Electron impact scattering by SF₆ molecule over an extensive energy range

Abstract

The present article reports the theoretical total cross sections for e-SF₆ scattering over the energy range 0.1–5000 eV. For low-energy calculations up to the ionization threshold of the target, the ab initio R-matrix formalism is employed, and beyond that energy the spherical complex optical potential method is used. Elastic, electronic excitation, rotational excitation, momentum transfer and total cross sections were calculated and the results of the low-energy calculations are presented. Differential elastic cross sections for various energies are also reported here. We have identified and detected two resonances at energies of 5.43 and 17.02 eV with the possible formation of anions. The present results show reasonable agreement with the existing theoretical and experimental results. The current work is the first report of the rotational excitation cross sections for the e-SF₆ scattering system.

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I. Introduction

Sulphur hexafluoride (SF₆) is one of the most important fluorine-containing feed gases¹ in current use. This molecule is suitable for, and used in, the semiconducting industry for dry plasma etching in material processing.² The insulating properties of this molecule make it in demand in electric power technology, particularly as a gaseous dielectric material.³ However, SF₆ is a potent greenhouse gas with a global warming potential 23 900 times as strong as that of CO₂ and a 3200 year residence lifetime in the environment.⁴ SF₆ is also an efficacious infrared absorber and hence plays a key role in the photochemistry of the atmosphere as an ozone-depleting molecule.⁵ This molecule is used appreciably in the magnesium and aluminium industry as a heavy and inert cover gas to isolate molten magnesium from oxygen and to reduce the porosity of cast aluminium, particularly in casting operations.⁶ SF₆ is very suitable as an interrupting medium, since it is a ‘self-healing’ dielectric and is stable in the decomposition process.⁷ Hence, determinations of electron collision cross sections for SF₆ over a wide energy range are indispensable, particularly to study its physio-chemical processes in an electron-rich gaseous medium.

In recent years, various theoretical and experimental investigations of electron-impact scattering by SF₆ have been reported by different groups owing to this molecule's environmental impact and considerable applications in industry as discussed earlier. Makochekanwa et al.,⁸ Dababneh,⁹ Kennerlya et al.,¹⁰ Kasperski et al.,¹¹ Zecca et al.,¹² Wan et al.,¹³ Ferch et al.,¹⁴ Trajmar et al.,¹⁵ Limão-Vieira et al.¹⁶ and Rohr¹⁷ measured total cross sections of electron impact on SF₆, while Cho et al.,¹⁸ Srivastava et al.,¹⁹ Johnstone and Newell²⁰ and Sakae et al.²¹ reported the experimental differential and integral elastic cross sections. Benedict and Gyemant²² calculated electron impact total elastic cross sections using a multiple scattering method. The momentum transfer cross sections were obtained by Phelps and Van Brunt²³ and Christophorou and Olthoff.²⁴ Winstead and McKoy²⁵ calculated differential, elastic and momentum transfer cross sections for electron scattering of the SF₆ molecule using the Schwinger Multichannel (SMC) method in the energy range 0.5–75 eV, while Dehmer et al.²⁶ reported theoretical cross section results for elastic collision impact energies between 0–40 eV. Limão-Vieira et al.¹⁶ calculated total cross sections for the 100–10 000 eV energy range using an independent atom model approximation and a modified single-center additivity rule. Calculations for differential, elastic, and momentum transfer cross sections have also been performed by Gianturco et al.,²⁷ Gianturco and Lucchese,²⁸ Jiang et al.,²⁹ Johnstone and Newell³⁰ applying different theoretical formalisms. Shi et al.³¹ and M. Vinodkumar et al.³² calculated total cross sections between 30–5000 eV and 15–2000 eV for electron scattering by the SF₆ molecule using the modified additivity rule and a modified single-centre additivity rule (MSC-AR) respectively. For energies of 10–2000 eV, Joshipura et al.³³ used spherical complex optical potential (SCOP) formalism to calculate total cross sections for electron scattering of the SF₆ molecule (“e-SF₆ scattering”). Fabrikant et al.³⁴ calculated integral, elastic, and differential cross sections between 0.2–5 eV using the effective-range theory. A survey of the literature about electron scattering of SF₆ is given in Table 1.

Table 1 Review of literature for e-SF₆ scattering

Energy range (eV)	Cross section	Reference	Method (Exp-experimental; Th-theoretical)
1–500	TCS	Dababneh^9	Beam transmission technique (Exp)
0.5–75	Elastic, DCS, MTCS	Winstead and McKoy^25	SMC (Th)
0–40	Elastic	Dehmer et al.^26	Multichannel model (Th)
0.5–100	TCS	Kennerlya et al.^10	Time-of-flight analysis (Exp)
0.8–100	TCS	Makochekanwa et al.^8	Linear transmission type time-of-flight instrument (Exp)
0.036–1	TCS	Ferch et al.^14	Time-of-flight mass spectrometry (Exp)
0.03–1	TCS	Trajmar et al.^15	Time-of-flight technique (Exp)
2.7–75	Elastic, DCS	Cho et al.^18	Crossed electron-molecular beam spectrometer (Exp)
100–10 000	TCS	Limão-Vieira et al.^16	Transmission beam system (Exp)
			Independent-atom approximation and a modified single-center additivity rule (Th)
meV–100 eV	Elastic, DCS, MTCS	Gianturco and Lucchese^28	Ab initio static exchange correlation polarization (SECP) potential approach (Th)
100–700	Elastic, DCS, MTCS	Jiang et al.^29	Independent-atom model with partial waves (Th)
75–700	Elastic, DCS, MTCS	Sakae et al.^21	Crossed beam method (Exp)
75–4000	TCS	Zecca et al.^12	Ramsauer-type electron spectrometer (Exp)
30–5000	TCS	Shi et al.^31	Modified additivity rule (Th)
15–2000	TCS	M. Vinodkumar et al.^32	MSC-AR (Th)
10–2000	TCS, ionization, excitation	Joshipura et al.^33	SCOP formalism (Th)
0.2–5	Integral, DCS	Fabrikant et al.^34	Effective range theory (Th)
0–30	Elastic, DCS	Gianturco et al.^27	Ab initio exact static exchange plus polarization (SEP) approach with close-coupling (CC) formulation (Th)
5–75	Elastic, DCS, MTCS	Johnstone and Newell^30	Hemispherical electron spectrometer (Exp)
0.3–10	Integral, DCS	K Rohr^17	Crossed-beam technique (Exp)
0–12	TCS	Wan et al.^13	Electron transmission spectrometer (Exp)
10–60	Elastic	Benedict and Gyemant^22	Multiple scattering method (Th)

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From Table 1, it is clear that there is a lot of interest in the electron-impact cross section of the SF₆ molecule. Total cross-section (TCS) studies of this molecule have been both theoretical and experimental, and have been comprehensive. However, there have not been any results reported for its electronic-excitation or rotational-excitation cross sections, and only fragmentary reports for its differential cross section (DCS) and momentum-transfer cross section (MTCS). Besides, most of the previous studies were confined to a small energy range. Different cross-section data for various scattering channels covering a wide energy range are still lacking.

In this article we report total cross section for e-SF₆ scattering for impact energies from 0.1–5000 eV. The differential, electronic excitation and rotational excitation cross sections are also calculated for low energies. Resonances are located at two different energies with the possibility of anion formation.

The organization of the paper is as follows: Section II explains the theoretical methodologies employed in the calculations, Section III describes and discusses the results obtained and Section IV summarizes and concludes this work.

II. Theoretical methodology

In this article, electron-impact cross sections of the SF₆ molecule are reported. For this purpose two formalisms were employed: ab initio R-matrix³⁵ calculations using the Quantemol-N³⁶ module for low energies and SCOP^37–40 formalism for intermediate to high energies. These methods are separately explained in the following subsections, after a description of the target model employed for the low-energy calculations.

A. Target model

SF₆ is an octahedral molecule with an S–F bond length of 1.561Ų.⁴¹ A 6-31G Gaussian basis set is employed for the representation of the target wave function, since the wave function of the system becomes converged up to the ionization threshold of the target by using the current basis set. The SF₆ molecule is considered to be in the D_2h point group (subgroup of the O_h point group) for the presented low-energy calculations. The ground-state Hartree–Fock electronic configuration for the SF₆ molecule is represented as 1A_g², 2A_g², 1B_2u², 1B_3u², 1B_1u², 1B_1g², 3A_g², 4A_g², 2B_3u², 2B_2u², 2B_1u², 5A_g², 3B_3u², 3B_2u², 3B_1u², 2B_1g², 6A_g², 7A_g², 4B_2u², 4B_3u², 4B_1u², 8A_g², 1B_2g², 1B_3g², 1A_u², 5B_3u², 5B_2u², 6B_3u², 6B_2u², 5B_1u², 3B_1g², 2B_2g², 2B_3g², 4B_1g², 9A_g². Out of the 70 electrons we have frozen 66 electrons in the 1A_g, 2A_g, 3A_g, 4A_g, 5A_g, 6A_g, 7A_g, 8A_g, 1B_3u, 2B_3u, 3B_3u, 4B_3u, 5B_3u, 6B_3u, 1B_2u, 2B_2u, 3B_2u, 4B_2u, 5B_2u, 6B_2u, 1B_1g, 2B_1g, 3B_1g, 1B_1u, 2B_1u, 3B_1u, 4B_1u, 5B_1u, 1B_2g, 2B_2g, 1B_3g, 2B_3g, and 1A_u molecular orbitals, while the remaining 4 electrons were allowed to move freely in active space of the 9A_g, 10A_g, 11A_g, 12A_g, 7B_3u, 8B_3u, 7B_2u, 8B_2u, 4B_1g, 6B_1u, 7B_1u, 3B_2g, and 3B_3g orbitals. A total of 349 configuration state functions (CSFs) are considered in the close-coupling expansion for the representation of seventeen target states.

For the static exchange plus polarization (SEP) calculations using the ab initio R-matrix method, we first generated target properties by constructing the transition density matrix utilizing GAUSPROP and DENPROP⁴² modules of the UK R-matrix software suite. The multipole transition moments in inner-region calculations were obtained using second-order perturbation theory and the property integrals computed by GAUSPROP.⁴²

The SEP calculation in the present study resulted in a ground-state energy of −993.5344 hartree for the SF₆ molecule, which is in good agreement with the theoretical value −993.786672 hartree.²⁸ The rotational constant of 0.09102 cm⁻¹ in the current study matches very well with the experimental value of 0.09107 cm^-1 (ref. 43) and the calculated value of 0.090686 cm^-1.⁴⁴ The computed dipole moment for SF₆ is zero, which agrees with the previously measured dipole moment.⁴⁵ The first electronic excitation energy for SF₆ was found to be 11.5942 eV, showing good agreement with the calculated value 11.19.²⁵ These target properties, along with available results from the literature, are given in Table 2.

Table 2 Target properties calculated in the current study, along with previous results

Properties of SF6	Present	Experimental	Theoretical
Ground-state energy (hartree)	−993.5344	—	−993.786672 (ref. 28)
First excitation energy (eV)	11.5942	—	11.19 (ref. 25)
Rotational constant (cm^−1)	0.09102	0.09107 (ref. 43)	0.090686 (ref. 44)
Dipole moment (D)	0	0 (ref. 45)	—

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B. Formalism for low-energy calculations (1–18 eV)

The ab initio R-matrix method is one of the most useful formalisms for low-energy calculations. The low-energy scattering problem can also be represented by Green's function, which is a useful tool in mathematical physics. For the representation of electron scattering in a general scattering geometry, Wong⁴⁶ introduced the properties and solution of Green's functions. Recently, Altunata et al.⁴⁷ applied an ab initio R-matrix method using an iterative Green's function for calculating the molecular reaction matrix of scattering theory with smooth energy dependence. This method mostly takes care of all the polarization effects, considers both polar and nonpolar ion cores in a unified fashion, and is equally valid for both short- and long-range potentials. The closed-form analytic expressions for one- and two-electron integrals of Cartesian Gaussian orbitals outside the R-matrix sphere is given by Wong et al.,⁴⁸ which can be used in ab-initio molecular scattering calculations. The R-matrix method is the most widely used of the ab initio methods.

In the present work we have employed the R-matrix method in the Quantemol-N³⁶ package. The fundamental concept behind R-matrix formalism is based on the division of configuration space into two specified regions: namely the inner and outer regions. The inner region is chosen such that it fits all the target wave functions of the molecule. For the present molecule we have taken the inner-region radius as 13a₀. This value is chosen such that the result becomes convergent. All N+1 electrons are confined to this region, which makes the inner-region calculations complex, but definite. In this region, various short-range potentials viz. static, exchange, absorption and electron–electron correlation polarization become influential as the wave functions are compact in an inner-region problem. Whereas in the outer region, exchange and correlation potentials are assumed to be negligible and only long-range multipolar interactions between target and scattering electrons become dominant. For simple and fast computations, the outer-region problem is approximated as a single centre and for the present calculations the outer-region radius is expanded up to 100a_0.

The wave function for the system using close-coupling approximation⁴⁹ for the inner-region problem can be specified as

where A is the anti-symmetrization operator imposed on electrons in the inner region to obey the Pauli principle, x_N is the spatial and spin coordinate of the N^th electron, Φ_i^N is the i^th state of the N-electron target, which is represented using a configuration integration (CI) expansion, and ζ_j is the continuum orbital spin coupled with the target states. The coefficients a_Ijk and b_mk are variational parameters. The second summation in eqn (1) contains functions χ_m, which describes all the N+1 electrons as L² configurations that disappear at r = a. They are included to relax the scattering and target orbitals of equal symmetry in the molecule. The single excited L² term incorporates the polarization effect to the Hartee–Fock (HF) ground-state wave function. This scattering model is denoted as the static exchange plus polarization model. Here, lowest numbers of target states are used for the close-coupled calculations. This is included to account for the orthogonality relaxation and short-range polarization effects by a CI expansion in the first summation and over a hundred configurations in the second.

For the molecular electronic-structure calculations, the target wave functions are represented by basis-function expansion. The Gaussian-type orbitals (GTOs) and the continuum orbitals of Faure et al.⁵⁰ are utilized in the Quantemol-N quantum-chemical package. The advantage of GTOs is that the multi-centred integral can be treated analytically to achieve fairly improved accuracy in the calculations. The inner-region calculation is propagated to the outer-region potential, until its solutions agree with the asymptotic functions given by the Gailitis expansion.⁵¹ Hence, the R-matrix maintains a bridge between the inner and outer regions. In the outer-region problem, the coupled single-centre equations are integrated to deduce all the observables by employing K matrices. The K matrices are employed to obtain T matrices using the definition

The T matrices are then used to calculate the cross sections in the outer region. To identify the position and width of the resonances, the eigenphase sum is fitted to a Breit–Wigner form.⁵² The differential cross-section calculations are performed by processing of K matrices using the approach reported by Sanna and Gianturco.⁵³

C. Formalism for high-energy calculations

The elastic and inelastic processes play crucial roles in electron–molecule interaction systems at high-impact energies too. In the present work the high-energy calculations are performed using the spherical complex optical potential (SCOP) formalism.^37–40 In the SCOP method, a complex potential is constructed and used to solve the Schrödinger equation. The potential may be expressed as

V_opt(r, E_i) = V_R(r) + iV_I(r, E_i) (3)

where the real part is given by

V_R(r, E_i) = V_st(r) + V_ex(r, E_i) + V_p(r, E_i) (4)

The static potential V_st(r) is a function of the radial vector (r) only, whereas exchange and polarization potentials depend on both r and incident energy of the particle (E_i). Utilizing the unperturbed Hartree–Fock wave function, we can calculate the static potential, V_st(r). The short-range correlation and the long-range polarization effect is expressed by V_p(r, E_i) and the electron exchange interaction is given by V_ex(r, E_i). All the potentials in eqn (4) depend on the electronic charge density of the molecule. The parameterized Hartree–Fock wave functions of Salvat et al.⁵⁴ are applied here to find the radial electron charge density of the molecule.

The exchange potential V_ex is formulated using the ‘Hara free-electron gas exchange' model⁵⁵ and the polarization potential V_p is formulated using the parameter-free model of correlation-polarization potential given by Zhang et al.⁵⁶ For consistent results in the intermediate region, Zhang et al.⁵⁶ have incorporated various non-adiabatic corrections that approach the correct asymptotic form at large ‘r’. The two non-spherical terms, vibrational and rotational excitations of the target, are not included for high-energy calculations in our model potential. This is justified because the time of interaction of the incident electron with the target in the intermediate- to high-energy range is sufficiently small compared to the vibrational and rotational times, and hence the cross sections due to these processes are negligible.

The absorption potential V_abs accounts for the total loss of flux due to excitation and ionization through these scattering channels. To represent this we have used a model potential given by Staszewska et al.⁵⁷ This absorption term is represented as

where the local kinetic energy of the incident electron is

T_loc = E_i − (V_st + V_ex + V_p) (6)

In eqn (5), p² = 2E_i, k_F = [3π²ρ(r)]^1/3 is the Fermi wave vector and A₁, A₂ and A₃ are dynamic functions that depend differently on θ(x), I, Δ and E_i. I is the ionization threshold of the target, θ(x) is the Heaviside unit step-function and Δ is an energy parameter below which V_abs = 0. So Δ is an important factor that determines the values of total inelastic cross section, and below this energy the ionization or excitation is not allowed. This is one of the notable features of the Staszewska model.⁵⁷ However, fixing Δ = I will restrict all the inelastic processes with a threshold lower than the ionization potential. This is a serious drawback of the theory and to fix this we have considered Δ as a slowly varying function of E_i around I. Also, if Δ is much smaller than the ionization threshold, then V_abs becomes unexpectedly high near the peak position. So we have introduced a minimum value of 0.8I to Δ and varied it as a function of E_i around I by the following formula:

Δ(E_i) = 0.8I + β(E_i − I) (7)

The parameter β is determined by considering that Δ = I(eV) at E_i = E_p, the value of incident energy at which Q_inel is maximized. E_p can be found by calculating Q_inel and by keeping Δ = I. Beyond E_p, Δ is kept as I. The expression given in eqn (7) is meaningful, as it would allow electronic excitations below the ionization potential as well.

The radial Schrödinger equation is solved by using the full complex optical potential, given in eqn (3). The solutions of the asymptotic scattering equation are obtained in the form of complex phase shifts (δ_l) for each partial wave. The phase shifts carries all the necessary information regarding the scattering event. The knowledge of the phase shift is utilized to compute the scattering amplitude as

where S_l(k) = exp(2iδ_l) are S-matrix (scattering matrix) elements constructed using δ_l. Hence eqn (8) can be written as

Thus the total inelastic cross section Q_inel and total elastic cross section Q_el can be calculated employing scattering amplitudes as above through standard relations,⁵⁸

andwhere η_l = exp(−2 Im δ_l) is the inelasticity or absorption factor for each partial wave. The total cross section Q_T is the sum of these two cross sections.

III. Results and discussion

In this article, a comprehensive computational study of electron collision with SF₆ in the gas phase is reported. The primary aim of this work is two-fold: (1) to detect the positions of resonances, if any, by studying the eigenphase diagram and (2) to present elastic and total cross sections for a wide energy range (0.1–5000 eV). Besides, we have also reported elastic DCS, electronic excitation cross section, MTCS and rotational excitation cross sections. The ab initio R-matrix method is used at low energies using the Quantemol-N³⁶ module, as it gives reliable results up to the ionization threshold of the target. For high-energy calculations, the spherical complex optical potential (SCOP) formalism^37–40 is used. The Q_T obtained from these two theories matches smoothly at the overlap energy (∼18 eV) and thus allows us to predict total cross sections for this wide energy range.

The eigenphase diagram of various doublet states (2A_g, 2B_3u, 2B_2u, 2B_1g, 2B_1u, 2B_2g, 2B_3g, 2A_u) for the e-SF₆ system is plotted in Fig. 1. The eigenphase diagram shows shape resonances at 5.43 eV belonging to the T_1u symmetry of the O_h group, which splits into the 2B_1u, 2B_2u, and 2B_3u symmetries of the D_2h group. The energy of the resonance of the current work is lower than the 7 eV resonance reported by the measurement of Dababneh⁹ and Kennerlya et al.¹⁰ and calculations of Fabrikant et al.³⁴ as given in Table 3. The resonance at 5.43 eV is confirmed by a sharp peak in the total cross section (see Fig. 3) at the same energy. Another sharp feature is seen at around 17 eV due to the 2A_u state, contributing significantly to the hump appearing in the total cross section around that energy.

Fig. 1 Eigenphase diagram for e-SF₆ scattering.

Table 3 TCS for e-SF₆ scattering

Energy (eV)	TCS (Å^2) (QMOL)	Energy (eV)	TCS (Å^2) (SCOP)
0.1	356.69	18	29.89
0.2	203.71	20	29.47
0.3	142.65	22	29.03
0.5	89.74	24	28.71
1.0	48.80	26	28.44
1.5	36.08	28	28.32
2.0	30.40	30	28.27
2.5	27.38	32	28.33
3.0	25.64	34	28.47
3.5	24.57	36	28.64
4.0	23.79	38	28.86
4.5	23.19	40	28.99
5.0	24.92	42	29.17
5.2	28.73	44	29.25
5.5	37.94	46	29.26
5.7	40.48	48	29.19
5.8	40.31	50	29.06
6.0	38.77	60	27.60
6.5	34.82	70	25.95
7.0	32.55	80	24.62
8.0	30.51	90	23.57
9.0	29.81	100	22.80
10	29.55	200	16.33
12	29.13	500	9.91
15	30.47	1000	6.37
16	32.81	2000	2.78
17	31.74	5000	1.40

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Fig. 2 Symmetry components of the total cross section for electron scattering by SF₆.

Fig. 3 (a) Total cross section for e-SF₆ scattering with theoretical comparisons. Solid line: present Qmol (SEP), dashed line: present Qmol (SE), dotted line: present SCOP, dash dotted line: Winstead and McKoy (elas),²⁵ dash-dot dotted line: Gianturco et al.,²⁷ short dashed line: Dehmer et al. (elas),²⁶ short dotted line: Vinodkumar et al.,³² dash-plus line: Shi et al.³¹ (b) Total cross section for e-SF₆ scattering with experimental comparisons. Solid line: present Qmol (SEP), dashed line: present Qmol (SE), dotted line: present SCOP, star: Makochekanwa et al.,⁸ solid circle: Dababneh,⁹ solid pentagon: Cho et al.,¹⁸ solid circle-dot: Kennerlya et al.,¹⁰ solid triangle: Zecca et al.,¹² rumbas-cross: Ferch et al.,¹⁴ hexagon-dot: Sakae et al. (elas).²¹

The contribution of various symmetries to the total cross section in SEP calculations for SF₆ is presented in Fig. 2. It is quite obvious that the contributions are quite different for different symmetries. At the low impact energy studied here, the collisions tend to be dominated by the contribution of the s-wave channel. The very high cross section at low energies is due to the s-wave (2A_g symmetry) as shown in Fig. 2. From the above figure, it is clear that the shape resonances appearing at about 5.43 eV are mainly due to the contributions from 2B_1u, 2B_2u and 2B_3u symmetries.

Fig. 3(a) and (b) show the total cross sections from 0.1–5000 eV impact energies in the current study compared with the available theoretical and experimental results respectively. To display the relevant details of the resonance peaks, both figures display an expanded view of the plots between y-axis values of 0 and 50 Ų. The results shown in Fig. 3(a) agree reasonably well with previous theoretical studies. At very low energies (<1 eV), the current SE calculations display a peak at 1.36 eV, matching quite well with the position of the peak by Dehmer et al.,²⁶ while the total cross sections for the current SEP calculations show asymptotic behaviour. The SEP calculation of the current work depicts sharp peaks at 5.4 eV of about 17 Ų due to the contribution of shape resonances from the 2B_1u, 2B_2u and 2B_3u states. The peak is found to be shifted to slightly higher energy (6.54 eV) for the SE calculations. The peak in TCS for elastic cross sections of Winstead and McKoy²⁵ is at 7.98 eV. A sharp peak is also reported by Gianturco et al.,²⁷ which is at much lower energy than that from Winstead and McKoy. The plot of the elastic cross section reported by Dehmer et al.²⁶ also shows structure similar to the present data, but with much lower magnitude. The data by Vinodkumar et al.³² shows similar shape and magnitude throughout the energy range. Shi et al.³¹ have used a modified additivity rule to report the total cross section and it falls very close to our data. It is interesting to note that the current results calculated by the R-matrix and SCOP methods agree at about 20 eV, which helped us to predict cross sections from 0.1 eV to 5000 eV.

In Fig 3(b) we can see that, at below 1 eV in the present work, the SEP results are much higher than the SE results. However, the shape of the SEP plot is similar to those from the measurements by Kennerlya et al.¹⁰ and Ferch et al.¹⁴ The energy of the peak from the present work shows a reasonable agreement with that of the experiments by Kennerlya et al.¹⁰ (7 eV), Dababneh⁹ (6.7 eV) and Cho et al.¹⁸ (8.5 eV). However, the magnitude of the present cross section is higher than their measurements. Another maximum is seen at 16.14 eV for the current SEP calculations, which is due to the excited states 1A_g and 1B_1g at that energy. Each of the measurements also indicate a secondary peak, albeit at different energies. Above 20 eV the current results nicely match the measurements of Dababneh⁹ and Kennerlya et al.¹⁰ The measurements reported by Zecca et al.¹² are higher than all other data presented here. In general, the overall trends seen in our theoretical data agree with previous experimental results. The current total cross section data from 0.1–5000 eV are presented in Table 3.

The electronic excitation cross sections from the ground state 1A_g to the eight low-lying excited states 3A_g, 3B_1g, 1Ag, 1B_1g, 3B_3u, 3B_2u, 1B_1u and 1B_2u are shown in Fig. 4. The threshold of vertical excitation energies for both triplet states 3A_g and 3B_1g are 11.594 eV and 11.597 eV, respectively, showing agreement with the first electronic transition at 11.19 eV of Winstead and McKoy.²⁵ However, calculation of the first excitation energy in the current work displays good agreement with the first strong continuum reported at 11.6 eV in the absorption spectrum of Trajmar and Chutjian.⁵⁹ There is also a prominent feature at around 17 eV in the excitation curves for 1A_g, which coincides with the resonance due to 1B_1g reported earlier.

Fig. 4 Electronic excitation cross sections for e-SF₆ scattering.

The differential cross sections are very sensitive to the use of different scattering formalisms and can be accurately measured by experiments, hence allowing scattering theories to be tested. This has prompted us to calculate differential cross sections of elastic scattering by electrons for the SF₆ molecule, particularly from energies of 1–10 eV. To facilitate comparisons, we report, in Fig. 5(a)–(d), only those energies where previous data are available. The DCS for 2.7 eV impact energy that we calculated shows a similar shape to those obtained experimentally by Rohr¹⁷ and Cho et al.,¹⁸ although not all the data are consistent with each other. At 5, 7, and 10 eV the DCS matches quite well with the calculation of Winstead and McKoy²⁵ and experiments of Rohr¹⁷ and Trajmar et al.¹⁵ The humps in the DCS are shifted backward for higher impact energies and follow the similar trend of the measurements by Rohr¹⁷ and Cho et al.¹⁸ for most of the energy levels.

Fig. 5 (a–d) Differential cross section for e-SF₆ scattering system from 2.7 –10 eV. Solid line: present, dashed line: Gianturco and Lucchese,²⁸ dotted line: Winstead and McKoy,²⁵ stars: Rohr,¹⁷ circles: Cho et al.,¹⁸ triangles: Trajmar et al.,¹⁵ hexagons: Johnstone and Newell,²⁰ rhombuses: Srivastava et al.¹⁹

The momentum-transfer cross sections (MTCS) for electron collision with SF₆ are shown in Fig. 6. MTCS is an important factor in modeling the plasma. The MTCS reported by Phelps and Van Brunt²³ seems to diverge from present results and that of Christophorou and Olthoff²⁴ in the low-energy region. The measurements of Christophorou and Olthoff²⁴ show a similar trend as that of present results, except the sharp rise at around 5–6 eV. The appearance of the sharp peak is due to the resonance at about that energy, which is missing in both previous results.

Fig. 6 Momentum-transfer cross sections for e⁻-SF₆ scattering system.

In Fig. 7 we have plotted the electron impact rotational excitation cross sections for the SF₆ molecule. The maximum contribution to the total rotational excitation cross sections comes from the j = 0 → j′ = 0 state. At low energy the rotational cross section for the j = 0 → j′ = 0 state becomes quite large due to a long-range effect. The 16-pole moment of SF₆ exerts a very strong long-range force on the scattering electron. Hence the target is subjected to torque even when the incident electron is at a distance, and the rotational force is imparted on the molecule for a relatively long period of time. It is noticeable that the doublet and quartet rotational excitations (j = 0 → j′ = 4 and j = 0 → j′ = 2) match quantitatively, whereas the triplet and quintet excitations (j = 0 → j′ = 3 and j = 0 → j′ = 5) also share similar features. For the j = 0 → j′ = 4 and j = 0 → j′ = 2 states, a hump is observed at about 5.5 eV, which is near the position of the shape resonance at 5.43 eV.

Fig. 7 Rotational excitation cross sections for e-SF₆ scattering system.

IV. Conclusion

A comprehensive study to calculate differential elastic, total elastic, electronic excitation, rotational excitation, momentum transfer and total cross sections for electron collision with the SF₆ molecule has been performed and reported in this article. The target properties obtained from close-coupling calculations with the 6-31G Gaussian basis set and considering the molecule in the D_2h point group show reasonable agreement with the available experimental and theoretical results. Hence, the target representation is considered to be justified. The position and width of resonances at low energies are detected using the eigenphase diagram. From Fig. 1, we can see noticeable features for the 2B_1u, 2B_2u and 2B_3u symmetries at 5.43 eV, confirming the presence of a shape resonance around that energy. The eigenphase presented here tends to π/2 instead of zero at low impact energies. At 17.02 eV, a short-pulsed resonance is also found to appear for the 2B_1g symmetry. The low-energy cross sections become sufficiently large due to the s-wave contributions (2A_1g symmetry). The high elastic cross section at low energy is due to the large positive s-wave scattering length of the SF₆ molecule. This indicates that the well-known virtual-state effect does not occur in SF₆ as observed for the C₆F₆ molecule from the calculations of Field et al.⁶⁰ Furthermore, for the virtual-state scattering at low energy, the s-wave phase shift should be high and negative. Hence the elastic cross section for 2A_1g symmetry is much higher (Fig. 2) than other symmetries as the s-wave scattering length of SF₆ is positive at low energies.⁶⁰ The total cross section for different symmetries also reflects a clear enhancement of cross sections at these resonant energies as shown in Fig. 2. The results presented here show reasonably good agreement quantitatively and qualitatively with the available previous theories and experiments, including a very good match between the DCS calculated in our study and the existing results.

As discussed earlier there have been many previous attempts to study electron collision-induced chemistry with SF₆ due to this molecule's importance in various applications. However, the results have been fragmentary and inconsistent as they were independent studies with varying degrees of approximations and differences in experimental set up. A complete study under a common umbrella was lacking. Hence, we have undertaken this task for calculating various cross sections under a hybrid methodology (R-matrix + SCOP), which can deliver the cross sections for a wide energy range (0.1–5000 eV). This work should be of great significance for various applications, particularly for the modeling of industrial plasma and for atmospheric research.

Acknowledgements

The authors would like to acknowledge W. J. Brigg, Department of Physics and Astronomy, University College London, UK for helpful discussions at the early stage of the calculations. BA is pleased to acknowledge the support of this research by the Department of Science and Technology (DST), New Delhi through Grant no. SR/S2/LOP-11/2013.

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