The Binary-Encounter-Bethe Model for Computation of Singly Differential Cross Sections Due to Electron-Impact Ionization

Abstract

In the present work, we assess the effectiveness of singly differential cross sections (SDCS) due to electron-impact ionization by invoking the binary-encounter-Bethe (BEB) model on various atomic and molecular targets. The computed results were compared with the experimental and theoretical data. A good agreement was observed between the present and the available results. This agreement improves as the incident energy of the projectile increases. The model can be applied to compute the SDCS for the ions produced due to the electron-impact dissociative ionization process and the average energy due to the secondary electrons. Both these quantities are of interest in plasma processing and radiation physics.

1. Introduction

The electron-impact ionization is one of the most elementary processes in basic and applied physics [1,2]. The ionization cross sections due to electron-impact are used as input parameters in the study of various physical phenomena such as radiation damage, [3] planetary atmospheres [4,5], and plasma processing [6,7]. Ionization SDCS are also the input parameters for the Monte-Carlo based simulations to model the track of energetic secondary particles in biological matter [8]. The secondary electrons produced during ionization can inflict significant damage to biomolecules [9,10]. Hence, reliable models are required to estimate the energy deposition due to electron-impact ionization. This requires computation of SDCS for a variety of targets over a wide energy range.

Various ab initio and semiempirical formulations that exist to compute electron-ionization cross sections have their own merits and limitations. The ab initio approaches such as the convergent close-coupling (CCC) approach [11,12] and the R-matrix with pseudostates [13] aim to compute the scattering cross sections directly from the quantum mechanical first principles. The applicability of these methods has so far been confined to simple diatomic targets because the computational complexity becomes prohibitive due to the target size and the basis set used [14]. Furthermore, these methods suffer from issues such as linear dependency [13] and ill conditioning [14].

The semiempirical approaches [15], on the other hand, are relatively easy to apply and perform quite well even for complicated targets. The models such as the binary-encounter-dipole (BED) and binary-encounter-Bethe (BEB) of Kim and Rudd [16], the Jain-Khare (JK) model and its modified form [17,18], the Deutsch-Märk model [19,20], and several modified forms of the BEB model [21,22] are examples of semiempirical formulations commonly employed to compute ionization cross sections. The BED and JK models require differential oscillator strength (DOS) values of the target as an input parameter. This hinders the applicability of these methods, as calculating DOS even for simple targets is a tedious process [23]. The BEB model is a simplification of the BED model in which the DOS term is approximated by a simple function of W (secondary electron energy). The BED and BEB models are briefly discussed in the next section. A detailed review of the BED model and related techniques can be found in [24]. This paper is organized as follows: Section 2 discusses the BEB and BED models. The computational aspects, resultsm and observations are presented in Section 3. The discussion is presented in Section 4.

2. Methodology

2.1. The BED Model

The binary-encounter-dipole, or BED model of Kim and Rudd [16] combines the aspects of binary-encounter theory with the dipole interactions of the Bethe theory and the electron exchange effects proposed by L. Vriens [25]. The overall accuracy of this model is accepted to be within $10\%$ or closer to the experimental values. The SDCS in the BED model is given by [16]:

$$\begin{array}{cc}\hfill \frac{d\sigma (W,T)}{dW}& =\frac{S_i}{B_i(t_i+u_i+1)}\{\frac{(N_i/N)-2}{t_i+1}\left(\frac{1}{w_i+1}+\frac{1}{t_i-w_i}\right)\hfill \\ \hfill & +(2-N_i/N)\left[\frac{1}{{(w_i+1)}^2}+\frac{1}{(t_i-w_i){)}^2}\right]+\frac{ln{t}_i}{N(w_i+1)}\frac{df\left(w_i\right)}{d{w}_i}\}\hfill \end{array}$$

(1)

Here, $\frac{d\sigma (W,T)}{dW}$ denotes the SDCS, and T refers to the energy of the projectile electron. The $w_i=W_i/B_i$, $t_i=T/B_i$, $u_i=U_i/B_i$, $S_i=4\pi a_0^2{N}_i{(R/B_i)}^2$, and $\frac{df\left(w\right)}{dw}$ is the differential oscillator strength. The $N_i$ is the occupation number of the i-th orbital. $B_i$, $W_i$, and $U_i$ are the binding energy, ejected electron energy, and the average kinetic energy of the electron in the i-th orbital, respectively. The total SDCS is obtained by summing over all those orbitals that satisfy the condition $T\ge B_i$.

2.2. The BEB Model

The SDCS in the BEB model are obtained by simplifying Equation (1), where $\frac{df\left(w_i\right)}{d{w}_i}$ is replaced by $\frac{N_i}{{(w_i+1)}^2}$ [16]. Thus, the SDCS is given by:

$$\begin{array}{cc}\hfill \frac{d\sigma (W,T)}{dW}& =\frac{S_i}{B_i(t_i+u_i+1)}\{\frac{(N_i/N)-2}{t_i+1}\left(\frac{1}{w_i+1}+\frac{1}{t_i-w_i}\right)\hfill \\ \hfill & +(2-N_i/N)\left[\frac{1}{{(w_i+1)}^2}+\frac{1}{(t_i-w_i){)}^2}\right]+\frac{N_{i}ln{t}_i}{N{(w_i+1)}^3}\}\hfill \end{array}$$

(2)

This leads to a very simple and convenient analytical form of total ionization cross sections (TICS):

$$\begin{array}{cc}\hfill {\sigma }_i\left(t_i\right)& =\frac{S_i}{t_i+u_i+1}\{\frac{Q_i}{2}\left(1-\frac{1}{t_i^2}\right)ln{t}_i\hfill \\ \hfill & +(2-Q_i)\left[\left(1-\frac{1}{t_i}\right)-\frac{ln{t}_i}{t_i+1}\right]\}\hfill \end{array}$$

(3)

Here, $Q_i=N_i/N$. In the BEB approximation, $Q_i$ is taken to be unity. Despite being simple, Equation (3) shows excellent agreement with the experimental data [26]. The advantage of invoking the BEB model is that the expressions for SDCS or TICS are given by a very simple analytical expression and require input parameters such as B, U, and N for different occupied orbitals. These parameters obtained at Hartree-Fock level have yielded excellent results for a large number of targets even with the modest basis set. The BEB model is found to give reliable results for the TICS but its effectiveness has not yet been investigated established for SDCS. The SDCS for some targets have been presented previously but not in a manner that highlights and compares the accuracy of BEB. For example, the SDCS for H-atom at 60 eV has been presented in [16]. The SDCS are useful in the calculation of the average energy of the secondary electrons $W_{av}$. Hence, it is important to evaluate the performance of the SDCS given by the BEB model.

2.3. Average Secondary Electron Energy

The average energy of secondary electrons, $W_{av}$, is an extremely important quantity in fields such as biophysics and plasma processing and is associated with the SDCS. The secondary electrons produced due to electron-impact ionization are responsible for the damage in tissues and genetic material [27]. The $W_{av}$ can be obtained using the SDCS:

$$W_{av}={\int }_0^{(T-B)/2}W\frac{d\sigma }{dW}dW/{\int }_0^{(T-B)/2}\frac{d\sigma }{dW}dW$$

(4)

In this work, we have used the BEB model to compute the SDCS and, hence, the $W_{av}$.

3. Computational Details and Results

The targets considered in this work were optimized using the Gaussian software [28] at Hartree-Fock (HF) level using 6-311G${}^{*}$ basis set. This generated the parameters (B${}_i$, U${}_i$, and N${}_i$) required to compute the SDCS. We present the SDCS for simple targets (H, He, H${}_2$, N${}_2$, H${}_2$O, and C${}_2$H${}_2$) and a few complex targets (C${}_2$H${}_5$OH, pyrimidine, tetrahydrofuran, etc.). Wherever possible, comparisons with available experimental and theoretical data have been made to illustrate the extent to which the BEB model is accurate and reliable.

3.1. Hydrogen Atom (H)

The electron-impact ionization of the H atom is relevant in the modeling of plasmas [29], stellar atmospheres [30], and planetary atmospheres [31]. The simplicity of the H atom makes it an ideal system to test any theory attempting to predict ionization cross sections. The BEB-SDCS for the H-atom at various energies are shown in Figure 1a–f. The results were compared with the experimental values of Shyn [32] and the CCC approach-based results of Mori et al. [33]. The experimental results of Shyn had an uncertainty of $\pm 20\%$. The CCC results of Mori et al. [33] were obtained after including 173 states in the close-coupling expansion. These are the most accurate theoretical results for the H-atom to date. We see a good agreement between the BEB and CCC results at different incident energies and secondary electron energies (W). However, both the theoretical results were lower than experimental results but still lay close to the experimental uncertainties at different energies.

3.2. Helium Atom (He)

The BEB-SDCS for the He-atom at various energies are shown in Figure 2a–d. The BEB model results are compared with theoretical results of Mori et al. [33] at 40 eV and 100 eV. We again notice a satisfactory agreement between the present and CCC results [33]. This is encouraging, as the CCC approach is quite complex, sophisticated, and requires large computational resources. A concordance was observed with the experimental data of Opal et al. [35] at 100 eV, 200 eV, and 500 eV and Röder et al. [36] at 40 eV.

3.3. Molecular Hydrogen (H${}_2$)

The electron impact ionization SDCS for H${}_2$ given by the BEB model for various incident energies are shown in Figure 3a–d. The experimental measurements are from Shyn et al. [32], with a reported uncertainty of $\pm 16\%$. We have also compared the present results with the CCC theory-based results of Mori et al. [33]. The SDCS given by the BEB model for H${}_2$ showed satisfactory agreement with both the experimental measurements and the CCC results at different energies [33].

3.4. Molecular Nitrogen (N${}_2$)

Ionization cross sections for N${}_2$ are relevant in modeling atmospheric plasmas and the interactions of high energy charged particles with the atmosphere (for example, the northern lights) [37,38]. The BEB-SDCS for various impact energies are shown in Figure 4a–d along with the experimental as well as the theoretical data. The experimental cross sections are from Opal et al. [35] and the theoretical results are from Pal et al. [39]. Pal and coworkers used a Jain-Khare semiempirical approach for the calculation of SDCS in their work. The SDCS given by the BEB model were in better agreement with the experimental values than Pal et al. [39].

3.5. Molecular Oxygen (O${}_2$)

Oxygen ionization cross sections are of significance in modeling atmospheric plasmas and the interaction of high energy charged particles with the atmosphere [40]. Electron impact ionization also plays a role in the inter-conversion between ozone and oxygen in the atmosphere [41]. The BEB-SDCS for an oxygen molecule at various incident energies of 50 eV, 100 eV, 200 eV, and 500 eV are shown in Figure 5a–d. The experimental cross sections are from Opal et al. [35], and the theoretical data is from Pal et al. [39] computed using a JK semiempirical approach. The SDCS given by the BEB model were in good agreement with the experimental values and better than those determined from the JK semiempirical approach [39].

3.6. Water (H${}_2$O)

The BEB-SDCS for H${}_2$O at various incident energies are shown in Figure 6a–d. The electron-impact SDCS are of use in determination of the rotational abundance of water in cometary atmospheres and the modeling of combustion products from fossil fuels [42]. Since it is present in living tissues, the ionization cross sections for water can be used to obtain a rough estimate of the energy deposition and damage caused by the ionizing radiation in biological matter [43,44]. A good agreement is observed between experimental results of Bolorizadeh et al. [45], Opal et al. [35], and the present results. These results improved as the energy of the primary electron increased.

3.7. Acetylene (C${}_2$H${}_2$)

Acetylene plasmas find use in various fields such as the production of fullerenes [46] and in nanoscience for the production of carbon nanoparticles [47]. Several kinds of simple hydrocarbons, including acetylene are present in interstellar clouds [48]. The fact that simple hydrocarbons form as a result of astrophysical processes has important implications for astrobiology [49]. The availability of accurate electron-impact cross sections of these simple hydrocarbons aids the study of the chemical evolution of the matter present in interstellar meda. Electron-impact ionization data for acetylene are also important in simulations of fusion plasmas for reactors, which use graphite as the wall material [50]. The BEB-SDCS for C${}_2$H${}_2$ at incident energies are shown in Figure 7a,b. We have also included the recent theoretical work of Pal et al. [51].

3.8. Ethanol (C${}_2$H${}_5$OH)

Ethanol is considered an alternative to traditional combustion fuels. Reliable ionization cross sections are required in plasma simulations of the combustion of ethanol in alternative fuel engines [52]. Ethanol is also found in interstellar environments and plays a significant part in the formation of organic compounds [53]. The BEB-SDCS for electron-impact ionization of ethanol are presented at various primary energies in Figure 8a,b. No experimental data were available to compare the present and other theoretical results [51]. Therefore to assess the quality of present results, we integrated the SDCS to obtain the TICS at specific energies. The integrated values of SDCS at 100 eV and 200 eV were in excellent agreement with the experimental results of Nixon et al. and Rejoub et al. [54,55]. The TICS of Pal et al. from the modified JK model [51] were higher than the experimental values at 100 eV and 200 eV. This provides us with the confidence to conclude that the present results can serve as a benchmark for future studies. The computed TICS are displayed in Figure 9.

3.9. DNA Constituents

Presented below are the scattering cross sections for pyrimidine (Py), tetrahydrofuran (THF), trimethylphosphate (TMP), and purine (PU). We compared the BEB-SDCS to the recommended data from the literature. Reliable cross sections for biomolecules such as these are important for track structure simulations such as PTra and the Geant4-DNA toolkit [56,57].

3.9.1. Pyrimidine (C${}_4$H${}_4$N${}_2$)

Pyrimidine is the precursor to the three nucleic bases (cytosine, thymine, and uracil) of DNA and RNA. Thus, pyrimidine is a target of tremendous interest in the study of the interaction of radiation with biological matter [58]. The electron-impact ionization SDCS for pyrimidine are shown in Figure 10a–c. We compared our results with points that were derived by Bug et al. from experimental data with an error of the order 25% [59].

3.9.2. THF (C${}_4$H${}_8$O)

THF is a heterocyclic compound also known as oxolane. The THF molecule is a heterocyclic organic compound, which is a structural analogue of the deoxyribose group of DNA [59]. Thus, electron-impact cross sections for THF can be used to model the effect of ionizing radiation on genetic matter [60]. The results for THF are shown in Figure 11a–c. The values derived by Bug et al. from experimental data with an error in the order of 25% [59] are also shown.

3.9.3. TMP (C${}_3$H${}_9$O${}_4$P)

Trimethyl ester of phosphoric acid (TMP) is a structural analogue of the phosphate group of DNA, which is one-half of the sugar–phosphate backbone of the DNA double helix structure. The electron-impact ionization SDCS for TMP are shown in Figure 12a–c. Values derived by Bug et al. from the experimental data with an error in the order of 25% [59] are also shown.

3.9.4. Purine (C${}_5$H${}_4$N)

Purine is the structural precursor to adenine and guanine (two of the five canonical nucleobases). Thus, in the context of the ionization of DNA, purine is a very important target. Purine derivatives are also present in other biomolecules such as ATP and caffeine. The electron impact ionization SDCS for TMP are shown in Figure 13a. Since, experimental measurements of the SDCS were not available, we evaluated TICS in the primary energy range from threshold up to 5000 eV via numerical integration of the SDCS. The TICS then were compared to the recommended data of Bug et al. [59] as an indirect check on the SDCS. The TICS are shown in Figure 13b, and it can be seen that from 200 eV onwards the agreement with the experiment was excellent. The maximum deviation in the magnitude of the BEB and experimental cross sections at the peak was about 20%, which is within the accepted range of uncertainty for BEB results.

3.10. Average Secondary Electron Energy

Average secondary electron energies, $W_{av}$, calculated via Equation (4) for the targets discussed throughout this work are shown in Figure 14a–d. It seems that there is some dependence of $W_{av}$ on the geometrical size of the targets. Furthermore, it appears that at around 100 eV energy of the primary electron, the average energy of the secondary electron in almost all cases was around 10 eV with the exception of He. This artifact is due to the fact that the graphs are plotted on a large scale along the x-axis. In fact, the W${}_{av}$ as a function of incident energy, depends upon several factors such as the threshold energy, ionization cross sections, and SDCS. The inputs to calculate SDCS or TICS were B, U, N, and the experimental value of the ionization threshold. All the targets showed rising trends in W${}_{av}$ with an increase in incident energy from their respective thresholds. The SDCS were continuously decreasing with the increase in primary electron energy or secondary electron energy, whereas the TICS showed rising trends from the threshold until a maximum was reached; then, they start falling with further a increase in incident energy of the primary electron.

A better way to explain the variation in W${}_{av}$ with the number of electrons in the target is by invoking the Thomas-Reiche-Kuhn rule [61] for dipole transitions in a closed shell N-electron molecular system, according to which

$$\sum _i{f}_i+{\int }_0^{\infty }\frac{df}{dw}dw=N$$

(5)

where i ranges for ground to dipole-allowed excited states. It means that the ground and each excited state must be of opposite parity, which is due to the fact that a photon has one unit of angular momentum. The continuum oscillator strength integrated over W, is proportional to $N-\sum f_i$. This dependence is reflected in the variation of W${}_{av}$ with the number of electrons in the target in Figure 15.

The SDCS computed for different targets showed a decreasing trend with an increase in secondary and primary electron energy. It means that the low energy electrons were produced due to electron impact even if the incident electron energy increased. The present SDCS results showed a good agreement with the ab initio convergent close-coupling data and experimental measurements. The deviations in the BEB and experimental results were observed near the threshold values. However, as the energy of the secondary electron increased, the present results showed a converging trend with the experimental data. The present BEB-based SDCS have limitations as well. They cannot reproduce the features such as resonances or estimate SDCS due to the dissociative ionization process [16]. However, the modified-BEB model [21,62,63] can be invoked to compute thge partial SDCS of cations and the energy deposited by them. These results were deliberately not shown, as there are no data to compare with the computed results. The present semiempirical approach not only yielded reliable results but was extremely simple to work with. It did not suffer from linear dependence or computational issues. The necessary input parameters were easily computed without posing a major challenge using any quantum chemistry code. It would be interesting to apply the BEB model to compute the SDCS due to positron impact as the model has shown promising results for TICS [64,65,66,67].

4. Discussion

The BEB model is fundamentally a high energy approximation. Thus, the results were expected to be in good agreement with the experimental measurements for high electron impact energies. This is in line with what we have shown in the present work. An encouraging aspect of the study has been that the BEB-SDCS compared very well with the latest CCC based results of Mori et al. [33]. The CCC theory relies on large basis sets, and the scattering wavefunction requires a large number of bound and continuum states. These features make the CCC approach extremely complicated and expensive to work with. For these reasons, its implementation on polyatomic targets has not been tested. The semiempirical approaches such as the BED and the JK both require the DOS as an input parameter. This hinders their general applicability.

In comparison, the BEB model has a simple analytical form, which works well with even modest computing resources. Thus, the present investigation establishes that the BEB model can be used for calculating the SDCS and various quantities of interest such as partial ionization cross sections (PICS) and W${}_{av}$.