthetime.ByorderingtheelementsusingZratherthanrelativeatomicmass,aswasdonepreviously,severalinconsistenciesintheperiodictablewereresolved.Therewerestillgapsthatwerelaterlledbythediscoveryofnewelements.Inparticular,fortherare-earthelementsthathavesimilarchemicalpropertiesandarethereforediffculttodistinguish,itwassaid'inanafternoon,Moseleycouldsolvetheproblemthathadbaedchemistsformanydecadesandestablishthetruenumberofpossiblerareearths'(Segre1980).Moseley'sobservationscanbeexplainedbyarelativelysimplemodelforatomsthatextendsBohr'smodelforhydrogen.第22页/共33页第二十二页，共34页。ShanxiUniversityAtomicPhysics

1.5Moseleyandtheatomicnumber_2AnaturalwaytoextendBohr'satomicmodeltoheavieratomsistosupposethattheelectronslluptheallowedorbitsstartingfromthebottom.Eachenergylevelonlyhasroomforacertainnumberofelectronssotheycannotallgointothelowestlevelandtheyarrangethemselvesinshells,labelledbytheprincipalquantumnumber,aroundthenucleus.ThisshellstructurearisesbecauseofthePauliexclusionprincipleandtheelectronspin,butfornowletussimplyconsideritasanempiricalfactthatthemaximumnumberofelectronsinthen=1shellis2,then=2shellhas8andthen=3shellhas18,etc.Forhistoricalreasons,X-rayspectroscopistsdonotusetheprincipalquantumnumberbutlabeltheshellsbyletters:Kforn=1,Lforn=2,Mforn=3andsoonalphabetically.ThisconceptofelectronicshellsexplainstheemissionofX-raysfromatomsinthefollowingway.MoseleyproducedX-raysbybombardingsamplesofthegivenelementwithelectronsthathadbeenacceleratedtoahighvoltageinavacuumtube.Thesefastelectronsknockanelectronoutofanatominthesampleleavingavacancyorholeinoneofitsshells.Thisallowsanelectronfromahigher-lyingshellto'falldown'tollthisholeemittingradiationofawavelengthcorrespondingtothedierenceinenergybetweentheshells.第23页/共33页第二十三页，共34页。ShanxiUniversityAtomicPhysics

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Fig.1.2Moseley'splotofthesquarerootofthefrequencyofX-raylinesofelements

againsttheiratomicnumber.Moseley'sworkestablishedtheatomicnumberZas

amorefundamentalquantitythanthe'atomicweight'(nowcalledrelativeatomic

mass).Followingmodernconventiontheunitsofthehorizontalscaleswouldbe

(108

(Hz)1/2)atthebottomand(10-10m)forthelogscaleatthetop.第25页/共33页第二十五页，共34页。ShanxiUniversityAtomicPhysics

1.5Moseleyandtheatomicnumber_3ToexplainMoseley'sobservationsquantitativelyweneedtomodifytheequationsinSection1.3,onBohr'stheory,toaccountfortheeffectofanucleusofchargegreaterthanthe+1eoftheproton.ForanuclearchargeZewereplacee2=40byZe2=40inalltheequations,resultinginaformulafortheenergieslikethatofBalmerbutmultipliedbyafactorofZ2.Thisdependenceonthesquareoftheatomicnumbermeansthat,forallbutthelightestelements,transitionsbetweenlow-lyingshellsleadtoemissionofradiationintheX-rayregionofthespectrum.ScalingtheBohrtheoryresultisaccurateforhydrogenicions,i.e.systemswithoneelectronaroundanucleusofchargeZe.Inneutralatomstheotherelectrons(thatdonotjump)arenotsimplypassivespectatorsbutpartlyscreenthenuclearcharge;foragivenX-rayline,saytheK-toL-shelltransition,amoreaccurateformulais(1.21)第26页/共33页第二十六页，共34页。ShanxiUniversityAtomicPhysics

1.5Moseleyandtheatomicnumber_4ThescreeningfactorsKandLarenotentirelyindependentofZandthevaluesofthesescreeningfactorsforeachshellvaryslightly(seetheexercisesattheendofthischapter).Forlargeatomicnumbersthisformulatendstoeqn1.20(seeExercise1.4).Thissimpleapproachdoesnotexplainwhythescreeningfactorforashellcanexceedthenumberofelectronsinsidethatshell,e.g.K=2forZ=74althoughonlyoneelectronremainsinthisshellwhenaholeisformed.Thisdoesnotmakesenseinaclassicalmodelwithelectronsorbitingaroundanucleus,butcanbeexplainedbyatomicwavefunctions|anelectronwithahighprincipalquantumnumber(andlittleangularmomentum)hasaniteprobabilityofbeingfoundatsmallradialdistances.ThestudyofX-rayshasdevelopedintoawholefieldofitsownwithinatomicphysics,astrophysicsandcondensedmatter,butthereisonlyroomtomentionafewbrieffactshere.WhenanelectronisremovedfromtheK-shelltheatomhasanamountofenergyequaltoitsbindingenergy,i.e.apositiveamountofenergy,anditisthereforeusualtodrawthediagramwiththeK-shellatthetop,asinFig.1.3.Thesearetheenergylevelsoftheholeintheelectronshells.Thisdiagramshowswhythecreationofaholeinalow-lyingshellleadstoasuccessionoftransitionsastheholeworksitswayoutwardsthroughtheshells.第27页/共33页第二十七页，共34页。ShanxiUniversityAtomicPhysics

1.5Moseleyandtheatomicnumber_5Thehole(orequivalentlythefallingelectron)canjumpmorethanoneshellatatime;eachlineinaseriesfromagivenshellislabelledusingGreekletters(asintheseriesinhydrogen),e.g.K,K;….ThelevelsdrawninFig.1.3havesomesub-structureandthisleadstotransitionswithslightlydierentwavelengths,asshowninMoseley'splot.ThisisnestructurecausedbyrelativisticeffectsthatweconsideredforSommerfeld'stheory;thesubstitutione2/40Ze2/40,asabove,(orequivalentlyZ)showsthatnestructureisoforder(Z)2timesthegrossstructure,whichitselfisproportionaltoZ2.ThusrelativisticeffectsgrowasZ4andbecomeverysignificantfortheinnerelectronsofheavyatoms,leadingtothefinestructureoftheL-andM-shellsseeninFig.1.3.ThisrelativisticsplittingoftheshellsexplainswhyinMoseley'splot(Fig.1.2)therearetwoclosely-spacedcurvesfortheK-line,andseveralcurvesfortheL-series.第28页/共33页第二十八页，共34页。ShanxiUniversityAtomicPhysics

Fig.1.3Theenergylevelsoftheinnershellsofthetungstenatom(Z=74)andthetransitionsbetweenthemthatgiverisetoX-rays.Thelevelschemehasseveralimportantdifferencesfromthatforthehydrogenatom(Fig.1.1).Firstly,theenergiesaretensofkeV,ascomparedtoeVforZ=1,becausetheyscaleasZ2(approximately).Secondly,theenergylevelsareplottedwithn=1atthetopbecausewhenanelectronisremovedfromtheK-shellthesystemhasmoreenergythantheneutralatom.

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1.5Moseleyandtheatomicnumber_6NowadaysmuchoftheX-rayworkinatomicphysicsiscarriedoutusingsourcessuchassynchrotrons;thesedevicesaccelerateelectronsbythetechniquesusedinparticleaccelerators.Abeamofhigh-energyelectronsc