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Charge Correlation Effects in the Broadening of Spectral Lines from Highly Charged Radiators

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................iv LISTOFFIGURES................................vii ABSTRACT....................................viii CHAPTER 1INTRODUCTION..............................1 2OVERVIEWOFPLASMASPECTROSCOPY..............5 2.1ExperimentalDetailsofICF.....................7 2.2ElectronandIonBroadening.....................8 2.3EvolutionofSpectralLineTheory..................12 3LINESHAPEFORMULATION.......................14 4SEMI-CLASSICALMOLECULARDYNAMICS..............23 5LINESHAPEFORMULAFORQUASI-STATICIONS..........30 6QUASI-STATICEQUATIONOFMOTION................36 6.1First-orderElectronBroadening...................37 6.2Second-OrderStaticElectronBroadeningTerm..........42 6.3DynamicElectronBroadeningTerm.................44 7CHARGECORRELATIONEFFECTSONPLASMAPROPERTIES..46 7.1PlasmaStructure...........................46 7.1.1LinearTheory.........................48 7.1.2Nonlinearmodel........................50 7.2ScreenedIonField..........................53 7.3IonMicroeldDistribution......................54 7.4ConstrainedElectronDensity....................55 7.5ElectricFieldCovariance.......................56 7.6Dynamics...............................57 7.7Summary...............................62 8INCORPORATINGIONDYNAMICS...................63 v

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9SUMMARYANDOUTLOOK........................73 9.1StaticIonLineShapeFunction...................75 9.2IonDynamics.............................77 APPENDIX AHOOPER'SGROUPCOMPUTERCODES................80 A.1AtomicWavefunctions........................80 A.2Electric-MicroeldCalculation....................80 A.3Electron-BroadeningOperator....................81 A.3.1Second-Order,FullCoulomb,QuantumTheory.......82 A.3.2All-Order,Full-Coulomb,Semi-ClassicalTheory......84 BDOPPLERBROADENINGINTHEQUASI-STATICAPPROXIMATION86 CPROJECTIONOPERATOREQUATIONOFMOTION.........89 DAVERAGEDIPOLEEQUATIONOFMOTION|QUASI-STATICCASE91 ECALCULATIONALDETAILSFOR B ( ).................96 E.1FirstOrderStaticShift B (1) Calculation...............96 E.2ConstantPerturberDensityAssumption..............101 FKINETICEQUATIONFORCOUPLEDIONDYNAMICSCASE....104 F.1AtomicLiouvilleOperator......................107 F.2PlasmaLiouvilleOperator......................108 F.2.1RadiatorCenterofMassPosition..............109 F.2.2RadiatorCenterofMassMomentum.............109 F.2.3ElectricField..........................110 F.3InteractionLiouvilleOperator....................111 F.3.1InteractionLiouvilleOperatoron D .............112 F.3.2InteractionLiouvilleOperatoron .............113 F.3.3Summary{DeterministicPart.................114 F.3.4DynamicsTerm........................115 GTHEKELBGANDDEUTSCHPOTENTIALS..............118 G.1DerivationoftheQuantumPotential................118 G.2CoulombPotentialandKelbg/DeutschResults...........121 REFERENCES...................................122 BIOGRAPHICALSKETCH............................126 vi

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LISTOFFIGURES Figure page 2{1PlasmaParameters..............................13 7{1PlasmaElectron-RadiatorPairCorrelationFunction...........51 7{2PlasmaElectron-RadiatorPairCorrelationFunction...........52 7{3ElectricFieldCovariance...........................58 7{4NormalizedElectricFieldAutocorrelation.................60 7{5IntegralofElectricFieldAutocorrelationFunction............61 E{1CapturedBoundCharge...........................101 E{2TemperatureDependenceof B (1) ......................103 E{3DensityDependenceof B (1) .........................103 vii

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2.1 Experimental Details of ICFInICFexperiments,smallmicrosphereswithallgasareimplodedwithlasers[13].Incurrentexperiments,thesemicrospherestypicallyhaveadiameterofabout1mm,andconsistofaplasticshellofabout20micronsthickness.Allgasisintroducedintothemicrosphere,whichisthenplacedinthetargetchamber.Twomainclassesofexperimentsareperformed:directdriveandindirectdriveexperiments.InadirectdriveICFexperiment,themicrosphereissuspendedaloneinthetargetchamber[20,21].Laserbeamsstrikethespherefrommanydirectionswhichcausetheouterportionoftheshelltoexplodeaway.Thereactionforcedrivestheinnerportionoftheshellinwards,compressingthefuelgastoformaplasma.Inanindirectdriveexperiment,themicrosphereisplacedinsideasmallcylindricaltube(calledahohlraum)whichisplacedinsidethetargetchamber[21].Thelaserbeamsenterthehohlraumfromeachendandilluminatetheinnerwalls.Theseinnerwallsthenradiate,anditisthisradiationthatleads

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2.2 Electron and Ion BroadeningHistorically,plasmaspectroscopictechniquesweremostimportantinthestudyofstellaratmospheres.Stellaratmosphereshaveelectrondensitiesandionizationsthatarerelativelylow;thuspressurebroadeningbyneutralperturbersbecomes

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2.3 Evolution of Spectral Line TheoryModernspectrallinetheoryincludesmanyeects,someofwhichmustbedealtwithconcurrentlywiththederivationofthelineshape.Probablythemostimportantoftheseeectsistheformoftheinteractionbetweentheplasmaelec-tronsandtheradiator.Theearliermodelsofspectrallineshapesdealtwithneutralradiators[27].Withtheseplasmas,itwascommonfortheinteractionbetweentheradiatorandtheplasmaelectronstobemodeledbyadipoleinteraction.Afundamentalchangeoccurredwhentheplasmaexperimentsreachedtemperatureshighenoughtoionizetheradiator.Ionizedradiatorsattractedtheplasmaelec-trons,andwiththesmallerseparation,thedipoleapproximationwasnolongeradequate[6].Todescribetheseionizedplasmas,thefullCoulombinteractionisrequiredinthetheory.Whenthiswasdone,itwasfoundthatlineshiftsarosefromtheelectron-radiatorinteractions.Therearealsoseveraleectsthatarepartofspectrallineshapeanalysis(andpartofthelineshapecodesdescribedinAppendixA)thatarenotincorporatedexplicitlyintoourstudy.Oneoftheseeectsisspectrallinemerging[8].Asden-sitiesincrease,thepositionsofneighboringlinesmovetogetherandmerge.Themergingofspectrallinesathighdensitiesrequiresarelaxationoftheno-quenchingapproximationandamuchlongercalculationtime.Opacitybroadening,especiallyrequiredforstellarapplications,hasbeenincorporatedintothelineshapecodes[3].

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Figure2{1:PlasmaParametersThetemperatureanddensityrangeofplasmasstudiedhereiscomparedwithotherexperimentalconditions.ParametersusedinMDsimulationsareshown.Theobservanceofsatellitesrequiredtheuseofmulti-electronradiatortheories[19].Inaddition,theexperimentalequipmentalsogivesabroadeningtothespectralline,whichneedstobeaccountedfortodirectlycomparetheoreticalandexperi-mentalresults.Theseeects,whileimportanttolineshapetheoryasawhole,canbeneglectedinourstudy.Ourstudywillformulatealineshapeofasingleionizedradiatorinaplasma.

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2NX6=V(;)!+1 2X6=0NX=1N0X=1V0(;)(3.8)InEq.(3.8),representsthetypeofparticle(freeelectron,ion,orradiatorcenterofmass)andNisthenumberofparticleoftype.ThersttermisthekineticenergywithK()p2=2m.Thesecondistheinteractionbetweenparticlesofthesametype,andthethirdistheinteractionsbetweenparticlesofdierent

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2nRX6=Vee(;)+nRX=1Ven()(3.9)withnRbeingthenumberofelectronsboundtotheradiator.ThethreetermsinEq.(3.9)describethekineticenergyoftheboundelectrons,theinteractionsbetweenboundelectrons,andtheinteractionbetweentheboundelectronsandthenucleusoftheradiator.Withthecenterofmassdegreesoffreedomabsentfromtheseterms,HRisidenticaltoacaseofelectronsboundtoamotionlesschargedparticle,andisthereforeastraightforwardatomicphysicsproblem.ThenalpartoftheHamiltoniandescribestheinteractionoftheatomicandplasmasystems.TheseinteractionsareCoulombinteractions,andcanbewrittenasfunctionsofthepositionoperatorsofthequantumparticles.Becauseofthemathematicalmanipulationstobedonelater,itwillbeconvenienttowritetheseinteractionsintermsofchargedensities.Theplasmaparticleandboundelectronchargedensitiescanbedenedas:(x)=NX=1Qe(xq)b=nRX=1e(x0r)(3.10)(InEq.(3.10),=fe;ig,indicatingtheplasmaelectronsorions,respectively;incontrasttoEq.(3.8),thepointradiatorisnotincluded.)Withthesedenitions,theinteractionstaketheformoftheabovechargedensitiesmultipliedbytheformofthepairpotentialfortheirinteraction.Forexample,theinteractionbetweenthefreeplasmaelectronsandtheatomicsystembecomesU(electrons)pR=Zdxdx0e(x)b(x0)W(x;x0)(3.11)

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3r30n=1!r0=3 4n1=3(3.17)wherenistheionchargedensity.TheprobabilitythattheionwillbeacertaindistancefromtheradiatorcanbeestimatedbytheBoltzmannfactor.AlowerlimitontheprobabilitycanbefoundbyestimatingtheBoltzmannfactorfortheionattheBohrradius.ThisgivesexpZiZRe2

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@tiHaiUpR(t)(t)=0(4.5)withUpR(t)=XZdxdx0(x;t)b(x0)W(x;x0)(4.6)(x;t)=eiHpt(x)eiHpt(4.7)where(x;t)istheplasmaperturberchargedensityandb(x0)istheboundchargedensity.NotethatinEq.(4.6),theonlydependenceontheplasmadegreesoffreedomoccurscompletelywithin(x;t).Once(x;t)isspecied,theproblemrepresentedbyEq.(4.5)ispurelyanatomicphysicsproblemofthecouplingofanatomtoanexternalpotential.Thedicultmany-bodyproblemresidingin(x;t)isthereforeofgreatimportancetocalculateaccurately,whichiswhyMDisrelevanthere.MDsimulationsclaimtobeabletoperformanaccuratecalculationoftheplasmachargedensitywithallchargecorrelationsintact.However,thereareseveraldicultiesinapplyingMDtothisproblem,andthischapterwillexaminethevalidityofMDsimulationsapproach.ThersttopictostudyisthefactthatMDisinherentlyaclassicalmethod.InMD,Newtonianequationsaresolveddirectlytoprovideparticletrajectories[31,32].Themotionoftheparticlesisthenfollowedforsometimeinterval,andthepropertiestobecalculatedaredeterminedbythetimedependentcoordinatesandmomentaoftheparticles.TheearlyusesofMDforplasmasimulationsinvolvedaonecomponentplasmaconsistingofpositiveparticles,plusaconstantnegativebackgroundforoverallchargeneutrality.Forthiscase,MDisveryusefulandclearlyapplicable,sincetherepulsivenatureoftheinterparticleinteractionskeepstheparticlesfarapart.Thismeantthat,withhightemperature,thedistancebetweentheparticlesismuchlargerthanthedeBrogliewavelength.Thereforea

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h1erfr i(4.14)AsecondformfortheregularizedpotentialistheDeutschpotential[41].ThisisaphenomenologicalversionthatisoftenusedinMDsimulationsandhastheformUD(r)=V(r)1er=(4.15)BoththeKelbgandDeutschpotentialshavesimilarqualitativefeatures;however,insidethedeBrogliewavelengththeKelbgpotentialispreferable.Oneadditionalfactormustbeevaluated.Theplasmasconsideredherearehighlycharged.Thishasthepotentialforfurtherdicultiesastheelectrontrajectoriesarepulledcloser,andthereisthepossibilityformetastableboundstates.Morestudyneedstotakeplacetodeterminetheeectsforagivenradiatorcharge,temperature,andelectrondensity.Withtheuseoftheseregularizedpotentials,MDsimulationscancalculatedynamicpropertiesforaplasma.Forexample,asimpletheorymighthavetheradiator-plasmainteractionbetotallydescribedbythemonopoleanddipoleinteraction.Inthiscase,thetimedependentelectriceldoftheplasmaisrequired.ByusingMDtodeterminetheelectriceld,allcorrelationsbetweentheplasmaparticlescanberetained.Thenextchapterwilllookataspecicplasmatodeterminethepropertiesneededthatwillgivethelineshapefunction.Itwillbefoundthatsomeofthesepropertiesneededincludetheconstrainedaverageoftheelectrondensity,andtheelectroneldautocorrelationfunction.Withstatisticalmechanicsmethods,uncontrolledapproximationsarerequiredtoevaluatethese

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mr(5.11)

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@t+LX=0:(6.2)WiththeHamiltonianseparatedintothreeterms(seeEq.(5.22)),theLiouvilleoperatorassociatedwiththeentiresystemseparatesintothreeanalogousterms,orL=Lp+L(a;)+LpR(6.3)36

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@t+L(a;)+B()D(;t)+tZ0dM(;t)D(;)=0(6.4)TheoperatorsB()andM(;t)representtheeectsofthecouplingoftheradiationtotheplasmaLpR.SincethestaticionbroadeninghasalreadybeenaccountedforinL(a;)itisexpectedthattheseareelectronbroadeningterms.However,theseelectrontermsretaintheircorrelationswiththeionsandadepen-denceonthexedionvalue.TheoperatorB()issimplythemeanvalueofLpRfortheconstrainedplasmaB()X=Trp()LpRX=Trp()i 6.1 First-order Electron BroadeningTheresultfortheelectronbroadeningB()containstheLiouvilleoperatorLpRorinteractionHamiltonianUpR.Inthissection,thiswillbeexpandedinpowersoftheinteractionenergyoftheplasmaandradiatorinternalstates.Asdescribedabove,theremovalofthemonopoleandiondipoletermsforUpRresultsinasofteningoftheinteractionsothatanexpansioninpowersofUpRisvalid.

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4Zdxhe(x);ip(6.20)b(x0)=Zd0xb(x0)(6.21)Forthemonopoletermoftherstorderelectronbroadening,theeectsofincludingallplasmacorrelationsexactlyhasbeenlocalizedinthetermhe(x);ip.Thistermrepresentsthefreeelectronchargedensity,averagedoverthoseplasmastatesinwhichtheionelectriceldESisconstrainedtohavethevalue.Thisfreeelectronchargedensityisnotuniformbecausethechargedradiatorattheoriginattractstheperturbingelectrons.Furthermore,itisnotisotropicbecauseoftheconstrainedaverage:withaspeciedeld,theiondensitywillbelargerononesideoftheradiatorthanontheother.Thefastmovingelectronswouldthenbeattractedtothemorepositiveside,leadingtoanasymmetryinthefreeelectronchargedensity[44].Theimportanceoftheseeectscanbeexploredwithasimpleestimatecalculation.Ifitisassumedthattheperturbingelectronchargedensitye(x)isconstantinsidetheboundelectronwavefunction,e(x)canberemovedfromthe

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6.2 Second-Order Static Electron Broadening TermAtthispointitisusefultomorecloselyexaminehowthechoiceoftheionmicroeldformhasaectedthephysicalinterpretationofB()asastaticelectronbroadeningterm.Notethatinitiallytheplasma-radiatorinteractionUpRrepresentedbothelectronandionperturbations,withboththemonopoleinteractionandaiondipoleinteractionESdremoved.WiththeabovechoiceforESandtheneglectofhigherorderionmultipolestheradiator-plasmacouplingenergycanbewrittenUpR=EedZdxdx0e(x)W(x;x0)b(x0);(6.23)Ee=EehEeie(6.24)wherehEeieistheelectroneldaveragedovertheelectrondegreesoffreedomhEeie1iTrepEe(6.25)Thus,thersttermoftheinteractionisjusttheuctuationoftheelectrondipoleinteractionwhoseaveragevalueinB(1)()iszerobydenition.Clearly,UpRnowrepresentsanelectronperturbationoftheradiator.Consequently,bothB()andM(;t)cannowberecognizedaselectronbroadeningoperators(althoughstillretainingthecorrelationsbetweenelectronsandions).TheprevioussectionexaminedthatpartofB()thatwasrstorderintheinteractionLiouvilleoperator.Theremainder,B()B(1)(),canbeexpandedintoatermthatissecondorderintheinteractionLiouvilleoperator.Thefullstatic

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6.3 Dynamic Electron Broadening TermTheprevioussectionshavedealtwiththetermB(),whichdescribesthestaticbroadeningbytheplasmaelectrons.ThedynamicalelectroneectsaredescribedbytheoperatorM(;t).InAppendixD,thisoperatoriscalculatedandshowntohavetheformM(;t)=f1(a)Trp(a;)(LpR)eQLtQ(LpR)(6.32)HereQisaprojectionoperatorsuchthatQXgivesthedeviationofXfromitsconstrainedplasmaaverage.QX=XTrp()X(6.33)TheprevioussectionaccountedfortheaveragevalueofLpRanditstimeindependentuctuationstosecondorder.Whatisleftisthetime-dependentuctuations,whichveriesthepreviousidenticationofM(;t)asadynamicelectronbroadeningterm.Morephysically,M(;t)describesatomictransitionsoftheradiatorcausedbycollisionswiththeplasmaelectrons.TosecondorderinLpRthisoperatorbecomes(seeAppendixDfordetails)M(2)(;t)=DLpRe(LR(+Lp)tLpRhLpR;ip;Ep(6.34)Thisformofthedynamicelectronbroadeningoperatorissimilartothoseobtainedpreviously.Typically,however,suchresultsneglectmanyoralloftheelectron-radiator-ioncorrelations.Thesecorrelationsenterthebroadeningtermintwoplaces.ThedenitionofESitselfincludedsomeofthesecorrelationsinthatthecorrelationsleadtoscreeningoftheionmicroeld,andareincludedonlyapproximatelyinmostprevioustheories.However,thetermabovealsoincludesallcorrelationswhenperformingthetrace.Insteadofperformingthetraceover

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7.1 Plasma StructureWewillrstconsiderthetime-independentstructuralpropertiesoftheplasma.Theplasmaisconsideredtobeatwo-componentplasmawithionsofchargeZandelectrons.Inaddition,adiluteconcentrationofanotherspecieswillbeintroducedastheradiator.Astheequationsarederived,theconcentrationoftheradiatorwillbesettozero,reectingourmodelofasingleimpurityionradiator.Tostudythestructureoftheplasma,anappropriatestartingpointisthepaircorrelationfunctiong;(r),orthesimplyrelatedquantityh;(r)=g;(r)1[45,46].Thefunctionh;(r)isameasureofthetotalcorrelationbetweentwoparticlesoftypeand,andhasthebehaviorh;!0forlarger.Thetotalcorrelationincludes46

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7.1.1 Linear TheoryWenowturntoanevaluationofthemeanpotentialU;R(r).First,weexaminetheresultsoftheprecedingsectionintheregimeofweakcoupling.ThentherelationofthepaircorrelationfunctionandthepotentialofmeanforceU;RfromEq.(7.4)becomesgR(r)=eU;R!1UR(7.9)Also,whenEq.(7.2)islinearized,theresultisc!V(7.10)

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7.1.2 Nonlinear modelThelastsubsectioncalculatedtheeectivepotentialtodeterminethechargedensitiesaroundtheradiatoratweakcouplinggR(r)!1UR(r)(7.24)withUR(r)givenby(7.23)fromthelinearizedequations.AnestimateforstrongercouplingisobtainedbyexponentiatingthisresultgR(r)!eUR(r)(7.25)withthesameformforUR(r).ThisisanuncontrolledapproximationbutisconrmedtobequalitativelycorrectevenatstrongcouplingbycomparisonwithMDsimulation.

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Figure7{1:PlasmaElectron-RadiatorPairCorrelationFunctionCom-parisonusingthehypernettedchainapproximation(HNC),thenonlinearDebyemodel(NLD),thelinearDebyemodel(LD),andwithmoleculardynamics(MD)results.[47]InFig.(7.1.2),resultsforgeR(r)areshownarisingfromHNC,non-linearDebye,andDebyetheoriesforanidealizedplasmaofelectronsinauniformpositivebackground.AlsoresultsfromMDsimulationsareshown[47].ThisgureshowsthatatZ=8,thenon-linearDebyeandHNCresultsagreewitheachotherandwithMDresults.Figure(7.1.2)showsthesamequantityfordieringvaluesofZ,usingthenon-linearDebyemodelandHNC.WithincreasingZ,theelectrondensityattheradiatorisenhanced.NotethattheagreementismuchbetteratlowerZ.AtZ=30,thereisconsiderabledierencebetweenthetworesults.However,thefunctionalformoftheHNCresults,evenatthishigherZremainsthesameasthenon-linearDebyeform.Ifthechargeistakentobeattingparameter,goodagreementisfoundbetweentheHNCresultswithZ=30andthenon-linearDebyeresultswithZ=25.

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Figure7{2:PlasmaElectron-RadiatorPairCorrelationFunctionCompar-isonofresultsfromthehypernettedchain(HNC)approximationandnonlinearDebye(NLD)modelforseveralradiatorcharges.

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7.2 Screened Ion FieldTheionmicroelddistributionisdenedby[44]Q()=h(Es)i=(2)3ZdeieiEs(7.26)Thebracketsdenoteanequilibriumensembleaveragefortheplasmaconsistingoftheelectrons,ions,andradiatormonopole.Thescreenedeldisdenedby(seeEq.(6.14))Es=Ei+RdepEe

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7.3 Ion Microeld DistributionIfthescreenedeldistakentobeasumofsingleparticleeldsEs=Xes(qi)(7.38)thentheexpressioninEq.(7.26)Q()=h(Es)i=(2)3ZdeieiEs(7.39)

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7.4 Constrained Electron DensityTheaverageelectrondensityaroundtheradiator,forgivenelectriceldvaluecanbeestimatedinthefollowingway.Consideragaintheelectrondistributioninthepresenceofagiveniondistributionasin(7.31)U(q1ejfqig)=VeR(q1e)+XiVie(jq1eqij)1neZdr0cee(jrr0j)eU(r0)1:(7.43)TheionelectronpotentialcanbewrittenintermsoftheiondensityXiVie(jq1eqij)=ZdrVie(jq1erj)ni(r):(7.44)Now,considertheiondensitytobetheaveragedensityforthegivenconstrainteld,ni(r)!ni(r;)=iondensityforagivenioneldattheradiatorsatisfyingZdr(rVie(j0rj))ni(r;)=(7.45)

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7.5 Electric Field CovarianceTheelectriceldautocorrelationfunctionC(t)=hEE(t)ihasaspecialemphasisinthisstudy.ThisintegralofC(t)isdirectlyrelatedtoseveraltransportproperties,suchasthelowvelocitystoppingpowerS,thefrictioncoecient,andtheself-diusioncoecientD,aswellastheimpact(fastuctuation)limitforspectrallinebroadeningbyelectrons[47].Thetransportpropertiesarerelatedby[47]m0=1

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7.6 DynamicsWenowturnfromstructuralpropertiestodynamicalpropertiesfortheplasmaelectronsinthepresenceofahighlychargedradiator.Fortheconditionsofinteresttothisstudy,theplasmaelectronsarealwaysweaklycoupled.ThenonlinearVlasovequation[26]isthenvalidandgives@ @t+vrr+1

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Figure7{4:NormalizedElectricFieldAutocorrelationThisquantityisnor-malizedtoaninitialvalueofone.ThedecreaseofthecorrelationtimeandtheincreaseinanticorrelationwithincreasingZisobserved.andanincreaseofanticorrelation,allcorrespondingtoanincreaseofZ.Figure(7.5)displaysthetotaleectontheintegralofC(t).ThelinearVlasovequationdisplaysthesethreeproperties,butneglectsthedynamicscreening.Thus,forC(t),thefeaturesofinteresttouscanbecapturedbycalculatingexactlytheinitialcorrelations,andthenusingasingleparticlemodelforthedynamics.Weproposethatthisprogramcanbeextendedtotheplasmadynamicsingeneralandthatasingleparticlemodelforthedynamicscanbeused.Then,theinitialcorrelationsarecalculatedexactlyfromequilibriumconditions,andthe

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Figure7{5:IntegralofElectricFieldAutocorrelationFunctionThetrueandnormalizedquantities,physicallyrelevantbecauseoftheirrelationtotransportpropertiesandlineshapes,areshown.TheconditionsareidenticaltothatofFig.(7.4).

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7.7 SummaryInthischapterwehavemadeestimatesforseveralimportantstructuralanddynamicalpropertiesforhighlychargedplasmasinconditionsrelevanttothisstudy.ByconsideringtheregularizedDeutschpotentialintheHNCapproximation,paircorrelationfunctionweredeterminedandfoundtobeinagreementwithresultsfromthenon-linearDebyemodelandfromMDsimulations.Theionmicroelddistribution,whichhaslongbeenavitalpartoflineshapetheorieshasbeenreformulatedinamannerthatincludesionandelectroncorrelationsconsistently.Finally,theelectriceldautocorrelationfunctionC(t),relatedtoseveralinterestingtransportandradiativeproperties,isfoundtohaveseveralimportantqualitativefeatures.TheintegralofC(t)ismodiedqualitativelyincompetingwaysastheradiatorchargeincreases.Wendthatoncetheinitialcorrelationsaredealtwithexactly,thesubsequentdynamicscanbeaccountedforbyusingoneparticledynamics.Thisdiscoveryisusedasmotivationfortreatingdynamiccorrelationfunctionsingeneralbythesameprocedure.TheseestimatesareintendedtocomplementtheresultsfromMDsimulations.TheMDresultsareexpectedtoprovideinformationastotherelevantdomainoftheseestimates.Then,onceagreementhasbeenreachedbetweenourestimatesandMDresults,wecanconsiderthephysicalbasisbehindthemodelsusedinthischaptertoprovideinsightintoMDresults.

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@t+L(a;)+B()D(;t)+tZ0dM(;t)D(;)=0(8.7)

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@t+L(a;)+B()D(;t)(8.8)totheiondynamicsform@ @t+L(a;)+B()+D^_bEb@ @bD(k;b;t)(8.9)Here^b=(^qR;^pR;^)representsthesetofplasmapropertieschosen,andD^_bEbistheconstrainedplasmaaverageofthetimederivativeof^b.Thus,whenincludingiondynamics,theeectonthedeteministicpartoftheequationofmotionistheadditionofseveralterms.Thepreviousdenitionoftheplasmaaverageddipoleoperatorwasconstrainedonlywiththevalueoftheionmicroeld.Here,however,thedenitionincludesconstraintsoverthevaluesofallthreepropertiesrepresentedbybasindicatedinEq.(8.6).Besidesthesechangesindenition,thedeterministicpartoftheequationofmotionincludesanentirelynewsetofterms,givenbyh_biB(@=@b)actingontheaverageddipole.Thisformhasastraightforwardinterpretation.Theextenttowhichchangesinthedegreesoffreedombaectthe

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@b=)p @pjpi @j(8.10)Thetypesoftermsappearin,forexample,theBoltzmannequationandcanbethoughtofinthesamemanner.Thersttermontherighthandsiderelatesthechangeintheaverageddipoleduetotheradiatormomentum;inthecontextoflineshapetheory,thistermisresponsiblefordopplerbroadening.Thenexttwotermsarisefromtheforceactingontheradiatorfromtheionmicroeldcausingtheradiatormomentumtochange.RecallfromdeningthesystemHamiltonianthatthechargescomprisingtheradiatorwereseparated,sothatthatcenterofmassdegreesoffreedomwerecollectedintheplasmaHamiltonian,whiletheinternalatomicstatesoftheradiatorcomprisedtheradiatorHamiltonian.ClassicalelectrodynamicsgivestheresultthattheforceonachargeQRinanelectriceldisQR,andtheforceonadipoledinanelectriceldEis(dr)E.Thatiswhatthesenexttermsrepresent.Therstistheforceonthecenterofmassdegreesoffreedomfromtheelectriceld,andthesecondistheforceontheatomicstatesrepresentedbythedipoleoperator.Thetotalforcefromthesetwotermswillthereforechangetheradiatormomentumwithtime.Thenaltermrepresentsthechangeintheionmicroeldvalueattheradiatorduetothemomentumoftheradiator.Thephysicalcontentofthistermcanbeunderstoodbythefactthatthequantityinanglebracketsisrelatedtothegradientoftheeld.Itisinstructivetorelatethismoregeneralcasetothepreviousresultsforthequasi-staticions.Thersttermontherighthandsideoftheaboveequation

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@t+L(a;)+B()+D_bEb@ @bD(k;b;t)+tZ0dZdb0M(b;b0;t)D(k;b0;)=0(8.11)whereM(b;b0;t)isfoundinAppendixFtohavetheformM(b;b0;t)X=f1(a;b)Trpe(b)LU(t)QL(b0)X(8.12)HereU(t)=exp[QL(t)],and(b)=(b^b)aredeltafunctionsthatconstraintheplasmatohavespeciedvaluesforthesetofpropertiesb.ThecomplementaryprojectionoperatorQisdiscussedinAppendixC.Asinthequasi-staticionapproximation,Qprojectsoutdeviationsfromtheconstrainedplasmaaveragedvalues.WediscussM(b;b0;t)byrstanalyzingtheeectsoftheLiouvilleoperatorL=LR+Lp+L,whichoccursexplicitlytwiceintheexpressionforM(b;b0;t).InAppendixFwefollowtheseeects.TheradiatorLiouvilleoperatorcancels

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@Trpe_^(b)eQLtQ_^0(b0)@ @0(8.15)Still,thisexpressioniscomplexandratherthananalyzingitdirectly,weuseastochasticapproach.TheprimaryfeatureofEq.(8.15)isachangingvalueof,thescreenedioneldvalueattheradiator.Forpracticalpurposesthishasbeenmodeledbyamasterequation[43,51].AreasonablerepresentationfortheiondynamicstermisgivenbythemasterequationZd0Mi(;0;t)X()=Zd0(W(;0)X(0)W(0;)X())(8.16)ThetransitionratesW(;0)settheamplitudeandtimescaleforchangesintheeld.Fortimeintervalsshorterthanthisnewtimethestaticionbroadeningoftheprevioussectionoccurs.Thenthiseldchangestoanewvalueduetotheiondynamicsandthenewstaticionbroadeningoccurs.Thisphenomenologicalpicturecanbeextractedfromareformulationoftheexactstatisticalmechanicsmotivatedbythisphysicalpicture.Theobjectiveofthissectionistogiveanoverviewofthis\stochastic"descriptionofiondynamics.Intheend,practicalapproximationsarerequired.However,inthespiritoftheinquiryherethoseapproximationsarestatisticalratherthanperturbativewithrespecttoanyofthechargecorrelationsstudiedhere.Astochasticmodelisthenrequiredforthetransitionrates.Forthesetran-sitionrates,theionmicroeldtransitionwillbemodeledasakangarooprocess,

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@t+L(a;)+B()+D_bEb@ @bD(k;b;t)+tZ0dZd0M(;0;t)D(k;b0;)(8.19)()D(k;b;t)Zd0Q(0)D(k;b0;t)=0Withthisresult,wecancomparetheformalismofourapproachwiththatofpreviousspectrallinetheories.WiththeexceptionofD_bEb@ @bwhichcouplesthecomponentsofb,allthetermsaboveappearandhavesimilarphysicalinterpre-tationsinthoseprevioustheories.Themoregeneralexpressionsderivedherecanthereforebeadirectguideintoextendingthetheorytoincludethesechargecorrelationsinregimeswhereitisnecessarytotreatthemcorrectly.

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9.1 Static Ion Line Shape FunctionForconditionsofstaticionsthetheoreticalanalysisofthelineshapefunctionissimpliedinseveralrespects.First,theDopplerbroadeningisdecoupledfromtheplasmabroadening.Second,theprimaryeectoftheionsbecomesastatisticaldistributionofStarkbroadeningbyeldssampledfromthemicroelddistributionQ().Ourrstnewcontributionappearsatthispoint,withthismicroelddistributiondenedoveratwocomponentplasmapluspointradiatorwithallthecorrelations.Furthermore,thescreenedioneldEsisdenedintermsoftheexactmany-bodyscreeningbytheelectrons.Underconditionsofweak

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9.2 Ion DynamicsTheextensionofthisworktoincludeiondynamicsismoreformalandlesscomplete.ItfollowstheinitialworkofBoercker,Iglesias,andDufty[14]whoprovidedtheformalismbutdidnotanalyzetheeectsonelectronbroadening.Animportantfeatureoftheformalism,discussedatthebeginningofChapter8,istheself-consistenttreatmentoftheradiatorcenterofmassmotionandtheelectriceldoftheperturbingions.Previousworkinthisgrouphasincludedthestochasticchangeintheelectriceld,asdescribedhereaswell,butneglectedtherelationshipofthiseldtotheradiatormotionandtheDopplerprole.Thisconsistencyproblemisamatterofkinematicsduetothefactthattherateofchangeofthecenterofmasspositionisproportionaltothecenterofmassmomentum,andthechangeinthismomentumisproportionaltothetotaleldattheradiator.Thusanyformulationofaspectrallineshapeincludingiondynamicsmustincludethedeterministicformoftheequationgivenhere.

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A.1 Atomic WavefunctionsTheatomicwavefunctionstobecalculatedareformulti-electron,ionizedradiators[7].Tocalculatethese,amodiedversionofacollectionoffourprogramsdevelopedbyRobertCowanofLosAlamosNationalLaboratoryisused[17].Thestartingpointiscalculatingtheone-electronrelativisticradialwavefunctionfortheelectroncongurationofinterest.TheHartree-Fockapproximationisusedforthiscalculation.Someotherneededquantities,suchasenergylevels,transformationmatrices,anddipoleandquadrupoleinteractionstrengths,arealsocalculatedandstoredatthisstage. A.2 Electric-Microeld CalculationTheelectricmicroelddistributioniscalculatedusingtheAdjustableParam-eterExponentialApproximation,orAPEX[53].Thisapproximationissimilarto80

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A.3 Electron-Broadening OperatorMuchoftherecentworkinplasmaspectroscopyhasbeenfocusedonaccurateevaluationsoftheeectsoftheplasmaelectrons[7,8].Thecodescalculatetheelectronbroadeningoperatorusingthreedierentmethods.Todescribethesedieringmethods,thefollowingterminologyisused.Thersttermistowhatorderterminperturbationtheorythatthemethodusestoevaluatetheelectronbroadeningterms.Thesemethodseitherevaluatethesetermstosecond-orderortoallorder.RecallthatthisstudyevaluatedtheelectronbroadeningtosecondorderintheinteractionpotentialLpR.Thesecondtermiswhattypeofdynamicstousefortheelectrons.Bothquantummechanicsandclassicalmechanicsareused.Thenaltermreferstowhattypeofinteractioniscalculatedbetweentheelectronsandtheradiator.Recallthatinthisstudy,theion-radiatorinteractionwasdealtwithasadipoleinteractionduetothelargerepulsionoccurring.Inthepastandespeciallywithneutralradiators,theelectron-radiatorinteractionwasalsoconsideredtobeadipoleinteraction.RecenttheoriestreattheelectronradiatorinteractionwiththefullCoulombexpression.Withthisterminologyexplained,thethreemethodsaredescribednext.

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A.3.1 Second-Order, Full Coulomb, Quantum TheoryAstheplasmasunderconsiderationachievehigherelectrondensities,thedipoleapproximationgivesinaccurateresultsfortheelectronbroadening[7].TheeectoftheperturbingelectronsthenmustbeevaluatedusingthefullCoulombinteraction.Thereasonthatthedipoleapproximationisstillvalidfortheplasmaionsbutnottheplasmaelectronsisduetotheattractionbetweenthepositive

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A.3.2 All-Order, Full-Coulomb, Semi-Classical TheoryWhentheperturbingelectronsundergostrongcollisionswiththeradiator,anall-orderclassicaltheoryfortheelectronscanbeused[8].Thestructureofthederivationissimilartothesecond-ordertheory,inthattheradiator-perturbingelectroneectsaredescribedbyanelectronbroadeningoperatorsplitintoastaticpartandadynamicpart.Thestaticpartdescribesthetime-independentinitialcorrelations,whichisseparatedintoatermthatisrstorderintheinteractionpotentialandatermthatcontainsallhigherordereects.Theapproximationsmadeinthistheoryarealsosimilartothesecondordertheory.Theno-quenchingapproximationisused,anditisassumedthattherearenodynamicalcorrelationsbetweentheelectronsandtheions.Thustheelectronsonlyaecttheionbroadeningbyscreeningtheioneld.Itisalsoassumedthatthereisonlyonestrongelectroncollisionduringtheradiationtime,sothattheelectronbroadeningeectonaradiatorisgivenbyconsideringonlyoneelectron.Thisapproximationwouldalsoindicatethattherearenoelectroncorrelations.Tocorrectforthis,aDebyelengthcutoisusedtoaccountforelectronscreening.Theinteractionofasingleperturbingelectronwiththeradiatorsystemishandledwithamultipoleexpansion.The(modied)monopoletermandthedipoletermaredominanthere,andangularmomentumrulesallowonlyalimitednumberofnonzeroterms.Thissemi-classicaltheorytreatstheperturbingelectronasclassicalobjects.Thisassumptionplacescertainlimitsonthevalidityofthetheory.Thereareseveralmainchangesneededtoevaluatetheelectronbroadeningterminthisway.First,thatpartofthedensitymatrixwhicharisesfromtheperturbingelectronHamiltonianisreplacedwithitsclassicalanalogue.Thephysicalinterpretation

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mr(B.10)andJ(!)isthestaticionlineshapewithJ(!)=1

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@tX=@ @tPX.TheLiouvilleequationforsomegeneraloperatorX(t)is@ @t+LX(t)=0(C.2)ActingfromtheleftwithPandinserting1=P+QgivesP@ @t+L(P+Q)X(t)=0(C.3)LettingtheprojectionoperatorcommutewiththederivativeandseparatingthetermthatincludesQgives@ @t+PLPPX(t)=PLQX(t)(C.4)89

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@t+QLQQX(t)=QLPX(t)(C.5)or@ @t[QX(t)]=QL[QX(t)]QLPX(t)(C.6)Ageneralsolutionforanequationoftheform@X @t=AX+B(t)(C.7)is[43]X(t)=eAtX(0)+tZ0deA(t)B()(C.8)AsolutionforQX(t)isthenQX(t)=eQLtQX(0)tZ0deQL(t)QLPX(t)(C.9)UsingthisresultinEq.(C.4)gives@ @t+PLPPX(t)=PL0@eQLtQX(0)tZ0deQL(t)QLPX(t)1A(C.10)Thespecicformoftheprojectionoperatorandthedetailsofthesystemunderconsiderationisusedtomodifythisfurther.

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@t+L(a;)+BD(;t)+tZ0dM(;t)D(;)=0(D.3)WedeneaprojectionoperatorPPXTrp(a;)X(D.4)FirstnotethatPsatisesthedenitionofaprojectionoperator,sinceP2=PP2X=PPX=Trp(a;)PX=Trp(a;)PX=PX(D.5)Intheabove,thefactthatPXisindependentofplasmacoordinates(becauseitisanaverageoverplasmacoordinates)wasusedtobringPXoutoftheplasmatrace.Then,theconstrainedtraceofjusttheconstraineddensitymatrixequalsunity,givingthelaststep.Anotherconsequenceistheactionoftheprojectionoperatorond(t)Pd(t)=Trp(a;)d(t)=D(;t)(D.6)91

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@t+Ld(t)=0(D.7)TheresultfromEq.(C.10)isthen@ @t+PLPPd(t)=PL0@eQLtQd(0)tZ0deQL(t)QLPd()1A(D.8)UsingEq.(D.6),wethenhavetheequationofmotionforD(;t)@ @t+PLPD(;t)=PL0@eQLtQd(0)tZ0deQL(t)QLD(;t)1A(D.9)Theinitialvaluetermforthedipoleoperatorwillcancel,sinceQd(0)=0.Thisisbecaused(0)doesnotdependonplasmacoordinates,soQd(0)=(1P)d(0)=d(0)Trp(a;)d(0)=d(0)Trp(a;)d(0)=d(0)d(0)=0(D.10)sotheformfortheequationofmotionis@ @t+PLPD(;t)=PLtZ0deQL(t)QLD(;)(D.11)ThespecicformoftheLiouvilleoperatorforthequasi-staticionapproxima-tioncannowbeusedtowritetheequationofmotioninaforminwhichdierentplasmaeectsareseparated.RecallthatthetotalLiouvilleoperatorwaswrittenasL=Lp+LR()+LPR(D.12)whereLpincludedtheplasmadegreesoffreedomandtheirinteractions,LR()wastheLiouvilleoperatorforaradiatorinanelectriceld,andLpRistheLiouville

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@t+LR()+B()D(;t)=tZ0dM(;t)D(;)(D.16)B()=Trpf(a;)LPR)g(D.17)M(;t)=PLeQL(t)QL(D.18)Thisisinacompactform;however,partsofthisdenitionofM(;t)canbeshowntobezerowhentheLiouvilleoperatorisonceagainseparatedintoL=Lp+LR()+LPR.SinceQ2=Q,thereforeM(;t)canbewrittenM(;t)=P(Lp+LR()+LPR)QeQL(t)QL(D.19)

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E.1 First Order Static Shift 4n1=3(E.3)rt=rdeBroglie

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Z2expZ1=2 3a30n0(E.8)isthenumberofelectronsinsideahydrogenBohrsphereassumingconstantdensity.Also,Zisthechargeoftheradiator,and=e2=kTr0istheusualstrongcouplingplasmaparameter.Finally,n`isageometricfactorthatdependsonlyonthequantumnumbersofthelineinquestion.Thefollowingassumptionsaremadeinthecalculation:Theplasmaelectronchargedensityisassumedtobeconstantwithintheboundelectronorbital,andequalinvaluetothevalueatthecenter.Theelddependenceofthefreechargedensityisneglectedhere.TheDeutschpotentialisusedtodeterminetheplasmaelectronchargedensityattheorigin.Theseleadtoaverysimpleforminwhichanalyticmanipulationcanbedone.However,weshowlaterthattheassumptionofconstantplasmachargedensitymaybeunwarranted.Wethenredothecomputationnumerically,lettingthechargedensityvary.

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E.2 Constant Perturber Density AssumptionThepreviouscalculationassumedthatthefreeelectronchargedensitycouldbetakentobeconstantovertheeectiveintegrationoftheboundelectronwavefunction.ThissectionexaminesthatclaimbycalculatingthechargedistributionatdistancesontheorderofthegeneralizedBohrradius.Thequantitytocalculateish(r)iexphZr0

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@t+Ld(k;t)=0(F.5)NotethatPd(k;t)=Zdb(b)D(k;b;t)(F.6)104

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@t+PLPPd(k;t)=PL0@eQLtQd(k;0)tZ0deQL(t)QLPd(k;t)1A(F.7)SimilarlytoEq.(D.10),theinitialvaluetermgoestozerosinceQd(k;0)=0(F.8)giving@ @t+PLPPd(k;t)=PLtZ0deQL(t)QLPd(k;)(F.9)HerethetermPLPPd(k;t)describesthedeterministicdynamicsoftheplasma,whichiswhatwouldoccurifthesetbweretheonlydegreesoffreedomoftheplasma.Theremainingintegraltermdescribestheeectfromtheremainingdegreesoffreedom,whichaccountforthechangesintheionmicroeldandthedynamicelectronbroadening.Thisappendixwillrstderiveaformforthedeterministicpartoftheaboveequation.Itwillbefoundthattheinclusionofmoredegreesoffreedomforthesetbwillrequirethepresencehereoftermsthatcouplethesedegreesoffreedom.Thenwewilltreatthenewpartofthenon-deterministicterm.Firstexaminethetimederivativeterm.Notingthatthetimederivativeandtheintegraloverbcommute,theresultis@tPd(k;t)=@tZdb(b)D(b;t)=Zdb(b)@tD(k;b;t)(F.10)Thistermhastheformofanintegraloverb.Allthetermsinthisequationwillbeputintothisform,andtheintegrandfromthesingleintegralthatresultswillbesettozero.Thedeterministicterm,afterexpandingtheprojectionoperators,

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F.1 Atomic Liouville OperatorTheeectofLainthedeterministictermisZdb0Trp((b)La(b0))D(k;b0;t)=Zdb0Trp1

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F.2 Plasma Liouville OperatorTheeectfromtheLpterminthedeterministicpartoftheequationofmotionisfoundtobeZdb0Trp((b)Lp(b0))D(k;b0;t)(F.20)HereD(b0;t)doesnotdependontheplasmaproperties,soLponlyactson(b0).Thefunctions(b)areassumedtobeclassical,sothatwhenLpactsonthem,LpwillactliketheclassicalLiouvilleoperator,andgiveaPoissonbracketLp(b0)=@H @p@(b0) @q@(b0) @p@^b @q@(b0) @q@^b @p@(b0)

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F.2.1 Radiator Center of Mass PositionFor^b=^rTrp((r;p;)(Lp^r))rrD(k;b;t)(F.26)TheLiouvilleoperatoranditsargumentbecomesLp^r=@H @p@^r @q@H @q@^r @p=@H @p@^r @q=^p m(F.27)Sothe^rtermbecomesTrp(r;p;)^p mrrD(k;b;t)(F.28)Becauseofthedeltafunction(p^p)inthedensitymatrix,theexpressionwillbezerounless^p=p,soTrp(r;p;)^p mrrD(k;b;t)=Trp((r;p;))p mrrD(k;b;t)=p mrrD(k;b;t)(F.29) F.2.2 Radiator Center of Mass MomentumFor^b=^pTrp((r;p;)(Lp^p))rpD(k;b;t)TheLiouvilleoperatoranditsargumentbecomesLp^p=@H @p@^p @q@H @q@^p @p=@H @q@^p @p=QREwhereEisthetotaleldexperiencedbytheradiatorofchargeQR.(ThepartoftheHamiltoniandependentupontheradiatorpositionisthepotentialenergybetweentheradiatorandeachplasmaparticle,sothederivativegivesthenegativeofthetotalelectricforceonthepointradiator.)Thiseldisproducedbyionsandelectrons,whichcanbewrittenasE=Ei+Ee.ThisleadstoTrp((r;p;)(Qr(Ei+Ee)))rpD(k;b;t)

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F.2.3 Electric FieldFor^b=^ETrp(r;p;)(Lp^E)@D(k;b;t) @q@Hp @p=p @qbecausetheeldsareCoulombic.Thepandtheqintheabovearethoseforallparticlesintheplasma,soexplicitlythisbecomesLp^E=Xionspi

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@^q+@D(k;b;t) @^Thisactuallystandsforthesumofthreeterms.ThextermofthissumisTrpb(Lp^Ex)@ @^xEvaluatingtheLiouvilleoperatorforjustthisxtermgivesTrpb(p @^x=(F.30)Trpb(px @^xTrpb(py @^xTrpb(pz @^xDoingthesamefortheyandztermsgivesatotalofnineterms.Inallcases,theradiatormomentumcanberemovedfromthetrace,sotheresultcanbewrittenpi F.3 Interaction Liouville OperatorTheinteractionLiouvilletermisZdb0Trp((b)LI(b0))D(k;b0;t)TheinteractionLiouvilleactslikeLI(b0)D(b0;t)=[H;D]=[H;D]+[H;]D=(LID)+(LI)D

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F.3.1 Interaction Liouville Operator on

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F.3.2 Interaction Liouville Operator on

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Summary{Deterministic PartWehavecalculatedthevariouspartsofthedeterministicpartoftheequationofmotion.FromEq.(F.15),wehavefoundthatZdb0(b;b0)D(k;b0;t)=(F.34)"La+p @jdi*@^Ej @pj+LI()+hLIi#D(k;r;p;;t)=0

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F.3.4 Dynamics TermWenowturntoadetailedderivationofM(b;b0;t).WestartwithPLQeQL(t)QLPdk(t)(F.35)PerformingthePprojectionoperators:Zdb(b)Zdb0Trp f(a;b)(b)LQeQL(t)QL(b0)D(k;b0;t)(F.36)FromEq.(F.13),therelevantpartisZdb0Trp f(a;b)(b)LQeQL(t)QL(b0)D(k;b0;t)(F.37)ExpandtherightmostinstanceofL,andusingQLR(b)D!0Zdb0Trp f(a;b)(b)LQeQL(t)Q((Lp+L)(b0))D(k;b0;t)+Zdb0Trp f(a;b)(b)LQeQL(t)Q(b0)LD(k;b0;t)(F.38)Zdb0Trp f(a;b)(b)LQeQL(t)Q(Lp+L)^b@(b0) f(a;b)(b)LQeQL(t)Q(b0)LD(k;b0;t)(F.39)Performinganintegrationbyparts:Zdb0Trp f(a;b)(b)LQeQL(t)Q(Lp+L)^b(b0)@D(k;b0;t) f(a;b)(b)LQeQL(t)Q(b0)LD(k;b0;t)(F.40)

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f(a;b)(b)LQeQL(t)Q_b0(b0)@D(k;b0;t) f(a;b)(b)LQeQL(t)Q(b0)LD(k;b0;t)(F.41)ToexpandtheotherinstanceofL,notethattheabovecanbewrittenasZdb0f1(a;b)Trpf(b)LQXg(F.42)WethencanwriteZdb01

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G.1 Derivation of the Quantum PotentialThetwoparticlepartitionfunctionZ2forthecanonicalensembleisZ2=TreH(1;2)=Zdr1dr2hr1;r2jeH(1;2)jr1;r2i1 2m1+p21 2m1Zdr1dr2eVq(jr1r2j)(G.1)Hereweassumetheparticlesaredierent,andspinisneglected.Thelastequalityof(G.1)denesthequantumpotentialeVq(jr1r2j).WenexttransformtorelativeandcenterofmassvariablestogetZdRhRjeHcmjRiZdrhrjeHrjri 2mcmZdrhrjeHrjri

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ZdpZdp0e()p2 2mrep02 2mrhpjV(r)jp0iei(pp0)r=~)=1 2mrep02 2mreV(pp0)ei(pp0)r=~)(G.8)whereeV(p)=Zdreipr=~V(r):(G.9)ScalingoutthetemperatureandmassgivesVq(r)!(2)62mr ~2)=Zdr0(jrr0j)V(r0)(G.10)

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2p r2(r)3Z10dZ10dppsinp0r `e(1)p2Z10dp0p0sinp0r `ep02=1 ((1))3=2e1 (1)(r ((1))3=2e1 (1)(r 43rZ11=2d1 ((1))3=2e1 (1)(r ((1))3=2e1 (1)(r (1);=1 21r x2(G.14)(r)=2 `)21 23rZ10dx 3r` re(r `)2(G.15)Thequantumpotential(G.10)isnowgiveninthesimpleformVq(r)!Zdr0p 3r` r0er0

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G.2 Coulomb Potential and Kelbg/Deutsch ResultsForthespecialcaseoftheCoulombpotentialthequantumpotentialbecomesVq(r)!q1q2Z10dr0r02(r0)Zd1 3r`1 `Z1r=`dr0er02!=q1q2 2e(r `)2+1 2+r `1 2p `)2+p `1erfr `(G.17)ThisistheKelbgresult.Forsmallxthisbehavesasf(x)!p


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Title: Charge Correlation Effects in the Broadening of Spectral Lines from Highly Charged Radiators
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Title: Charge Correlation Effects in the Broadening of Spectral Lines from Highly Charged Radiators
Physical Description: Mixed Material
Language: English
Creator: Wrighton, Jeffrey Michael
Publication Date: 2004
Copyright Date: 2004

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CHARGE CORRELATION EFFECTS IN THE BROADENING OF
SPECTRAL LINES FROM HIGHLY CHARGED RADIATORS

















By

JEFFREY MICHAEL WRIGHTON


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2004
































Copyright 2004

by

Jeffrey Michael Wrighton














I dedicate this work to my wife, Jennifer Lee Wrighton.














ACKNOWLEDGMENTS

I would first like to thank Dr. James Dufty, my advisor at the University of

Florida, for the guidance he has provided. His patience and dedication during

difficult times is truly appreciated, as well as the constant encouragement he has

given me.

I would also like to acknowledge my previous advisor, the late Dr. C'! i 1!

Hooper, Jr. I did not know him nearly long enough, and yet I know I will ah--iv-i

remember our meetings, where physics talk was ah--iv-i intermixed with lessons he

had learned from life.

I also thank Dr. Mark Gunderson and Dr. Donald Haynes, Jr. In recent years,

distance has made communication difficult, yet they have ahv-- taken time to

answer questions I have had. I thank the members of my committee, Dr. David

Reitze, Dr. David Micha, and Dr. Fredrick Hamann, for the guidance and advice

they have given me.

iM !1 thanks go to the incredible staff at the Department of Physics. I

especially would like to thank Ms. Darlene Latimer, who during my entire stay at

the University of Florida has done so much for the students of this Department.

I thank my parents, Arthur Wrighton III and Marjorie Wrighton, for instilling

in me a love of learning as I grew up. Finally, I thank my wife, Jennifer Wrighton,

for her patience and love during these past years.














TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

LIST OF FIGURES ..................... .......... vii

ABSTRACT ...................... ............ viii

CHAPTER

1 INTRODUCTION .................... ....... 1

2 OVERVIEW OF PLASMA SPECTROSCOPY ...... ........ 5

2.1 Experimental Details of ICF ......... ........ ... 7
2.2 Electron and Ion Broadening .......... ....... ... 8
2.3 Evolution of Spectral Line Theory ......... ........ 12

3 LINE SHAPE FORMULATION .......... ............ 14

4 SEMI-CLASSICAL MOLECULAR DYNAMICS ....... ..... 23

5 LINE SHAPE FORMULA FOR QUASI-STATIC IONS ......... 30

6 QUASI-STATIC EQUATION OF MOTION ....... ....... 36

6.1 First-order Electron Broadening ......... ......... 37
6.2 Second-Order Static Electron Broadening Term ...... 42
6.3 Dynamic Electron Broadening Term ....... ........ 44

7 CHARGE CORRELATION EFFECTS ON PLASMA PROPERTIES 46

7.1 Plasma Structure ................... ..... 46
7.1.1 Linear Theory ......... ............... 48
7.1.2 Nonlinear model ........... ............ 50
7.2 Screened Ion Field .......... ............... 53
7.3 Ion Microfield Distribution .......... ............ 54
7.4 Constrained Electron Density ............. ... .. .. 55
7.5 Electric Field Covariance .................. .. 56
7.6 Dynam ics .. .. .. ... .. .. .. ... ... .. .. .. .. 57
7.7 Sum m ary .. .. .. ... .. .. .. .. .. .. .. .. .. .. 62

8 INCORPORATING ION DYNAMICS ............. .. 63








9 SUMMARY AND OUTLOOK ................ .... 73

9.1 Static Ion Line Shape Function ........ .......... 75
9.2 Ion Dynamics ................... ..... 77

APPENDIX

A HOOPER'S GROUP COMPUTER CODES ................. 80

A.1 Atomic Wavefunctions .................. ... .. 80
A.2 Electric-Microfield Calculation ............ .. .. .. 80
A.3 Electron-Broadening Operator ....... .. 81
A.3.1 Second-Order, Full Coulomb, Quantum Theory ...... 82
A.3.2 All-Order, Full-Coulomb, Semi-Classical Theory ..... 84

B DOPPLER BROADENING IN THE QUASI-STATIC APPROXIMATION 86

C PROJECTION OPERATOR EQUATION OF MOTION ... 89

D AVERAGE DIPOLE EQUATION OF MOTION-QUASI-STATIC CASE 91

E CALCULATIONAL DETAILS FOR B(e) ...... ......... 96

E.1 First Order Static Shift B(1) Calculation. ...... .... ... 96
E.2 Constant Perturber Density Assumption .............. 101

F KINETIC EQUATION FOR COUPLED ION DYNAMICS CASE .... 104

F.1 Atomic Liouville Operator ........ .............. 107
F.2 Plasma Liouville Operator ......... ............. 108
F.2.1 Radiator Center of Mass Position .... 109
F.2.2 Radiator Center of Mass Momentum ... 109
F.2.3 Electric Field. .................. ... 110
F.3 Interaction Liouville Operator .... ... 111
F.3.1 Interaction Liouville Operator on D 112
F.3.2 Interaction Liouville Operator on .. 113
F.3.3 Summary-Deterministic Part ..... 114
F.3.4 Dynamics Term .................. ... 115

G THE KELBG AND DEUTSCH POTENTIALS .... 118

G.1 Derivation of the Quantum Potential ..... 118
G.2 Coulomb Potential and Kelbg/Deutsch Results ... 121

REFERENCES .................. ........ .. 122

BIOGRAPHICAL SKETCH ............. .. 126














LIST OF FIGURES
Figure page

2-1 Plasma Parameters ............... .......... 13

7-1 Plasma Electron-Radiator Pair Correlation Function ... 51

7-2 Plasma Electron-Radiator Pair Correlation Function ... 52

7-3 Electric Field Covariance .................. ........ .. 58

7-4 Normalized Electric Field Autocorrelation ................ .. 60

7-5 Integral of Electric Field Autocorrelation Function 61

E-1 Captured Bound C(! .ige .................. ........ .. 101

E-2 Temperature Dependence of B() ................... .. 103

E-3 Density Dependence of B(1) .................. ..... .. 103














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CHARGE CORRELATION EFFECTS IN THE BROADENING OF
SPECTRAL LINES FROM HIGHLY CHARGED RADIATORS

By

Jeffrey Michael Wrighton

May 2004

C(! r: James Dufty, C!ii 1! s F. Hooper, Jr. (deceased)
Major Department: Physics

The theory of spectral line broadening is reformulated to examine the validity

of standard approximations for the expected increase in the range of experimental

conditions. The new effects studied here are the correlations that exist among

the plasma perturbers and with a charged radiator, which are usually neglected

or approximated in current theories. The first analysis here assumes quasi-static

ions, which is a relevant condition for many lines of experimental interest. The

electron broadening operator, or width and shift operator, is calculated to second

order in the interaction of the plasma perturbers with the bound electrons of

the radiator. All other charge correlations are included without approximations.

A semi-classical representation of these results provides the necessary physical

quantities required for recently proposed methods of molecular dynamics simulation

in plasma spectroscopy. An extension to include the effects of ion dynamics is

described.














CHAPTER 1
INTRODUCTION

The theory of plasma line broadening has been used for many years to study

hot, dense plasmas [1, 2]. With advances in experimental equipment, more rigorous

theoretical developments are required in order to explore the broader range of

state conditions being made accessible. Over time, this has led to the inclusion of

effects previously neglected (e.g., ion dyr i' ~-[:]) as they became important for

specific conditions of higher density and temperature. Therefore, as experimental

techniques continue to improve, opportunities for new exploration expand, and

continual evaluation of current line broadening theories is needed.

One area of the theory that needs evaluation now is the treatment of charge

correlations for the spectroscopy of charged radiators. Present theories account

for the correlation between the various plasma constituents and between the

plasma constituents and the total charge of the radiator in inconsistent v--i-. Some

correlations, such as ion-ion correlations, are treated quite rigorously through a

detailed study of their interactions[4, 5], but an ad hoc screening length is used to

account for electron-ion correlations. Similarly, electron correlations may be treated

by assuming independent particles restricted to a Debye sphere[6, 7, 8].

We study the charge correlations for an ionized radiator in a two component

fully ionized plasma of ions and electrons. To simplify this first analysis, we focus

on those lines and conditions for which the heavy ions behave in a "quasi-static"

manner. In this regime the plasma ions and the radiator both have negligible

acceleration during the radiation time. The line shape function is formally written

in terms of a single operator, the broadening operator, describing all effects of

the radiator's environment. The radiator is represented as a point charge plus its








residual bound electron distribution. Our main result for this case is an evaluation

of the broadening operator that is exact to second order in the interactions of the

plasma with the bound electron distribution. All correlations among the plasma

constituents are included, as are the correlations induced by the point charge of the

radiator.

In addition to this general result following from quantum statistical mechanics,

several other approaches will be examined. First, the inherent quantum effects of

the electron-ion interactions will be represented by a semi-classical i i ,l! 1. 1

Coulomb interaction[9, 10]. This accounts for diffraction effects at separations of

the order of the de Broglie wavelength, and these regularized potentials are well

studied for similar plasma conditions in other contexts. A motivation for this

representation is to make contact with recent studies of plasma charge correlations

using classical molecular dynamics (\!1)) simulations [1]. An outcome of these

studies has been the proposal to study plasma spectroscopy by MD as well[12].

Theoretical analyses of the type given here are necessary precursors for any

such MD study, so as to identify the appropriate properties to simulate. This is

discussed in some detail in C'! plter 4.

Another limit for both quantum and semi-classical representations is obtained

by approximating the full Coulomb interaction between perturbing electrons and

the bound electron distribution by dominant multipoles[6]. It is proposed that

monopole and dipole interactions are sufficient following the results of two closely

related theoretical formulations for charged radiators developed by the Florida

group over the past 10 y, i -[7, 8, 13].

The quasi-static ion analysis properly addresses the correlations. However,

under some conditions and for some lines, the time for ion motion becomes

comparable to times of interest on the line profile. To address this more general

case, C! iplter 8 shows how ion dynamics can be incorporated into the theory.








Since the ions are generally strongly coupled, the ion dynamics must be described

nonperturbatively to avoid inconsistent assumptions about the charge correlations

described in the static limit. For practical purposes, following our formal analysis

the ion dynamics is modeled using stochastic approximations that allow the

inclusion of correlations[14]. The resulting descriptions can then be used, after

some modification to allow for the inclusion of experimental effects, to determine

characteristics of a broad class of the plasmas, radiators, and lines.

This work was begun under the guidance of Professor ('! i i!. F. Hooper, Jr.

who led the Florida plasma spectroscopy group for more than 30 years. During

that time, a code for predicting spectral line shapes was developed, with continual

improvement as the field evolved. It remains one of the only codes general enough

to describe multi-component, multi-temperature plasma broadening of 1r ii', -

electron atomic lines. In his memory and in an attempt to record the content

of that code, we provide an overview of the computer codes developed for this

problem at the University of Florida. These programs have been written to

calculate line shapes, mainly for near-hydrogenic, highly charged radiators in hot,

dense plasmas. Initial work done on the programs focused on accurate calculation

of the ion microfield distribution, while more recent work has centered on accurate

evaluations of the electron broadening operator. Details are given in Appendix A.

The formal analysis given here is an extension of earlier work begun by Dufty

and Iglesias for neutral radiators and static ions[15]. Its relevance to highly charged

radiators is outlined in the recent thesis of Gunderson[8]. Important differences

with the work here are noted in ('! i pter 9. The use of stochastic models for the

radiative and transport properties of impurities in plasmas was introduced by

Boercker, Iglesias, and Dufty[14].

We hope that a more rigorous treatment of the theory of spectral line broad-

ening will provide better experimental diagnostics and theoretical understanding.






4


Before developing the theoretical framework, C'! lpter 2 gives an overview of the

importance of plasma spectroscopy in general, and gives some physical concepts

that characterize the radiative process in a plasma environment.













CHAPTER 2
OVERVIEW OF PLASMA SPECTROSCOPY

A spectral line shape function I(w) gives the distribution of radiation emitted

or absorbed by an atom in the presence of a photon field. For an isolated atom

making a simple transition between two atomic states, this is a Lorentzian function

with a half-width given by the natural broadening that arises from the interaction

of the radiating atom or ion with its own radiation field[1]. This width can be given

to an order of magnitude by Fnatural ~ a(aZ)2Z2 in Rydberg units, where a is

the fine structure constant [16]. It is approximately a factor of a smaller than the

fine structure splitting of atomic levels. For cases considered here, this is negligible

in the presence of other broadening mechanisms[1]. One of these more important

mechanisms is Doppler b ... 1,1~ ii- which occurs if the radiator has thermal motion

due to its environment[17]. If a Maxwellian distribution for the radiator velocity is

assumed, the resulting Doppler profile is


IDoppler () -exp G 2 (2.1)
D7Vp Fdoppler

where the Doppler half-width is


FDoppler { j 1C2 ( 0 (2.2)

Here T is the kinetic temperature of the radiator, M is the radiator mass, and wo

is the frequency of the unperturbed transition. In C'i lpter 5 we shall show how

Doppler broadening is incorporated in a general line shape.

The more interesting and informative broadening mechanisms are those due

to the direct interaction of the radiator with the other particles of its environment.

Through a careful analysis of how these mechanisms determine the shape of I(u),








experimental determination of the line shape can provide detailed information

about the environment. Plasma spectroscopy is thus a unique tool to serve as a

diagnostic and as an exploratory probe of complex environments.

Spectroscopic techniques have been used for many years to study the char-

acteristics of plasmas. The noninvasive character of spectroscopy makes it an

ideal tool for the study of plasmas in stellar interiors and in inertial confinement

fusion (ICF) experiments, which are the focus of our study. The radiators in these

plasmas emit spectral radiation in a manner that is significantly affected by the

details of the plasma environment. Each radiating ion has its atomic structure

level disturbed by the electric fields of all other plasma constituents. For diagnostic

purposes, e.g., in laser produced plasmas, a careful theoretical analysis of a complex

line shape often allows reliable determination of key physical properties such as

average electron density and the temperatures of the different plasma constituents

that could not be obtained otherwise.

The accuracy of the calculated line shapes is increased by including more

detailed information about the exchange of energy between the radiator and the

plasma. The shape of the spectral emission or absorption also depends on the

detailed atomic structure of the radiator and the characteristic properties of the

plasma constituents (such as proton vs. deuteron concentration). The widths,

shifts, merging, and satellites are all features of the spectral line shape that we

wish to accurately determine so as to best compare theoretical and experimental

results[18].

At the National Laser User Facility (NLUF), part of the Laboratory for

Laser Energetics (LLE) located at the University of Rochester in New York,

inertial confinement fusions (ICF) experiments are currently being run[19]. Plasma

spectroscopy (matching the radiation spectrum with theoretical line shapes) is an

important tool to analyze the outcome of these experiments. These spectroscopic








techniques will also be important at the National Ignition Facility (NIF) which

is currently being made operational at Lawrence Livermore National Laboratory.

It is expected that plasmas created at NIF will reach temperatures and densities

approximately seven times greater than currently achieved values, which will be

conditions beyond the temperature and density of the solar core. (The expected

temperature is about 100 million degrees, compared with 15 million at the center

of the sun; the expected density is about 300 g/cm3, roughly twice that of the solar

core.) The theoretical approximations currently being used for spectroscopy must

be r( i, &i1. -1 in the context of these extreme conditions. A primary motivation

for our study is the new plasma state conditions and new types of highly charged

radiators made possible by such experiments. To better appreciate the complexity

of these experiments, and therefore the need for accurate spectroscopic analysis, a

brief description is now provided.

2.1 Experimental Details of ICF

In ICF experiments, small microspheres with a fill gas are imploded with lasers

[13]. In current experiments, these microspheres typically have a diameter of about

1 mm, and consist of a plastic shell of about 20 microns thickness. A fill gas is

introduced into the microsphere, which is then placed in the target chamber.

Two main classes of experiments are performed: direct drive and indirect

drive experiments. In a direct drive ICF experiment, the microsphere is suspended

alone in the target chamber[20, 21]. Laser beams strike the sphere from many

directions which cause the outer portion of the shell to explode away. The reaction

force drives the inner portion of the shell inwards, compressing the fuel gas to

form a plasma. In an indirect drive experiment, the microsphere is placed inside

a small cylindrical tube (called a hohlraum) which is placed inside the target

chamber[21]. The laser beams enter the hohlraum from each end and illuminate

the inner walls. These inner walls then radiate, and it is this radiation that leads








to the compression of the microsphere in an indirect drive experiment. Using a

hohlraum leads to a more even radiation distribution striking the microsphere. This

is important since the most common problem is instabilities causing a nonuniform

implosion. However, more energy can be delivered to the microsphere in a direct

drive experiment.

After the fill gas forms a plasma, spectroscopic techniques can be used to

determine density and temperature. As an example of typical results, densities of

1.9 x 1024 cm-3 were reached at temperatures of 1.15 keV. For ICF experiments,

other diagnostic tests can be used (such as neutron emission[22]) and these show

agreement with the spectroscopic method to within experimental uncertainties.

The choice of the fill gas determines the theoretical approach. We consider

two main v- -v- to fill the microsphere: a heavy-ion dopant and a heavy-ion fill

gas[3, 23]. For example, the microsphere can be filled with a deuterium gas

that is doped with a small amount of argon; alternately, the microsphere can

be completely filled with argon. The actual choice of radiator (argon, krypton,

etc.) depends on the expected experimental results. The plasma will then consist

of radiating ions surrounded by free electrons and other ions. The temperature

reached by the plasma will determine the relative percentages of different ionization

levels of the li, i,. -ion radiator. The probability of nonhydrogenic radiators can be

accounted for and increases the accuracy of the diagnostics.

The radiation is captured by streak cameras and is then analyzed. In the

ain 1 ,-i- the effects of the experimental equipment are also taken into account in

the line shape. A set of line shapes is used and a computer code finds the best fit.

2.2 Electron and Ion Broadening

Historically, plasma spectroscopic techniques were most important in the study

of stellar atmospheres. Stellar atmospheres have electron densities and ionizations

that are relatively low; thus pressure broadening by neutral perturbers becomes








important [24]. This is essentially collisional bi ...l. 1. 11- and differs considerably

from broadening by the long range Coulomb forces of fully ionized matter. The

most exciting new developments of the field are therefore in the direction of

characterizing the opposite extreme of hot, dense, ionized plasmas. Furthermore,

since these conditions support radiators of large charge number, the coupling to

this environment can be considerably enhanced over that for neutral radiators.

Consequently, the attention here is directed at a careful treatment of interactions

of the radiator with the surrounding positive ions and negative electrons as the

dominant source of line broadening.

A dominant effect of the surrounding charged particles is Stark b .... 1. i.-[1],

a form of pressure broadening that can be described in terms of the Stark effect.

It is also referred to as pressure broadening because it is sensitive to the density

of (and thus the pressure from) the surrounding plasma particles. For Stark

broadening, the atomic energy levels of the radiator are shifted by the electric

fields of the surrounding plasma particles coupling to the radiator dipole. For

hydrogenic ions, the bound electron wavefunctions can be solved for exactly using

either spherical or parabolic coordinates, and a first-order perturbation calculation

gives the energy shift of a parabolic wavefunction in the presence of an electric

field. The energy shift is found to be AE = 3aoenqc/2Z, where ao is the Bohr

radius, e is the electron charge, Z is the nuclear charge of the radiator, and the

electric field magnitude is e[25]. The numbers n and q are quantum numbers of the

wavefunction, and the parabolic coordinate system is oriented along the perturbing

electric field. The timescales come into p1l iv here, as the energy shift from the ions

changes little during the radiation time, while the energy shift from the electrons'

electric field does change. Hence, different theoretical approaches are used for the

effects of electrons and ions.








As discussed in the next section, the line shape function can be described in

terms of a time autocorrelation function for the radiator's dipole moment[1]. A

characteristic decay time can be associated with this function to give a radiation

time[13]. The characteristic times for various processes can then be estimated

and used to determine how they should be handled theoretically. One of the basic

assumptions used in the first part of our study is that the perturbing ions are

massive enough so that during the radiation time they do not move far enough to

alter appreciably their force on the radiator. Consequently, they can be treated as

static. In contrast, the electrons, because of their smaller mass, will move a great

deal during the radiation time. This means that the effect they have on the atomic

energy levels changes during the radiation time, so the electrons require a dynamic

treatment. This section provides some order-of-magnitude calculations to check for

the validity of these approximations.

Several approaches can be taken to estimate the relative effect of perturber

motion on the line shape [7]. The first approach is to calculate the relative change

in the perturber electric field during the typical radiation time Tr A typical

radiation time can be approximated by the experimentally determined line width.

One example line shape relevant to our study gives a full width at half maximum

(fwhm) of 40 eV. The quantum energy-time relation gives a corresponding time of

Tr 10-17 s, which we use here to estimate the relative field changes.

The electric field from a perturber is e = qr/r3 with r being the displacement

from the radiator to the perturber. The fractional change in c due to a change in

this displacement is then I6e/e = 26r/r a 2vaveTr/ro, where ave = /2kBT/m and

the values of m (and possibly T) are different for each species of perturber. Also

ro is a characteristic distance from the radiator. For the ions, this characteristic

distance can be chosen to be the average ion distance from the radiator ro =

( T,-. /3)1 Using typical plasma quantities of temperature of 1000 eV and








electron density of 1 x 1024 /cc, the estimated value of 6e/ec for a hydrogen

perturber is 6e/e w 0.07, and for argon ions as perturbers 6e/e = 0.01. Clearly,

these massive ions can be treated quasi-statically for the conditions considered.

Using the same density given above for the electrons, this leads to an esti-

mated fractional change in the electric field of the perturbing electrons of 6e/e w 3.

This estimate can be assumed to be too small, since the highly charged positive

radiator will tend to pull electrons in closer. Because of the possibility of large field

changes, the electrons must be treated with a dynamic theory.

An alternate method for determining the need for dynamic theories is by

considering the Debye length. An electron separated by more than a Debye length

from the radiator has its effect on the radiator screened by that part of the plasma

between the two[26]. Using this length and the average velocity of the electron,

a characteristic time can be calculated that represents the time that an electron

can travel and not be screened by the plasma environment. The Debye length

is AD = VkBT/47ne2 and the average thermal velocity was given before. The

characteristic time can then be calculated and related to the electron plasma

frequency t = AD/Vave = m/8m-/ne2 = 1/V2Wp. Therefore, a rough estimate leads

to the requirement that the electrons be treated dynamically for energy widths

less than the electron plasma frequency. Similar considerations apply to the other

perturbing particles of the plasma.

The time scales of the plasma under consideration also determine whether

Doppler broadening can be calculated separately from Stark broadening. By

assuming that it can, the assumption made is that the radiator velocity is constant

during the radiation time. The momentum change of the radiator is equal to the

impulse delivered to the radiator during some time t, and this impulse can be

estimated using characteristic values that were used above. An estimate of the

force during this time can be made using the average interparticle spacing of the








ions. The resulting change in momentum can be used to determine the fractional

change in momentum by using the average momentum determined by the thermal

velocity. For the density and temperature given above, these fractional changes

are 0.002 and 0.038 for hydrogen ions and hydrogenic argon perturbers. Therefore,

neglecting this small acceleration is valid. Then Doppler broadening effects become

statistically independent of the Stark broadening by the ions and electrons, as

indicated in Chapter 5.

2.3 Evolution of Spectral Line Theory

Modern spectral line theory includes many effects, some of which must be

dealt with concurrently with the derivation of the line shape. Probably the most

important of these effects is the form of the interaction between the plasma elec-

trons and the radiator. The earlier models of spectral line shapes dealt with neutral

radiators[27]. With these plasmas, it was common for the interaction between

the radiator and the plasma electrons to be modeled by a dipole interaction. A

fundamental change occurred when the plasma experiments reached temperatures

high enough to ionize the radiator. Ionized radiators attracted the plasma elec-

trons, and with the smaller separation, the dipole approximation was no longer

adequate[6]. To describe these ionized plasmas, the full Coulomb interaction is

required in the theory. When this was done, it was found that line shifts arose from

the electron-radiator interactions.

There are also several effects that are part of spectral line shape analysis (and

part of the line shape codes described in Appendix A) that are not incorporated

explicitly into our study. One of these effects is spectral line merging[8]. As den-

sities increase, the positions of neighboring lines move together and merge. The

merging of spectral lines at high densities requires a relaxation of the no-quenching

approximation and a much longer calculation time. Opacity broadening, especially

required for stellar applications, has been incorporated into the line shape codes[3].










109

10- 7- r= 0.03
10'-
SICF r= 0.1

107- r -= 0.5

106
I-



104- 1

1015 1020 1025
no (cm3)



Figure 2-1: Plasma Parameters The temperature and density range of plasmas
studied here is compared with other experimental conditions. Parameters used in
MD simulations are shown.


The observance of satellites required the use of multi-electron radiator theories[19].

In addition, the experimental equipment also gives a broadening to the spectral

line, which needs to be accounted for to directly compare theoretical and experi-

mental results. These effects, while important to line shape theory as a whole, can

be neglected in our study. Our study will formulate a line shape of a single ionized

radiator in a plasma.













CHAPTER 3
LINE SHAPE FORMULATION

The line shape function I(w) that is the starting point for the theoretical

,in ,i -; here follows directly from an analysis of the radiated power spectrum P(w)

of a quantum mechanical system undergoing a transition from initial states a to

final states b

P() = (- L b de -ikq a)2 PaW ab), (3.1)
a,b
where d = erb is the total dipole moment operator associated with the

radiator, with the sum taken over all bound electrons [1, 2, 28]. The position vector

rb = b qR is the position relative to the center of mass position qR of the
radiator. The vector k in Eq. (3.1) points from the radiator to the detector and

has a magnitude of k = w/c. Integrating this vector over all directions captures

all of the radiation. The delta function ensures that the transition considered

conserves energy. This term contains the quantity jab = (E, Eb)/h, which is

the frequency associated with the energy difference between the initial and final

state. Finally, this power is found by averaging over the initial states and summing

over the final states. The weighting factor pa produces the average over the initial

states. For many line shapes, the quantity w4 is approximately constant for the

energy range of a line shape[1], and therefore it is convenient to define a line shape

I(w) as

i(w) = VP() = b de -kq-R )l 2 6(- ab) (3.2)
a,b








For an equilibrium system, this can be written as an integral of the dipole autocor-

relation function I(t)
0C
I(U)= -Re dt (3.3)
0


1(t) Kd. dk(t)jp, (3.4)

with the following definitions


dk de-ikqR dk(t) = eiHt/dke-iHtlh (3.5)


The brackets in the autocorrelation function denote an equilibrium ensemble

average:

(X) = Tr peX. (3.6)

The density operator p, is from the equilibrium Gibbs ensemble for the radiator

and its environment. In this form, the line shape is seen to be determined by the

dynamic correlation of the dipole with itself. The decorrelation time Td is therefore

a measure of the relevant radiation time or half-width of the line shape.

The system under consideration here is a fully ionized plasma of point elec-

trons and ions, plus a single impurity radiator ion of charge number ZR. Overall

charge neutrality is assumed. Unlike the plasma electrons and ions, the radiating

ion has internal structure and therefore a charge distribution about its center of

mass. In the following analysis, the system will be decomposed into two subsys-

tems, the plasma and atomic systems. The radiator ion will be itself decomposed

into two parts, which will be separated into the above groups. The center of mass

degrees of freedom of the radiator, which include the kinetic energy of the center

of mass and the total charge of the radiator located at the center of mass, will be

grouped with the plasma system. The internal degrees of freedom of the radiator,

which include the potential energies of the bound electrons with the nucleus and








each other and the kinetic energies of the bound electrons about the center of

mass, will comprise the atomic system. Because of this grouping, the interaction

between the atomic system and the plasma is only due to the distribution of the

bound electrons relative to their monopole representation. This is an important

point to emphasize here, since the analysis in the next section treats interactions

among plasma constituents exactly. That includes exact coupling to the radiator as

a monopole. Approximations only occur for coupling to the excess bound electron

distribution.

The Hamiltonian for this system therefore describes a radiator ion with bound

electrons, and also free electrons, free ions, and all interactions among them. The

Hamiltonian can be expressed as the sum of a plasma Hamiltonian Hp, an atomic

radiator Hamiltonian HR, and an interaction Hamiltonian UpR that describes the

interaction between the two systems.


H = Hp + HR + Up (3.7)


As stated above, a point charge representing the radiator center of mass degrees

of freedom will be included in the plasma subsystem. Then, the plasma Hamil-

tonian Hp contains the kinetic energies of the free electrons, free ions, and the

radiator center of mass, as well as all interaction energies between these three

components. The result is

(N, N17 N, N,'
S= K S K( 2) 2 E 2 5 5'(a ,3) (3.8)
17 (a-=l a743 q747l a- 13=1

In Eq. (3.8), rl represents the type of particle (free electron, ion, or radiator center

of mass) and N, is the number of particle of type ty. The first term is the kinetic

energy with K,((a) p2/2m,. The second is the interaction between particles

of the same type, and the third is the interactions between particles of different








types. The interactions V,,' are Coulomb pair interactions. Then Hp describes a

two component plasma of electrons and ions, plus a point impurity ion.

For the atomic radiator system, the Hamiltonian is written as

2 '2p t
2mHR + (a,/3) + V.(a) (3.9)
a a7/3 a=l

with nR being the number of electrons bound to the radiator. The three terms

in Eq. (3.9) describe the kinetic energy of the bound electrons, the interactions

between bound electrons, and the interaction between the bound electrons and the

nucleus of the radiator. With the center of mass degrees of freedom absent from

these terms, HR is identical to a case of electrons bound to a motionless charged

particle, and is therefore a straightforward atomic physics problem.

The final part of the Hamiltonian describes the interaction of the atomic and

plasma systems. These interactions are Coulomb interactions, and can be written

as functions of the position operators of the quantum particles. Because of the

mathematical manipulations to be done later, it will be convenient to write these

interactions in terms of charge densities. The plasma particle and bound electron

charge densities can be defined as:

N np
p,() = Qe6(x q) Pb = e6(x'- r3) (3.10)
a=l a=l

(In Eq. (3.10), r = {e, i}, indicating the plasma electrons or ions, respectively; in

contrast to Eq. (3.8), the point radiator is not included.) With these definitions,

the interactions take the form of the above charge densities multiplied by the form

of the pair potential for their interaction. For example, the interaction between the

free plasma electrons and the atomic system becomes

(ectrons) dx dx' p(x (')p ( ) (3.11)








The actual form of W(x, x') can be found as follows. Before the introduction of

charge densities, the explicit form of the plasma electron-atomic system interaction

is

Ne RR 2 Ne N 2 N e 2
(electrons) e e R Ze N C12
P/ R I q,, qR rTp [q1 qR I aQ qRj (3.12)
a=1 \=1 a=1 a=/

The first term describes the interaction between the free and bound electrons,

and the second term is the interaction between the free electrons and the radiator

nucleus. The interaction between the free electrons and the radiator center of mass

was included in the plasma Hamiltonian, so it must be subtracted, giving the last

term. The other quantities in the Eq. (3.11), qg, qR, and rp, are the positions of

the free electrons, radiator nucleus, and bound electrons respectively. Finally, NR is

the nuclear charge and ZR is the radiator charge. Since ZR = NR nR, this can be

simplified to

Ne R ( 1
/(electrons) = 1 2 1 (3.13)
pR (J \- -r,3 q|- (R


This shows that the form factor in Eq. (3.11) is


V(x, x') = -x qR x Ix qRI (3.14)



The remaining part of the interaction Hamiltonian is the interaction of the free

ions with the atomic system. The ion-interaction Hamiltonian can be written in

the same form as above, with the only difference being that the charge of the free

ions replacing the charge of the electrons. When these are added together, the total

interaction Hamiltonian is


UpR = dx dx' pf(x)pb(x') W'(x,x') (3.15)
'7








W(x,x') x q 1- (3.16)

where the sum over Tl is the sum of plasma species.

The form factor for the ion-radiator interaction can be simplified by consid-

ering the various length scales involved in the integration. In the denominators

of the form factor, the two vectors of interest are x qR and x'. The first vector

represents the separation vector of an ion perturber from the radiator, and the

second vector represents the bound electron position relative to the radiator center

of mass. The integral over x and x' are formally taken over all space. However,

physical considerations limit the effective range of each integral. First, the integral

over x' is weighted by the bound electron wave function. Since the wave function

has a characteristic length scale given by the Bohr radius, the upper integration

limit at infinity can be replaced by a factor on the order of the Bohr radius for that

species of radiator. Second, when comparing the ion-radiator separation x qR and

the relative position of the bound electron x', it is found that the ion-radiator sep-

aration is much larger than the relative bound electron position, or x qR >> x'.

To see why this is so, we can use as a characteristic length the ion sphere radius ro.

This is the separation of the plasma ions assuming uniform charge density, and is

given by
4 / 3 13
-rn n = ro = ) (3.17)
3 47n
where n is the ion charge density. The probability that the ion will be a certain

distance from the radiator can be estimated by the Boltzmann factor. A lower limit

on the probability can be found by estimating the Boltzmann factor for the ion at

the Bohr radius. This gives


exp -+ZiZe2 exp -ZLZFroP (3.18)








where Zi and ZR are the charges of the plasma ion and radiator, ao is the Bohr

radius, ro is the ion sphere radius, and F is the electron plasma parameter given by

F /3e2/r0. For typical plasmas under consideration, ro/ao is on the order of the

radiator charge, and the plasma parameter lies in the range 0.01 $ F $ 0.1. With

these values, the exponential factor is small at the Bohr radius, and approaches

zero as the exponential of 1/r. Since the radiator typically has a much greater

charge than the plasma ions, the average radiator-plasma ion separation is typically

larger than the ion sphere radius. The vector x', representing the distance from the

bound electron to the radiator center of mass, is limited by the effective cutoff of

the bound wavefunction. This distance is on the order of several Bohr radii.

The conclusion from this is that in the denominator of the first term of

W(x, x'), the vector x qR is much larger than x'. Because of this, the form
factor can be expanded in terms of the first quantity. To first order in x', W(x, x')

becomes:
( 1 xIx 1
W(x, x')= + + 3 (3.19)
x qR x qR x qRj

VW(, X') = ( x' (3.20)
|x qR
For the ion-radiator interaction, the form factor is now separated. The interaction

Hamiltonian for the plasma ions is then, from Eq. (3.13)


U[J = [ dx pj(x) (x pb(x')x' (3.21)
(x- qR)l J

S e (ri qR) ] y = E[ d (3.22)
ions |(ri qR rb

where on the last line the quantity in the first brackets was identified as the electric

field of the plasma ions at the radiator, and the quantity in the second brackets

is the definition for the dipole moment of the radiator. It is important to note

here that this ion electric field is the bare Coulomb field. This is a long range

force, and it will later be convenient to take advantage of the screening by the








plasma electrons to define a modified ion field by including certain averaged plasma

electron-radiator interactions. We will return to this topic in C'! lpter 5.

For the plasma electron-radiator interaction, the situation is quite different.

The distant electrons will give a contribution similar to Eq. (3.22) due to the

electron's electric field. However, the highly charged radiator also pulls the plasma

electrons in close, so that the plasma electron charge density is appreciable even

within the bound electron orbitals. We will now explore the form factor for these

close plasma electrons. To estimate the effect of the close electrons we make the

assumption that the plasma electron charge density is spherically symmetric about

the radiator, and that the radiator is at the origin. Then Eq. (3.15) for the plasma

electrons is
Selections 2 x dx p b(x') di W(x, x') (3.23)

with dQ being the angular integration over the plasma electron coordinates. The

form factor can be expanded in a multiple expansion


W(x, x') (-1 1 ) 4 Y, (Q) (Q/) _J 1 (3.24)


where x> and x< is the greater and lesser, respectively, of {|xl, x'}1, and .Y(Q) is

a spherical harmonic[29]. If the plasma electron charge distribution is spherically

symmetric as assumed, then only the = 0 term in the sum will be nonzero due

to the orthonormality of the spherical harmonics. Using Yo0() 1/v4r, the form

factor for this case is therefore


WV(x, x') x')- + x' x) (3.25)
x> x x x' x

where O(x x') has the value of 1 if x > x' and 0 for x < x'. Since O(x x') +

O(x' x) = 1, this factor can be introduced into the last term and the theta








functions combined to give

W(x, x') = e(x' x) (1 1 e(x' x) (3.26)
G x' x xx')

This indicates that in addition to the dipole interaction the plasma electron-atomic

radiator interaction includes a "monc.lp. term from the penetration of the bound

charge density by the plasma electrons.

In summary, the line shape has been identified in terms of the radiator dipole

autocorrelation function. After a separation of center of mass and internal degrees

of freedom for the radiator, the dynamics of the dipole is determined from a

Hamiltonian of these internal degrees of freedom coupled to a plasma of point

charges. This coupling is through a dipole coupling to the ions and electric fields of

the plasma charges. In addition there is a coupling to the plasma electrons within
the atomic structure of the radiator.













CHAPTER 4
SEMI-CLASSICAL MOLECULAR DYNAMICS

In this chapter, the application of molecular dynamics (\!1)) simulations to the

line shape problem will be examined. Recall that in C!i lpter 3, an expression for

the line shape was expressed in terms of the dipole autocorrelation function as

I(t) =(dt dk(t))= Z-Trdt iHtdke-H(tt- (4.1)

where the angle brackets indicate an average over an equilibrium ensemble for the

plasma, and Z = Trexp[-3H] is the partition function. It is important here to

determine where the most difficult part of this calculation lies. To this end, we will

approach the calculation of the dipole autocorrelation function by direct solution of

the Schr6dinger equation. The Hamiltonian for the system is, from C(i plter 3

H = Hp + H + UpR (4.2)

where Hp is the isolated plasma Hamiltonian (including the radiator center of

mass), Ha is the Hamiltonian describing the atomic degrees of freedom of the

radiator, and UpR is the interaction between the two. Going to the interaction

representation[30] defined by

eiHt eiH (t) (4.3)

gives the autocorrelation function in the form

(dt dk(t) Z- Tre-3Hptd U(t)dkUt (+ i/) (4.4)

A basis set {|c)} to perform the trace can then be chosen, and interaction picture

states can be defined with ',(t) = U(t)la). The correlation function can be








calculated by solving the Schrodinger equation


H,- iUpR(t)) '(t) 0 (4.5)

with

UpR(t) dx dx' p x(, t)pb(x')W(x, x') (4.6)

p(x, t)= eiHtpx)e-iH't (4.7)

where p,(x, t) is the plasma perturber charge density and pb(X') is the bound

charge density. Note that in Eq. (4.6), the only dependence on the plasma degrees

of freedom occurs completely within p,(x, t). Once p,(x, t) is specified, the problem

represented by Eq. (4.5) is purely an atomic physics problem of the coupling of

an atom to an external potential. The difficult n i-1 -body problem residing in

p,(x, t) is therefore of great importance to calculate accurately, which is why MD is

relevant here. MD simulations claim to be able to perform an accurate calculation

of the plasma charge density with all charge correlations intact. However, there are

several difficulties in applying MD to this problem, and this chapter will examine

the validity of MD simulations approach.

The first topic to study is the fact that MD is inherently a classical method.

In MD, Newtonian equations are solved directly to provide particle trajectories

[31, 32]. The motion of the particles is then followed for some time interval, and

the properties to be calculated are determined by the time dependent coordinates

and moment of the particles. The early uses of MD for plasma simulations

involved a one component plasma consisting of positive particles, plus a constant

negative background for overall charge neutrality. For this case, MD is very useful

and clearly applicable, since the repulsive nature of the interparticle interactions

keeps the particles far apart. This meant that, with high temperature, the distance

between the particles is much larger than the de Broglie wavelength. Therefore a








classical description is a good approximation to the one component plasma, since

in such a case there is small probability for the particles to be close enough for

diffraction effects to be important.

For this study, we would like to use MD to study a plasma that would include

positive plasma ions, plasma electrons, and a highly charged radiator impurity.

One initial problem is the mass difference between the plasma ion and electrons.

The electrons require a short characteristic time scale. Any simulation would have

to move with time steps less than this time scale, or else the simulation would

not capture the effects of the electron motion. However, the ions move much

more slowly, and thus have a time scale much larger than that of the electrons.

Therefore, the time steps have to not only be small enough to capture the electron

motion, but the simulation has to run long enough to capture the effects of the ion

motion. This situation requires the recent advances in computer calculation speed

to have an effective simulation tool.

There is an additional difficulty, however, with the basic validity of the

simulation itself. For cases with attractive particles, it is discovered that there are

theoretical difficulties with modeling an electron-ion interaction with a classical

interaction energy. For the ions, and especially the radiator impurity, the attractive

interactions cause the electrons to approach very closely. Using a bare Coulomb

potential for the radiator-electron interaction causes a singularity to appear in the

ensemble. For this classical case, the partition function includes Boltzmann factors

with the form
e-3H ~ = e6/r (4.8)


where Z, is the ion charge, and r is the separation from the electron and ion. For

small r, this factor diverges, leading to a singularity in the Gibbs ensemble. This

reflects the fact that there is no classical limit for a plasma of oppositely charge

particles. It has been shown that systems with attractive Coulomb interactions








are unstable unless quantum mechanics is used [33]. The result of this is that MD

seems to not be a valid tool if bare Coulomb interactions are used for oppositely

charged particles, as a collapse of electron-ion pairs is the inevitable result.

Recently a method has been used to resolve this difficulty of using an inher-

ently classical method to give quantum results. To this end, the potential between

electrons and ions is modeled by a form that does not diverge for small distances,

but gives the Coulomb potential at larger distances. These are called ;,! i.. 1

Coulomb potentials. To understand this procedure, consider the quantum partition

function for two particles [9, 10]

Z =Tr e- 3H(1,2) dri dr2 rr2 3H(1,2) 1 rr2) (4.9)
2(4.9)

where H(1, 2) = H(1) + H(2) + V(1, 2) is the two particle interaction. A classical

expression for the partition function is then written down which defines the

regularized potential

Z Tr e-3H(1,2) d drl 2 e-U(|i-r2) (4.10)

where A = 2V h2/mkBT. This defines the regularized potential to be exactly


U(Irl r2a) -- In (A2 rl, r2 e- H(1,2) l, r2)) (4.11)

This potential resolves the singularity problem, as it is finite even in the limit of

Irl T2 -i 0. It also agrees with the Coulomb potential for distances farther
away from the ion. This characteristic distance is on the order of the de Broglie

wavelength AD = 27h/p. The results are somewhat sensitive to the specific distance

used, and later in this chapter we will discuss some recent work that shows the

optimum distance to use for certain cases.

With the regularized potential defined in this way, the expectation value of

a quantum operator that depends only on the coordinates of two particles can be








given exactly by using this classical potential along with the classical form of the

operator

(A Tr e-3H(,2)A(r, r2) d62Z dr2 e-_U(rI-r)A(rl, 1 2) (4.12)


Because this last equation is exact, the regularized potential must be accounting

for all of the quantum effects. Therefore, MD simulations for many particles using

these potentials will include quantum effects to some degree, but will also be a

well-defined classical problem.

At this point, we must examine these results and determine how 'correct' they

are, i.e. in what regimes do classical calculations using regularized potentials agree

with full quantum calculations. In simulations involving hydrogen plasmas, it is

found that good results are obtained for high temperatures. As the temperature

lowers, hydrogen atoms appear that are also described well [34, 35]. However,

at even lower temperatures, regularized potentials do not adequately describe

molecular hydrogen. These results agree with the primary assumption made

in using regularized potentials, that quantum effects are most important with

pairwise interactions. This holds true for the high temperature plasma and atomic

hydrogen systems. At very low temperatures many-body quantum effects become

more important. In this study, the system considered is a hot, dense plasma, and

MD simulations with regularized potentials would appear to be a good choice for

simulations.

A form for the regularized potential still has to be found from Eq. (4.11).

There are several different methods for evaluating this. In Appendix G one

method is shown that results in the Kelbg potential for an arbitrary potential

V(|ri r2 )

UK(r) dx -(/) V(Irl r2 x) (4.13)
iXJ *








In this form it is straightforward to see that the regularized potential can be

interpreted as a smoothing of the bare potential over a volume characterized by the

de Broglie wavelength. For the specific case of the Coulomb interactions, the Kelbg

potential takes the form [36, 37, 38, 39, 40]:


UK(r) V(r) ( e(r 2 + 1 [ erf ()} (4.14)

A second form for the regularized potential is the Deutsch potential [41]. This is a

phenomenological version that is often used in MD simulations and has the form


UD(r) V(r) ( e- ) (4.15)

Both the Kelbg and Deutsch potentials have similar qualitative features; however,

inside the de Broglie wavelength the Kelbg potential is preferable.

One additional factor must be evaluated. The plasmas considered here are

highly charged. This has the potential for further difficulties as the electron

trajectories are pulled closer, and there is the possibility for metastable bound

states. More study needs to take place to determine the effects for a given radiator

charge, temperature, and electron density.

With the use of these regularized potentials, MD simulations can calculate

dynamic properties for a plasma. For example, a simple theory might have the

radiator-plasma interaction be totally described by the monopole and dipole

interaction. In this case, the time dependent electric field of the plasma is required.

By using MD to determine the electric field, all correlations between the plasma

particles can be retained. The next chapter will look at a specific plasma to

determine the properties needed that will give the line shape function. It will be

found that some of these properties needed include the constrained average of the

electron density, and the electron field autocorrelation function. With statistical

mechanics methods, uncontrolled approximations are required to evaluate these






29


properties. MD simulations can be shown to be valid for many cases, and they can

provide accurate methods to determine these properties.













CHAPTER 5
LINE SHAPE FORMULA FOR QUASI-STATIC IONS

In C'!I ipter 3, an expression for the line shape function was given, and in

('!C ipter 4, a numerical method to evaluate certain properties important to the

line shape was discussed. In this chapter and the next, statistical mechanics will

be used to reformulate the problem in terms of those few properties that are most

important in determining the line shape function. Recall that in C'! lpter 3, the line

shape was written in the form


( (t) K)dk () (5.1)

Now an expression for this will be derived that separates the various mechanisms

affecting the line shape. Consider first the physical aspects of the plasma particles.

The electron and ion components of the plasma affect the dipole in different v--v

due to their different charges and masses. For positively charged radiators (the only

case considered here) the ions are repelled by the radiator and their dominant effect

is Stark broadening by their electric field coupled to the dipole of the radiator.

In many cases of physical interest, the change in this field is small during the

radiation time (relaxation of I(t)). Under such conditions the ions behave as static

perturbers and this will be referred to as the quasi-static ion approximation. The

quasi-static ion approximation is physically relevant because of the large mass of

the plasma ions as the example in ('C! lter 2 illustrates. It is important to note

that the quasi-static ion case does not indicate that the ions are motionless, only

that the effect of their motion on the forces of interaction is negligible.

We want to isolate the effects of the ions from the other processes of the

plasma. Shortly we will examine under what conditions this is useful. Going to








an interaction representation in which the effect of the ion motion (including the

radiator) is separated leads to


CiHT CiKTU() (5.2)


where U(t) is the desired propagator for the plasma that includes all processes

except ion (including radiator ion) motion. This propagator obeys the following

equation of motion


(O< + iH. + iV({q, + vjt}, {q}, {q}))U(t) = 0 (5.3)

We can now examine the conditions under which the ion motion effectively

decouples from the other degrees of freedom of the plasma. The important quantity

to consider is the fractional change in the ion and electron interaction energy

caused by ion motion during the characteristic time under consideration. The

fractional change is AV/V, with a differential change given by dV = (VV) (v)dt.

With an average speed of vo and a characteristic time r, the conditions required

have
rvo VV({qiJ, {qeJ) Tvo
(- < 1 (5.4)
V({qi}, {Jq}) ro
Under these conditions, the kinetic energy for the ions commutes with the

other terms, and the equation above can be written

eiHt eiKteiHt (5.5)


where

H, = H, + K + Ve({qe}) + IV({qi}, {q,}) + UpR (5.6)

The contribution VI~({qj}, {e}, {qb}) denotes the total interaction of both bound

and free electrons with the ions. The ion-ion potential cancels in d(t) for this

static-ion approximation (but not in the Gibbs density matrix), and H, becomes

the Hamiltonian for electrons in the frozen field of the ions.








A remark here about the quasi-static case and the absence of certain cor-

relations is in order. The introduction stated that this work would neglect no

correlations. Yet clearly correlations between the radiator center of mass and other

plasma particles are absent in going from the general case to the quasi-static ion

approximation. There is no discrepancy here, however. It is the time scale of the

quasi-static ion approximation, not an approximation made in the calculation of

the quasi-static ion case, which causes the lack of these correlations. Therefore

there will be no need at the end of the calculation to introduce any theoreti-

cal device to correct the result. All correlations present in the quasi-static ion

approximation are included.

We can now decompose the average for the autocorrelation function into a

product form, consisting of a static dipole autocorrelation function and a doppler

term, resulting in


d dk(t)) (d d(t)) eik VRt (=d* d(t)) J dv3/2 e /2ikvt (5.7)


= (d d(t)) }(k, t) (5.8)

where K(k, t) is the time dependent Doppler line shape. The frequency dependent

line shape can be written as a convolution of a doppler line shape and a quasi-

static ion line shape[42].


I(w) = d'ID ( /')J(/w) (5.9)


where
1(w (w ')2
ID () exp (5.10)
2V2 T( 2a2

2 11 /2kBT (5.11)
c m, )








and
OO
J(w) = Re dt ei (d d(t)) (5.12)
7 J
0
The quasi-static ion dipole autocorrelation function is now


(d d(t)) = (d(-t) d) = Tr Trpd(-t) d (5.13)
a p

and the plasma-atomic Hamiltonian is


UpR({qi}, J{qe, qb}) = E({q}, ,{q d) -

+ d ddx' pe(X)pb(X') (W(x, x') .- ) (5.14)


This form in Eq. (5.13) for the autocorrelation function still has a difficult

ii ii v-body problem in performing the trace over the plasma degrees of freedom.

Because the ions are taken to be quasi-static their effect is easier to analyze.

First, the large mass of the ions not only justifies the quasi-static approximation

but also implies a quasi-classical approximation for the conditions of hot dense

matter considered here. This means the ion thermal de Broglie wavelength A

2/h2/mrkBT is short compared to the distance between ions. Therefore, the trace

over ion degrees of freedom can be converted to an integral over configurations for

the ions. Furthermore, as was discussed in C'! lpter 3, the large separation between

the radiator and plasma ions leads to the dominant ion-radiator interaction being

the coupling of the ion electric field to the radiator dipole. In other words, the only

relevant property of the ions is their electric field value at the radiator. Finally, for

the quasi-static ions the ion electric field is constant during the radiator time.

In performing the integration over ion configurations it is recognized that

effect of the plasma ions on the line shape is determined by the value of their

instantaneous electric field. Also, many points in phase space correspond to

ion arrangements with the same electric field at the radiator. This si-l-:. -1- the

following conceptual method to calculate the trace. For each possible value of








the ion electric field, -v- e, pick out the group of all those configuration points

which have their ion microfield equal to e at the radiator. Then average the dipole

operator over just that group of points and then perform a weighted average over

all possible ion field values. This gives


Trpd(-t) = dc Trp6(e Es)d(-t) (5.15)
p J P

-= dc f (a, )D -t) (5.16)

which defines a primary quantity to consider, the constrained average dipole

operator with an ion electric field constraint Es = e at the radiator


D(, -t) = Trp(a, e)d(-t) (5.17)
P

p(a, ) =f-l(a, )pb( Es) (5.18)

f(a, e) = Trp( Es) (5.19)
p

The quantity Es is the screened ion field at the radiator site. However, at this

point in the study the form of the screening is arbitrary. Later in the analysis a

particular form will be chosen that simplifies the calculations.

With the average over plasma states ahv-- i constrained, the interaction

Hamiltonian can be redefined in such a way that it does not depend on the ion

degrees of freedom. The Hamiltonian for this system is


H = H+ HP + UpR (5.20)


The ions interact with the atomic system solely through a dipole interaction. To

explicitly extract these effects, we add and subtract a term based on the screened

ion field Es from the Hamiltonian


H= Hp+HR+E, d+ UpR Es, d


(5.21)








Here, the screened ion field Es only depends on the ion coordinates. However,

the specific form is still not defined at this point, since the form will be chosen to

simplify the analysis later. Now in the calculation of D(e, -t), the Hamiltonian

alv-i-, appears in a constrained average, where the trace is restricted by a delta

function to only include terms in which the screened ion-field is equal to some given

value e. In the quasi-static ion approximation, the plasma dynamics do not affect

this restriction, so the Hamiltonian has the following useful form


H =Hp+ HR(e) + 6UpR (5.22)


where HR(c) = HR + e d is the Hamiltonian for a radiator in an external field e,

and UpR = UpR E8 d is the interaction between the plasma and the atomic

system in this new Hamiltonian. The new interaction JUpR is given by


6UpR({q,}, {qj, qb}) = (E({q}, {q} ) E({qJ})) d (5.23)

+ Jdx dx'p(x)pb(x') (W(x,x') -x. x'


A suitable choice of the screened ion field removes all ion dependence from JUpR

and gives a form convenient for analysis. The static-ion line shape is then
OO
J(Uc) 1= Re fdt eiwt de Trf(a, e)D(e, -t) d (5.24)
7 JJ a
0

Note that this has the form of an effective atomic physics problem. Here all of

the effects from the plasma degrees of freedom have been localized in the quantity

D(e, -t). Once D(e, -t) has been evaluated, the remaining dynamics are that of

an atom in an external field e. This problem is well studied, and many results,

especially atomic wavefunctions and various matrix elements, are available [17].

The next chapter will evaluate the equation of motion for D(e, -t), and use

perturbation theory to calculate various plasma effects to second order in the

plasma-radiator interaction.













CHAPTER 6
QUASI-STATIC EQUATION OF MOTION

The previous chapter derived an expression for the line shape in which the

most difficult many body calculations were contained within the constrained

average dipole operator D(e, -t). As was mentioned in C'!i pter 5, an equation

of motion for D(e, -t) will be derived that will give exact dynamics, including all

correlation effects, for the case of quasi-static ions. This equation of motion will

contain terms that can be identified with various broadening mechanisms for the

line shape.

The derivation is more straightforward with the introduction of a Liouville op-

erator [43]. A brief overview is given here, and more details are given in Appendix

D. A Liouville operator L can be associated with any Hamiltonian operator H. For

some quantity X whose time evolution is determined by the Hamiltonian H, the

associated Liouville operator is given by


X(t) = eiHt/hX(O)e-iHt/ = LtX(O) (6.1)


or equivalently


LX =-[H,X] and + L X =0. (6.2)


With the Hamiltonian separated into three terms (see Eq. (5.22)), the Liouville

operator associated with the entire system separates into three analogous terms, or


L Lp + L(a, c) + 6LR (6.3)








Using a projection operator technique detailed in Appendix D and this form of the

Liouville operator, the time evolution equation for D(e, t) takes the form


+ L(a, e) + B(c) D(e, -t) + dr M(e; t T)D(e, -r) 0 (6.4)
0

The operators B(e) and M(e, -t) represent the effects of the coupling of the

radiation to the plasma 6LpR. Since the static ion broadening has already been

accounted for in L(a, e) it is expected that these are electron broadening terms.

However, these electron terms retain their correlations with the ions and a depen-

dence on the fixed ion value. The operator B(e) is simply the mean value of 6LpR

for the constrained plasma


B(e)X = Trp(e)6LpRX = Trp() [6UpR, X] (6.5)
p P

It is seen from the equation of motion that B(e) describes the short time dynamics

of D(c, -t). The operator M(e; -t) describes the dynamical electron broadening

and is more complex. Its detailed form is given in Appendix D. These quantities

can be expanded in powers of 6LpR as


B(c) = B(1(c) + B(2)(c) +

M(e,t) = M(2(c,t)+.


In the remainder of this chapter these electron broadening operators are brought

to a more practical form by an expansion to second order in 6LpR.

6.1 First-order Electron Broadening

The result for the electron broadening B(e) contains the Liouville operator

6LpR or interaction Hamiltonian 6UpR. In this section, this will be expanded in

powers of the interaction energy of the plasma and radiator internal states. As

described above, the removal of the monopole and ion dipole terms for JUpR results

in a softening of the interaction so that an expansion in powers of JUpR is valid.








This will give a form of the first order electron broadening ready for calculation,

and also allows us to show that this term arises solely from the effect of the plasma

electrons on the radiator.

The evaluation of this trace occurring in B(c) is complicated by the presence of

6UpR in the density matrix p(e). However, since the interaction Hamiltonian SUpR

itself occurs in the trace, the first order part is found by writing the density matrix

to zeroth order in 6UpR. The first order contribution to B(E) is then

B (e) = Trp(e)6Lp RX Trp,(e) [6UpR,X] = [(6UpR) ,X] (6.6)
p h

where

(X), Trpp,()X (6.7)
P
is the constrained plasma average. In this formula, we have


pp(C) Q-l()pp6( Es) pp = (6.8)
Tre- 3Hp

where pp is the plasma equilibrium density matrix and pp(c) is the constrained

density matrix.

Up to this point the definition of Es has not been specified. It will now be

shown that a suitable definition of Es will remove all ion dependency from the

average interaction Hamiltonian. To get this result, the electric field will be defined

to be the field of the ions screened by the electrons. First, the form of SUpR is
rewritten so that the dipole contribution for distant electrons is also made explicit

6UpR (E E) d- f dx dx' pe(X)pb(x') W(xa, x') (x ro) (6.9)

where d = Y: ers and

E = E + E, = E + dx pe,(x) (X ro) (6.10)
/ x 3








where Ei is the ion field and E, is the electron field. The constant ro is chosen to

be a characteristic size of the bound charge distribution outside of which the dipole

interaction is well-defined. The screened ion field E8 is now defined such that this

dipole contribution from JUpR gives no contribution to B(1)(e)


Trppp()(E- E)= Trppp(e)(E, + E- E,) (6.11)

Trip4(e)(E, + p~-TreppE E) (6.12)


In the second equality use has been made of the fact that the screened ion field and

the total ion field have no dependence on the electron coordinates, so the trace over

the electron coordinates can be carried out. Also, pi is the reduced density operator

for the ions and pi(c) is the corresponding constrained operator


Pi Trepp pi(C) Q-1()p6(c E,) (6.13)

The definition of Es is now chosen such that this dipole contribution is zero


Es E, + p~TrppE, (6.14)

This is a primary result of our general analysis that includes all charge

correlations. As noted above, the primary effect of the static ions is a Stark

broadening of the line. This is described by L(a, c) in Eq. (6.3) above. The field

values c are sampled from the probability distribution Q(e). The probability

distribution is defined here as an average over the two component plasma, in

contrast to current theories which approximate the effect of electrons on the ions

using a One Component Plasma (OCP) of ions only. In addition, however, we now

see that the choice of microscopic ion field Es whose probability is being computed

in Q(c) also must include the effects of the electrons in order that L(a, c) should

give all of the ion Stark broadening (i.e., that there be no additional effects from

B(1) ()). The cancellation of this additional ion broadening leads to two effects.








First, the relevant fields of the ions at the radiator are screened (below it is shown

that the second term of Eq. (6.11) provides Debye screening in the weak coupling

limit). Secondly, the dipole interaction of the electrons is entirely accounted for by

this screening effect. The definition of the screened ion field in this theory includes

all plasma correlations. Therefore, the claim that it accounts for all of the dipole

interaction between the plasma particles and the internal radiator states is an exact

claim for quasi-static ions.

The remaining non-zero contributions to B(1) () comes from the second term

on the right of Eq. (6.9). This represents the interaction of the perturbing electrons

with the bound electron distribution, now with both the monopole and dipole

interactions separated out. Using this, the nonzero part of B(1)(e) takes the form

B(1c (')X= dx dx' W(x, x') (pe,(x)) [pb(x'),X] (6.15)


where
( 1 1 x o( )) (6.16)
W(x, x') =( ) (6.16)

and (pe(x)), indicates a constrained average of the free electron charge density, and

the theta function is defined by

0 ro > x
O(x ro) = (6.17)
1 x > ro

This form of W(x, x') will be used again for the second order parts of the

calculation. However, let us now expand the first term in the form factor W(x, x')

using a multiple expansion, and keep just the monopole term. The monopole

term is an important piece to study, because in many theories, the assumptions

made lead to all other terms going to zero. This term is due solely to the monopole

interaction arising from free electrons penetrating the bound electron orbitals

(x < x' < ro). The form factor then becomes (including the factor of x2x'2 from the








radial integrals)

W(x, x') -xx'(x- x')0(x' x) (6.18)

The theta function will change the upper limit on the integration over x. Also,

the integration over x' is effectively limited by the bound electron wavefunctions.

To represent this, they will be given an upper limit of some characteristic length

related to the Bohr radius. The angular integrals can be absorbed in the charge

density calculations, giving


B((e)X i47 dx' dx (p(x); ) [pb(x'),X] xx'(x x') (6.19)
0 0

(P W(x);e } ( dQ p (p, (x); (6.20)

Pb(X) J d6 pb(x') (6.21)

For the monopole term of the first order electron broadening, the effects of

including all plasma correlations exactly has been localized in the term (pe(x); e)p.

This term represents the free electron charge density, averaged over those plasma

states in which the ion electric field Es is constrained to have the value e. This

free electron charge density is not uniform because the charged radiator at the

origin attracts the perturbing electrons. Furthermore, it is not isotropic because of

the constrained average: with a specified field e, the ion density will be larger on

one side of the radiator than on the other. The fast moving electrons would then

be attracted to the more positive side, leading to an .,-mmetry in the free electron

charge density [44].

The importance of these effects can be explored with a simple estimate

calculation. If it is assumed that the perturbing electron charge density pe(x) is

constant inside the bound electron wavefunction, pe(x) can be removed from the








integral and the integration over x performed, leading to



0
B ( pe(Oe} dx'4 [pb(X'),X] (6.22)



6.2 Second-Order Static Electron Broadening Term

At this point it is useful to more closely examine how the choice of the

ion microfield form has affected the physical interpretation of B(e) as a static

electron broadening term. Note that initially the plasma-radiator interaction

6UpR represented both electron and ion perturbations, with both the monopole

interaction and a ion dipole interaction Es d removed. With the above choice

for Es and the neglect of higher order ion multipoles the radiator-plasma coupling

energy can be written


sUpR = E, d dx dx' p,(x)W(x,X')bX'pb ), (6.23)


E, = E, (E,), (6.24)

where (E,), is the electron field averaged over the electron degrees of freedom


(E,), p7 TrppE, (6.25)


Thus, the first term of the interaction is just the fluctuation of the electron dipole

interaction whose average value in B(1)(E() is zero by definition. Clearly, 6UpR now

represents an electron perturbation of the radiator. Consequently, both B(e) and

M(e; t) can now be recognized as electron broadening operators (although still

retaining the correlations between electrons and ions).

The previous section examined that part of B(e) that was first order in the

interaction Liouville operator. The remainder, B(c) B(1) (), can be expanded into

a term that is second order in the interaction Liouville operator. The full static








broadening term explicitly is


B(e)X = Trp'(a, e) 6LRX (6.26)
P

and the density matrix is given by


p(c) = f-(a, )pb(c Es), f(a, ) = TrpS(e Es) (6.27)
P

p =Z e-3(HR()+Hp+6UpR) (6.28)

where Z is the associated partition function. Define the quantity Ho = HRn() + Hp.

Then, to first order in 6UpR, this density matrix is


p(e) pp(e) 1 dr (eHO p6R) e-'HO Trpp(c)eHo (6UpR) e (6.29)
0 P

The first term leads to the first order static shift as described in the previous

section. The second term gives a second order contribution of

13
B(2) (eX = dr Trpp()6Upn(-ihr) (6UpR (UpR; c) X] (6.30)
0

where UpR(-ih-r) denotes the Heisenberg operator at an imaginary time.

It is expected that the dominant contribution to B(2)(E) should come from the

electron dipole interaction of 6UpR. Then B(2)(E) simplifies to

13
B(2)(c)X = dr (6E (- ihr)6Ee; c d(-ihr) [d, X] (6.31)
0

This form includes all correlations present in the quasi-static ion case. Also, in

this form the atomic and plasma physics have been separated. The electric field

autocorrelation function is a plasma physics problem and is solved separately from

the atomic dipole operators.








6.3 Dynamic Electron Broadening Term

The previous sections have dealt with the term B(e), which describes the

static broadening by the plasma electrons. The dynamical electron effects are

described by the operator AM(e; t). In Appendix D, this operator is calculated and

shown to have the form


M(e; t) f-'(a)Tr [p(a, e) (6LpR) e-QtQ (6LpR)] (6.32)

Here Q is a projection operator such that QX gives the deviation of X from its

constrained plasma average.


QX = X Trp(c)X (6.33)
P

The previous section accounted for the average value of 6LpR and its time

independent fluctuations to second order. What is left is the time-dependent

fluctuations, which verifies the previous identification of AM(; t) as a dynamic

electron broadening term. More physically, AM(; t) describes atomic transitions of

the radiator caused by collisions with the plasma electrons.

To second order in 6LpR this operator becomes (see Appendix D for details)


M (2)(, t) = 6LpRC (LR(e+L) (J6LpR (6LpR; c) P (6.34)


This form of the dynamic electron broadening operator is similar to those obtained

previously. Typically, however, such results neglect many or all of the electron-

radiator-ion correlations. These correlations enter the broadening term in two

places. The definition of Es itself included some of these correlations in that

the correlations lead to screening of the ion microfield, and are included only

approximately in most previous theories. However, the term above also includes

all correlations when performing the trace. Instead of performing the trace over







separate ion and electron subsystems, the trace above is over a two-component
plasma including the radiator monopole.
This result can be made more explicit using the form for the perturbation

UpR 6E d- dx dx' pe(x)W(x,x')pb(x') (6.35)

to get the action of M(c, -t) on some radiator operator. Clearly, it will be deter-
mined by the time dependent correlations between 6Ee and pe(x)

e(L(x)tM(, x-t'X = CPt, e)2 [d, )(t) [d X]



[2 b b 1),X]]

+ dxdx'W(xx')Dox,t,c) ) [dJ(t), [pb(x',X]]

Sd Dx (x, -t, c) J dx'W(x, x') ) [pb(', t)), [do,X]]

where the time dependence of the bound electron density and dipole are given by
that for the free radiator in the Stark field c

pb(x', -t) = -(Le ( )tPb (), d (-t) = -(Lun ()t

The plasma properties are contained in the correlation functions

S(x, Xl, t' C) P(', t) pe(XI) (pe(ai);E)p) ; e

D,(x, t, e) = 6Ec,(t) (pe(X) (pe(X;E));e) ;e

Cop(t, e)= (6E,,(t)6Eep; E),














CHAPTER 7
CHARGE CORRELATION EFFECTS ON PLASMA PROPERTIES

In the previous chapters, a formulation for the spectral line shape has been

derived to second order in the plasma-radiator interaction potential, and keeping

the charge correlations between the components of the plasma. We now turn to

examining the effects that retaining the charge correlations has on various plasma

properties. Here the focus is on studying how these properties can be calculated,

and giving estimates as to their values. We consider the plasma with the semi-

classical representation from C!i lpter 4, and use classical statistical mechanics.

Quantities of interest include charge densities near the radiator, pair correla-

tions among perturbing ions and electrons, the screened ion field, the microfield

distribution, the reduced density operators for the electrons and ions, and the time

correlation functions for the electron charge density and electric field near the

radiator. The estimates also provide guidance for more accurate MD simulations as

the ultimate test of their range of validity.

7.1 Plasma Structure

We will first consider the time-independent structural properties of the plasma.

The plasma is considered to be a two-component plasma with ions of charge Z and

electrons. In addition, a dilute concentration of another species will be introduced

as the radiator. As the equations are derived, the concentration of the radiator

will be set to zero, reflecting our model of a single impurity ion radiator. To study

the structure of the plasma, an appropriate starting point is the pair correlation

function g,,a(r), or the simply related quantity h,a(r) = ga,a(r) 1[45, 46]. The

function h,,s(r) is a measure of the total correlation between two particles of type

a and /, and has the behavior h,,3 -i 0 for large r. The total correlation includes







the direct influence from one particle to another, plus the indirect influence caused
by a other participating particles. A simpler quantity, which emphasizes the direct
influence between particles, is the direct correlation function co,a(r) defined by the
Ornstein-Zernicke equation[45, 46]

h,,a(r) cr,(r) + n7 J dr' (c,,,(r r'\)h-,(r') (7.1)

where n, is the density of particles of type 7. Another relation in terms of these
values can be found from the hypernetted chain (HNC) approximation[45, 46].
This approximation is found to work well for ionic systems. The hypernetted chain
approximation results from a specific approximation in terms of ca,p and ha,beta

ha,(r) -1 + exp [-pVa,(r) + hc,a3(r) c,,(r)] (7.2)

or, using the relation h,p(r) = g,a(r) 1 and Eq. (7.1)we get

ln g~(r) -Va3(r) + n j dr'c( r -r' |) (r') 1) (7.3)

The two equations Eq. (7.1) and Eq. (7.3) provide the basis for evaluating the
pair correlation functions between the various plasma constituents. To proceed, we
introduce the "potential of mean force" by defining

In9 Sg(r) = -3UaP(r), (7.4)

Then Eq. (7.3) immediately becomes a nonlinear integral equation for Up(r)

Uc,(r) = V,(r) -/3-1 jn dr'ca ( r r' |) (e-U(r') 1) (7.5)

These results are quite general. These equations represent the correlation functions
for a two-component plasma of ions and electrons, with no impurity radiators
present. The presence of the monopole of charge number ZR from the radiator does








not change these equations for the electron and ion correlations within the plasma

away from the radiator because nR = 0.

However, we are interested also in the structure of the ion and electron

configuration about the radiator. The equations for the densities of electrons

and ions about the radiator do couple to these results. We first specialize the

above results for the case of a radiator correlated with some other particle. The

Ornstein-Zernicke equations for the correlation function between the radiator and a

particle

hcR(r) = ca(r) + Z n J dr'c%(| r r' |)hR(r') (7.6)

and the corresponding HNC equations can be written


In gR(r) = -/U3R(r), (7.7)


UaR(r) VaR(r) -1> / dr'co(I r r' ) (e-3U(r') 1) (7.8)

Here a and 7 refer to an electron or ion of the two component plasma. The

plasma direct correlation function co,(r) is provided independently from the two

component plasma HNC equations.

7.1.1 Linear Theory

We now turn to an evaluation of the mean potential Uo,R(r). First, we

examine the results of the preceding section in the regime of weak coupling. Then

the relation of the pair correlation function and the potential of mean force Ua,R

from Eq. (7.4) becomes


gaR(r) = e-U. -, 1 /3UR (7.9)

Also, when Eq. (7.2) is linearized, the result is


Ca-y V V.1


(7.10)







and the linear HNC equations for UJR in Eq. (7.8) become

UR(r) V R(r) + n7 J dr'c,(I r r' 1)UR(r') (7.11)

In particular, for the electron-radiator mean potential, Eq.(7.11) is

UeR(r) VeR(r) + ne dr'cee( r- r' 1)UeR(r')+ ni dr'cei(I r r' |)UR(r') (7.12)

Fourier transformation gives

(k) ( k) + nie(k)UFR(k)
UeR([1 n (k)(713)

and the same procedure for the ion radiator mean potential is

(R(k) k) + n (k)U(k)4)
[1 nijCi(k)

Solving these equations gives

(feR(k) (1 nih(k)) + niea(k)VR(k))
Ue -() = (7.15)
(1 necee,,(k) nicii(k) + nnie (c,,ee(k)a(k) Cei(k)Cei(k)))
(fR(k) (1 ne-ee(k)) + nei(k)VR(k))
Uin(k~) =(7.16)

A dimensionless form for c,, is obtained by scaling the distance with respect to the
average electron spacing given by 47rr/3 = 1/ne. Then

fe2


Here V* is the dimensionless pair potential and F is the plasma coupling constant
taken to be small here. In that case Eq. (7.15) and Eq. (7.16) can be simplified to

Vo R( k)
UJR(k) = V (k) = 1 na(k) ni(k) (7.17)

Here c(k) is the dielectric function for a weakly coupled two component plasma.








To be more specific it is necessary to choose the form for the potential VaR(r).
A simple and convenient form is the Deutsch potential (see Appendix G)


V(r}) Ze ) '(7.18)

where Za carries the sign as well as the magnitude of the charge number. The
Fourier transforms are

V7,(k) 47wZ~Z e2k-2 (k2A + )-1 (7.19)

(k) = 1 + 3F(1 + Zj) (kro)-2 (7.20)

UCR(k) = 4rZZ~e2 (k2 + ) ((kro)2 + 3F (1 + Z,)) (7.21)

and the inverse Fourier transform gives the desired screened potential

/3UR(r) ZaZ7F (1 3F (1 + Zi) 62) -1 x- 1-\ 1 -Ic)x (7.22)

x = r/ro, 657 = A7/ro (7.23)

7.1.2 Nonlinear model

The last subsection calculated the effective potential to determine the charge
densities around the radiator at weak coupling

gaR(r) -1 /3UaR(r) (7.24)

with UaR(r) given by (7.23) from the linearized equations. An estimate for stronger
coupling is obtained by exponentiating this result

gaR(r) e-3UCR(r) (7.25)

with the same form for UaR(r). This is an uncontrolled approximation but is
confirmed to be qualitatively correct even at strong coupling by comparison with
MD simulation.












Z=8,
r=0.1, 6=.4


C *L A HNC
S-- NLD
----LD
o MD
MD
2


0 1 2 3 4
r



Figure 7 1: Plasma Electron-Radiator Pair Correlation Function Com-
parison using the hypernetted chain approximation (HNC), the nonlinear Debye
model (NLD), the linear Debye model (LD), and with molecular dynamics (M!1))
results. [47]


In Fig.(7.1.2), results for geR(r) are shown arising from HNC, non-linear

Debye, and Debye theories for an idealized plasma of electrons in a uniform positive

background. Also results from MD simulations are shown[47]. This figure shows

that at Z = 8, the non-linear Debye and HNC results agree with each other and

with MD results. Figure (7.1.2) shows the same quantity for differing values of

Z, using the non-linear Debye model and HNC. With increasing Z, the electron

density at the radiator is enhanced. Note that the agreement is much better at

lower Z. At Z = 30, there is considerable difference between the two results.

However, the functional form of the HNC results, even at this higher Z remains the

same as the non-linear Debye form. If the charge is taken to be a fitting parameter,

good agreement is found between the HNC results with Z = 30 and the non-linear

Debye results with Z = 25.




















1=0.1 and 8=0.4
Z=1,4,8,20,30


Z=30


10- o HNC
NL









0 1 2 3 4
r



Figure 7 2: Plasma Electron-Radiator Pair Correlation Function Compar-
ison of results from the hypernetted chain (HNC) approximation and nonlinear
Debye (NLD) model for several radiator charges.








7.2 Screened Ion Field

The ion microfield distribution is defined by[44]

Q(e) (6( E,)) (27)-3 iJ dA e iA.E) (7.26)

The brackets denote an equilibrium ensemble average for the plasma consisting of

the electrons, ions, and radiator monopole. The screened field is defined by (see Eq.

(6.14))
E E + fdppE (7.27)
f dFpp
where dF, denotes an average over the phase space for the electrons. The momen-

tum dependence can be integrated out leaving

f dql,..dqN e-C(ve+Vi++VR)Ee
E = Ei +
f dql,..dqNee-3(Vee+Vi+VeR)

SE, + J dqlee (qie) n, (qie {qj}). (7.28)

All coordinates have been taken relative to the radiator which can be assumed at

the origin. The electron density n, (qle I{qi}) is the number of electrons around

the radiator for a given configuration of ions

N f dq2e..dqNe6-e3(Vee+VKe+Ven)
n, (q {qi}J) f dq e .dqNvee,3(ee+V+v) (7.29)

If the ion-electron interaction is neglected then the second term of (7.28) van-

ishes because n, (qle I{qi}) becomes spherically symmetric. This many-particle

correlation function can be obtained from a generalization of Eq. (7.8)


ne (qij I{qi) = ne exp (-/U (qle I {qj)) (7.30)

U(r) Ve(r) + Vie (Ir r,|) -i, J dr'cee( r r' |) (e-t ') ) (7.31)
J k








For weak coupling, we apply an analysis similar to that from above that leads
to
VeR(k) + VI (k) kE t "
Ue(k) e() e(k) = 1 neee(k) (7.32)

E, E,j + dqleee (qle) n' (qe) (Ce-3U(lj-qie ) 1 (7.33)

Ue' (r) (2x)-3 Idke-ikr vs (k) k
JU(k(r)- de (7.34)
Se ( k)()

(ql,) nee U, (r) = (27) -3 e-ikr (7.35)

To linear order in UR (r)

E, = (e (qj) -3 dqle, (qle) e (qle) Uj (I q1- q)) (7.36)

S ees (qi) (7.37)

The screened field E, in this weak coupling approximation is the sum of single
particle screened fields

eis (qi) e (qi) --3 dqleee (qle) We (qle) Ule (Iqie qil)

This electron screening of the ion field is enhanced by the presence of the radiator,
due to the factor of n' (ql). If in addition n' (ql) is linearized then the usual

electron Debye screened field is recovered.
7.3 Ion Microfield Distribution

If the screened field is taken to be a sum of single particle fields

E, es (qi) (7.38)
a
then the expression in Eq. (7.26)

Q(c) (c Es)) (27)-3 dAe-i ~X eiAE) (7.39)
(7







can be written

Q(c) (27)- / dieX (i (1 + (qa)))

(27)-3 ie die- cG[ (7.40)

where
S(q) =eie(qi) -. (7.41)

A functional expansion of G[O] in powers of Q has as its leading term

G[QI] J dqi4( (qi) giR(qi) + orderQ2 (7.42)

This is the Baranger Mozer approximation for a two component plasma[48, 49,
50]. Note that the screening length for e8 (qi) is determined by the electrons while
that for gaR(qi) is for both ions and electrons.
7.4 Constrained Electron Density
The average electron density around the radiator, for given electric field value
c can be estimated in the following way. Consider again the electron distribution in
the presence of a given ion distribution as in (7.31)

U (ql {qi}) = VeR (ql) + Vie ( q9e q|)

i-n, J dr'ce(| r r' |) (e-(r') 1). (7.43)

The ion electron potential can be written in terms of the ion density

SVie (qie qi) J drVe (qie r|) i (r). (7.44)

Now, consider the ion density to be the average density for the given constraint
field c, ni (r) -- ni (r,c) = ion density for a given ion field at the radiator satisfying

I dr (VVe (|0 r)) n (r,c) (7.45)








Once the ion density is known the average electron charge density can be deter-

mined from the solution to (7.43)


U (qe | e) = VeR ( ) + J drVi (|e r|) n (r,e)

-/3- n dr'cee( r r' 1) (e-3U(r') 1 (7.46)

This can be solved by the above method of linearization followed by ad hoc

exponentiation.

The determination of the constrained ion charge density has been discussed

by Lado and Dufty, with practical methods available based on the above Baranger-

Mozer calculation of the microfield distribution[44].

7.5 Electric Field Covariance

The electric field autocorrelation function C(t) = (E E(t)) has a special

emphasis in this study. This integral of C(t) is directly related to several transport

properties, such as the low velocity stopping power S, the friction coefficient ,

and the self-diffusion coefficient D, as well as the impact (fast fluctuation) limit for

spectral line broadening by electrons[47]. The transport properties are related by

[47]
mo 1 S(v) o Z2r4 fdt C(t) (7.47)
tip
0
As a first step towards the evaluation of the autocorrelation function, we evaluate

the covariance C(O). The electric field covariance in dimensionless form is defined

by

C(0) (E E) (7.48)
C2
where E is the sum of electron microfields at the radiator site. A direct calculation

can be performed in terms of the one and two electron charge densities

C(0) = 1 dr e(r) ne(r)e(r) + dr' ne(r, r')e(r')] (7.49)
C2 \








with
f drodrT2 N drNe-ou
nle (di) dN-3U = ne9eR (7.50)
f dro drNe-p3
f drodr3 drde-u
e(rT, r2) N(N 1) fdro dre-3U (7.51)
f dro drNCe- "
However, an expression can be obtained in terms of a mean force field emf defined

in terms of the potential of mean force in Eq.(7.4). Writing the covariance as

4 4
C(0) (V, Ui. E) r= (Vo E) (7.52)
Ze3 3Ze3
0 7 dr n (r) Vnne(r)) dr ne(r)emf(r) e(r)
PZe3 pze e

where the last expression defines the mean force field as

1 1
emf(r) V In ne (r,t) VUeR (7.53)
pZe Ze

Comparing Eq.(7.49) and Eq.(7.52) indicates that the two electron effects have

been included in the mean force field. Figure (7.3) shows how the covariance

increases monotonically with increasing Z for the idealized one component electron

plasma.

7.6 Dynamics

We now turn from structural properties to dynamical properties for the plasma

electrons in the presence of a highly charged radiator. For the conditions of interest

to this study, the plasma electrons are alv--i- weakly coupled. The nonlinear

Vlasov equation[26] is then valid and gives

+v V, + -(F, + e + F i) V) fe(r, v, t) 0 (7.54)
\ L- ^ '7-'e.


























15







10
0






5




10 20 30 40
Z

Figure 7-3: Electric Field Covariance Shown here as a function of Z. Tempera-
ture is 1000 eV, and density is 1024/cc.








where


FeR = rVeR (7.55)

Fee V r dr' VeR(r- r')ne(r',t) (7.56)
7 (7.56)
Fe V Jdr' Vi(r- r')nr', t) (7.57)


and the number density for species a is n, = f dv f,(r, t). The quantities

FeR, Fee, and Fe, are the forces calculated from the electron-radiator potential,

electron-electron correlations, and electron-ion correlations, respectively.

For a hydrogen plasma, the ions will be as weakly coupled as the ions due

to the identical charge. The coupling increases with ionic charge, so here we

assume that the plasma ions are hydrogen to ensure the Vlasov equation is still

valid. An ionic equation analogous to the electron formulation in Eq. (7.54) is

thus found. With these two equations and the semi-classical electron-radiator

interaction, equilibrium solutions to the distribution functions for the electrons

and ions can be determined. The equilibrium solutions are equivalent to those of

the HNC equations; therefore strong electron-radiator coupling is included in this

formulation.

Time correlation functions can be evaluated in the following manner. The

dynamics in this case is determined from the linear Vlasov equation, and the

time correlation function can be determined exactly in terms of the initial corre-

lations (determined from the equilibrium solution), the single particle dynamics

arising from potential of mean force for the electron around the radiator, and the

dynamical screening due to electron and ion correlations.

However, we propose a simpler method based on conclusions drawn from the

electric field autocorrelation function C(t) = (E(t) E). MD results for the C(t)

expose several relevant qualitative features[47]. Figures (7.3) and (7.4) show an

increase of the initial value due to correlations, a decrease in the correlation time,









1.0
MD
0.8- o Theory
r=0.1, c=0.25Z
06 Z=4,8,20,30
0.6


o 0.4 Z=4
Z=30 ,
0.2





-0.2
0.01 0.1 1
cot
P


Figure 7-4: Normalized Electric Field Autocorrelation This quantity is nor-
malized to an initial value of one. The decrease of the correlation time and the
increase in anticorrelation with increasing Z is observed.

and an increase of anticorrelation, all corresponding to an increase of Z. Figure

(7.5) di-l'1,i-v the total effect on the integral of C(t). The linear Vlasov equation

di-pl-,l- these three properties, but neglects the dynamic screening. Thus, for C(t),

the features of interest to us can be captured by calculating exactly the initial

correlations, and then using a single particle model for the dynamics.

We propose that this program can be extended to the plasma dynamics in

general and that a single particle model for the dynamics can be used. Then, the

initial correlations are calculated exactly from equilibrium conditions, and the


















0.8
.8- Area C(t)/C(0)

\ --- Areat C(t)


0.6-




S0.4-




0.2 E-




0.0
10 20 30 40
Z



Figure 7-5: Integral of Electric Field Autocorrelation Function The true
and normalized quantities, pi'.- il11y relevant because of their relation to transport
properties and line shapes, are shown. The conditions are identical to that of Fig.
(7.4).








dynamics following are that of a single particle in a self-consistent field, or


CAB(t) -- Jdr lp/.,.)a(r(t),p(t))b(r,p) (7.58)

where r(t) and p(t) are single particle trajectories.

7.7 Summary

In this chapter we have made estimates for several important structural and

dynamical properties for highly charged plasmas in conditions relevant to this

study. By considering the regularized Deutsch potential in the HNC approximation,

pair correlation function were determined and found to be in agreement with

results from the non-linear Debye model and from MD simulations. The ion

microfield distribution, which has long been a vital part of line shape theories

has been reformulated in a manner that includes ion and electron correlations

consistently. Finally, the electric field autocorrelation function C(t), related to

several interesting transport and radiative properties, is found to have several

important qualitative features. The integral of C(t) is modified qualitatively in

competing v-i as the radiator charge increases. We find that once the initial

correlations are dealt with exactly, the subsequent dynamics can be accounted for

by using one particle dynamics. This discovery is used as motivation for treating

dynamic correlation functions in general by the same procedure.

These estimates are intended to complement the results from MD simulations.

The MD results are expected to provide information as to the relevant domain of

these estimates. Then, once agreement has been reached between our estimates

and MD results, we can consider the physical basis behind the models used in this

chapter to provide insight into MD results.














CHAPTER 8
INCORPORATING ION DYNAMICS

The previous chapters have dealt with the calculation of the line shape using

the quasi-static ion approximation. This is sufficient for many lines of interest when

the time scales for changes in the ion field are larger than the decay time for the

dipole autocorrelation function. However, this relation does not ahlv-P- hold[3]. The

ion microfield at the radiator will then change enough during the radiation time to

affect the line shape. In this chapter, an analysis accounting for these ion dynamics

effects will be outlined and discussed. The starting point once again is the general

line shape formula written as a dipole autocorrelation function (Eq. (3.4))


I(t) = Trpdtdk(t) dk = de-ik (8.1)


Recall that in using the quasi-static ion approximation, we extracted the screened

ion field as the dominant interaction and separated the trace into separate traces

over the atomic and plasma subsystems. Once this separation was effected, the

trace over the plasma subsystem was constrained. It was not the entire phase

space for the plasma subsystem, but only over a surface of that phase space which

corresponded to those plasma states in which the screened ion field at the radiator

had some specified value. Now, when considering ion dynamics, we perform

the same separation, but we extract not a single property but a set of plasma

properties as being most important to treat exactly. The set we choose here is the

radiator position qR, radiator momentum p^R, and the screened ion microfield c.

Denoting this set by b = {qRpR, p }, we have


I(t) f db TrdTre-ikRpe(b b)d(t) (8.2)
a p








We have therefore separated the calculation into several steps, with the plasma

trace being constrained over surfaces of constant b = {q, PR, e}. We then write the

line shape in the form of an atomic physics problem


I(t) [ dpR d(p)de Trf(a, )D(k,pR, -t) d (8.3)


where

f(a, b) = Trp 6(b- b) = (p)f(a, e) (8.4)
P

and


D(k, pR, ,-t) dqRe-ikq D(k, b, -t) (8.5)

D(k,b,-t) f-l(a,b)Trpe(b- b)d(k,-t) (8.6)
p

is the transform D(k, p, E, -t) of the constrained plasma averaged dipole

D(k, b, -t). In the same manner as before, once we find an expression for

D(k, p, E, -t) the problem reduces to an atomic physics problem. Again we

use the projection operator formalism introduced in Appendix C to derive an

equation of motion for D(k, b, -t) and its Fourier transform. In Appendix F the

calculations and the detailed form are shown for this ion dynamics equation of

motion. In this chapter we use those results and the results from the quasi-static

ion equation of motion to highlight the differences brought about by the inclusion

of ion dynamics.

Recall the static-ion equation of motion for the constrained plasma averaged

dipole operator. This equation was derived using the projection operator technique

as outlined in ('i plter 6 and is given by


+ L(a, e) + B(e) D(, -t) + dr M(, t )D(e, -r) 0 (8.7)
0








Note here that if the value of the ion microfield were the only degree of freedom to

the quasi-static ion plasma, then the integral term would be zero, and the terms

in parenthesis would completely specify the dynamics of D(e, --). In this sense,

the part in parenthesis can be referred to as the deterministic part of the time

evolution equation, since it is completely specified by the ion microfield. Then the

integral can be referred to as the non-deterministic part, as it describes the other

degrees of freedom of the plasma not accounted for by the ion microfield. (Here the

term "non-deterministic p 11 refers to not being determined by the ion microfield,

which we chose as the most important plasma information.) In the derivation of an

analogous equation for the case when ion dynamics are important, changes to both

the deterministic and non-deterministic parts occur.

The changes to the deterministic part of the equation will be explored first.

In the quasi-static ion approximation, the radiator momentum was, from the

definition of quasi-static ions, independent of the ion microfield. The dependence

of D(e, -t) on the radiator momentum could then be factored out, and thus the

Doppler broadening of the line could be handled with a convolution of the Doppler

line shape and the stark broadened line shape. The different time scale of the more

general ion dynamics case makes this approach invalid due to the requirements of

Newtonian mechanics. Consider the radiator momentum. The force on the radiator

from the electric field will change the radiator momentum. Also, the momentum

propagating the radiator will take the radiator to different spatial positions, leading

to a change in the electric field experienced by the radiator. Therefore, even

without yet considering the motion of the plasma ions, consideration of the physics

from the different time scale for the ion dynamics case leads to a coupling of the

radiator momentum and ion microfield.

As shown in Appendix F, there are several changes to be made in the projec-

tion operator technique to derive an equation of motion for the averaged dipole








operator. For purposes of the outline here, it will suffice to -iv that the constrained

averages need to be modified. For the quasi-static ion case, the plasma averages

are constrained so that the trace is over those configuration points with a specified

value of the ion microfield. The reasoning is that for the quasi-static ion case, the

screened ion field is the dominant interaction. Once that is accounted for exactly in

the equation of motion, the other effects can be treated with perturbation theory.

For the ion dynamics case, the constraint is over a greater set of physical values.

We choose this set to be three plasma values: the radiator position qR, the radi-

ator momentum PR, and the ion microfield e. When these degrees of freedom are

chosen and used in the projection operator technique described in Appendix F, the

deterministic part of the equation of motion for the average dipole operator changes

from the quasi-static form


+ L(a, e) + B(e) D(e,-t) (8.8)

to the ion dynamics form

+ L(a, ) + B() + b D(k, -t) (8.9)


Here b = qR,pR, ) represents the set of plasma properties chosen, and () is

the constrained plasma average of the time derivative of b. Thus, when including

ion dynamics, the effect on the deteministic part of the equation of motion is

the addition of several terms. The previous definition of the plasma averaged

dipole operator was constrained only with the value of the ion microfield. Here,

however, the definition includes constraints over the values of all three properties

represented by b as indicated in Eq. (8.6). Besides these changes in definition, the

deterministic part of the equation of motion includes an entirely new set of terms,

given by (b)B(a/0b) acting on the averaged dipole. This form has a straightforward

interpretation. The extent to which changes in the degrees of freedom b affect the








averaged dipole operator, multiplied by the constrained averaged value of the time

rate of change of these degrees of freedom, gives the time rate of change of the

averaged dipole operator due to those degrees of freedom. With the calculations

performed in Appendix F, these terms can be shown to represent the coupling

between the various degrees of freedom represented by b. Explicitly, these terms are


( P V, QRE VP di / (8.10)
\ /b 9b m .] ) Op, m ( ). Oj1

The types of terms appear in, for example, the Boltzmann equation and can be

thought of in the same manner. The first term on the right hand side relates the

change in the averaged dipole due to the radiator momentum; in the context of

line shape theory, this term is responsible for doppler broadening. The next two

terms arise from the force acting on the radiator from the ion microfield causing

the radiator momentum to change. Recall from defining the system Hamiltonian

that the charges comprising the radiator were separated, so that that center of

mass degrees of freedom were collected in the plasma Hamiltonian, while the

internal atomic states of the radiator comprised the radiator Hamiltonian. Classical

electrodynamics gives the result that the force on a charge QR in an electric field

e is QRE, and the force on a dipole d in an electric field E is (d V)E. That

is what these next terms represent. The first is the force on the center of mass

degrees of freedom from the electric field, and the second is the force on the atomic

states represented by the dipole operator. The total force from these two terms

will therefore change the radiator momentum with time. The final term represents

the change in the ion microfield value at the radiator due to the momentum of the

radiator. The physical content of this term can be understood by the fact that the

quantity in angle brackets is related to the gradient of the field.

It is instructive to relate this more general case to the previous results for

the quasi-static ions. The first term on the right hand side of the above equation








arises from qR; the next two terms arise from pR; the last term comes from c.

Therefore, for the quasi-static ion case, the first term would still be present as

doppler bN ..1. .iii- but the time scale defined by the quasi-static ion case would

have the other terms equal to zero. This equation shows directly that this gives

an upper limit to the ion microfield strength and also the gradient. The repulsion

between the radiator and plasma ions is what makes the quasi-static ion case

suitable for many situations.

Up to this point, the dynamics of the plasma ions have still not been taken

into account; all that has been handled is the coupling of the radiator dynamics.

Considering points in phase space is worthwhile here. Consider the equation of

motion for the constrained average dipole operator with a specific set of values

for the plasma properties b. The set of values b picks out a set of phase space

points, each of which correspond to a plasma configuration that has the values

of b. Now assume that the chosen properties completely specified the plasma, so

that the only degrees of freedom to the plasma are b. Suppose one phase point

with properties measuring b evolves to a state with properties b'. Then, if these

properties completely specified the plasma, all the phase points with b would

evolve to some point with values b'. In other words, all the points in the subspace

constrained to have values b will move to the subspace constrained to have values

b'. It is this sense that the part of the equation of motion above is referred to as

the deterministic part.

However, the chosen degrees of freedom do not completely specify the plasma,

since these other degrees of freedom destroy this simple behavior. The dynamic

electron broadening operator described in Chapter 4 is an example of this, and the

ion dynamics described in this chapter is another. The properties b were chosen

because it is expected that they would dominate the evolution of the phase points,

and yet the n ,iv-b1ody effects from the plasma electron and ion dynamics will








cause some points to end up in the b' subspace that would not be there under the

deterministic motion, and some points to not end up in the b' subspace that would

have been there. The non-deterministic part of the equation will account for these

other degrees of freedom. The dynamical electron broadening term from before will

again be used. What is now needed is terms to represent the changing of the ion

microfield due to ion dynamics.

The ion dynamics terms will first be examined formally for their physical

content, and then modeled approximately with a stochastic approach. This

is in contrast to a direct ni ,i-lbody approach, in which the exact interaction

between the plasma ions and the radiator internal states is written, and then

approximations are made until a tractable form suitable for calculations is found.

The exact form for the equation of motion is

+ L(a, c) + B(c) + D(k, b, -t)
t
+ dr Jdb' M(b,b';t D(k, b', ) 0 (8.11)
0
where M(b, b'; t ) is found in Appendix F to have the form

M(b,b';t-7)X= f- (a,b)Ti,. (b)LU(t r)QL(b')X (8.12)
P

Here U(t 7) = exp [-QL(t 7)], and &(b) = 6(b b) are delta functions

that constrain the plasma to have specified values for the set of properties b. The

complementary projection operator Q is discussed in Appendix C. As in the quasi-

static ion approximation, Q projects out deviations from the constrained plasma

averaged values.

We discuss M(b, b'; t 7) by first analyzing the effects of the Liouville operator

L = LR + Lp + 6L, which occurs explicitly twice in the expression for M(b, b'; t r).

In Appendix F we follow these effects. The radiator Liouville operator cancels








in both places, and several terms result from the remaining parts. One piece is

related closely to the dynamic electron broadening term found in the quasi-static

ion approximation


3 (b,b',t) f-'(a,b)Ti1, ,(b)6LRQe-(LP+L)t Q (b')LpR

= LpnQe-(L+L)tQ (b'6LpR (8.13)

The above expression is written to second order in the interaction Liouville oper-

ator, which is why the density matrix and U(t r) are written to zeroth order in

6LpR. This expression differs from that found in the quasi-static ion approximation

in that the constraining delta functions appear twice. The correlation between the

free charge densities present through 6L at different times are thus calculated for

different possible condition of constrained conditions. That is, in the quasi-static

ion approximation, the relationship was between 6LpR at two different times, both

of which were constrained to have the same screened ion field. In the ion dynamics

case, M(b, b'; t T) is related to the correlation between 6LpR in a plasma state

constrained to have the value b, and 6LpR a time t T later in a plasma state con-

strained to have the value b'. If this change in the value of b during the correlation

time is small, then the dynamic electron broadening form found in the quasi-static

ion approximation is recovered.

Of the remaining terms of M(b, b; t r), the most important is

1 (b,b', t) f -(a, b)Trpe (( )(b) ) e-QLt ((b')i l) (8.14)

where the definition of the Liouville operator as a differential operator has been

used. (This operation is shown in Eq. (F.21).) Equation (8.14) actually represent

a number of terms, since b is from a set of properties. However, due to our choice

of plasma properties b to project onto, this second term can be simplified consider-

ably. Only the changes in the screened ion field is included. This results from the








fact that the time derivative of the momentum is directly related to the electric

field force on the monopole, and so projecting onto states in which the electric field

is specified account for the collisional variations in the momentum. The resulting

expression is


3i (b,b', t) f -l(a, b) Tl ()e-b (8.15)
a' P(815

Still, this expression is complex and rather than analyzing it directly, we use a

stochastic approach. The primary feature of Eq. (8.15) is a changing value of e,

the screened ion field value at the radiator. For practical purposes this has been

modeled by a master equation[43, 51].

A reasonable representation for the ion dynamics term is given by the master

equation


J de'M (, e', t)Xe) de' ( (e, e')X(e') We', e)X(e)) (8.16)


The transition rates W(c, c') set the amplitude and time scale for changes in the

field e. For time intervals shorter than this new time the static ion broadening of

the previous section occurs. Then this field changes to a new value due to the ion

dynamics and the new static ion broadening occurs. This phenomenological picture

can be extracted from a reformulation of the exact statistical mechanics motivated

by this physical picture. The objective of this section is to give an overview of this

"stochastic" description of ion dynamics. In the end, practical approximations

are required. However, in the spirit of the inquiry here those approximations are

statistical rather than perturbative with respect to any of the charge correlations

studied here.

A stochastic model is then required for the transition rates. For these tran-

sition rates, the ion microfield transition will be modeled as a kangaroo process,








which is defined by

W(e, ')= A(e)B(') (8.17)

so that in the transition rate there are no correlations between the initial and final

electric fields[51]. With this model, the ion dynamics terms becomes

S(e)D(e, -t) -v(e)D(, -t) + v(e) de' Q(e')D(e', -t) (8.18)

Here, the first term represents those phase points leaving the configuration space

with a specific value e due to the ion dynamics. The transition rate specifying the

rate of departure due to ion dynamics only depends upon the field strength under

consideration. The second term represent points which began at some value e', with

probability Q(c'), which end up with field strength e due to the ion dynamics. The

frequency v(e) is a parameter of the model, and at this point is arbitrary.

With this stochastic model chosen, the formal equation of motion Eq. (8.11)

becomes the approximate representation

+ L(a, e) + B(c) + ( ) D(k, 6, -t)
t
+ dT de' M(e,';t T)D(k,b', r) (8.19)
0
-v(c) (D(k, -t) de Q(e')D(k, b', -t) = 0

With this result, we can compare the formalism of our approach with that of

previous spectral line theories. With the exception of 6) which couples the

components of b, all the terms above appear and have similar physical interpre-

tations in those previous theories. The more general expressions derived here

can therefore be a direct guide into extending the theory to include these charge
correlations in regimes where it is necessary to treat them correctly.














CHAPTER 9
SUMMARY AND OUTLOOK

In this study, we have reexamined the problem of spectral line broadening in

plasmas. Our objective has been to explore the charge correlations between the

plasma electrons, the plasma ions, and a highly charged impurity ion. To pursue

this objective, we therefore were required to treat these correlations consistently

and to derive exact expressions to contrast with current semi-phenomenological

results. Existing models for prediction of spectral line profile have been quite

successful in general and it is expected that they should be recovered in many cases

(e.g., neutral radiators or radiators with small charge).

Two important external influences were felt during this study: previous

theoretical work and MD simulations. There exists a large body of theoretical

work and computer codes done previously by the plasma spectroscopy group at

the University of Florida. These have been developed over several decades, and

are believed to provide accurate diagnostics for experimental conditions currently

achieved[7, 8]. However, due to the complexity of the analysis, the difficult problem

of charge correlations are dealt with in different and possibly inconsistent v--i-.

These correlations are in some cases simply neglected, and in other cases treated

by using uncontrolled approximations. Despite the aforementioned success in

their use as experimental diagnostics, there are several clear reasons why a more

consistent treatment of the charge correlations is desired. The first reason is

that by dismissing the correlations or using uncontrolled approximations, the

process to extend the theoretical results to new regions of plasma conditions is

not made explicit. By deriving results that include these correlations consistently,

the ingredients needed in a more general theory are made clear. Rather than








replace, our aim is to build upon them and to provide a guide into consistently

extending them to treat different plasma conditions. A second reason for a more

consistent treatment of charge correlations is related to the purpose of plasma

spectroscopy. We have discussed its use as an experimental diagnostic. However,

plasma spectroscopy is also a tool for exploring the plasma system. From this point

of view, we treat the charge correlations in a consistent manner to give physical

insight into the properties of the plasma.

Another aspect that has driven our approach is the recent advances in MD

simulations. It is no surprise that computing technology has continuously advanced

and allows more detailed and longer running simulations. In addition to this, there

have been theoretical advances in the classical description of inherently quantum

systems through the use of regularized potentials. Thus there is new interest in

the simulation of two component plasmas with highly charged radiators, which is

the system of interest in this study. Specifically, the application of MD to electrons

is an evolving new area. However, the plasma properties to determine through

these simulations is not clear without a rigorous treatment of the expression for

the line profile. We have determined here the appropriate expressions that treat

charge correlations correctly. An example is the presence throughout this study of

constrained averages of quantities. In many cases, an average quantity is needed to

determine the line shape. This study shows that many of these quantities should

not be averaged over all plasma states, but only over a set of states that obey a

particular constraint (on the screened ion field, for example). This is a difficult

theoretical problem but one that can be studied readily by MD.

In addition to our formal expressions, we have provided simple estimates of

various structural and dynamical properties of the plasma. These estimates are

designed to be used in conjunction with MD simulations. The MD simulations

provide the criteria of validity in these estimates, while the estimates provide








simple physical insight into the MD results. When calculating these estimates,

we followed the constraints brought about by the inherent classical nature of MD

simulations. Thus we used regularized semi-classical potentials in our calculations

to more closely follow the path taken in performing the simulations. With these

potentials, we found that the free charge density did build in the region of the

radiator as expected. However, the divergence in the free charge density that would

have come about from using the Coulomb potential was avoided. Another example

of the physical insight into the plasma processes brought about by our estimates

is found in our calculation of dynamical correlation functions. When calculating

the plasma correlations functions, it was found that the charge correlations among

the plasma electrons and ions was most important in determining the initial value

of these correlation functions. Once that initial value was found, the resulting

dynamics was mostly due to one-particle dynamics.

More specifically, the primary new contributions of this work are two fold: a

detailed derivation of a practical form for the line shape function under conditions

of static ions, and an exposition of the more complex structure to be encountered

in the treatment of dynamic ions. In the following, the primary results are summa-

rized in each case with comments on the outlook for their future applications.

9.1 Static Ion Line Shape Function

For conditions of static ions the theoretical analysis of the line shape function

is simplified in several respects. First, the Doppler broadening is decoupled

from the plasma broadening. Second, the primary effect of the ions becomes a

statistical distribution of Stark broadening by fields sampled from the microfield

distribution Q(e). Our first new contribution appears at this point, with this

microfield distribution defined over a two component plasma plus point radiator

with all the correlations. Furthermore, the screened ion field E8 is defined in terms

of the exact ni liiv-1 ody screening by the electrons. Under conditions of weak








coupling this field becomes a sum of single particle Debye screened fields and the

microfield distribution becomes essentially that for a screened one component

plasma as used in current theories. More generally, for highly charged radiators

the electron radiator coupling becomes strong and this approximation should be

revisited. The general forms given here provide the basis for this study.

The remaining broadening due to the electrons was treated to second order in

the interaction between the bound state electron distribution and the electrons of

the plasma, in the presence of the ions. With the exception of similar early work

by Iglesias and Dufty[15] for neutral radiators, this is among the first treatments

of electron broadening in a two component plasma. The perturbation expansion

is justified since the dominant monopole part of the radiator-plasma interaction

has been extracted in the reference state. The broadening operator to first order

describes shifts in the spectral line due to electrons penetrating the bound state

distribution, discovered recently by the Hooper group[52]. This shift operator has

now been described without approximation at this order of perturbation theory and

related to the average electron density around the radiator. A new feature is that

this average is constrained: the static ion distribution yields a specified value for

the field at the radiator. The calculation of the equilibrium electron density around

the radiator for large Z is a difficult problem that can be addressed in the semi-

classical form by the HNC integral equations. The field-constrained calculation

is a new problem, never addressed before. An approximate means to consider the

effects of the constraint is described in C'! lpter 7.

The second order contribution includes the dynamical electron broadening

and is given in terms of the autocorrelation function for the electron density about

the radiator. Again, the average in the correlation function is constrained by the

given ion field value. Even without this constraint, the autocorrelation function in

the presence of a charge at the origin has been studied only in recent years. These








studies have been for the idealized semi-classical case of electrons in a uniform

positive background (jellium) with the radiator at the origin. The primary features

of the MD study, limited to the electron field autocorrelation function, are captured

by a simple mean field model for the dynamics of a single electron around the

radiator. We have proposed this model as a practical method for calculating the

charge density dynamics as well.

In summary, C'! Ilpters 5-7 give a complete and practical form for the calcula-

tion of spectral lines under conditions for which the ion motion is negligible. The

input for the formulation are electron structure and dynamics around the radia-

tor that require new theoretical methods for their analysis. First approximations

suitable for practical implementation and assessment of correlations have been

provided. It is expected that MD simulations of these quantities will be performed

soon.

9.2 Ion Dynamics

The extension of this work to include ion dynamics is more formal and less

complete. It follows the initial work of Boercker, Iglesias, and Dufty[14] who

provided the formalism but did not analyze the effects on electron broadening. An

important feature of the formalism, discussed at the beginning of ('! Ilpter 8, is the

self-consistent treatment of the radiator center of mass motion and the electric field

of the perturbing ions. Previous work in this group has included the stochastic

change in the electric field, as described here as well, but neglected the relationship

of this field to the radiator motion and the Doppler profile. This consistency

problem is a matter of kinematics due to the fact that the rate of change of the

center of mass position is proportional to the center of mass momentum, and the

change in this momentum is proportional to the total field at the radiator. Thus

any formulation of a spectral line shape including ion dynamics must include the

deterministic form of the equation given here.








The ion Stark broadening is again extracted explicitly as in the static ion case,

with the same screened ion field. The first order perturbation term is also similar

to that in the static ion result, with the same dependence on the constrained av-

erage electron distribution around the radiator. The electron broadening operator

is now quite different. It still entails an autocorrelation function of the charge

density around the radiator, but now with a double constraint: the field and radi-

ator momentum must have specified values in the initial and final states. If these

quantities do not change during the correlation time the previous static ion results

are regained. More generally, nothing is known about such correlation functions.

Perhaps some guidance is possible from MD simulation. The main conclusion is a

warning that when ion dynamics is important, the electron broadening can become

quite complex.

In addition to the Stark broadening of the ions, a dynamical ion broadening

operator is also present. This is essentially another doubly constrained autocor-

relation function for the rate of change of the field. The effects of this operator

have been modeled for plasmas with good success using a master equation. It is

proposed here for practical purposes as well. In particular, the Kangaroo process

is based on two plasma properties as input, the microfield distribution as its sta-

tionary state and a jump frequency for the fields. Both can be provided for a two

component plasma without compromising the correlations among charges.

In summary, this part of the thesis has exposed the important consistency

conditions of ion dynamics and provided the beginnings for the controlled analysis

necessary for practical model building.

With new experiments, containing extreme regimes of temperature, density,

and radiator charge, and the recent increasing interest in using MD simulations

for attractive, highly charged systems, the problem of treating charge correlations

consistently is important. Our study provides practical results to guide MD






79

simulations in the quasi-static ion approximation. For the more general case of ion

dynamics, we have provided a foundation for future work to build on.














APPENDIX A
HOOPER'S GROUP COMPUTER CODES

A set of codes has been developed based on recent theories of plasma spec-

troscopy. There are three main calculations to be done for the calculation of a line

shape: the atomic wave functions of the bound radiator electrons, the electric mi-

crofield due to the screened ions, and the electron broadening term, which accounts

for the effects of the free electrons beyond their screening of the ions. Accurate

theories to deal with the first two have been in place for a number of years, while

recent research has largely focused on the electron broadening term.

In addition to the calculation of theoretical line shapes, other codes have been

developed which apply experimentally relevant effects, such as doppler bN ,ii.I-

opacity, and instrumental b .. 1. iii.- so that the theoretical results can be most

directly compared with experimental line shapes.

A.1 Atomic Wavefunctions

The atomic wavefunctions to be calculated are for multi-electron, ionized

radiators [7]. To calculate these, a modified version of a collection of four programs

developed by Robert Cowan of Los Alamos National Laboratory is used[17]. The

starting point is calculating the one-electron relativistic radial wavefunction for the

electron configuration of interest. The Hartree-Fock approximation is used for this

calculation. Some other needed quantities, such as energy levels, transformation

matrices, and dipole and quadrupole interaction strengths, are also calculated and

stored at this stage.

A.2 Electric-Microfield Calculation

The electric microfield distribution is calculated using the Adjustable Param-

eter Exponential Approximation, or APEX[53]. This approximation is similar to








the exponential approximation used in fluid theory for the pair correlation function.

In the present case, the approximated function has the interpretation as a pair

correlation function for a fluid with a complex potential energy, which introduces

difficulties in a physical understanding. However, the approximation gives good and

quick results that agree well with more rigorous theories for strongly-coupled and

weakly-coupled plasmas.

Once the APEX routine generates a microfield distribution, the .,-vmptotic

part of the distribution is calculated using a nearest-neighbor approximation and

the two are matched for a final result. The integral of the field-constrained line

shape is taken over this distribution for the final result.

A.3 Electron-Broadening Operator

Much of the recent work in plasma spectroscopy has been focused on accurate

evaluations of the effects of the plasma electrons[7, 8]. The codes calculate the

electron broadening operator using three different methods. To describe these

differing methods, the following terminology is used. The first term is to what

order term in perturbation theory that the method uses to evaluate the electron

broadening terms. These methods either evaluate these terms to second-order or

to all order. Recall that this study evaluated the electron broadening to second

order in the interaction potential 6LpR. The second term is what type of dynamics

to use for the electrons. Both quantum mechanics and classical mechanics are

used. The final term refers to what type of interaction is calculated between the

electrons and the radiator. Recall that in this study, the ion-radiator interaction

was dealt with as a dipole interaction due to the large repulsion occurring. In the

past and especially with neutral radiators, the electron-radiator interaction was also

considered to be a dipole interaction. Recent theories treat the electron radiator

interaction with the full Coulomb expression. With this terminology explained, the

three methods are described next.








The first method is a second-order, quantum mechanical dipole theory[54, 55].

In this case, the interaction between the plasma electrons and the radiator is

taken to be a pure dipole interaction, in the same way that the plasma ion-

radiator interaction was handled. The dipole interaction was then expanded using

perturbation theory to second order for the result. This type of electron dipole

interaction was originally used for neutral radiators, in which case the electrons

were not pulled close to the radiator by the radiator charge. The second method is

a second-order, full-Coulomb quantum ti., ii [1, 7]. With this method, the plasma

electron-radiator interaction is taken to be the Coulomb interaction between

the plasma electrons and the radiator. (In the analysis, this interaction may be

separated into several parts, such as removing the monopole interaction from

the interaction Hamiltonian, but all of the Coulomb interaction is present in the

i,: ,1i--i See Eq. (3.7) and Eq. (5.22) for how similar modifications were made

in this study.) With this full Coulomb interaction, a perturbation expansion is

again used to calculate the electron broadening operator to second order in this

interaction. A third method is an all-order, full-Coulomb, semi-classical tli., .i []

Here, the same interaction is used as in the previous method. However, the electron

broadening operator is calculated to all orders in this interaction. In addition, the

plasma electrons are modeled as classical objects with well defined trajectories. The

following sections give more details about the full Coulomb theories, as they are

most relevant to the type of highly charged radiators considered in this study.

A.3.1 Second-Order, Full Coulomb, Quantum Theory

As the plasmas under consideration achieve higher electron densities, the

dipole approximation gives inaccurate results for the electron broadening [7]. The

effect of the perturbing electrons then must be evaluated using the full Coulomb

interaction. The reason that the dipole approximation is still valid for the plasma

ions but not the plasma electrons is due to the attraction between the positive








radiator and the electrons, which is strong enough to allow the electron perturbers

to penetrate the radiator orbitals. This theory was able to calculate the line shifts

due to the electrons from a relaxation theory in a consistent manner.

In this calculation, the electron broadening operator is split into two parts:

a time independent term corresponding to initial correlations that shift the

line shape, and a time dependent term that both shifts and broadens the line

shape. These two terms are calculated to second order in the radiator-perturber

interaction potential. Limiting the calculation to second order places two main

constraints on this calculation. First, the calculation is valid in a range that

corresponds to less than twice the electron plasma frequency[7]. Second, it is valid

only for weak collisions. This weak collision constraint means that the kinetic

energy of an average plasma electron must be large enough so that it spends a

small amount of time near the radiator so that its momentum change is negligible.

The valid temperature region is given by

z(z 1)
kT> Z( (Ryd) (A.1)


which is determined by comparing the perturber's kinetic energy far away from the

radiator and at its closest approach to the radiator[7].

The no-quenching approximation is applied to the theoretical derivation. This

indicates that there are no non-radiative transitions between initial and final states,

which is valid when the energy difference is large. The perturbing electrons are

described using positive energy Coulomb wavefunctions, and the screening that

arises from electron correlations is accounted for by using a cutoff at the Debye

length. (This method of using a cutoff is not the only available method, but it

allows for convenient comparison with other theories.)

The static shift is split into two parts, one that is first order in the interaction

potential, and a term that includes all higher order effects. When this is done, it








is found that the first order shift arises solely from the monopole interaction of an

electron that has penetrated the radiator orbitals.

A.3.2 All-Order, Full-Coulomb, Semi-Classical Theory

When the perturbing electrons undergo strong collisions with the radiator,

an all-order classical theory for the electrons can be used [8]. The structure of the

derivation is similar to the second-order theory, in that the radiator-perturbing

electron effects are described by an electron broadening operator split into a static

part and a dynamic part. The static part describes the time-independent initial

correlations, which is separated into a term that is first order in the interaction

potential and a term that contains all higher order effects.

The approximations made in this theory are also similar to the second order

theory. The no-quenching approximation is used, and it is assumed that there are

no dynamical correlations between the electrons and the ions. Thus the electrons

only affect the ion broadening by screening the ion field. It is also assumed that

there is only one strong electron collision during the radiation time, so that the

electron broadening effect on a radiator is given by considering only one electron.

This approximation would also indicate that there are no electron correlations. To

correct for this, a Debye length cutoff is used to account for electron screening.

The interaction of a single perturbing electron with the radiator system is handled

with a multiple expansion. The (modified) monopole term and the dipole term

are dominant here, and angular momentum rules allow only a limited number of

nonzero terms.

This semi-classical theory treats the perturbing electron as classical objects.

This assumption places certain limits on the validity of the theory. There are

several main changes needed to evaluate the electron broadening term in this way.

First, that part of the density matrix which arises from the perturbing electron

Hamiltonian is replaced with its classical analogue. The physical interpretation








of this is that those plasma electrons outside the interaction range are in thermal

equilibrium and their velocity distribution can be handled using a Maxwell-

Boltzmann velocity distribution. For this to be valid, it is required that


EFermi << kBT (A.2)

where EFermi is the plasma Fermi energy given by


Fermia { 32e }23 (Ryd) (A.3)

which ensures that the degeneracy effects are small for the plasma electrons [8].

A second change is to replace the quantum time-dependent electron-radiator

interaction operator with an interaction term that depends on the coordinates

of the perturbing electron as it moves along its path. This hyperbolic path is

calculated in a rotating frame. A third change is that the quantum trace must be

replaced by a classical trace. Finally, to account for the quantum diffraction effects

for distances close to the radiator, a minimum distance cutoff is used.

The results from the second-order and all-order theories are in agreement with

each other for current experimental results.














APPENDIX B
DOPPLER BROADENING IN THE QUASI-STATIC APPROXIMATION

In C'!i pter 5, it was stated that for the quasi-static ion case, the radiator

center of mass degrees of freedom were not coupled to the plasma degrees of

freedom, which allows a doppler term to be factored out in the time dependent

equation. This section will provide the details of those calculations.

At the beginning of C'! lpter 5, the line shape formula was written in the form

of

I(t) dt dk (t) (B.1)

where

dk(t) eiHt/hde-ik-q"e-iHt/h (B.2)

was defined in Eq. (3.5). The details of the quasi-static ion approximation were

then used to separate out the radiator center of mass degrees of freedom from the

other plasma degrees of freedom. In that approximation, the motion of the ions in

the plasma (and the radiator in particular) was considered to be independent of the

rest of the system. Then the density matrix factors into


P = Pc.o.m.Pp (B.3)

where pc.o.m. depends on the position and momentum of the radiator, and pp

depends on all other degrees of freedom. Then Eq. (B.1) can be written


I(t) Tr [p..m.ppeik*qdd(t)e-ik-(t)]

STr [pc.o.m.ik-q Tr d (t)]
c.o.m. p
= (k, t)Tr [ppd d(t)] (B.4)
P








In the above, the trace was separated and performed separately over the

center of mass degrees of freedom and the degrees of freedom of the rest of the

plasma; 4(k, t) describes the effect of the radiator motion on the line shape and

is responsible for doppler broadening. We now turn to evaluating this expression,

as the quasi-static ion approximation allows K{(k, t) to be considerably simplified.

Recalling that the quasi-static ion approximation predicts no acceleration for the

ions during the time scale of interest, and also that the ions are treated classically,

the position of the radiator center of mass at the time -t is


qR(t) = (O) + Pt (B.5)

where PR is the (constant) momentum of the radiator. Using this in K(k, t) gives


S(k, t)= Tr [pc.o.m.-ik-(pt/m)] [ dp O(p)e-ik.(pRt/I) (B.6)
c.o.m.

If the frec q', i-,-i-dependent line shape is considered, it is found that the

Doppler effect can be incorporated into the line shape through a simple convo-

lution. Recall that the frc qu-ii'-in,-dependent line shape is related to the above

time-dependent line shape through

I(w) = Re dt eilI(t) (B.7)

We have shown that I(t) is a product of a doppler line shape (k, t) and the dipole

autocorrelation function for static ions. Taking the transform converts this product

into the following convolution

I(w) = dw'ID( o')J(w') (B.8)

where ID(w) is the Doppler line shape given by[17]


ID(') exp G-') (B.9)
V2o-2 \, 2j2





88


2j.2 /2 (2kBT (B.10)

and J(w) is the static ion line shape with


J()) = Re dt eit (d d(t)) (B.11)
0

With this the doppler broadening is fully accounted for in the quasi-static ion

approximation.













APPENDIX C
PROJECTION OPERATOR EQUATION OF MOTION
In this study a projection operator technique is used to derive an equation of
motion for both the quasi-static ion line shape and the more general ion dynamics
line shape. The definition of the projection operator is different in each case, yet
the general derivation is the same in both cases. Here the general derivation of an
equation of motion for an arbitrary operator is presented. The results are later
used in Appendix D and Appendix F.
We choose an operator P. The actual definition of P will vary for different
applications. If we assume that it is a projection operator, then P2 = P. We also
define a complementary operator Q 1 P. Note that QP = 0, since

QP= (1 ) p p2 =p p (C.1)

and also that Q is a projection operator itself (2 = Q). Finally, we assume that
for all cases of interest, P(O) = 0 and P aX = jPX.
The Liouville equation for some general operator X(t) is

+ L X(t) 0 (C.2)

Acting from the left with P and inserting 1 = P + Q gives

P + L (P + Q)X(t) 0 (C.3)

Letting the projection operator commute with the derivative and separating the
term that includes Q gives

S+ PLP PX(t) -PLQX(t) (C.4)








The procedure is to calculate an expression for QX(t) and substitute it into Eq.

(C.4). We follow the same method as above to derive an analogous expression for

the complementary projection operator Q, resulting in


+ QLQ QX(t) -QLPX(t) (C.5)


or

S[QX(t)] = -QL [QX(t)] QLPX(t) (C.6)

A general solution for an equation of the form

8X
= AX + B(t) (C.7)


is[43]
t
X(t) = eAtX(O) + dr eA( (--)B() (C.8)
0
A solution for QX(t) is then

t
QX(t) = e-QLtQX(O) dr e- L(t-) QLPX(t) (C.9)
0

Using this result in Eq. (C.4) gives


( + PLP PX(t) -PL e-QLtQX(O)- dre-QL(t-7) QLPX(t)) (C.10)
0

The specific form of the projection operator and the details of the system under

consideration is used to modify this further.













APPENDIX D
AVERAGE DIPOLE EQUATION OF MOTION-QUASI-STATIC CASE

This chapter uses the projection operator results from Appendix C to derive

an equation of motion for the time dependent constrained average dipole operator.

Here the focus is on finding the result in the quasi-static ion approximation.

In Eq. (5.17), the constrained plasma averaged dipole operator was defined as

D(, -t) Tr [p(a, )d(-t)] (D.1)
P

p(a, c) =f-l(a, c)pb(, Es) (D.2)

The equation of motion for D(e, -t) will be put into the form


S+ L(a, ) + B D(, -t) + dr M(c, t -T)D(, -r) 0 (D.3)
0
We define a projection operator P

PX Trp(a, e)X (D.4)

First note that P satisfies the definition of a projection operator, since P2 p

p2X = PX ( Trp(a, c)PX Trp(a, e) PX= PX (D.5)
P ) (P

In the above, the fact that PX is independent of plasma coordinates (because it is

an average over plasma coordinates) was used to bring PX out of the plasma trace.

Then, the constrained trace of just the constrained density matrix equals unity,

giving the last step. Another consequence is the action of the projection operator

on d(-t)

Pd(-t) (Trp(a, e)d(-t)} D(E, -t) (D.6)
\P /








We also define the operator Q 1 P.

The Liouville equation for d(-t) is written as


( +L d(-t)= 0 (D.7)

The result from Eq. (C.10) is then


( + PLP Pd(-t) = -PL e-QLtQd() J d -QL(- T)QLPd(r) (D.8)
o 0

Using Eq. (D.6), we then have the equation of motion for D(e, -t)


( + PLP) D(, -t) -PL e-LQd() -J dr e-Q(t-)QLD(e, -t)
0
(D.9)

The initial value term for the dipole operator will cancel, since Qd(0) = 0. This is

because d(0) does not depend on plasma coordinates, so


Qd(0) (1 P)d(0) d(0)- Trp(a, )d(0)
P
Sd(0) Trp(a, e) d(0) d(0) d(0) = 0 (D.10)

so the form for the equation of motion is


+ PLP )D(, -t) PL dr e-QL(t-) QLD(E, -r) (D.11)
0
The specific form of the Liouville operator for the quasi-static ion approxima-

tion can now be used to write the equation of motion in a form in which different

plasma effects are separated. Recall that the total Liouville operator was written as

L = Lp + LR(E) + 6LpR (D.12)

where Lp included the plasma degrees of freedom and their interactions, LR(E) was

the Liouville operator for a radiator in an electric field e, and 6LpR is the Liouville