ELECTRON ELASTIC COLLISIONS WITHC 3 F 6 MOLECULE PAWEŁMOŻEJKO 1,2 ANDCZESŁAWSZMYTKOWSKI 1

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1 TASK QUARTERLY 7 No 2(2003), ELECTRON ELASTIC COLLISIONS WITHC 3 F 6 MOLECULE PAWEŁMOŻEJKO 1,2 ANDCZESŁAWSZMYTKOWSKI 1 1 AtomicPhysicsDivision, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics, Gdansk University of Technology, G. Narutowicza 11/12, Gdansk, Poland czsz@mif.pg.gda.pl 2 DepartmentofNuclearMedicineandRadiobiology, Faculty of Medicine, University of Sherbrooke, 3001,12 th AvenueNord,Sherbrooke(Québec)J1H5N4,Canada pawel.mozejko@courrier.usherb.ca (Received 10 August 2002) Abstract: We report calculations of differential and integral cross sections for intermediate- and high-energy( eV)elasticcollisionsofelectronswithhexafluoropropene(C 3F 6)molecules. The calculations have been carried out using the independent atom model with static-polarization model potential. The present results are compared with elastic cross sections estimated from total and ionization experiments. Agreement between present calculations and the experiment is good for energies above 70eV. Keywords: electron scattering, elastic cross section, hexafluoropropene, independent atom method 1. Introduction Fullyfluorinatedhydrocarbons(e.g.CF 4,C 2 F 6,C 3 F 8 )andsf 6 aregaseswith wide technological applications: plasma etching, deposition and cleaning, pulse power switching, gaseous dielectrics[1, 2]. These perfluorides are also of atmospheric and environmental concerns due to their rather long residence time in the environment and, in consequence, a high global warming potential(gwp)[3]. Therefore, there is an increasing demand for environmentally friendly alternatives(substitutes) for compounds used so far. One of the replacement candidates, with a relatively low GWP, ishexafluoropropene(c 3 F 6 )[4].Thee - C 3 F 6 scatteringprocesseshavereceivedalittle experimental and theoretical attention. A few experimental works concentrate on the dissociative electron attachment[5 9] and electron-induced ionization processes[7, 10 12]. Jiang et al.[13] calculated the total(elastic + inelastic) cross section at intermediate and high impact energies( eV) using a simple additivity rule TQ207B-E/ X 2003 BOP s.c.,

2 172 P. Możejko and Cz. Szmytkowski (AR) and an energy-dependent geometric additivity rule(egar). Very recently, Szmytkowski et al.[14, 15] have presented an absolute total cross section for electron collisionswithc 3 F 6 moleculemeasuredwithatransmissiontechniqueforincident electron energies between 0.5 and 370eV. Preliminary results of integral elastic cross section calculations for energies ranging from 30 to 300eV have been also reported[14]. We are not aware of any other calculated or measured elastic cross sections for electron scatteringfromc 3 F 6 molecules. 2. Theory The present calculations have been carried out using the independent-atom method(iam)[16, 17] with static-polarization model potential. In this approximation, the electron-molecule collision problem is reduced to electron-atom collision, assuming that each atom of the molecule scatters independently, any redistribution of atomic electrons due to the molecular binding is unimportant, and multiple scattering within the molecule is neglected[16]. This approach offers reasonable approximations to elastic, momentum transfer and total cross sections for intermediate- and highenergy electron and/or positron scattering from many polyatomic molecular targets (see e.g.[13, 17 28] and references therein). The electron-impact ionization cross sectionsforamoleculeareobtainedasasumoftherelevantatomiccomponents[29]. In the simple form of independent atom approximation, the differential cross section (DCS) for elastic electron scattering on a molecule, taking into account all possible orientations of the intermolecular axis, is given as: N dσ dω = i N j f i (θ,k)f j(θ,k) sin(sr ij) sr ij, (1) wherenisthenumberofatomsinthemolecule,θisthescatteringangle,andkisthe incidentelectronwavenumber;f i (θ,k)andf j (θ,k)arecomplexscatteringamplitudes duetothei th andj th atomofthemolecule,respectively,ands=2ksin(θ/2)isthe magnitudeofthemomentumtransferduringthecollision.r ij istheinternuclear distancebetweenthei th andj th atomofthetargetmolecule.inthisinvestigation, thedistancesr ij inthec 3 F 6 moleculeareobtainedusinganoptimizationprocedure with GAMESS code[30], taking as the starting point the experimental geometry of thec 3 F 6 moleculepresentedbyloweryetal.[31].inallequations,atomicunitsare usedinwhiche=m e = h=1,althoughthefinalresultsofthecalculationsaregiven insiunits. It follows from the optical theorem that the integral cross section(ics) for electron elastic scattering on the molecule in the IAM approximation is given by: σ(e)= 4π k Imf(s=0,k)= = 4π k N N Imf i (θ=0,k)= σ i (E), (2) i=1 whereσ i (E)istheintegralcrosssectionofthei th atomofthemoleculeatenergy E=k 2 /2. i=1 TQ207B-E/ X 2003 BOP s.c.,

3 ElectronElasticCollisionswithC 3F 6Molecule 173 To obtain the atomic scattering amplitudes and elastic electron-atom cross sections, we have employed partial wave analysis and solved numerically the radial Schrödinger equation [ d 2 dr 2 l(l+1) under the boundary conditions ] r 2 +k 2 2(V stat (r)+v polar (r)) u l (r)=0 (3) u l (0)=0, u l (r) r A l ĵ l (kr) B lˆn l (kr), (4) whereĵ l (kr)and ˆn l (kr)arethesphericalbessel-riccattiandneumann-riccatti functions,respectively.v stat (r)isthestaticpotentialoftheatomdeterminedfollowing theprocedureofsalvatetal.[32]: V stat (r)= Z 3 a n exp( β n r), (5) r n=1 wherezisthenuclearcharge,anda n andβ n aretheparametersdeterminedbyan analytical fitting procedure to Dirac-Hartree-Fock-Slater self-consistent data[32]. The polarizationpotentialv polar (r)isexpressedintheformproposedbypadialetal.[33]: { v(r) r rc V polar (r)= α/2r 4, (6) r>r c where v(r) is the free-electron-gas correlation energy[34], and α is the static electric dipolepolarizabilityoftheatom.ther c isthefirstcrossingpointofthev(r) and α/2r 4 curves[35].theexchangeeffectsaresupposedtobesmallatthe incident energies considered in the present investigation and hence are neglected. The scattering amplitudes for electron scattering on the atom are obtained using the following equation: f(θ,k)= 1 l max 2ik +παk l=0 (2l+1)(e 2iδ l 1)P l (cosθ)+ ( lmax sinθ 2 l=1 ) P l (cosθ), (7) (2l 1)(2l+3) wherep l (cosθ)arelegendrepolynomialsandthesecondterminequation(7)isthe Born scattering amplitude for potential of the form(6). In the presented calculations l max =100.Thephaseshiftsδ l areconnectedwiththeasymptoticformofthewave function,u l (r),by: tanδ l = B l. (8) A l The differential cross section for elastic electron scattering from a particular atom was calculated according to: dσ dω = f(θ,k) 2, (9) while atomic integral elastic cross section was derived from the following expression: σ= 4π k 2 l max (2l+1)sin 2 δ l + l=0 (2l+1)sin 2 δ (B) l l=l max+1. (10) Since the above approach has yielded quite encouraging results for molecular targets oftetrahedralsymmetrylikexy 4 (X=C,Si,Ge;Y=H,F,Cl)andforC 2 F 6 [36,37],it TQ207B-E/ X 2003 BOP s.c.,

4 174 P. Możejko and Cz. Szmytkowski is expected that the present differential and integral cross sections for electron elastic scatteringbyc 3 F 6 mayalsobefairlyreliable. 3. Results The present differential cross sections(dcss) for electron elastic-scattering fromc 3 F 6 moleculecalculatedatincidentenergyrangingfrom50to1000evare showninfigures1and2. Figure 1. Differential cross section for elastic electron collisions withc 3F 6moleculesatintermediateenergies The values of calculated DCS and integral cross sections(icss) at various energies are also presented in Table 1. There are no elastic measurements and calculations to compare with the present results. It is evident from both figures that, for all investigated energies, the qualitative nature of the calculated DCSs is almost the same. DCS exhibits one distinct minimum TQ207B-E/ X 2003 BOP s.c.,

5 ElectronElasticCollisionswithC 3F 6Molecule 175 Figure 2. Differential cross section for elastic electron collisions withc 3F 6moleculesathighenergies at the scattering angle located between 80 and 90. For lower scattering angles, in the angular range, some additive weak features are barely discernible. With the increasing incident energy, the DCS increases at near-zero degree scattering angles and decreases for backward scattering. The integral elastic cross section is presented in Figure 3 together with experimental elastic cross section estimated as a difference of the absolute grand total cross section[14] and the total ionization cross section data[12]. Although, at the investigated energies, inelastic processes are dominated by ionization, the experimental elastic cross section evaluated in this way can differ slightly from the true elastic cross section, as we have neglected the contribution from non-ionizing inelastic collisional processes. In the investigated energy range, the elastic cross section is a monotonically decreasing function of the incident electron energy. At the lowest energies studied, viz. below 70eV, the applied approximation gives values of TQ207B-E/ X 2003 BOP s.c.,

6 176 P. Możejko and Cz. Szmytkowski Table 1. Calculated differential(dcs) and integral(ics) cross sections (in10 20 m 2 /srand10 20 m 2,respectively)fore C 3 F 6 elasticcollisions Angle Energy E[eV] [deg] σ(e) integral elastic cross section which distinctly exceed the experimental estimation. This disagreement may be due to neglecting the bond distortion and multiple scattering at impact energies at which electron wavelength becomes comparable to the internuclear distances. Above 70eV, agreement between theory and experiment is satisfactory. Based on our calculations, we have estimated that, for energies ranging from 100 to 370eV,theelasticprocessesintheC 3 F 6 moleculesconstituteupto70 60%ofthe total cross section, which seems to be characteristic for perfluorinated targets[14]. It is well known that total cross sections measured with a transmission technique can TQ207B-E/ X 2003 BOP s.c.,

7 ElectronElasticCollisionswithC 3F 6Molecule 177 Figure 3. Comparison of integral cross sections: elastic, present calculations(iam); experimental elastic, estimated as a difference of absolute total[14] and ionization[12] cross sections; experimental absolute total cross section[14]; theoretical total cross section (EGAR model)[13]; theoretical total cross section(ar model)[13] be lowered by incomplete discrimination of electrons which are scattered throughout small angles in the forward direction[38]. This is mainly due to electrons which undergo elastic scattering, as the electrons which are scattered inelastically in the very-near-forward direction can be discriminated using an electron-energy filter. The uncertainty in the total cross section, related to this effect, increases with the impact energy.theamount, σ tot (E),bywhichthetotalcrosssectionisloweredduetothis effect can be roughly estimated as follows: σ tot (E)= dσ dω Ω, (11) θ=0 wheredσ/dω θ=0 isthedifferentialcrosssectionforelasticelectronscatteringatzero degree angle and Ω is the angular acceptance of the detector. Having in hand angular distributions of scattered electrons calculated in the present work, we can use them to estimatethe σ tot (E)forabsoluteexperimentalC 3 F 6 data[14].respective σ tot (E) valuesdonotexceed2%for eV,2.5%at350eVand3%at370eV. 4. Conclusions In this work the differential and integral cross sections for the elastic scattering of electrons from hexafluoropropene have been calculated, for impact energies from 50 to 1000 ev. The present results, obtained using the independent atom approximation with static-polarization model potential, are in reasonable agreement with elastic cross sections derived from the experimental total and ionization cross sections. TQ207B-E/ X 2003 BOP s.c.,

8 178 P. Możejko and Cz. Szmytkowski Acknowledgements This work was partly sponsored by Polish State Committee for Scientific Research(KBN). References [1]ChristophorouLG,OlthoffJKandRaoMVVS1996J.Phys.Chem.Ref.Data [2]ChristophorouLGandOlthoffJK2001Adv.At.Mol.Opt.Phys.4459 [3] Abatement of Emissions of Global Warming Gases, Praxair Inc., Danbury, 1997 [4]NakamuraM,HoriM,GotoT,ItoMandIshiiN2001J.Vac.Sci.Technol.A [5]BibbyMMandCarterG1966Trans.FaradaySoc [6]HarlandPWandThynneJCJ1972Int.J.MassSpectrom.IonPhys.9253 [7]LifshitzCandGrajowerR Int.J.MassSpectrom.IonPhys.1025 [8]HunterSR,ChristophorouLG,McCorkleDL,SauersI,EllisHWandJamesDR1983J. Phys.D16573 [9]JarvisGK,PeverallRandMayhewCA1996J.Phys.B29L713 [10]ChelobovFN,DubovSS,TikhomirovMVandDobrovitskiiMI1963Dokl.Akad.Nauk SSSR (in Russian) [11]LifshitzCandLongFA1965J.Phys.Chem ;ibid.3741 [12]BartM,HarlandPW,HudsonJEandVallanceC2001Phys.Chem.Chem.Phys.3800 [13]JiangY,SunJandWanL2000Phys.Rev.A [14]SzmytkowskiCz,MożejkoPandKwitnewskiS2002J.Phys.B [15] Szmytkowski Cz, Kwitnewski S, Możejko P and Ptasińska-Denga E 2002 Phys. Rev. A [16]MottNFandMasseyHSW1965TheTheoryofAtomicCollisions,Oxford [17]RajD1991Phys.Lett.A [18]KhareSP,RajDandSinhaP1994J.Phys.B [19]SunJ,JiangYandWanL1994Phys.Lett.A19581 [20]JiangY,SunJandWanL1995Phys.Rev.A52398 [21]JoshipuraKNandVinodkumarM1997Z.Phys.D41133 [22]LiuY,SunJ,LiZ,JiangYandWanL1997Z.Phys.D4245 [23]RajDandTomarS1997J.Phys.B [24]RaizadaRandBalujaKL1997Phys.Rev.A [25]MajiS,BasavarajuG,BharathiSM,BhushanKGandKhareSP1998J.Phys.B [26]JoshipuraKNandVinodkumarM1999Eur.Phys.J.D5229 [27]ReidDDandWadehraJM1999Chem.Phys.Lett [28]dePablosJL,KendallPA,TegederP,WilliartA,BlancoF,GarciaGandMasonNJ2002 J.Phys.B35865 [29]MargreiterD,DeutschH,SchmidtMandMärkTD1990Int.J.MassSpectrom.IonProc [30]SchmidtMW,BaldridgeKK,BoatzJA,ElbertST,GordonMS,JensenJH,KosekiS, MatsunagaN,NguyenKA,SuS,WindusTL,DupuisMandMontgomeryJrJA1993J. Comp. Chem [31]LoweryAH,GeorgeC,D AntonioPandKarleJ1979J.Mol.Structure53189 [32]SalvatF,MartinezJD,MayolRandParelladaJ1987Phys.Rev.A36467 [33]PadialNTandNorcrossDW1984Phys.Rev.A [34]PedrewJPandZungerA1981Phys.Rev.B [35]ZhangX,SunJandLiuY1992J.Phys.B [36] Możejko P, Żywicka-Możejko B and Szmytkowski Cz 2000 Uzhhorod Univ. Sci. Herald, Physics [37] Możejko P, Żywicka-Możejko B and Szmytkowski Cz 2002 Nucl. Instr. and Meth. in Phys. Res.B [38]SzmytkowskiCz,MożejkoPandKasperskiG1998J.Phys.B TQ207B-E/ X 2003 BOP s.c.,