Elastic differential cross sections for electron collision with ... - Unicamp

J. Phys. B: At. Mol. Opt. Phys. 32 (1999) L539–L545. Printed in the UK

PII: S0953-4075(99)04395-3

LETTER TO THE EDITOR

Elastic differential cross sections for electron collision with CF3 and CF3 H Roberto B Diniz, Marco A P Lima and Fernando J da Paix˜ao Instituto de F´ısica ‘Gleb Wataghin’, UNICAMP, 13 083–970 Campinas SP, Brazil Received 18 May 1999, in final form 13 July 1999 Abstract. We report theoretical differential elastic cross sections for scattering of electrons by CF3 radical (trifluoromethyl or carbon trifluoride) and CF3 H (fluoroform). The calculations were performed with the Schwinger multichannel method at the static-exchange level. Our results for CF3 are the first cross sections for open-shell molecules with nonlinear geometry. There is no experimental data for e− –CF3 scattering but the theoretical data for CF3 H are in good agreement with recent experimental results of Tanaka. Our results show interesting similarities between elastic cross sections of e− –CF3 and e− –CF3 H scattering.

The understanding of collisions between electrons and molecules has important applications for several areas. For instance, a better understanding of cold plasma processing requires elastic, excitation, ionization and dissociation cross sections for a large number of species, including very reactive radicals (Garscadden 1992). CF4 plasma has important industrial applications in polymer surfaces deposition and as an etching material in microelectronics (Manos and Flamm 1989, d’Agostino 1990, Beenakker et al 1981). In a CF4 plasma radicals are produced by electron–molecule collisions. The first excited state of CF4 is dissociative, and as a result inelastic electron collision breaks the molecule producing several chemical species, such as CF3 , CF2 and CF radicals, F atoms, plus ionic radicals (Bonham 1994). Recently, Schwarzenbach et al (1997) measured the concentration of CFx radicals (x = 1, 2, 3) in a CF4 plasma, showing that CF3 has the greatest concentration among CFx . This result is in agreement with early measurements of total dissociation cross sections made by Nakano and Sugai (1992). They indicate that neutral production cross sections for CF3 are the highest when compared with other dissociations (10–300 eV). These results indicate that CF3 plays an important role in the dynamics of the CF4 plasma. Knowledge of the cross sections of the intermediate species is also important for plasma diagnosis techniques, such as the determination of radical concentrations (Schwarzenbach et al 1997), and to study the dynamics of discharges (Garscadden 1992). However, most radicals have a very short lifetime and are highly reactive. As a result, experimental measurement of the cross sections and even the detection of the radicals are a very difficult task (Tarnovsky and Becker 1993, Yamada and Hirota 1983). For example, Tarnovsky and Becker (1993) have measured absolute partial electron-impact ionization cross sections for CFx radicals. The neutral CFx beam was obtained from an ion beam, originating, in turn, from an electric discharge through CF4 . The presence of the radicals caused complex reactions that corroded the instruments. Consequently, the data reported on that work were obtained through combinations of several individual measurements. With such experimental complexities, 0953-4075/99/190539+07$30.00

© 1999 IOP Publishing Ltd

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there is a considerable lack of data on general electron–radical cross sections measurements. Consequently, theoretical results may play an important role in providing the necessary data. Several molecular radicals are open-shell systems. Electron–molecule scattering by these systems requires a theoretical treatment of spin coupling and unfilled orbitals different from that used for closed-shells molecules. Several simple rules used for closed-shell systems are no longer valid. As a result, electron open-shell molecule collision calculations are more complex and there is a small number of ab initio theoretical calculations (Noble and Burke 1992, da Paix˜ao et al 1992, 1996). In this letter we present theoretical results for elastic cross sections for scattering of electrons by CF3 radicals at the static-exchange level of approximation. To our knowledge this is the first ab initio result for open-shell molecules with arbitrary geometry (previous works were restricted to linear molecules). We also present theoretical results for elastic e− – CF3 H cross sections at the static-exchange level. In fact, the lack of experimental results on CF3 motivated the present comparative study with CF3 H, a stable closed-shell molecule with similar geometry. As discussed below, we have found interesting similarities between the two molecules and very good agreement between our calculated cross sections and Tanaka’s experimental data for CF3 H (Tanaka et al 1997). We use the Schwinger multichannel method (SMC) (Takatsuka and McKoy 1981), a multichannel version of the original Schwinger variational principle (Lippmann and Schwinger 1950). Details of the formulation have been given elsewhere (Lima et al 1990), and here we only present the formula for the scattering amplitude: 1 X hS E |V |ψm i(d −1 )mn hψn |V |SkEi i (1) fkEf kEi = − 2π mn kf where dmn = hψm |A(+) |ψn i

(2)

and A(+) =

 1 ˆ 1 (P V + V P ) + H − 21 (N + 1)(Hˆ P + P Hˆ ) − V G(+) P V. 2 N +1

(3)

Here, ψm are (N + 1)-particle Slater determinants in which the total wavefunction of the target + electron system is expanded. kEf and kEi are the wavenumber of the scattered and incident electron, respectively, and V is the potential due to the electromagnetic interactions of the incident electron with the electrons and the nuclei of the target. Also, SkEi is the product between a target wavefunction and a plane wave, P is a projector operator that projects a wavefunction onto the space spanned by the open-channel wavefunctions and G(+) P is the Green’s function projected by this operator. Finally, Hˆ = E − H , where H is the total Hamiltonian and E is the total energy of the system. To treat open-shell molecules we use the general expression for the scattering amplitudes for particles with spin, within a non-relativistic approximation (Rodberg and Thaler 1967, p 299): fkEf ,kEi (MSf mf , MSi mi ) =

X S,MS

1 2

S

CMfS

f

S Si 21 S (2S+1) C f mf MS Msi mi MS kEf ,kEi

(4)

where MSi (MSf ) is the spin component of the initial (final) target state, mi (mf ) is the spin S

1

component of the incident (scattered) electron and CMfS m2f MSS is a Clebsch–Gordan coefficient. f It is convenient to work with scattering amplitudes for the system (electron + target) with a

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defined total spin S. In this form the spin-irreducible scattering amplitudes f (2S+1) can be calculated directly. Codes have been developed to provide these spin-irreducible scattering amplitudes f (2S+1) , using formula (1). The differential cross section for elastic scattering of unpolarized electrons is given by  dσ = 41 3|f (3) |2 + |f (1) |2 . d

(5)

The code used in this letter is a new version of the open-shell code initially designed to study linear molecules (da Paix˜ao et al 1996). The present code is able to study molecules of arbitrary geometry. It also allows the use of pseudopotentials to describe the core of the target (Bettega et al 1993). This technique reduces substantially the computational effort and it will enable us to calculate cross sections of targets with a large number of electrons. We choose the basis functions as being Cartesian Gaussians, all of them taken from Varella et al (1999). Our calculation was performed in the following way. For the CF3 we used a set of 104 uncontracted Cartesian Gaussians. For both the carbon and the fluorine atoms we used 5s, 4p and 1d functions. For the CF3 H, we used a set of 111 contracted Cartesian Gaussians. For the carbon and the fluorine atoms we used the same basis of the CF3 and for the hydrogen atom we used 5s and 1p functions, contracted to 4s and 1p. The combinations of the contracted functions that represent both the target bound orbitals and the orbitals on the continuum were obtained by a Hartree–Fock self-consistent method. Table 1. Coordinates of the nuclei of CF3 H. Atom C H F1 F2 F3

x

y

z

0 0 0.701 279 386 0 0 2.776 199 125 0 2.351 419 93 −0.196 883 108 −2.036 3894 −1.175 709 97 −0.196 883 108 2.036 3894 −1.175 709 97 −0.196 883 108

Table 2. Coordinates of the nuclei of CF3 . Atom C F1 F2 F3

x

y

z

0 0 0.641 0088 0 2.367 5344 −0.135 0843 −2.050 3449 −1.183 7672 −0.135 0843 2.050 3449 −1.183 7672 −0.135 0843

The geometry of the CF3 H molecule was taken from the CRC Handbook of Chemistry and Physics (1990–1991). The Cartesian coordinates of the atoms in atomic units, with the origin on the carbon atom, is shown in table 1. The geometry of the CF3 molecule was taken from Yamada and Hirota (1983) and is shown in table 2, also in atomic units. Both of the molecules have pyramidal symmetry. The convergence of the numerical integrations necessary for the calculations was ensured by raising the quadratures on k, θ, φ until the results between two different quadratures differ by less than 3%. We also compared the integral cross sections calculated in the body frame and in the laboratory frame, to ensure convergence. In figure 1 we present our calculated differential cross sections for CF3 and CF3 H, together with Tanaka’s experimental results for CF3 H at incident electron energies of 6.5 7.0, 8.0 and

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Figure 1. Elastic differential cross sections for scattering of electrons by CF3 (full curve, present results); CF3 H (broken curve, present results) and CF3 H ( , experimental results by Tanaka et al 1997). Energies from 6.5 to 9 eV.

•

9.0 eV. Figure 2 is similar to figure 1 but with incident electron energies of 10.0, 15.0, 20.0 and 30.0 eV. In general, our theoretical results for CF3 H agree very well with those obtained experimentally by Tanaka, in particular for angles smaller than 60◦ . Although not shown, our results for CF3 H compare quite well with those obtained by Natalense et al (1999) using the same method but with a smaller basis set. For larger angles the agreement between theory and experiment improves as the incident electron energy increases. We interpret these results in the following way. CF3 H is a molecule with a permanent dipole, so this interaction should be the dominant effect at small scattering angles. This will mask other important effects at small angles. As a result, long- and short-range correlation effects, will play a more important role at larger scattering angles. For instance, the effect of the target polarization is more pronounced at smaller energies and lower partial waves. As the energy increases, this effect becomes relatively less important because there are a larger number of partial waves which are important for describing the scattering amplitude. The combination of these effects explain the trend for a better agreement at higher energies shown with our theoretical static-exchange calculation. There is no experimental results for the differential cross sections for CF3 . However, there is a striking similarity between the elastic differential cross section for CF3 and CF3 H. The similarity increases as the energy increases. This may indicates to us that the hydrogen atom has little influence on the differential cross sections of the CF3 H molecule at these energies. The reason for this may be the orientational averaging inherent in molecular targets in gas

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Figure 2. Same as figure 1, for energies from 10 to 30 eV.

phase collisions. Both CF3 and CF3 H have the same (pyramidal) symmetry and the distances between an F and a C atom differ by 1%. Then the effect of the H atom is reduced by the orientational average. This is also in agreement with the energy dependence of the differential cross section. Our results display a better agreement as the energy increases. Smaller energy means fewer partial waves important to the scattering calculation. Lower partial waves have smaller centrifugal barriers and as a result they are more sensitive to the molecular potential. As the energy increases more partial waves becomes important and they reduce the relative importance of each one. These considerations may indicate a way of predicting differential elastic cross sections for very reactive radicals from those of less reactive species. In figure 3 we present partial integral cross sections (PICS) for a given outgoing partial waves, for singlet and triplet scattering of electrons by CF3 . They are calculated in a fashion presented by Natalense et al (1999). PICS are obtained by summing over l and ml for the incident electron, summing over ml for the scattered electron, and taking an average over all molecular orientations. The angular momentum l is not a good quantum number for molecular scattering. However, partial cross sections are more sensitive to the presence of structures than the integral cross section. In addition, l may limit the irreducible representations present in a particular PICS which then helps to identify negative ion state. The PICS was calculated at energies starting from 2.5 eV up to 9.0 eV with steps of 0.5 eV, from 9.0 eV up to 18.0 with steps of 1.0 eV, and then steps of 2.0 eV up to 30 eV. The region around of 4.0 eV was studied from 3.6 up to 4.5 eV in steps of 0.05 eV. Our results show several structures below 15 eV. We have a singlet structure around 4.0 eV, in the S, P and D PICS. The presence of the

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Figure 3. Partial integral cross sections for elastic scattering of electrons by CF3 , for singlet and triplet scattering. Energies from 2.5 to 30 eV.

S outgoing channel implies the 1 A1 state of the negative ion. Indeed, we calculated the bound state energy of CF3− filling the last half-occupied orbital. The result was an energy close to the position of this structure. Shape resonances of this kind may be a very important mechanism of molecular dissociation. Although a static-exchange calculation overestimates the energy of a resonance, the obtained value of 4.0 eV is high enough to suggest that it remains in a more accurate calculation. We have structures of triplet character at energies of 6.5 and 9.0 eV in the P, D and F exit channels, but none at the S triplet. We also observe that singlet and triplet cross sections become very close at high energies, which is consistent with the reduction of the exchange interaction. We would like to thank the Brazilian computational center CENAPAD-SP (Centro Nacional de Processamento de Alto Desempenho em S˜ao Paulo), and also the Brazilian financial support agencies FAPESP (Funda¸ca˜ o para o Amparo a` Pesquisa do Estado de S˜ao Paulo) and CNPq (Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico). References Beenakker C I M, van Dommelen J H J and van de Poll R P J 1981 J. Appl. Phys. 52 480 Bettega M H F, Ferreira L G and Lima M A P 1993 Phys. Rev. A 47 1111 Bonham R A 1994 Japan. J. Appl. Phys. Part 1 33 4157 1991 CRC Handbook of Chemistry and Physics 72nd edn (Boca Raton, FL: Chemical Rubber Company) d’Agostino R (ed) 1990 Plasma Deposition, Treatment and Etching of Polymers (New York: Academic) da Paix˜ao F J, Lima M A P and McKoy V 1992 Phys. Rev. Lett. 68 1698 ——1996 Phys. Rev. A 53 1400 Garscadden A 1992 Z. Phys. D 24 97 Lippman B A and Schwinger J 1950 Phys. Rev. 79 469 Lima M A P, Brescansin L M, Silva A J R, Winstead C and McKoy V 1990 Phys. Rev. A 41 327 Manos D M and Flamm D L (eds) 1989 Plasma Etching, an Introduction (New York: Academic)

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Nakano T and Sugai H 1992 Japan. J. Appl. Phys. Part 1 31 2919 Natalense A P P, Bettega M H F, Ferreira L G and Lima M A P 1999 Phys. Rev. A 59 1 Noble C J and Burke P G 1992 Phys. Rev. Lett. 68 2011 Rodberg L S and Thaler R M 1967 Introduction to the Quantum Theory of Scattering (New York: Academic) Schwarzenbach W, Tserepi A, Derouard J and Sadeghi N 1997 Japan. J. Appl. Phys. Part 1 36 4644 Takatsuka K and McKoy V 1981 Phys. Rev. A 24 2473 Tanaka H, Masai T, Kimura M, Nishimura T and Itikawa Y 1997 Phys. Rev. A 56 R3338 Tarnovsky V and Becker K 1993 J. Chem. Phys. 98 7868 Varella M T do N, Natalense A P P, Bettega M H F and Lima M A P 1999 Phys. Rev. A accepted for publication Yamada C and Hirota E 1983 J. Chem. Phys. 78 1703