"(Classical) binary encounter model in molecular ionization. Restricted Hartee-Fock calculations and target polarizability" 

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International Journal of Mass Spectrometry

j o u r n a l ho me p a g e :w ww . e l s e v i e r . c o m / l o c a t e / i j m s

Electron-impact ionization of fluoromethanes – Review of experiments and binary-encounter models

Grzegorz P. Karwasz

, Paweł Mo ˙zejko

, Mi-Young Song

aFacultyofPhysics,AstronomyandAppliedInformatics,NicolausCopernicusUniversity,Grudzi ˛adzka5,87-100Toru´n,Poland

bFacultyofAppliedPhysicsandMathematics,Gda´nskUniversityofTechnology,Narutowicza11/12,80-233Gda´nsk,Poland

cPlasmaTechnologyResearchCenter,NationalFusionResearchInstitute,814-2Osikdo-dong,573-540Gunsan-si,RepublicofKorea

a r t i c l e i n f o

Articlehistory:

Received17November2013 Receivedinrevisedform 27December2013 Accepted14January2014 Availableonline25January2014

Keywords:

Ionization Electronimpact Fluoromethanes

a b s t r a c t

Experimentsandrecommendeddataonelectron-impactionizationofmethaneandfluoromethanes (CH3F,CH2F2,CHF3,CF4)arereviewedandcomparedwithbinary-encountermodels(Gryzi ´nski’s,Deutsch andMärk’s,andKimandRudd’s).Agoodagreementbetweenrecentexperimentsandthetwolatter classical-likemodelsisshown.KimandRudd’smodel(calculatedpresentlyintherestrictedHartree- Fock6-31**Gorbitalbasis)predictswelltotalionizationcrosssectionsforallfivemoleculesconsidered.

However,counting-ionizationcrosssectionshavetobeextractedfromexperimentaldatatoshowthis agreement.TheadditivitymodelofDeutschandMärkperformsequallywell,onceallmolecularorbitals aretakenintoaccount.Themaximaoftotal(counting)ionizationcrosssectionscalculatedpresentlyin KimandRudd’sbinary-encounterapproximationcorrelatelinearlywiththemolecularpolarizability.

©2014ElsevierB.V.Allrightsreserved.

1. Introduction

OnehundredyearsfromBorn’sformulationoftheatomicmodel [1]theamountofexperimentaldataonelectron-atomscattering seemstobequitesatisfactory,seefore.g.reviewsbyMärkand Dunn[2],Karwaszetal.[3],LindsayandMangan[4].However,a morecarefulinsightshowsthatsomesimplesystematicsarestill missing,seefore.g.[5,6].Inthecaseofelectron-impactionization neitherthetotalionizationcrosssectionnorpartitioningintospe- cificionicchannelshavebeencorrelatedtosomeotheratomicor molecularfeatureslikeelectronaffinity,ionizationpotential,polar- izability.Numericalfittingofcrosssectionsforparticularprocesses likeionizationforavastchoiceoftargets(e.g.Refs.[7–9])and/or reviewsofcrosssectionsetsforseveralpossibleprocessesbutfor selectedtargetsonly,likemethane[10]orfluoromethanes[11,12]

wereundertaken,butnoconclusivepictureshavebeenobtained sofar.

Inthepresentworkweperformananalysisofrecentexperi- mentaldataonfluoromethanes(andmethane)andcomparethem withtwobinary-encounter models:DeutschandMärk[13] and KimandRudd[14].

∗ Correspondingauthor.Tel.:+48566112407.

E-mailaddress:karwasz@fizyka.umk.pl(G.P.Karwasz).

2. Deutsch–Märkandbinary-encounterBethe’smodels AsresumedbyMargreiteretal.[15],Thomson[16]wasthefirst toapplyclassicalmechanicstoderiveaformulafortheelectron impactionizationcrosssection

=



n

4a²₀n



In



t , (1)

wherengoesthroughatomsubshells,_nisthenumberofelectrons onthenthsubshell,Inistheionizationenergyofthenthsubshell, tisthenormalizedkineticenergyoftheincidentelectront=E/In, RtheRydberg’sconstantanda₀ istheBohr’sradius.Thompson assumedthatthevelocitiesoftheelectronofthetargetatomsare smallcomparedtothevelocityoftheincidentelectron.

Gryzi ´nski[17]includedexplicitlythecontinuousdistribution ofthevelocitiesoftheelectronsinthetargetatomderivingthe followingexpressionfor

=



n

4a²₀n



In



1 t



−1 t+1





1+2 3



1− 1 2t



ln[2.7+(t−1)^1/2]



(2)

DeutschandMärk[13]proposedamodificationofGryzi ´nski’sfor- mula,substitutingBohr’sradiusa0withradiirnofthenthsubshells andthefactor4inEq.(2)byweightingfactorsgn obtainedfrom

1387-3806/$–seefrontmatter©2014ElsevierB.V.Allrightsreserved.

http://dx.doi.org/10.1016/j.ijms.2014.01.010

Fig.1. DifferentapplicationsofDeutsch–Märkmodelformolecules:H2DMmodel [15];forCH4“DM1”and“DM2”modelsfrom[15]usingtwoalternativeMulliken populationsofmolecularorbitalsandorbitalsradii;forCF4“DM”from[15]and

“modifiedDM”from[24].Experiments:CF4,Nishimuraetal.[42]andCH4presently suggestedtotalcountingcrosssections,seeFig.3.Lowersymbolsareexperimental recommended[4]totalgrosscrosssectionsofH2andF2.

empiricaldataforelectronscatteringonnoblegases[18,19]

=



n

gnrnn1 t



t+1





1+2 3



1− 1 2t



ln[2.7+(t−1)^1/2]



(3) ThissocalledDeutsch–Märk(DM)approachwasextended [15]

tomolecularionizationassuminganadditivityrule,i.e.thatthe molecularcrosssection(i.e._AB)isasumofthecrosssectionsof constituentatoms(_AandB)withappropriateweightingfactors

Aand_b).

AB=AA+BB (4)

Theouterelectrons(i.e.thosefromthemolecularorbitals)bring themaincontributiontotheionizationcrosssection.Inorderto includethedetailsofthemolecularstructureintotheadditivity ruleEq.(4),theweightingfactorsnaremadeequaltothepartial contributionsoftheconstituentatoms(i.e.AandB)tothespe- cificmolecularorbitals.Letusrecallagainthatradiirn inEq.(3) andinconsequenceinEq.(4)aretakenfromquantummechani- calcalculationsforconstituentatomsandthefactorsg_nfromthe semiempiricalanalysis[18,19].Forexample,forH₂theweighting factorisgH=3,rH=a0(andH=2)[15].AcomparisonbetweenDM resultsfromRef.[15]andpresentlyrecommendedexperimental valuesforH₂,CH₄andCF₄areshowninFig.1.TheDMmodelforCH₄ predictsthemaximumofthetotalcrosssectionwithintheexisting spreadforrecentexperimentaldata.ForCF4theagreementofthe DMmodelwithrecentexperimentsisalsoquitegood.

For the DM additivity model for molecules, the knowledge of partial occupancies of molecular orbitals by electrons from constituentatoms(Mullikenpopulations)isneeded.Further,the molecularcrosssectionsdependontheionizationenergiesofthe constituentatoms.Obviously,thisisanindirectwaytoaccountfor thestructureofthemolecule.TheDMmodelpredictsreasonably wellthemagnitudeoftotalionizationcrosssections,inparticu- larintheregionbetweenthethresholdand themaximaofthe crosssection,i.e.intherangewhichismostimportantforpractical applicationsofplasmas.

TheDMmodelwasalsosuccessfullyappliedtomultipleioniza- tion[20]andinner-shellionizationofatoms[21].Amorerefined analysisoftheweightingfactors_iinformula(5),basedonabroad experimentalbasewasincorporatedintoa modifiedDMmodel

[22–24].ThemodifiedDMmodelpredictsquitewellamplitudes ofthetotalionizationcrosssectionsinmolecules,seeFig.1forCF4. NeitherGryzi ´nski’snorDeutschandMärk’smodeltakesinto accountexplicitlythevelocitiesoftheelectronsinthedifferent orbitalsandtheirbindingenergies(i.e.therealionizationenergies ofthemolecule).Afurtherstepinthesemiempiricalunderstand- ingoftheelectron-impactionizationwasdonebyKim[25]and Rudd[26]whoproposednewformulaenotonlyfortotalbutalso fordifferentialcrosssections(vs.angle,vs.collisionenergyand vs.energy-loss).Subsequently,usingMott’s[27]approximationfor lowandBethe’s[28]approximationforhigh-energycollisions,Kim andRudd[14,29]proposedthefollowingapproximationfortotal ionizationcrosssections

=



n

4a²₀n



In





1−1 t +

lnt 2



1− 1 t²





(5) whereunisanormalizedkineticenergyofanelectrononthenth orbital,u_n=U_n/I_n.Asthismodelincludesthebinary-encounterand Bethe’sideasitiscommonlycalledBEB.

InBEBbothionizationenergiesofelectronsonnthorbital(i.e.

theirbindingenergies)aswellastheiraveragekineticenergies havetobeevaluatedfromquantummechanicalcodesformolec- ularstructures.Asaconsequence,theBEBresultsdependtosome extentonthemolecularorbitalbasissetschosen.TheBEBmodel wassuccessfullyappliedtolightatoms(He),(Ne),molecules(H₂, H₂O),ions(Li⁺⁺)[14],metals[30]andtargetslikeN₂,O₂,CO,CO₂, NH3,C3H8,butrarelytoCH4[31].

TheDMandBEBmodelsoffluoromethanes(CH₃F,CH₂F₂,CHF₃ andCF₄)wereextensivelydiscussedbyTorresetal.[32].Inpar- ticulartheyexploitedtheoriginalDMapproachwiththeadditivity rule[15]andwiththemodifiedDMadditivityrule[23]calculat- ingMulliken’smolecular-orbitalpopulationsandweightingfactors foratomicelectrons(gn)directlyfromquantum-mechanicnumer- icalcodes (GAUSSIAN98W).Torreset al.[32] concludedfor all thefourgasesconsideredthattheD-MmodelwithHartree-Fock STO-3Gtendstooverestimatetotalionizationcrosssectionswhile withothermolecularbasesconsidered(HF/6-311G,HF/6-311**G, MP2/6-311G,MP4/6-311G,CC/6-311G,CISD/6-311G)theresults differ inalmostundistinguishableway (within 1%)and tend to underestimateslightlytheexperiments,seetheirFig.3in[32].

IntheBEBmodeltheinfluenceofthemolecularbasischosen issimilarlyinsignificant(withdifferenceswithin2%),apartfrom theHF/STO-3Gbasiswhichoverestimatestheexperiments[32].

InpresentworkwecalculatedBEBcrosssectionsusingrestricted Hartree-Fock6-31**Gorbitalbasissetfortheentireseries,CH₄, CH3F,CH2F2,CHF3,CF4andcomparedthemwithexperimentaldata inFigs.3–5.

3. Discussionofexperimentaldata

3.1. Methane

Methaneisoneofthemostextensivelystudiedmoleculartar- gets–itisimportantforglobalwarmingbalance,fortokomakedge plasmas[5] and for hydrogen production viaelectric-discharge pyrolysis[33].The BEBmodel was appliedtoCH₄ byKim and collaborators, [34] and [31], using vertical (14.25eV) and adia- batic(12.6eV) ionizationpotentials,respectively. Obviously,the useoflowerionizationpotentialrisesthetotalcrosssection,bring- ingthemaximumtoabout4.3×10⁻²⁰m²,i.e.abovethehighest experimentaldata[35,36]andshiftingthemaximumtowardlower energies.

Theresultsofdifferentmeasurementsoftotalionizationcross sectionsofCH4areshowninFig.2.At100eVthedifferentdata

Fig.2.ReviewofexperimentalgrossionizationcrosssectionsforCH4.Rappand Englander-Golden[37],Schrametal.[79],Chathametal.[80],OrientandSrivastava [35],NishimuraandTawara[39],Vallanceetal.[61],Straubetal.[40],TianandVidal [36],thedataofGluchetal.[41]havebeenre-normalizedtothemeasurementsof RappandEnglander-Golden[37].“L-B”withlinedrawnaseye-guidestandsfor recommendedvaluesofRef.[4].

showaspreadofabout20%whatmakesthecomparisonwithBEB modelsdifficult.Somevalidationofexperimentaltechniquesused indifferentmeasurementsisneeded.RappandEnglander-Golden [37]normalizedtheirdatatoownmeasurementsofH₂;Tianand Vidal[36]normalizedtheirdatatoAr⁺measurementsbyStraub etal.[38].Nishimura andTawara[39]measuredabsolutecross sections;theirdataalmostcoincidewiththosebyStraubetal.[40].

LindsayandMorgan[4]compareddifferentexperimentsincluding thatfromtheirlaboratory[40]andproducedasetofrecommended values;thissetdiffersby−9%at100eVandby−2%at1000eVfrom theexperimentbyStraubetal.[40].

Gluchetal.[41]normalizedthesumoftheirpartialcrosssec- tionstothetotalvalueofRappandEnglander-Golden[37].InFig.2 wehavere-normalizedthedataofGluchetal.byafactorofa⁰₂ totheoriginalvalueofRappandEnglander-Goldenat100eV.The mostrecentdata[39–41]andtherecommendedvalues[4]agree

Fig.3. SearchfortotalcountingionizationcrosssectionsforCH4.Experimentand

“LBtotal”,seeFig.2.“Wang”singleionization[50].“LBcounting”hasbeenobtained fromrecommendedgrosstotal“LBtotal”[4]bysubtractinghalfoftheH⁺yield(“LB:

H⁺ion”[4]),seetextforthediscussion.BEBKim[34]“Ward:double”isthesumof H⁺,CH⁺,C⁺,H2+,CH3+,CH2+yieldscomingfromdoubleionizationprocesses.“BEB 2a1”ispresentcalculationforionizationfromthe2a1molecularorbital.

within10%,seeFig.3.ThepresentBEBmodelremainslowerthan theseexperiments.

Theusualexplanation[42]ofthisdifferenceisthatBEBmodels includealsothedissociationintoneutrals.This,inturndisagrees withtheverybasisoftheclassicalmodels,Eq.(1)–(5):theirkine- maticsassumeanelectronleavingthemolecule.Measurementsof dissociationintoneutralsexistforonlyfewmolecules.Winters[43]

measuredthetotalyieldofdissociatedfragments(ions+neutrals) byadsorptionontitaniumgetterobtainingacrosssectionofabout 4.0×10⁻²⁰m²atthemaximumat80eV,comparabletothemax- imumin thegrossionizationcrosssection.At80eVtheparent ionization (i.e. into CH4+)is 1.55×10⁻²⁰m² [4] so BEB would underestimatethe sumof theionization and dissociation-into- neutralscrosssections.

ThedifferencebetweenpresentBEBcalculationsand experi- mentscanbeexplainedintermsofmultipleionization,andmore preciselyintermsofmultiplecountingionscomingfromthesame ionizationevent.Methodsinwhichthetotalchargeismeasured giveso-calledgrosstotalcrosssection.Thisquantitydiffersfrom thecountingionizationcross sectionsas multiplychargedions arecountedwithweightsoftheircharge.Forargonat100eVthe Ar²⁺/Ar⁺ratioamountsto6.6%,see[3].Inmolecules,dissociative ionizationisacompetingchanneltomultipleionization.Gener- allythedoubleionizationinmoleculesislow:inCF4at180eVthe summedcontributionfromCF22+andCF32+isabout1–2%ofthe totalgrossionizationcrosssection[44,45].

However,inmoleculesconsideredinthispaper,particularlyin CH4 and CF4 careful attentionmustbepaid todiscriminatefor doubleioncounts,like(CH₂⁺+H⁺)etc.Formethaneseveralrecent experimentalpapers[46–48]showedthatthedicationCH₄²⁺inits groundandexcitedstatesdissociatesimmediatelyanddecaysinto twoionizedfragments.Inparticular,usingthecoincidencetech- nique,Wardetal.[46]havereportedcrosssectionsforformation ofpairsofdissociatedionsandattributedthemtoaspecificpre- cursor(monocation,dication,trication);theirdataarerelativeto theCH₄⁺yield.ThistechniquewasalreadyusedbyLindsayetal.

[49],who reportedcross sectionsforproduction of(CH2++H⁺), (CH⁺+H⁺),(C⁺+H⁺)ionpairsbuttheirresultsareprobablyunder- estimated,comparewith[46].Intotal,at200eVasmuchas58%

ofthegrossionizationcrosssectioncomesfromdissociativeion- ization(i.e.fromotherchannelsthanformationoftheCH4+ion) [4];obviouslyinpartoftheseeventsthesecondfragmentcanbea neutral.

AlreadyWangandVidal[50]noticedarelativelyhighamount ofmultiplecountingsofionscomingfromdissociativechannels that contributes to the gross total cross section. They evalu- ated the cross section for the double ionization at 200eV as 0.23×10⁻²⁰m² (vs.2.89×10⁻²⁰m²forsingleionization).Wang and Vidal[50] estimatedroughly that theH⁺ ion is formed in abouthalfofthedissociativeionizationevents.AccordingtoWard et al. [46], at 200eV as much as 52% of H⁺ ionsresult from a doubleionization(and1.6%fromtripleionization).Theabsolute valuefortheH⁺overallyieldat200eVis0.328×10⁻²⁰m²accord- ing to therecommended values in [4] and 0.30×10⁻²⁰m² for thesummed-upyieldofsingle,doubleandtripleionizationchan- nelsreportedbyWardetal.[46](andnormalizationthemtothe CH4+ yield from[4]).Thesum ofall cross sectionsfor appear- anceofionpairsreachesamaximumof0.37×10⁻²⁰m²at100eV [46].

Foralltheseexperimentalevidencesitisreasonabletoassume thatthecountingcrosssection(i.e.withtheexclusionofdouble countingionpairs)ofCH₄canberoughlyevaluatedbysubtracting halfofthecrosssectionfortheformationofH⁺ion[4]fromthegross totalcrosssection[4].WedidsoinFig.3–weconsiderfullpointson thisfigureassemi-empiricallysuggestedvaluesforthetotalcount- ingionizationcrosssectioninCH4.NowtheBEBmodelcoincides

Fig.4.Ionizationcrosssectionsinfluoromethanes.CH3F(shiftedby−1):grosstotal Vallanceetal.[61],Torresetal.[32],Rejoubetal.[58,4];H⁺+F⁺yieldfrom[32].

CH2F2:Torresetal.[57].CHF3(shiftedby+1):BeranandHevan[60],Jiaoetal.[59], Torresetal.[45].BEBispresentRHF6-31**Gcalculation.

inthe40–1000eVrangewiththeexperimentwithinthecombined uncertaintyoftheexperimentalandevaluationprocedures.

Tovalidatefurtherourestimate,inFig.3weshow:(i)thediffer- encebetweentheL-Brecommended[4]grosstotalionizationcross sectionandthepresentBEBresults,(ii)theH⁺ ionyieldaccord- ingtotheL-Breview[4],(iii)thesumofH⁺,CH⁺,C⁺,H₂⁺,CH₂⁺, CH3+yieldscomingfromdoubleionizationprocesses(tableIIfrom Wardsetal.[46])normalizedpresentlytotherecommendedCH₄⁺ yield[4].Thesethreesetsshowasimilarrangeofamplitudes.

Thedifferencebetweentherecommendedgrossandpresent BEB cross section remains unexplained only around 30eV, i.e.

wheredissociationsintoneutrals(CH₂andCH₃signals)reachtheir maxima[51].However,wearenotabletojudgeontheamplitude ofpossiblecontributiontothegrossionizationcrosssectioncom- ingfrompossibledissociationintoaneutral+ionizedfragmentvia anexcitationtosomehigherelectronicstates.Existingexperiments [51–53]arefragmentaryanddisagreewiththeories[54].Addition- ally,theBEBmodelwiththeverticalionizationpotentialprobably underestimates crosssectionsin thenear-thresholdregion, see theBEBcalculationwiththeadiabaticpotential[31].Experimen- talchecks[55,56]of theenergydependencesin thisregionare thereforeprecious.

3.2. Fluoromethanes(CH₃F,CH₂F₂,CHF₃)

Torresetal.[32,45,57]performedrecentlyaseriesofmeasure- mentsof partialand total ionizationcross sectionsupto85eV withadeclared uncertaintyless than10%.Morgan andLindsay [4] gave recommendedcross sectionsfor CH3F coinciding with themeasurementsofRejoubetal.[58].Forallthreemixedflu- oromethanes,CH₃F,CH₂F₂andCHF₃theagreementbetweenthe presentBEBmodelandrecentexperiments[25,45,57–60]isvery good,seeFig.4.ForCH₃FweshowalsothesumofH⁺andF⁺ion production[57],withamaximumof0.4×10⁻²⁰m²,butit does notmeannecessarilythattheseionscomefromdoubleionization events.

3.3. Tetrafluoromethane(CF₄)

Tetrafluoromethaneisthegasmostcommonlyusedforplasma etching in semiconductor industries [62], therefore numerous experimentshavebeenperformed,startingfromthedetermina- tionofthedissociation(ionizationplusneutral-dissociation)cross

Fig.5.IonizationcrosssectionsinCF4.Experimentalgrosstotal:Polletal.[24], BruceandBonham[64],Nishimuraetal.[42],Torresetal.[45],Sieglaffetal.[65,4];

Bonhamcounting[70];Sieglaffdouble(CF3++F⁺andCF2++F⁺)from[65];BEBKim iscomplete-active-spacecalculation;BEBpresentisRHF6-31**Gcalculation.

sectionof5.5×10⁻²⁰m²atitsmaximum[63].Allionizationevents in CF4 lead tothe dissociation of the molecule, with the CF3+

ionpredominant(3.3×10⁻²⁰m²atitsmaximum[4]).Someearly experimentssufferedfromincompletecollectionoflightfragment ions but newer results[24,42,45,64] forma congruent dataset, agreeingwithin±15%,seeFig.5.Outofseveralrecentexperimen- taldata,theresultsfromInnsbrucklaboratory[24]thataccount for thecorrectionof iontrajectoriesin theion sourceformthe highest set,givea maximum of 6.0×10⁻²⁰m² at120eVwhile theresultofNishimuraetal.[42]is5.3×10⁻²⁰m²at125–175eV.

ThecoincidencemeasurementsbySieglaffetal.[65]doneinabso- lutewaywith±5%uncertaintygaveatmaximumalowervalue, 4.95×10⁻²⁰m².

ThepresentBEBmodelagreeswellwiththerecentexperiments [42,45,64]andcoincideswiththeBEBcalculationdoneinslightly higher molecularorbitalbasis, RHF 6-311+G(d)[42], seeFig.5.

Kimandcollaborators[42]triedtoevaluatetheeffectofmultiple ionizationperformingBEBcalculationswiththeuseofacomplete- active-space wave functionsset (CAS, in Fig. 5).Such a model overestimatesthegrossionizationmeasurements[42,45,64],see Fig.5.

WhilstallionizationprocessesinCF4leadtothedissociationof themolecule,thedataofSieglaffetal.[64]showthatthedou- ble ionization(CF₂⁺+F⁺ andCF⁺+F⁺)islowerthan forCH₄ and reachesamaximumof0.23×10⁻²⁰m²at200eV,seeFig.5.This canbeexplainedbythehighelectronegativityoffluorineandits componentswhatmakestheformationoftwopositiveionsless probableinCF4thaninCH4(i.e.thesecondfragmentinthedisso- ciativeionizationofCF₄shouldpredominantlybeaneutral).This isalsoindirectlyconfirmedbystudiesofdissociationintoneutrals [52,63].ThecrosssectionoftheformationofanFatominionizing andneutral-dissociationeventsisabout7×10⁻²⁰m²atitsmaxi- mum[52]comparedto0.74×10⁻²⁰m²fortheformationoftheF⁺ ion[4].

Note,however,thatthedistinctionbetweengrossandcounting ionizationcrosssectionsofCF₄ isfarlessdecisivethanforCH₄. Awholeseriesofre-measurementsandre-analysiscomingfrom severalgroups[24,44,66–70]testifiesthis.

4. Towardsystematicsoftotalionizationcrosssections Presentanalysisofexperimentaluncertaintiesandcorrections bringcredibilitytothesystematictendenciesobservedbothinBEB

Table1

Lineardependenceofthemaximumcrosssections_max[10⁻²⁰m²]inpresentBEB modelvs.moleculardipolepolarizability˛[10⁻³⁰m³],experimentalvaluesfrom Ref.[73].

Molecule ˛[×10⁻³⁰m³] max[×10⁻²⁰m²] max/˛

CH4 2.593 3.571 1.377

CH3F 2.97 4.02 1.354

CH2F2 3.27^a 4.5 1.376

CHF3 3.57 4.92 1.378

CF4 3.838 5.273 1.374

aValueinterpolatedbetweenCH3FandCHF3.

(presentand[32])andinDM[42]estimatesofionizationcrosssec- tions.Inparticular,wereputethattheBEBmodelwith6-31**G molecularorbitalbasis reproducesthetotal countingionization cross sectionsfor the wholeseries CH₄ – CF₄ within5% accu- racy.Inthisseriesthemaximumofthecrosssectionrisesfrom 3.57×10⁻²⁰m²forCH4to5.27×10⁻²⁰m²forCF4.Thepositionof thecrosssectionmaximumrisesinasimilarmanner,from75eV inCH₄to140eVinCF₄.Thisratherreflectstheincreaseinenergy depthoforbitallevelsthanthechangesinthresholdvalues,14.13, 13.63,14.01,15.47and17.08eVforCH₄,CH₃F,CH₂F₂,CHF₃ and CF₄,respectively.

Inseveralworkssystematicdependencesofionizationcrosssec- tionswereexamined.Harlandetal.[71]studiedthecorrelationof atomicandmolecularmaximumionizationcrosssectionsmaxon thedipolepolarizability␣,checkingbothmax∝˛andmax∝√˛ proportionalities.Kimandcollaborators[42]postulatedforfluoro- carbonsCF₄,C₂F₆,C₃F₈amax∝√

˛ZdependencewithZbeingthe totalnumberofelectronsinthetarget.Asimilardependenceonthe polarizabilityintherangeof50–100eVandtheapplicationofthe additivityrulewasalsonoticedfortotal-scatteringcrosssections [72].

MaximaoftheBEBtotalionizationcrosssectionsmaxvs.the dipolepolarizabilities[73] of thefivemolecules consideredare resumedinTable1.Themaxrises proportionallytothedipole polarizability of the target with the proportionality coefficient 1.37(±0.01),max=1.37˛,ifmaxisexpressedin10⁻²⁰m²and˛ in10⁻³⁰m³.Thecorrelationofthelinearfitis0.999.

Thisrathersimpledependencedeservessomecomments.All fourclassical-likeformulaeconsideredhereindicatethatthenum- berofelectronsonmolecularorbitalsdeterminestheamplitude oftheionizationcrosssection. Ontheother hand,themolecu- lar polarizabilitycan beconsideredas a sumof polarizabilities fromseparatemolecularbonds.Secondly,fromclassicalelectro- dynamics,the polarizability is themeasure of the deformation oftheelectroniccloudintheexternalelectricfield.Fromquan- tummechanicalpoint ofview,thepolarizabilityisthemeasure forthesumofallvirtualexcitationsofthetarget:electronic[74]

andvibrational[75].Again,thevibrationalexcitationsreflectprop- ertiesofthemolecularbonds;forelectronicexcitationsthesum of oscillator strengths is equal to the number of valence elec- tronsevenifthedetailedelectronic-excitationspectraofmolecules canbequitecomplex[76].Therefore, presentlyobservedinter- dependenceoftheionizationcrosssectionandthepolarizability canreflectthesame,somewhatsyntheticmolecularfeature:the numberofvalenceelectronsconvolutedwiththeirbinding(and kinetic)energies.

Weareawarethattheseargumentsarenotconclusivesofur- theranalysisareneeded.Notealsothatforsakeofcomparisonson anequalbasisweusedpolarizabilitiesfromthesametypeofopti- calmeasurements[73].Theories,includingthepresentHFmethod, tendtounderestimatethemolecularpolarizabilities,evenifthe generaltrendfortheseriesofmoleculesispreserved.Thequality ofcalculationdependsmuchonthechoiceofthemolecularorbital basis,comparefore.g.[77,78].

5. Conclusions

Binary-encountermodels[13,14,17]provetobequitesuccess- ful in predictingionization cross sections. Whatbecomes clear fromthecaseofmethane,theseareratherexperimentalsingle- ionization(i.e.counting)crosssectionsthatagreewithKimand Rudd’s[14] BEBmodel.Outofnumerous experimentsonlyfew allowtoestimatethesingle-ionizationcrosssectionwhichisthe outputofthebinary-encountermodels.

Formethane,Wardetal.[46]showedthatat200eVmoreH⁺ ionscomefromthedoubleionizationprocessthanfromthesingle ones.Subtractingtheestimateforthedoubleionizationfromthe recommended[4]grossionizationoneleadstoanalmostperfect agreementwiththepresentBEBmodel(6-31**Gorbitalsandthe verticalionizationenergyof14.128eV).

MoreuncertaintyexistsonsuchadistinctionforCF₄andother fluoromethanes.Quantumchemistrycalculationsoftheenergetics ofspecificdissociationchannels–intoneutrals,intoneutralsand ions,andintoionpairs,areimportantinthiscontext[32].

Fromexperimentalside,singlecoincidence[46] andkinetic- energyreleasemeasurements[41]areimportantforunderstanding possibleionization channels.Such a knowledge is essential for projectingand diagnosisofplasmaprocessesin manytechnical applications.

DeutschandMärk’smodel[13,15,24]underestimatestheposi- tionoftheionizationmaximumbutthisshouldberatherattributed tothesimplifiedapplicationsofthemethod,whereonlythelowest ionizationthresholdsareconsidered,asinFig.1.AsshownbyTor- resetal.[32]whoincorporatedamorecompletesetofquantum levels,includinginnerorbitals,intotheircalculation,theagreement oftheDMmodelwithexperimentsisnotworsethanforBEB.

AnextensionoftheBEBmodeltodoubleanddissociativeion- ization wouldbe desirable.Already Gryzi ´nski [17] showedtwo possiblemechanismofdoubleionization:thedirectprocess,i.e.

thesecondionizationbythe(scattered)incomingelectron,andthe recoilprocess,i.e.ionizationbythe(leaving)secondaryelectron.It isnottobeexcludedthattheinformationonthedissociativeion- izationcanbealreadyextractedfromthestandardBEBcalculation.

InFig.3wecomparetheexperimentalH⁺yield[4]withtheioniza- tionfromthe2a₁molecularorbitalinthepresentBEBmodel:the twocurvesseemtocoincide.

Presentcalculationsdonottakeintoaccountthepossibilityof ionizingtransitionscomingfromhighervibrationalstates.Inother words,weassumethatthevibrationaltemperatureofthetarget moleculesis T=0.Asalready noticedbyKim andcollaborators, thevibrationalstructureoftheelectroniclevelswouldmodifythe ionizationcrosssectionsinthenear-to-thresholdregion.Further calculationswouldbeneededtoexplorethisquestionindetail.

Acknowledgments

ApartofthisworkhasbeendoneinQuantumOpticsCenter (COK),UMKToru ´n.NumericalcalculationsintheGAUSSIANcode havebeenperformedattheAcademicComputerCenterinGda ´nsk (TASK).WethankdrA.Karbowskiforthepreparatoryworkonthe analyticalfitsforCH₄crosssections.GKthanksforthehospitality atNFRI,Gunsan.

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