Problems with Standard Semiclassical Stark Broadening Theory
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arXiv:atom-ph/9605003v1 10 May 1996PROBLEMSWITHTHESTANDARDSEMICLASSICALIMPACTLINE-BROADENINGTHEORY
SpirosAlexiouP2IM,URA773,Case232,UniversitedeProvence,CentreSt.Jerome13397Marseille,France(Received17March1995)
InthisworkwestudyindetailtheNeVII2s3p-2s3ssingletline,whichwasalsotheobjectofarecentexperiment.Thestandardperturbativeimpacttheorypredictionsaretestedagainstafullynon-perturbativesemiclassicalimpactcalculation,takingintoaccountdipoleandquadrupoleinter-actions.Potentiallyverysignificantproblemswiththestandardperturbativetheoryareencounteredanddiscussedandasimpleremedyisproposed.PublishedinPhys.Rev.Lett.75,3406(1995).
PACSnumber(s):52.70Kz,32.70Jz,32.30Jc,32.60+i
Thecalculationofplasma-broadenedlinespectra,pro-videsaveryusefuldiagnostictoolandadditionallyisanecessaryingredientforlarge-scalecomputationsinas-trophysicsandplasmaphysics.Amajorcornerstone,re-ducingthemany-bodyproblemofline-broadeningtothecomputationofone-bodyquantities,istheimpactap-proximation[1,2].Forpracticalcalculations,theimpacttheoryisusuallyemployedinitsperturbativeversion,andeventhensimplifiedformulasareoftenused.Isolatedlines[2],bybeingrelativelysimpleandusu-allyunaffectedbyionmicrofieldeffects,whetherstaticordynamic[3]areanexcellenttestinggroundforthetheoriesofelectroncollisionalbroadening.Suchtestsre-quirereliableexperimentalprofilesandmuchprogresshasbeenmadeinthisdirectioninrecentyearsmainlybytheBochumgroup[4–10].Thesestudieshaverevealedsignificantdiscrepancieswithsimplifiedexpressionsthatareoftenusedfortheelectroncollisionalbroadening[2,11–14].Furthermore,seriousdiscrepancieswithclose-coupling(cc)calculations[15]werefoundin[5].Evenmoreimpressiveisarecentlyobtainedfactorof2discrep-ancybetweencccalculations[15]andexperiment[10],foralineandparameterrangewhereccshouldbeatitsbest.Inbothcases,muchbetteragreement(roughlybyafactorof2)isobtainedbysemiclassical(sc)calcu-lations.Thismeansthatsccalculationsareinfactthebestavailabletoday,inthesenseofgivingagreementwithexperiment.Atthefoundationofanysophisticated[3,16–20]scper-turbativecalculationistherequirementthatunitarityisnotviolated.Thisisimportant,ofcourse,sinceunitar-ityviolationcanleadtoaseriousoverestimationofthewidth[17,21].Unitarityispreservedforover30yearsbyusingacriterion,thoughttobebothnecessaryandsuffi-cient.This“fact”hasgoneunchallengedoverthisperiodoftime.Hence,aminimumimpactparameterρmin(v)isdetermined,suchthatunitarityissatisfiedforlarger
1impactparameters,bynumericallysolvingtheequation|{a′∞−∞V′aa′(t1)dt1t1−∞dt2V′a′a(t2)+b′∞−∞V′bb′(t1)dt1
t1
−∞dt2V′b′b(t2)isananalyticsolutionundertheapproximationofne-glectingtime-orderingeffects,whilethesecond[23]isafullynumericalsolutionoftheSchr¨odingerequation.Toavoidambiguities,wehavechosenthesecondapproachhere.Thus,iftheρmin(v)requiredtosatisfyunitarityissignificantlylargerthanthenρmin(v)requiredtopre-servethescapproach,asisveryoftenthecase[6–10]aNPcalculationachievesaverysignificantreductionoftheerrorbarsgivenbythePRcalculation.Asareminder,thehalfwidth(HWHM)iswrittenas[18]
HWHM=2πnρdρdvvf(v)Q(ρ,v)(0.2)wherenistheelectrondensity,f(v)istheMaxwelianvelocitydistributionandQisgivenby
Q(ρ,v)={I−Sa(ρ,v)S−1b(ρ,v)}(0.3)wherethesubscriptsaandbdenotetheupperandlowerlevelsrespectively,Iistheunitmatrix,SistheS-matrixand{...}denotesangularaverage.Whenoneformulatestheproblemonthecollisionaxes,onefindsforisolatedlines[18]
Q(ρ,v)=(2Ja+1)−1M,ma,m′a,mb,m′bJbm′b1M|Jam′aJbmb1M|Jama[δmb,m′bδma,m′a
−Jam′a|Sc(ρ,v)|JamaJbm′b|Sc(ρ,v)|Jbmb⋆](0.4)
wherethesubscriptcdenotesthattheS-matrixhasbeencomputedforagivendirectionoftheperturbertrajectory(collisionaxes).Forionlines,thetrajectoryisparametrizedintermsofthe”time”variableu,definedby[3,18]
t=s(ǫsinh[u]−u)/v(0.5)withs=Zeme2theU-matricesremainclosetotheunitmatrix,thentheperturbationexpansioniscertainelyvalid.Thisisthecaseofthesolidline,whichrepresentsthelargereccen-tricity.However,wemaygetasmallQ(ρ,v)withoutthisbeingthecase,asisdemonstratedbythedashedline,whichrepresentsthesmallereccentricity.Inotherwords,itisonlyforsmalltimes(oru)thattheU-matricesevolveaccordingtoperturbationtheoryinthesmallerǫcase.Consequently,theunitaritycriterionisinsufficientandtheNPresultissubstantiallydifferentfromthePRone.WenotetheflatinitialandfinalregionsforbothcasesinFig.1,whichshowthattheU-matrixhasindeedconvergedtotheS-matrix.Fig.1goeshere
Thingsaredifferentinthequadrupolecasewherethedominantcontributioncomesfromsmallerimpactpa-rametersandvelocities.InPRcalculations,unitarityre-quiresacutoffatlargerimpactparameterscomparedtoapuredipolecalculation,withanassociated,oftenverysignificant,increaseinthestrongcollisionerrorbars.Insuchcalculationsincludingquadrupoleimpactbroaden-ingoften[18]onlythediagonalchannelsareincluded.In[20]thisassumptionwasdiscussedandaroughcriterionwasadoptedforincludingthe”nearhydrogenic”nondi-agonalchannels.Althoughtablesoftherelevantnondi-agonalfunctionsexist[24],analyticexpressionshavenotbeendeveloppedforthesefunctions,unlikethedipolecase,anditwouldbedesirabletodoso,inordertobeabletodeterminewhichquadrupolechannelsshouldbeincludedinthecalculation.Weherepresentconcreteex-ampleswherethe”nearhydrogenic”nondiagonalchan-nelsmustbeincluded:Forthelineinquestion,theonlydiagonalquadrupolechannelis3p-3p,since3s-3sisnotallowed.However,the3s-3dchannelisallowedandthisisoftenneglected[18].Fig.3showsacomparisonofpurequadrupolecalculations.Therearebasicallytworegimes,theperturbativeregimeandthestrongcollisionregimeatsmallimpactparameters.ThefirstthingtonoteisthateveninthePRregime,itmakesasignificantdif-ferencewhetherthe3s-3dchannelisincludedornot.Ontheotherhand,becauseperturbationtheoryisvalid,the3d-3dchannelmakesnodifferenceintheresults.Thischangesinthestrongcollisionregimeandtheadditionofthe3d-3dchannelmakesanimportantdifference,re-sultinginapeakratherthanatrough.Thisresultcon-firmsthataswemovetosmallerρ,progressivelymoreandmoreperturbinglevelsandchannelscomeintoplay.
