The collisions of electrons with atoms, molecules, ions, and surfaces are of fundamental importance in low-temperature plasmas (LTPs), with numerous applications in plasma science and technologies. LTPs represent a unique state of matter consisting of neutral atoms and molecules, radicals, ions, electrons, and excited states. They are used in the semiconductor industry for etching, depositing materials, synthesizing nanomaterials, and cleaning reaction chambers and in the field of plasma medicine [1]. Electron collision data are important in the fields of astrophysics, astrochemistry, aeronomy, radiation physics, gas discharge, electron beam lasers, and plasma processing devices for auroras and solar plasma [2]. Plasma–wall interactions are one of the major challenges in the realization of plasma reactors. These interactions are important for the physical and chemical processes in all plasmas with technological applications. Perfluorocarbons (PFCs) are extensively used as feedstock gases for various applications in plasma processing [3,4]. PFC gases, such as CF₄, C₂F₆, and C₃F₈, have been widely used for plasma etching applications [3]. SF₆ is widely used in high-voltage gas-insulated switchgears and gas-insulated lines [5] and as an etchant [6]. However, the global warming potential (GWP) of SF₆ is 22800 times larger than that of CO₂, and it is extremely difficult to decompose in the atmosphere [7]. This has led to interest in finding a suitable replacement for this gas. Moreover, owing to the requirement of the Kyoto Protocol [8] for the reduced emission of PFCs and SF₆, research is being actively conducted on new gases with a low GWP, which can be suitable substitutes for existing plasma gases in various plasma applications. Hence, it is desirable to find alternative gases that are environment friendly with a relatively low GWP but with a potential similar to SF₆ or other currently used PFCs. For this purpose, the detailed electron collision studies of possible substitute gases at the molecular level is important at energies ranging from 0–100 eV. In LTPs (3–5 eV), the energy of electrons can reach up to 100 eV. Hence, the collision cross-section data in the range of 3–100 eV are important. Moreover, to experimentally analyze electron heating mechanisms [9,10], scattering and excitation cross- section data are required up to a minimum of 25 eV but desirable up to a maximum of 100 eV.

The literature survey shows that there is still a lack of electron collision cross-section data for numerous PFCs and their fragments. The ionization cross sections (Q_ion) for various PFCs, such as CF, CF₂, CF₃, C₂F₄, C₂F₆, C₃F₆, C₃F₈, C₄F₆, C₄F₈, and c-C₅F₈, have been measured using various experimental techniques [11–24]. Tarnovsky et al. [25] have measured the absolute cross section for the electron impact ionization and dissociative ionization of C₂F₅ radicals from a threshold to 200 eV using the fast-neutral-beam technique. The theoretical predictions [11–15,26,27] for the above species have been made using the binary-encounter Bethe (BEB) [28,29], Deutsch-Mark (DM) [30,31] and complex scattering potential-ionization contribution (CSP-ic) [32,33] methods, and they have shown reasonable agreement with experimental measurements. In recent years, there has been an increase in the research on PFCs for various electron collision processes with the objective of their quantifying cross sections. A complete study of Q_ion has been performed for C₂F_x (x = 1 – 6), C₃F_x (x = 1 – 8) [34], and C₄F_x (x = 1–8) [35] neutral and radical species using the BEB method. The BEB and CSP-ic methods are briefly described in Section 2.

The literature survey for the low and intermediate energies for electron scattering elastic, differential, momentum, excitation, and total cross sections for PFCs shows the lack of complete cross-section data. Various elastic and total cross sections for PFCs and SF₆ have been experimentally measured [36-52]. Szmytkowski and Możejko have compiled the measurements of the total cross sections from a range of molecules from 2009–2019 [53]. The low and intermediate energies cross-section calculations for various collision processes for several fluorocarbons have been theoretically predicted [54–59]. In recent years, the scarcity of cross-section data for simple fluoro- carbons has led to studies on the low-energy cross section of fluoro- carbons, such as C₂F₂ [60], C₃F₄ [61], and C₄F₆ isomers [62], for various elastic and inelastic processes using the R-matrix method. At inter- mediate and high energies, differential and total cross sections have been computed using the single-center approach for C₄F₆ isomers [63].

Goswami and Antony [64] have recently calculated cross-section for elastic and inelastic processes for an extensive range of energies for SF₆ using the R-matrix and spherical complex optical potential (SCOP) [32,33] methods. They have provided detailed references for various cross-section studies on SF₆ in their article; hence, readers are directed to their article for more information about SF₆. In the aspect of finding potential substitutes for SF₆, fluoroketones and fluoronitriles have been extensively studied over the past few years owing to their low GWP compared to SF₆ [65–68]. These studies have focused on the insulation properties of these gases under different conditions. Furthermore, these gases have received considerable attention for the study of electron impact ionization because this is an important parameter for examining electron avalanche and gas breakdown mechanisms. Wang et al. [7] have calculated the total Q_ion of fluoroketones (C₅F₁₀O and C₆F₁₂O) and fluoronitriles (C₃F₅N and C₄F₇N) using the BEB method. Sinha et al. [69] have calculated the Q_ion and electronic excitation cross section for C₃F₆O, C₄F₈O, C₅F₁₀O, C₆F₁₂O, and C₄F₇N using the CSP-ic method from the ionization threshold to 5000 eV. Wang et al. [70] have calculated the Q_ion for C₅F₁₀O and C₆F₁₂O using the DM method. Xiong et al. [71] have calculated the Q_ion for C₅F₁₀O and C₄F₇N using the modified DM method. However, there are no studies on the elastic, excitation, differential, momentum transfer, and total cross sections for these species, which may be important for improving the understanding of these gases as potential substitutes for SF₆. In addition, it may be interesting to investigate larger PFCs, such as c-C₅F₈, c-C₆F₈, C₇F₈, C₇F₁₄, and c-C₁₀F₈, as potential substitutes for SF₆ and other PFCs with higher GWP. One of the primary motivations of the present review is to highlight the requirement for the detailed investigation of fluoroketones, fluoronitriles, and larger PFCs along with their frag- ment species and radicals for various electron collision processes that could be useful for plasma applications/modeling. A recommended set of databases for electron collision studies for multiple PFCs, such as CF₄, C₂F₆, C₃F₈, c-C₄F₈, and SF₆, has been provided by Christophorou and Olthoff [72–77]. A recent recommended set of data for electron collision for plasma relevant molecules has been provided by Yoon et al. [78] and Cho et al. [79].

In the current scenario, it is widely recognized that there are new technological advances in plasma processing and the utilization of plasmas in general, ranging from plasma medicine to material processing. These developments are based on the manipulation of plasma properties. This highlights the requirement for a detailed understanding of the atomic and molecular processes within plasmas. It is well known that theoretical calculations are important for providing data over a com- prehensive range of energies [80]. Even though there is active research worldwide on the experimental determination of electron collision cross sections for several atoms/molecules/radicals, the data for nu- merous other important targets are still unknown [81]. In the next section, a brief description of the theoretical methods used by the author is presented along with recent results obtained using each method.

2.1. R-matrix method for low-energy calculations (0–15 eV)

There are various theoretical methods for studying low-energy electron molecule collisions. This subsection presents a brief over- view of the ab-initio R-matrix method [82,83], which is used by the author for numerous calculations. This is the most widely used approach for calculating electron atom/molecule collision cross sections at low energies. There are other ab-initio methods, such as the Schwinger multichannel [84] and Kohn variation [85] methods, which can be used to calculate cross sections at low energies. All three methods are based on the variation principle, and they provide reliable cross sections for numerous systems. Furthermore, the results obtaining using each method with the same target and scattering models exhibit good agreement. The molecular convergent close-coupling [86] method based on the momentum-space formulation of the close-coupling theory is an emerging technique for calculating the electron collision cross sections of atoms and molecules at low energies.

The UK polyatomic R-matrix codes [87] have been implemented in the Quantemol-N software [88], which can be used by specialists and nonspecialists with a background in scattering theory. Quantemol- N has been used for several years for successfully calculating various low-energy cross sections for a variety of molecules [89–101]. The fixed nuclei approximation has been employed in all calculations performed by the author. The basis of the R-matrix method is the division of a configuration space into an inner region and outer region. The inner region should be selected such that it contains the wave function of a target, where the center of mass of the target coincides with the origin of the coordinate system. The interaction potentials originating from the static, exchange, polarization, and correlation between the target and projectile must be included in the inner region. The advantage of the inner region problem is that a precise solution can be obtained using quantum chemistry codes. The inner region problem is energy independent, and hence, it must be solved only once. This makes the R-matrix method computationally affordable and easy to use. All target properties are calculated in the inner region.

The wave function constructed inside the inner region for the (N+1) electron system is given by a close-coupling [102] approximation,

where A is an antisymmetrization operator, which ensures that Pauli’s exclusion principle is obeyed by a scattering electron, ${\Phi }_i^N$ is the wavefunction of the i^th target state, and x_N is the spatial and spin coordinate of the N ^th target electron. u_ij denotes the continuum orbitals used to represent the scattering electron. a_ijk and b_ik are variational coefficients determined by the diagonalization of the Hamiltonian matrix. In Eq. (1), the first summation runs over the target and continuum states. It accounts for one electron in the continuum state and the remaining electrons in the target states; this is known as a ‘target + continuum’ configuration. The electron correlations and polarization effects are accounted for by the second summation, where ${\chi }_i^{N+1}$ are referred to as the L ² configurations. These are multicenter quadratically integrable functions constructed by placing all N+1 electrons in target molecular orbitals (MOs).

A set of MOs is constructed from occupied and virtual target MOs using the Hartree–Fock self-consistent-field calculation augmented with Gaussian type orbitals and the continuum orbitals of Faure et al. [103]. The calculation is performed with the continuum orbitals up to g partial waves. The effect of correlation/polarization can be improved by including more excited states in the calculations. This improve- ment is indicated by the shifting of resonance positions to lower energies. The important aspect of the problem is to achieve the convergence of results. Once convergence is achieved, further improvement in the correlation/polarization effects has a minimal effect on results. After computing the target properties inside the inner region, the solution of the inner region is propagated to the boundary, where the R-matrix connects the inner region solution to the outer region. Depending on the size of the target, the inner region radius varies from 10a ₀ to 15a ₀ or a larger radius so that the molecular charge density of the target is fully contained inside this radius in general. In the R-matrix calcula- tions for C₂F₂ and C₃F₄, the inner region radius is chosen to be 10a ₀, whereas for C₄F₆ isomers it is kept at 12a ₀. In the outer region, the R-matrix is matched with asymptotic functions obtained from a Gailitis expansion [104] by extending it outwards up to 100a ₀ to the asymptotic region. The problem becomes simpler in the outer region because the scattering electron is quite far from the target charge cloud, where the exchange and correlation effects are negligible. Here, only the long-range multipolar interactions between the projectile and target must be considered. However, energy dependence is applicable in this region. Asymptotic K-matrices are obtained by integrating the coupled single-center equations that describe the scattering. These K-matrices are used to deduce all observables, such as eigenphases, cross sections, and resonances, using the T-matrices defined in the terms of the K-matrices, as follows:

Moreover, differential cross sections and momentum transfer cross sections can be acquired using the K-matrices in the POLYDCS program developed by Sanna and Gianturco [105]. Subsequently, the RESON resonance detection program [106] can be used for detecting and performing Breit-Wigner fits [107] to resonances. The resonances are an important feature of low-energy collision studies.

2.2. SCOP and CSP-ic methods for intermediate to high energy calculations (~10 eV to 5 keV)

The SCOP and CSP-ic methods are widely used theoretical formalisms for calculating elastic and inelastic cross sections at inter- mediate to high energies. The total cross section is the sum of elastic and inelastic cross sections. These methods have been successfully used to calculate various electron collision cross section for atoms [108–114] and molecules [115–124] at intermediate to high energies. In addition, they have been adapted to study positron collisions with atoms and molecules [125]. In the SCOP method, complex optical potential (V_opt ) is formulated for a projectile–target system consisting of real (V_R ) and imaginary (V_I ) parts. The potential is spherically symmetric such that it de ends on radial distance ‘r’ and incident energy E_i , and it is given by

The first term of Eq. (3) accounts for all elastic processes, and the second term represents absorption processes, which account for the loss of flux into various inelastic channels. Real potential consists of static (V_st ), exchange (V_ex ), and polarization (V_p ) potentials, and it is given by

The static potential and charge density of a target are obtained from the parameterized Roothan–Hartree–Fock wavefunction of the Cox and Bonham parameters [126]. Exchange potential is obtained from the parameter-free Hara [127] free electron gas exchange model, and the model potential of Zhang et al. [128] is used for polarization and correlation interaction. The imaginary term is given by the model potential of Staszewska et al. [129]. The improved or advanced versions of these potentials can be found in literature, and they can be easily used in the SCOP method to improve the quality of results.

Once the complete interaction potential given by Eq. (3) is obtained, the Schrödinger equation is solved using the partial wave approximation to obtain the complex phase shifts (δ_l ) for each partial wave using the Numerov method. These phase shifts contain all relevant information regarding projectile–target interactions. The obtained phase shifts are used to calculate elastic, inelastic, and momentum transfer cross sections, which are expressed as follows [130]:

Here, k is a wave vector. The “inelasticity” or “absorption” factor for each partial wave, ‘l’, is given by

The sum of elastic and inelastic cross sections from Eqs. (5) and (6) provides the total cross section, which is expressed as

The detailed derivation of the equations are provided in [130].

The inelastic cross section calculated using the SCOP method can be utilized to obtain Q_ion by employing the CSP-ic method. As Q_inel is the sum of all allowed electronic excitation and ionization cross sections, it can be written as

where the first term represents the total excitation cross section for all dipole-allowed electronic transitions and the second term represents all ionization processes (single, double, inner shell, etc.) induced by a projectile electron. The contribution of excitation decreases as the contribution of ionization increases with incident energies. Q_ion is derived by defining an energy dependent ratio of Q_ion /Q_inel . This subsection provides only a brief description of the CSP-ic method; the detailed description can be found in previous works [131-134].

2.3. BEB method for total ionization cross section

The BEB method is combination of the Bethe theory and Mott cross section. Kim and Rudd [28] and Hwang et al. [29] have combined these two theories for the dipole interaction between incident and target electrons at higher incident energies accounting for Bethe theory and the collision of two free electrons accounts for the Mott theory. The BEB method can provide cross sections for a wide range of targets from atoms to molecules, and the results of the method are in reasonable agreement with experimental data. The main inputs for the BEB method are the kinetic and binding energies of each target orbital. The total Q_ion of a target is obtained by adding the Q _ion of each target orbital. Q_ion is calculated using the following equations:

where

Here, B and U are the binding and kinetic energies of an electron on a given MO, respectively, T is the incident energy of the electron, N is the occupation number of electrons on the given MO, R is the Rydberg energy, and a₀ is the Bohr radius.

A recent article by Karwasz et al. [135] provides a fairly detailed description of the BEB and DM methods used for calculating Q_ion for fluoromethanes. The DM method is widely used for obtaining the Q_ion of atoms/molecules/clusters, and it requires the knowledge of the partial occupancies of MOs by electrons from constituent atoms.

This section discusses the recent results obtained by Gupta et al.for C₂F₂ [60], C₃F₄ [61], and C₄F₆ isomers [62] for elastic and inelastic scattering at low energies and C₂F₆ [34] for ionization.

Figure 1(a) shows the application of the R-matrix and SCOP methods at low, intermediate, and high energies to calculate the elastic cross section, as studied by Gupta et al. [60]. The result is shown for C₂F₂. The R-matrix method is employed up to an energy of 12 eV, after which the SCOP method is applied at intermediate and higher energies. A smooth transition between the two methods is observed at approximately 12 eV. A π* shape resonance for C₂F₂ is detected at 5.86 eV, and a sharp feature in the elastic cross section is observed at the same energy. As there are no other results available for C₂F₂, the aforementioned results have been compared with those obtained by Gauf et al. [136] for the C₂H₂ molecule, whose structure is similar to that of C₂F₂. The effect of fluorination is clearly seen in the elastic cross section, where the shape resonance for C₂F₂ is shifted to higher energies compared to C₂H₂. Figure 1(b) compares the total and elastic cross sections obtained by Gupta et al. [61] for C₃F₄ with the experimental results of C₃H₄ [137], in the absence of any other data for C₃F₄. Similar to C₂F₂, the shape resonance for C₃F₄ is shifted to higher energies compared to C₃H₄. The shifting of resonances to higher energies may be due to the shortening of the C-C bond in the case of C₂F₂ and C₃F₄ compared to C₂H₂ and C₃H₄. A detailed explanation for such behavior is given in our previous articles [60,61]. It should be noted that there are no experimental or theoretical data available for such small fluorocarbon species. Moreover, the dissociation of stable parent fluorocarbon molecules, such as CF₄, C₂F₆, C₃F₈, and C₄F₈, in plasma by electron impact leads to the formation of numerous such reactive species of C_xF_y (x = 1 – 3, y = 1 – 7). Hence, studying the microscopic properties of such fragment species and radicals is important for fluorocarbon plasmas [25]. As highlighted in the introduction, cross sections in a range of 0–100 eV are desirable for LTPs. The calculation of cross sections over a wide energy range using two different methods for C₂F₂ shows that such a hybrid methodology can easily be applied to other such cases. The study of these simple fluorocarbons, for which data are not available, highlights the requirement for further investigation into these systems.

Figure 1. (a) Elastic scattering cross section for e-C2F2 from low (0.1 eV) to high (5 keV) energies calculated using the R-matrix and spherical complex optical potential (SCOP) methods [60] and comparison with experimental data of Gauf et al. for C2H2 [136]. (b) Elastic and total cross section for C3F4 [61] calculated using the R-matrix method and comparison with the experimental data of Szmytkowski et al. for C3H4 [137].

Figure 2 shows the results for the R-matrix calculation of the elastic and excitation cross sections for the three isomers of C₄F₆, i.e., 1,3-C₄F₆, c-C₄F₆, and 2-C₄F₆. Even though the molecular formulae for all isomers are the same, their elastic scattering cross sections are quite different below 8 eV as depicted in Fig. 2(a) owing to the different molecular geometry of the targets. The shape resonance is detected for all three isomers, 1,3-C₄F₆, c-C₄F₆, and 2-C₄F₆ at 2.95, 2.57 and 3.20 respectively. The experimental results of the elastic cross section obtained by Hoshino et al. [45] do not indicate a resonance structure (not shown here). The comparison of the experimental data with theoretical R-matrix calculations is presented in the paper by Gupta et al. [62]. The cross section for 2-C₄F₆ is the largest, which may be owing to its larger molecular structure. Moreover, the isomer effect is quite strong below 8 eV, and it becomes weaker as incident energy increases.

Figure 2. Isomer effect in the (a) elastic cross section and (b) first excitation cross section of 1,3-C₄F₆, c-C₄F₆, and 2-C₄F₆ in a multistate configuration interaction (CI) calculation. Solid line: Present 1,3-C₄F₆; Short dashed line: Present c-C₄F₆; Dashed line: Present 2-C₄F₆ (reproduced from Gupta et al., Phys. Plasmas 25, 063504 (2018) with permission from AIP publishing) [62].

Figure 2(b) depicts the excitation cross section of the first excited state of the three isomers. The thresholds for the first exci- tation energy of 1,3-C₄F₆, c-C₄F₆, and 2-C₄F₆ are at 4.88, 5.84, and 6.74 eV for the ³B, ³B₂ and ³A'' states, respectively. The isomer effect is not sufficiently strong in the case of the electronic excitation cross section. The sharp feature is observed at approximately 9.6 eV for 1,3-C₄F₆ and 2-C₄F₆, which may be owing to the Feshbach resonances detected at the same energy range for these isomers. A recent study by Bharadvaja et al. [63] provides the differential and integral cross sections for these isomers at intermediate and high energies. C₄F₆ is quite important for plasma applications/processing, and hence, its physical and chemical reference data and cross sections must be further investigated to completely understand its nature and characteristics.

Figure 3 shows the Q_ion of C₂F₆ calculated using the BEB method for various orbital energies by Gupta et al. [34]. C₂F₆ has been experimentally and theoretically studied in detail. The data of Gupta et al. are further compared with available experimental and theoretical data. Orbital energies are calculated at various levels of theoretical approximation using the Gaussian 09 [139] package. Bull et al. [14] have recently suggested that the orbital parameters employed in the calculation of Q_ion using the BEB method should be obtained using a high level of theoretical approximation. They have utilized the partial third-order electron propagator theory (P3-EPT) level of approximation for obtaining orbital parameters for binding energies less than 20 eV. In BEB calculations, the orbital energies of lower orbitals are the most important and have the maximum contribution to the Q_ion . However, for a few fluorocarbons, the results obtained using such orbital parameters do not agree better with experimental data compared to Hartree–Fock (HF) orbitals and overestimate experimental results. For C₂F₆, Gupta et al. [34] have shown that the results obtained using the density functional theory (DFT) (ωB97X) functional for orbital binding and kinetic energies are in reasonable agreement with the P3-EPT calculations of Bull et al. [14], with slight variations at the peak of the cross section. The results obtained using the DFT (ω B97X-D) functional are slightly higher than those obtained using the DFT (ωB97X) functional, which may be owing to the inclusion of dispersion in the functional. The DFT (B3LYP) and DFT (PBE0) functionals provide low binding energies. Hence, the cross sections obtained using these functionals are large compared to other results. The Q_ion calculated using the BEB method is extremely sensitive to the binding energy calculations for lower orbitals.

Figure 3. Total ionization cross section for C₂F₆ in Å2 along with available comparisons. Solid line: Gupta et al. (restricted Hartree–Fock; RHF) [34]. Dashed line: Gupta et al. (ωB97X-D) [34]; Short dashed line: Gupta et al. (ωB97X) [34]; Line symbol: Gupta et al. [34] (PBE0); Short dotted line: Gupta et al. [34] (B3LYP); Dotted line: Recommended [73]; Dash-dot line: Antony et al. [26]; Dash-dot-dot line: binary-encounter Bethe (BEB)-NIST [138]; Short dash-dot line: Bull et al. (BEB) [14]; Triangle: Nishimura et al. [12]; Stars: Bull et al. [14]; Solid circle: Basner et al.[17]; Half circle: Poll and Meichsner [19] (adapted from Gupta et al., Eur. Phys. J. D 71, 88 (2017) with permission from the European Physical Journal) [34].

The BEB results of Gupta et al. [34] are also compared with the experimental data of Nishimura et al. [12], Bull et al. [14], Basner et al. [17], and Poll and Meichsner [19], the theoretical data of Antony et al. [26] and Bull et al. [14], and the cross sections recommended by Chirstophorou and Olthoff [73]. The Q_ion from NIST [138] calculated using the BEB method is also included for comparison. The restricted HF (RHF) results of Gupta et al. [34] show excellent agreement with all experimental data from the threshold to the peak. In addition, they show good agreement with the experimental data of Basner et al. [17] at higher energies. Furthermore, the RHF results of Gupta et al. [34] are in good agreement with all other theoretical data below 80 eV and less than the theoretical data above 80 eV. The theoretical data of Antony et al. [26] and the BEB data from NIST [138] show good agreement with the experimental data of Nishimura et al. [12]. Antony et al. [26] have used the CSP-ic method for calculating Q_ion, and the results agree well with the RHF BEB data of Gupta et al. [34]. Moreover, the CSP-ic method is effective for calculating Q_ion and provides the necessary reference to compare data with the BEB and DM methods. This enables us to find the consistent or recommended sets of data for Q_ion. The studies by Gupta et al. for the ionization of C₂F_x (x = 1 – 6), C₃F_x (x = 1 – 8) [34], and C₄F_x (x = 1 – 8) [35] neutral and radical species have shown that the use of the DFT (ωB97X) functional may be suitable for fluorocarbon containing species when compared with recent experimental and theoretical data in general. Hence, such functionals can be used for further studies of ionization.

The author would like to thank the Weizmann Institute of Science, Israel, for the prestigious Dean of Faculty Fellowship and Prof. Daniel Zajfman and Dr. Oded Heber from the Weizmann Institute of Science for their support. The author would also like to thank his collaborators, Dr. Jung Sik-Yoon, Dr. Mi-Young Song, Dr. Deuk Chul Kwon, and Dr. Heechol Choi from National Fusion Research Institute, South Korea, Prof. Bobby Antony from the Indian Institute of Technology (Indian School of Mines) Dhanbad, India, and Dr. Suvam Singh from the Max Plank Institute for Nuclear Physics, Germany.