Calculations of positron scattering from small molecules

Abstract

Recently, convergent close-coupling calculations have been completed for positron scattering from the carbon and oxygen atomic targets. These, together with previously completed calculations for atomic hydrogen, are utilized to perform positron scattering calculations for molecular hydrogen ($\hbox {H}_2$), molecular oxygen ($\hbox {O}_2$), diatomic carbon ($\hbox {C}_2$), carbon monoxide (CO), carbon dioxide ($\hbox {CO}_2$), ozone ($\hbox {O}_3$), water ($\hbox {H}_2\hbox {O}$), and methane ($\hbox {CH}_4$) through a modified independent atom approach. For these molecules, positronium-formation, direct ionization, electron-loss, elastic, total electronic excitation, total inelastic, and total cross sections are obtained for energies between 0.1 and 5000 eV. There is, in general, good agreement between the current results and past experiments for most transitions, particularly at high energies where this approach is expected to be most accurate.

Graphic Abstract

1 Introduction

Positron scattering from molecular targets is of significant interest to astronomical, atmospheric, and medical research. In medical research, positrons have become essential in medical imaging due to their use in positron emission tomography (PET) scans. Beyond PET scans, the other main use of positrons in medicine is positherapy, which utilizes positrons to destroy cancer cells [1]. The $\hbox {H}_2\hbox {O}$ molecule is by far the most important biologically active molecule, with it representing approximately 60% of the human body. Cross sections for this molecule are required to accurately model the energy deposition and transport of a positron as it loses energy and annihilates within the body. With a better understanding and accurate modeling of the processes these positrons undergo upon emission into the body, image blur can be reduced in PET scans [2], and damage to healthy tissue minimized in positherapy treatments. As 80% of the gamma rays detected by PET scans are emitted from the decay of positronium [3], the positronium-formation cross section ($\sigma _{\text {Ps}}$) is of particular interest.

Some of the molecules considered in this work ($\hbox {H}_2$, CO, O$_2$, $\hbox {O}_3$, $\hbox {CO}_2$, $\hbox {H}_2\hbox {O}$, and $\hbox {CH}_4$) are important components of the atmospheres of Earth, planets within our solar systems [4], several of the moons of Jupiter and Saturn [4, 5], and other exoplanets [6]. Due to the interactions of cosmic rays, thunderstorms, and other high-energy processes occurring in the upper atmospheres of these bodies, positrons are continuously produced [7,8,9,10]. Accurate scattering cross sections for these molecules, therefore, are required for atmospheric research of positron transport.

Diatomic carbon ($\hbox {C}_2$) is typically only found in extreme conditions, usually as a component of carbon vapor, as it forms at temperatures $>3500^{\circ }\hbox {C}$ [11]. Consequently, its presence in nature is constrained mainly to comets, stellar atmospheres, and the interstellar medium (ISM) [12]. In industry, this molecule typically forms when carbon is introduced to plasmas and is involved in the production of fullerenes [13]. Alongside $\hbox {C}_2$, all of the considered molecules, except for $\hbox {O}_3$, have been detected in the ISM [14, 15]. Substantial quantities of positrons are known to propagate through the ISM, and there is much interest in the transport, interactions, and source of these particles [16,17,18].

There exist numerous positron experiments for several processes for each of the considered molecules, except for $\hbox {C}_2$ and $\hbox {O}_3$. At high energies, the inelastic results for positrons and electrons are expected to become equivalent. For elastic cross sections, the electron results provide an upper bound on the positron results for the considered energies. As the elastic cross section at large energies is small, the total cross section for these projectiles will also be close at such energies. Therefore, electron experiments are presented either where appropriate or when a transition has no existing positron measurements. Due to the extreme conditions required to create and maintain $\hbox {C}_2$, there exists no experimental results for electron or positron scattering for any of the considered processes.

Previous calculations for positron scattering on the molecules considered in this work have been conducted in either the independent atom model (IAM) [19, 20], IAM with screening-corrected additivity rule (IAM-SCAR) [21, 22], IAM-SCAR with the inclusion of interference terms (IAM-SCAR+I) [23,24,25], model potential-based approaches [26,27,28,29,30,31,32,33], many-body theory (MBT) [34], binary-encounter Bethe (BEB) [35, 36], distorted-wave [37,38,39,40,41], Schwinger multichannel (SMC) [42,43,44,45,46], body-frame vibrational close-coupling (BF-VCC) [47, 48], R-matrix [49, 50], and close-coupling [51] methods. Of these, BEB and distorted-wave methods have been utilized solely for the calculation of direct ionization cross sections ($\sigma _{\text {ion}}$), whereas the close-coupling, BF-VCC, R-matrix, and SMC calculations have been conducted only for low energies. From the remaining approaches, almost all possible cross sections have been calculated previously. The exceptions to this are $\hbox {O}_3$, in which positron calculations have been completed for only the total and elastic cross section, and the total electronic excitation cross sections ($\sigma _{\text {exc}}$) of $\hbox {CH}_4$. As with experiment, current calculations are compared to electron calculations either where appropriate or in the absence of positron results.

For molecular systems, the molecular convergent close-coupling (MCCC) method is currently limited to molecular targets with a few electrons. This includes $\hbox {H}_2$ and $\hbox {H}_2^+$ for positron [52,53,54,55] and electron scattering [56,57,58,59], and HeH$^+$ for electron scattering [60]. The extension of this approach to the other molecules considered in this study would be a significant undertaking and calculations would require considerable computational resources. As CCC calculations have been completed for atomic hydrogen [61, 62], carbon [63], and oxygen [64], a simpler approach is instead to use the IAM-SCAR method [65].

The IAM-SCAR method allows us to approximate molecular cross sections with only the relevant atomic cross sections and interatomic distances of the molecules. It has previously been applied to a wide range of molecular systems where its accuracy was tested [22, 65,66,67,68,69]. Typically the atomic values utilized for these calculations are from model potential approaches. Here, we instead use the results obtained from the relevant atomic convergent close-coupling (CCC) calculations [61,62,63,64]. Unlike the CCC approach, some model potential approaches are not ab initio and rely upon the choice of parameters to model the scattering system. For C and O, there are no existing positron scattering experiments and a small number of theoretical calculations. Furthermore, for these atoms, notable discrepancies exist between the CCC and previous model potential calculations [63, 64]. Therefore, the benefit of using these CCC cross sections is that they provide highly accurate results across the entire energy range for which the errors are well-documented.

The present approach includes a modification to the IAM-SCAR approach that accounts for molecular inelastic thresholds by shifting the thresholds of inelastic processes. With this modified approach, a comprehensive set of cross sections for molecular hydrogen ($\hbox {H}_2$), molecular oxygen ($\hbox {O}_2$), diatomic carbon ($\hbox {C}_2$), carbon monoxide (CO), carbon dioxide ($\hbox {CO}_2$), ozone ($\hbox {O}_3$), water ($\hbox {H}_2\hbox {O}$), and methane ($\hbox {CH}_4$) are obtained for energies between 0.1 and 5000 eV. In the case of $\hbox {H}_2$, we are able to compare with the most accurate MCCC method [52]. From this comparison, we have introduced a linear scale for molecules containing H that fits to accurate high-energy results and is expected to improve the accuracy of the current $\hbox {H}_2$, $\hbox {H}_2\hbox {O}$, and $\hbox {CH}_4$ results.

The paper is organized as follows: First, in Sect. 2 the methodology of the CCC, CCC-pot, and IAM-SCAR approaches and the linear scaling of molecules containing H is described. In Sect. 3, the results for each of the considered molecules are provided, with Sect. 3.1 for H2, Sect. 3.2 for O2, Sect. 3.3 for CO, Sect. 3.4 for C2, Sect. 3.5 for CO2, Sect. 3.6 for O3, Sect. 3.7 for H2O, and Sect. 3.8 for CH4. In each case, total, elastic, direct ionization, total electronic excitation, electron-loss, and positronium-formation processes are considered, as well as an estimation of the errors of the current approach based on comparison to other calculations or available data. All units are in atomic units unless otherwise specified.

2 Method

2.1 CCC

The CCC method is well described within the literature for positron scattering from atoms in both the one- and two-center cases [61,62,63,64, 70,71,72,73]. Therefore, rather than provide a full mathematical description of this approach only a succinct description is provided.

The atomic structure is described through a configuration–interaction (CI) representation:

$$\begin{aligned} \Phi _n^{N}(x_{1},\ldots , x_{N_\text {e}})= \sum _{i=1}^{N_{c}} C_{i}^{(n)} \phi _{i}(x_{1},\ldots , x_{N_\text {e}}). \end{aligned}$$

(1)

Here, $\phi _i$ are the Laguerre-based antisymmetrized $N_{e}$ electron configurations, $C_{i}^{(n)}$ are the CI coefficients, $N_{c}$ is the number of configurations, and $x_{i} = (\varvec{{r}}_{i}, \sigma _{i} )$ represents the spatial ($\varvec{{r}}_{i}$) and spin ($\sigma _i$) coordinates of electron i.

Within both the one- and two-center approaches, the target Hamiltonian is then diagonalized with these configurations, which produces N target pseudostates ($\Phi _{n}^{N}$). In the two-center approach, a set of positronium pseudostates is used to account for positronium-formation.

The Schrödinger equation for the total scattering wave function is given by:

$$\begin{aligned} (H-E)|\Psi _i^{(+)} \rangle = 0, \end{aligned}$$

(2)

where E is the energy of the collision system and H is the total Hamiltonian for the positron scattering system. In the single-center approach, Eq. (2) is solved through the expansion of $\Psi _i^{(+)}$ in the set of target pseudostates. In the two-center approach, this expansion makes use of the set of both target and positronium pseudostates.

In the single-center approach, the expansion is then substituted into Eq. (2) leading to a set of coupled Lippmann–Schwinger equations for the T matrix

$$\begin{aligned}&\left\langle \varvec{k}_f^{} \Phi ^{N}_f|T|\Phi ^{N}_i \varvec{k}_i^{} \right\rangle = \left\langle \varvec{k}_f^{} \Phi ^{N}_f|V|\Phi ^{N}_i \varvec{k}_i^{} \right\rangle \nonumber \\&\quad +\sum _{n=1}^{N} \!\int d{\varvec{k}} \frac{\left\langle \varvec{k}_f^{} \Phi ^{N}_f|V |\Phi ^{N}_n \varvec{k} \right\rangle \left\langle \varvec{k}\Phi ^{N}_n|T |\Phi ^{N}_i \varvec{k}_i^{} \right\rangle }{E-\epsilon ^{N}_n - k^2/2 + i0}. \end{aligned}$$

(3)

Following a partial-wave expansion, these equations are then solved numerically for each total orbital angular momentum J [74]. To accelerate the rate of convergence with increasing J, an analytic Born completion technique [56] is utilized.

2.2 CCC-pot

For single-center calculations, it is impossible to distinguish between positronium-formation and direct ionization, obtaining only the electron-loss cross section ($\sigma _\text {EL}$). Furthermore, for energies between the positronium-formation and direct ionization thresholds, this approach is unable to obtain converged results due to mismatched boundary conditions. To overcome these shortfalls, the CCC-pot method was developed and has currently been applied to carbon [63], oxygen [64], and noble gas atoms [75]. The CCC-pot approach is described in detail within Refs. [63, 64, 75], and therefore, we only provide a brief description here.

In this approach, a model potential calculation is used to obtain cross sections across the considered energy range. The potential ($V_{\textrm{opt}}$) is described through,

$$\begin{aligned} V_{\textrm{opt}}(r,E_i) = V_{\textrm{st}}(r) + V_{\textrm{pol}}(r) + V_\textrm{abs}(r,E_{i}), \end{aligned}$$

(4)

where $V_{\textrm{st}}$ is the static potential, $V_{\textrm{pol}}$ the polarization potential, and $V_{\textrm{abs}}$ the absorption potential. The $V_{\textrm{st}}(r)$ is obtained using the same structure model as our single-center calculations. The $V_{\textrm{pol}}(r)$ includes a term which is modified at each energy to reproduce the elastic cross section from the single-center CCC calculations. For $V_{\textrm{abs}}$, we utilize the absorption potential of Staszewska et al. [76].

As positronium-formation cannot be explicitly included in the absorption potential, we rely upon the delta variational technique to calculate positronium-formation cross sections. This entails modifying the absorption threshold with the form given by Chiari [21],

$$\begin{aligned} \Delta (E) = \Delta _\text {e} - (\Delta _\text {e} - \Delta _\text {p})e^{-(E_i - \Delta _\text {p})/E_{m}}. \end{aligned}$$

(5)

Here, $\Delta _\text {e}$ and $\Delta _\text {p}$ are the electronic excitation and positronium-formation threshold energies, respectively. For the adjustable parameter $E_m$, we have used the energy at which the single-center CCC $\sigma _\text {tot}$ has a maximum.

In our calculations, a scaling factor ($F_{s}(E)$) is included in the $V_{\textrm{abs}}(r,E_{i})$ of the calculations containing Eq. 5. This factor increases the absorption potential and, therefore, the positronium-formation cross section for energies below its maximum. This factor was derived through comparison with two-center CCC calculations of atomic hydrogen [64] and is expected to improve the accuracy of this approach at low energies.

Relevant cross sections are then obtained from T-matrix elements calculated through the solution of the Lippmann–Schwinger equations for the potential scattering system. The final step of this process is the scaling of the obtained cross sections to better approximate the single-center CCC calculation; this process is provided in detail in Ref. [63].

2.3 IAM-SCAR

The IAM method for a molecule containing $N_{a}$ atoms is given by:

$$\begin{aligned} \sigma _{m} = \sum ^{N_{a}}_{i=1} \sigma _{i}, \end{aligned}$$

(6)

where $\sigma _{m}$ is the cross section of the considered molecule and $\sigma _{i}$ is the cross section of each independent atom i. This is a simple approximation and is expected to only be accurate for high incident energies, where the projectile “sees” the molecule as a sum of its independent atoms. For lower energies, the atomic cross sections are typically too large and, if they are geometrically visualized, will overlap the cross sections of the surrounding atoms within the molecule. Without accounting for these overlaps, the IAM method double-counts these interactions with the projectile and overestimates the cross section. The fundamental concept of the IAM-SCAR method [77], therefore, is the use of screening coefficients ($s_{i}$) that account for the geometry of a molecule and reduces the individual atomic contributions to the molecular cross section. This concept leads to a reformulation of Eq. 6 to

$$\begin{aligned} \sigma = \sum _{i} s_{i} \sigma _{i}. \end{aligned}$$

(7)

These screening coefficients are calculated through,

$$\begin{aligned} s_{i}=1-\frac{\epsilon ^{(2)}_{i}}{2!} + \frac{\epsilon ^{(3)}_{i}}{3!} -\frac{\epsilon ^{(4)}_{i}}{4!} + \cdots \nonumber \\ \end{aligned}$$

(8)

where

$$\begin{aligned} \epsilon _{i}^{(k)} = \frac{N_{a}-k+1}{N_{a}-1}\sum _{j\ne i} \frac{\sigma _{j}\epsilon _{j}^{(k-1)}}{\alpha _{ij}} \quad (k=2, \ldots , N_{a}), \nonumber \\ \end{aligned}$$

(9)

and $\alpha _{ij}= \text {max}(4\pi r_{ij}^2, \sigma _{i}, \sigma _{j})$. Here, $r_{ij}$ is the distance between the centers of the atoms i and j and $\sigma _{j}$ is the total cross section of atom j. The first coefficient is defined as $\epsilon ^{(1)}_{i}=1$. For each molecule, the molecular geometries were obtained from the CCCB database of NIST [78].

This approach does not account for the vibrational or rotational motion of the target molecule and is expected to be most accurate for energies above 30 eV [79]. Furthermore, rotational cross sections, if available, can be included through their summation to the IAM-SCAR result, a process referred to as the IAM-SCAR+rot method [80]. This approach was applied for $\hbox {H}_2\hbox {O}$ as rotational results are available in the literature [25] and rotational excitation is expected to greatly contribute to the total cross section.

2.4 CCC-SCAR

For atomic hydrogen, cross sections were obtained from existing two-center calculations. For carbon and oxygen, the total cross section is equivalent to the CCC-pot result from the positronium-formation threshold to 10 eV above the direct ionization threshold. Outside of this region, total cross sections are from the single-center CCC calculation. The positronium-formation cross section is calculated with the CCC-pot approach and used to extract the direct ionization cross section from the electron-loss results of the single-center approach. To obtain the cross sections for molecular systems with the CCC calculations for independent atoms, the IAM-SCAR method has been modified. First, we utilize the IAM-SCAR approach for the total cross section to obtain screening coefficient values for each atom at each considered energy. Following this, these screening coefficients are applied independently to all of the cross sections that compose the total cross section for each atom at that energy. For molecules, the thresholds for each process, besides elastic, are typically different to the atoms that compose them. Therefore, we shift the cross sections for each process so that its threshold is equal to its accepted threshold value for each molecule, which are given in Table 1. Direct ionization thresholds were obtained from the CCCB database of NIST [78]. The sources of the electronic excitation thresholds are provided in Table 1. These shifted cross sections can then be summed to obtain the final total cross section of the molecule at each energy. Calculations conducted with this approach and utilizing atomic CCC cross sections are referred to as CCC-SCAR.

Table 1 Direct ionization and electronic excitation thresholds for each considered molecule in eV

Fig. 1

Total cross section for positron scattering on molecular hydrogen. Theoretical CCC-SCAR, CCC-SCAR without the scaling described in text, and IAM results are shown alongside the MCCC results [52]

Fig. 2

Total cross section for positron and electron scattering on molecular hydrogen. MCCC results are from [52, 86] for electrons and positrons, respectively. The recommended electron results for $\hbox {H}_2$ are from Yoon et al. [87]. For the IAM results, we use the electron measurements of Zhou et al. [90], the recommended positron results of Ratnavelu et al. [89], and the CCC calculations for positrons [62] and electrons [88]

2.5 H scaling correction

For CCC-SCAR calculations of $\hbox {H}_2$, we found large discrepancies with existing MCCC calculations, with the CCC-SCAR substantially overestimating the total cross section. This is most transparent at high energies, see Fig. 1, where the IAM cross section is systematically larger than the MCCC cross section. To validate the current cross sections, we have also included Fig. 2. In this figure, we present the MCCC results for incident positrons [52] and electrons [86] alongside the recommended electron results of Yoon et al. [87]. There is near-perfect agreement between these results to 1000 eV. Also shown in this figure are the IAM results of the CCC for both of these projectiles [62, 88]. We have also included the recommended results for positron scattering from H [89] and the measurements of Zhou et al. [90] for electron scattering on H, both multiplied by a factor of two in accordance with the IAM. Comparing the IAM results, there is excellent agreement between the CCC and experimental cross sections across the presented energy range. Apparently, the IAM method fails to approximate $\hbox {H}_2$ cross sections with corresponding atomic cross sections for both calculated and measured results for both electron and positron impact. The origin of this failure is beyond the scope of this paper.

To correct this, we have introduced a scale factor of 0.77, which the H component of the CCC-SCAR cross section is uniformly multiplied by above 100 eV. Between the positronium-formation threshold and 100 eV, this factor is linearly reduced from 1 to 0.77. The linear scale was included to 100 eV as the CCC-SCAR approach is expected to be of high accuracy above these energies and should accurately reflect the high-energy behavior. We have chosen not to scale the CCC-SCAR results to the known MCCC values at all energies. The aim here is to rectify the obvious problem with the CCC-SCAR technique in a way that can be easily extended to other molecular scattering systems where often accurate close-coupling calculations are not available while it is often relatively easy to establish the high energy behavior of the total cross section through either existing electron measurements or Born approximation calculations.

The impact of this scale factor is demonstrated in Fig. 1 for $\hbox {H}_2$ where, after applying this scale factor, substantially better agreement is found between the CCC-SCAR results and the MCCC calculations. For other molecules containing H, similar disagreements were found at high energies with existing high-energy electron measurements for the total cross section. For molecules that did not contain H, no such discrepancies were found. This suggests that there is a failure in the IAM approach when including the current cross sections for H. Therefore, for the molecules containing H, i.e., $\hbox {H}_2\hbox {O}$ and $\hbox {CH}_4$, we have scaled the H component of these calculations. With this modification, we found a very good agreement with high energy electron measurements for the total cross section as will be demonstrated in what follows.

3 Results

3.1 $\hbox {H}_2$ scattering

For $\hbox {H}_2$, CCC-SCAR calculations are conducted with existing one- and two-center results for atomic H [61, 62]. Two-center results [52] are used for energies $\ge$ 10 eV and single-center [54] for energies below this. Through comparison of the CCC-SCAR calculations with the MCCC calculations for $\hbox {H}_2$, the error of the current approach is estimated.

3.1.1 Total cross section

In Fig. 3, we present the CCC-SCAR calculations for the total cross section of $\hbox {H}_2$ alongside previous MCCC calculations [52] and several experiments [90,91,92,93,94,95]. Close agreement is found between the MCCC and CCC-SCAR calculations for energies above 50 eV and below 3 eV. Between these two energies, the CCC-SCAR overestimates the MCCC calculations. Above 40 eV, good agreement is found with the presented experimental results. At low energies, there is little consensus between experiments; however, the MCCC and CCC-SCAR are closest to the most recent experiment of Machacek et al. [93].

Fig. 3

Total cross section for positron scattering on molecular hydrogen. Theoretical CCC-SCAR results are shown alongside the MCCC calculations [52] and the experiments of Hoffman et al. [91], Zecca et al. [92], Machacek et al. [93], Karwasz et al. [94], Deuring et al. [95], and Zhou et al. [90]

3.1.2 Elastic cross section

The elastic cross section for positron scattering on $\hbox {H}_2$ is shown in Fig. 4. The CCC-SCAR elastic cross section is close to the MCCC results [52] for energies below the positronium-formation threshold. From 15 to 1000 eV, however, the CCC-SCAR results are slightly lower than the MCCC calculations. The opening of the positronium-formation channel can result in resonant-like structures in the elastic cross section known as a Wigner cusp [96]. As the positronium-formation thresholds, and the magnitudes of this cross section, are different in H and $\hbox {H}_2$, this is likely the source of the discrepancy between the CCC-SCAR and MCCC results for energies between 4 and 15 eV.

Fig. 4

Elastic cross section for positron scattering on molecular hydrogen. Theoretical CCC-SCAR results are shown alongside the MCCC calculations [52]

3.1.3 Total electronic excitation cross section

The current CCC-SCAR results for the total electronic excitation cross section for $\hbox {H}_2$ are shown in Fig. 5 alongside the MCCC calculations [52] and the summed $B^{1}\Sigma _{u}^{+}$ and $C^{1}\Pi _{u}$ BSMC results of Arretche and Lima [45]. The measurements of Sullivan et al. [97] for excitation to the $B^{1}\Sigma _{u}^{+}$ state are also shown; there are no experimental results for the total electronic excitation. The CCC-SCAR results are close to the MCCC calculations for energies below 20 eV and both are within the error of Sullivan et al. [97] under 15 eV. Other excited channels open above 15 eV; therefore, this experiment will be smaller above this energy. Above 20 eV, the CCC-SCAR results are higher than the MCCC calculations to 5000 eV. However, above 100 eV, differences between these calculations decrease. Compared to the summed results of Arretche and Lima [45] for the first two excitations the MCCC and CCC-SCAR calculations are higher even toward threshold.

Fig. 5

Total electronic excitation cross section for positron scattering on molecular hydrogen. CCC-SCAR results are shown alongside MCCC calculations [52] and the summed $B^{1}\Sigma _{u}^{+}$ and $C^{1}\Pi _{u}$ BSMC results of Arretche and Lima [45]. The experiment of Sullivan et al. [97] for excitation to the $B^{1}\Sigma _{u}^{+}$ state is also shown

3.1.4 Electron-loss, direct ionization, and positronium-formation cross section

The positronium-formation cross section for $\hbox {H}_2$ is shown in Fig. 6. The CCC-SCAR results are shown alongside the MCCC calculations [52] and those of Singh and Anthony [29] and Biswas et al. [98]. The experiments of Machacek et al. [93], Zhou et al. [90], and Fromme et al. [99] are also presented. The CCC-SCAR method is in close agreement with the MCCC calculations for energies above 30 eV, although it slightly underestimates between 50 and 100 eV. Compared to the experiments, we find the two-center and CCC-SCAR to be lower than experimental uncertainty for energies between 40 and 80 eV. The two-center calculations lie below the uncertainty of both experiments for energies below 15 eV. In contrast, the CCC-SCAR is within experimental uncertainty below 9 eV and above 30 eV but above experimental points between these energies. The MCCC results follow a similar shape to the calculations of Biswas et al. [98] but are slightly lower across the energy range. The results of Singh and Antony [29] are close to the MCCC result to 15 eV but predict a much lower maximum. Above 40 eV, both of these theoretical calculations are of a higher magnitude than the MCCC and CCC-SCAR results.

Fig. 6

Positronium formation cross section for positron scattering on molecular hydrogen. Theoretical CCC-SCAR and MCCC [52] results are shown alongside the calculations of Singh and Antony [29] and Biswas et al. [98], and the measurements of Machacek et al. [93], Zhou et al. [90], and Fromme et al. [99]

The CCC-SCAR direct ionization results are presented in Fig. 7 alongside the MCCC calculations [52], other previous theory [29, 36, 100], and previous measurements [99, 101,102,103]. The CCC-SCAR are in excellent agreement with experiment for energies below 40 eV, whereas the two-center results are larger than experiment for this energy range. Between 30 and 90 eV, the CCC-SCAR is above the MCCC calculations, finding a sharper peak than these results. Above 80 eV, the CCC-SCAR slightly underestimates the MCCC and is in close agreement with experiment above 700 eV. For energies between 40 and 300 eV, there is little consensus among experiments and theory. In this region, the MCCC results are closest to the BEB calculations of Fedus and Karwasz [36], whereas the CCC-SCAR is similar to the SCOP results of Singh and Antony [29] to its peak. We expect that the two-center MCCC results are of higher accuracy than the CCC-SCAR results and that the CCC-SCAR approach is overestimating for energies between 30 and 80 eV.

Fig. 7

Direct ionization cross section for positron scattering on molecular hydrogen. Theoretical CCC-SCAR and MCCC [52] results are shown alongside the calculations of Singh and Antony [29], Campeanu et al. [100], and Fedus and Karwasz [36]. Experimental results are those of Knudsen et al. [101], Fromme et al. [99], Ashley et al. [102], and Jacobsen et al. [103]

The electron-loss cross section of $\hbox {H}_2$, presented in Fig. 8, is equivalent to the sum of the positronium-formation and direct ionization cross sections. Therefore, we find similar discrepancies as with these two components. There is agreement with the CCC-SCAR and MCCC [52] results for energies above 80 eV. For energies below 80 eV, the current approach is significantly higher than the MCCC results. Compared to the measurements of Fromme et al. [99] and Moxom et al. [104], good agreement is found below 10 eV, between 20 and 30 eV, and above 70 eV. With the CCC-SCAR results much higher than these experiments for energies between 10 and 30 eV and lower for energies between 50 and 70 eV. Compared to the only other theoretical calculation of Singh and Antony [29], we find substantial differences across the energy range.

Fig. 8

Electron-loss cross section for positron scattering on molecular hydrogen. Theoretical CCC-SCAR and MCCC [52] results are shown alongside the calculations of Singh and Antony [29] and the measurements of Fromme et al. [99] and Moxom et al. [104]

3.1.5 CCC-SCAR error analysis for $\hbox {H}_2$

For the total cross section, we find the largest disagreement between the CCC-SCAR and the MCCC for energies between the positronium-formation threshold and 50 eV. This discrepancy is directly related to the positronium-formation cross section, which the CCC-SCAR finds to be significantly larger than the MCCC calculation. Outside this region, the CCC-SCAR results are within 10% of the MCCC calculations. For energies above 30 eV, the positronium-formation cross section results for the CCC-SCAR are within 20% of the MCCC results to 100 eV, above which this cross section becomes negligible. Below 30 eV, the CCC-SCAR positronium-formation is significantly higher than the MCCC result and existing experiments. The total electronic excitation cross section is within 10–20% of the MCCC results at low and high energies, with larger discrepancies observed at intermediate energies. For the direct ionization cross section, the CCC-SCAR results are within 20% of the MCCC calculations for high and low energies but much larger at intermediate energies. For the electron-loss cross section, the CCC-SCAR results are within 10% of the MCCC results for energies above 50 eV.

This approach relies on simple approximations and does not account for vibrational or rotational processes or the different polarizabilities present within molecules. For diatomic non-polar molecules, we expect a similar error of approximately 20% as found here for all $\hbox {H}_2$ processes besides positronium-formation. For the positronium-formation cross section, we instead expect an error of 20% for energies above 30 eV. Below this energy, we estimate a higher error of 30%, based on the comparison with experiment. For molecules with more atoms or that are polar, we expect that the accuracy of the current approach will also decrease for low and intermediate energies. However, without accurate calculations or experiments for these molecules to compare with we are unable to estimate to what extent the accuracy will decrease.

3.2 $\hbox {O}_2$ scattering

3.2.1 Total cross section

The total cross section of a positron incident upon molecular oxygen is presented in Fig. 9. For energies above 1000 eV, the current calculations are slightly below the electron measurements of García et al. [105]. Therefore, as the current approach reproduces the expected high-energy behavior scaling is not required for this molecule to obtain accurate results. CCC-SCAR calculations are lower than the IAM-SCAR+I calculations of Ellis-Gibbings et al. [24], the IAM-SCAR calculations of Chiari et al. [21], and the SCOP calculations of Singh et al. [28]. Close agreement is found between the CCC-SCAR and the experiment of Chiari et al. [21] for energies between 1 and 10 eV. Although the results of Chiari et al. [21] are not corrected for forward scattering effects, as $\hbox {O}_2$ is a nonpolar molecule this correction is not expected to be significant. Therefore, the CCC-SCAR agrees fairly well with the experiment of Chiari et al. [21] but underestimates for energies below 1 eV. The IAM-SCAR approach is of least accuracy at these low energies where the impact of a molecule’s polarizability and structure are at their most significant.

Fig. 9

Total cross section for positron scattering on molecular oxygen. Theoretical CCC-SCAR results are shown alongside calculations by Singh et al. [28], Reid and Wadehra [19], Ellis-Gibbings et al. [24], Tenfen et al. [26], and Pinheiro et al. [51]. Experimental $\hbox {O}_2$ results for positron scattering are from Charlton et al. [106], Chiari et al. [21], and Dababneh et al. [107]. Electron $\hbox {O}_2$ results are by García et al. [105]

There is good agreement with the CCC-SCAR results and those of Tenfen et al. [26] and Pinheiro et al. [51] for energies between 1 eV and the positronium-formation threshold. Both of these calculations utilized model potential-based approaches in which higher-order polarizabilities are accounted for, with Pinheiro et al. [51] also including adiabatic corrections. The calculations of Singh et al. [28], on the other hand, employed the SCOP method, where the polarization component of the potential is obtained via the approach of Padial and Norcross [108]. It has been demonstrated by Jain [109] that model correlation potentials which model electron impact, such as the one utilized by Singh et al. [28], are inadequate to describe positron scattering, particularly at low energy. This is due to the fundamental differences in the correlation between both projectiles, which are more pronounced at lower energies. Chiari et al. [21] and Ellis-Gibbings et al. [24] calculations also utilized the IAM-SCAR approach, with the underlying atomic cross sections calculated through model potential calculations. These other theoretical calculations agree at 1 eV and are substantially higher than the current results and those of Tenfen et al. [26] and Pinheiro et al. [51].

The CCC-SCAR results are in agreement with positron experiment for energies up to 100 eV. In contrast, the other theoretical calculations predict substantially higher cross sections in this energy range. Above 100 eV, the CCC-SCAR results are slightly higher than the measurements of Dababneh et al. [107] and much higher than those of Charlton et al. [106]. The results of Singh et al. [28] are above this experiment, whereas the other calculations are below the CCC-SCAR result.

3.2.2 Elastic cross section

The elastic cross section for $\hbox {e}^+$-$\hbox {O}_2$ is shown in Fig. 10. CCC-SCAR results below the positronium-formation are equivalent to the TCS. The IAM-SCAR+I calculations of Ellis-Gibbings et al. [24], the IAM-SCAR calculations of Chiari et al. [21], and the IAM calculations of Reid and Wadehra [19] follow similar behaviors but at different magnitudes. The CCC-SCAR results, however, are notably different, with a more gradual descent. Compared to the experimental $e^-$ results of Iga et al. [110], our results lie within the error bars of the 1000 eV measurement. The CCC-SCAR results are also in close agreement with the theoretical $e^-$ calculations of Williart et al. [111] for energies above 2000 eV. As $e^+$ and $e^-$ scattering are expected to be close for these high incident energies, with the $e^-$ results providing an upper limit for the $e^+$ result, this lends validity to our results at high energy.

Fig. 10

Elastic cross section for positron scattering on molecular oxygen. Theoretical results include those of the CCC-SCAR, Reid and Wadehra [19], Ellis-Gibbings et al. [24], Chiari et al. [21], Tenfen et al. [26], Pinheiro et al. [51], and Williart et al. [111] for the electron case. Experimental results are from Chiari et al. [21], Dababneh et al. [107] and Iga et al. [110] for the electron case

3.2.3 Total electronic excitation cross section

The total electronic excitation cross sections for $\hbox {O}_2$ are presented in Fig. 11. The current CCC-SCAR results are lower than the experimental data of Katayama et al. [112], which measures positron excitations to the Schumann–Runge continuum, even if the CCCC-SCAR reproduces well the near-to-threshold resonant-like dependence of the cross section. This continuum represents excitations from the ground state $X ^3 \Sigma ^{-}_{g}$ to the $B ^3 \Sigma ^{-}_{u}$ state. The other existing theoretical results of Ellis-Gibbings et al. [24] are significantly larger than the CCC-SCAR results across the calculated energy range.

To facilitate comparison with $\hbox {O}_2$ excitations for incident electrons at high energies, we present the summed theoretical and experimental results of Suzuki et al. [113] and Newell et al. [114] for excitations to the Schumann–Runge continuum and the longest band state. The longest band state represents excitations of $\hbox {O}_2$ from the ground state to $E ^3\Sigma _{u}^{-}$. Compared to this summed experiment, the positron measurements and current calculations are within error for all values in their measured range of 15–200 eV. The CCC-SCAR is higher than high-energy theoretical results of Suzuki et al. [113] for energies above 100 eV. This difference is expected as the CCC-SCAR should approximate the sum of all electronic excitations, whereas the summed theoretical results of Suzuki et al. [113] contain only two transitions.

Fig. 11

Total electronic excitation cross section for positron scattering on molecular oxygen. Theoretical results include the CCC-SCAR and the IAM-SCAR calculations of Ellis-Gibbings et al. [24] for positrons and the calculations of Suzuki et al. [113] for the electron case. Experimental results for positrons are of excitations to the Schumann–Runge continuum from Katayama et al. [112]. Electron experimental results are from Newell et al. [114] and Suzuki et al. [113]

3.2.4 Electron-loss, direct ionization, and positronium-formation cross section

In Fig. 12, the results for the positronium-formation cross section of $\hbox {e}^+$-$\hbox {O}_2$ are presented. The theoretical calculations of Singh and Antony [115] and Ellis-Gibbings et al. [24] predict higher peaks than the CCC-SCAR. These calculations also predict that $\sigma _{\text {Ps}}$ increases to a maximum and then steadily decreases, smoothly in the case of Singh and Antony [115] and sharply in that of Ellis-Gibbings et al. [24]. All theoretical methods agree for energies above 100 eV; however, the CCC-SCAR results are lower than the other theory for energies between 10 and 100 eV but in better agreement with experiments of Marler and Surko [82], Archer et al. [116], and Laricchia et al. [117]. Compared to all of the experiments, the CCC-SCAR is in good agreement for energies below 7 eV, with all experiment, and above 30 eV, with that of Marler and Surko [82]. Experimental results predict a first maximum between 7 and 9 eV, which is expected to result from coupling between the positronium-formation and excitation to the Schumann–Runge continuum [117]. A second maximum is predicted by experiment to occur at 30 eV. The CCC-SCAR results are in a qualitative agreement with the more recent experiments.

Fig. 12

Positronium-formation cross section for positron scattering on molecular oxygen. Theoretical results include CCC-SCAR and the calculations of Ellis-Gibbings et al. [24] and Singh and Antony [115]. Experimental $\hbox {O}_{2}$ measurements are from Marler and Surko [82], Archer et al. [116], Laricchia et al. [117] and Griffith [118]

Results for the direct ionization cross section for the $\hbox {e}^+$-$\hbox {O}_2$ system are shown in Fig. 13. Excellent agreement is found between the calculations of Ellis-Gibbings et al. [24] and the CCC-SCAR results for energies above 150 eV. For energies above 500 eV, there is agreement between the CCC-SCAR, [24], and Franz et al. [35] calculations. The calculations of Singh and Antony [115], on the other hand, are almost double that of the other calculations for incident energies above 500 eV. At lower energies, there is no agreement between different theoretical methods, with all varying in their magnitude. For energies below 20 eV, the results of the CCC-SCAR and Ellis-Gibbings et al. [24] are within the range of experimental results. For energies between 20 and 60 eV, the experiments are typically between the calculations of Singh and Antony [115] and Franz et al. [35] with these calculations lying with the error of some points. For this same energy range, the CCC-SCAR results are above the calculations of Singh and Antony [115] and much lower than the calculations of Ellis-Gibbings et al. [24].

Fig. 13

Direct ionization cross section for positron scattering on molecular oxygen. Theoretical results include CCC-SCAR, Ellis-Gibbings et al. [24], Singh and Antony [115], Franz et al. [35] and Campeanu et al. [39]. Experimental results for positrons are from Marler and Surko [82] and Katayama et al. [112], electron results are from Krishnakumar and Srivastava [119]

Fig. 14

Electron-loss cross section for positron scattering on molecular oxygen. Theoretical results include CCC-SCAR and Singh and Antony [115] and Ellis-Gibbings et al. [24] calculations. Experimental $\hbox {O}_{2}$ measurements are from Marler and Surko [82] and Laricchia et al. [117]

The electron-loss cross sections are presented in Fig. 14. The $\sigma _\text {EL}$ is equivalent to the direct ionization cross section for high energies. The CCC-SCAR results predict a similar shape to the experiments of Marler and Surko [82] and Laricchia et al. [117] but have a larger magnitude for energies above 7 eV. The calculations of Singh and Antony [115] are also higher than experiment but predict a very different shape. The results of Ellis-Gibbings et al. [24] predict a rise in this cross section to a peak at 30 eV, approximately 1.5 times larger than the other calculations and experiment, followed by a rapid decrease after which there is excellent agreement with the CCC-SCAR results for energies above 200 eV.

3.3 CO scattering

3.3.1 Total cross section

The total cross section for positron scattering on carbon monoxide is shown in Fig. 15. Above 1000 eV, the CCC-SCAR is larger than this experiment but in close agreement with the measurements of García et al. [126]. From this, we can conclude the IAM calculation of this molecule is valid and scaling is not required for this system. The current CCC-SCAR calculations are shown alongside the theoretical results of Billah et al. [120], which utilize the IAM-SCAR+I approach, Gianturco et al. [47], that use the BF-VCC, and those of Singh et al. [27]. Experimental results for electrons are from Karwasz et al. [125] and García et al. [126] and for positrons are from Zecca et al. [121], Sueoka and Mori [122], Coleman et al. [123], and Kwan et al. [124]. For energies below the positronium-formation, our results find similar behavior to the measurements of Zecca et al. [121] but at a larger magnitude. As CO is polar, it is likely that the results of Zecca et al. [121] underestimate due to not accounting for forward scattering. The calculations of Billah et al. [120], on the other hand, predict a minimum at 4 eV and are significantly lower than the current result. The results of Gianturco et al. [47] were calculated only below the positronium-formation threshold and have a similar shape to the CCC-SCAR results but with a lower magnitude and steeper rise below 1 eV. For energies above the positronium-formation, our results are between the calculations of Billah et al. [120] and Singh et al. [27] to 60 eV. Above this energy, the current results are larger than these theoretical approaches. However, above 1500 eV, the current results are close to those of Singh et al. [27]. Our calculations are generally above the positron measurements across the calculated energy range. Agreement is found with the electron measurements of Karwasz et al. [125] and the CCC-SCAR results for energies between 200 and 1000 eV. Above 1000 eV, the results of Karwasz et al. [125] underestimate the current results and the electron measurements of García et al. [126]. Therefore, the current theory is expected to be more reliable at high energies.

Fig. 15

Total cross section for positron scattering on carbon monoxide. Theoretical results include those of the CCC-SCAR and those of Singh et al. [27], Billah et al. [120] and Gianturco et al. [47]. Experimental results for positrons are from Zecca et al. [121], Sueoka and Mori [122], Coleman et al. [123], and Kwan et al. [124]. For electrons, the measurements are from Karwasz et al. [125] and García et al. [126]

3.3.2 Elastic cross section

The elastic cross section for a positron incident on carbon monoxide is shown in Fig. 16. CCC-SCAR results are shown alongside the theoretical calculations of Billah et al. [120] and Singh et al. [27], and Gianturco et al. [47]. Very close agreement is found with the calculations of Billah et al. [120] above 100 eV. Below this energy, these results, and those of Gianturco et al. [47], exhibit very different behavior and are at a lower magnitude than the CCC-SCAR calculation. The calculations of Reid and Wadehra [19] are lower than the CCC-SCAR results for energies above 200 eV and, at high energies, are in agreement with the calculations of Kothari and Joshipura [127].

Fig. 16

Elastic cross section for positron scattering on carbon monoxide. Theoretical results include those of the CCC-SCAR, Singh et al. [27], Billah et al. [120], Kothari and Joshipura [127], Reid and Wadehra [19] and Gianturco et al. [47]

3.3.3 Total electronic excitation cross section

In Fig. 17, we present the $\hbox {e}^+$-CO total electronic excitation cross section. Current CCC-SCAR results are shown alongside the calculations of Kothari and Joshipura [127] for the total excitation, the calculations of Silva et al. [46] for excitation to the A$^1 \Pi$ state, and the measurements for excitation to the A$^1 \Pi$ state from Marler and Surko [82]. The calculations of Kothari and Joshipura [127] are below the CCC-SCAR results across almost the entire energy range. Over its calculated energies, the SMC results of Silva et al. [46] are slightly higher than the CCC-SCAR results. The current results, and those of Silva et al. [46], have a similar shape to the experiment of Marler and Surko [82]. However, both theoretical results predict a maximum that occurs at a higher energy, with the peak of the CCC-SCAR results is 20 eV above that predicted by this experiment. Toward threshold, the SMC and CCC-SCAR results predict lower values than that of the experiment.

Fig. 17

Total electronic excitation cross section for positron scattering on carbon monoxide. Theoretical results include those of the CCC-SCAR and Kothari and Joshipura [127] for the total excitation and Silva et al. [46] for excitation to the $\hbox {A}^1 \Pi$ state. Experimental results for excitation to the $\hbox {A}^1 \Pi$ state are from Marler and Surko [82]

3.3.4 Electron-loss, direct ionization, and positronium-formation cross sections

The CCC-SCAR calculations for the $\hbox {e}^+$-CO positronium-formation cross section are presented in Fig. 18 alongside the theoretical calculations of Singh and Antony [29] and experiment of Marler and Surko [82]. The current results are in agreement with Singh and Antony [29] for energies above 80 eV. For lower energies, the CCC-SCAR results, however, are significantly higher. As with $\hbox {O}_2$, agreement is found with experiment for energies above 30 eV. Below 30 eV, the current results are significantly larger than experiment.

For the direct ionization cross section, presented in Fig. 19, the CCC-SCAR results are larger than other theory and experiment for energies between 30 and 200 eV. Above 200 eV, the CCC-SCAR results lie within the uncertainty of the experiment of Bluhme et al. [128] and is between the calculations of Kothari and Joshipura [127] and Singh and Antony [29], and those of Tóth et al. [37] and Campeanu et al. [40]. Below 30 eV, the CCC-SCAR results are close to other presented theory and experiment.

Fig. 18

Positronium-formation cross section for positron scattering on carbon monoxide. Theoretical results include CCC-SCAR and the calculations of Singh and Antony [29]. Measurements are from Marler and Surko [82]

Fig. 19

Direct ionization cross section for positron scattering on carbon monoxide. Theoretical results include CCC-SCAR, Singh and Antony [29], Kothari and Joshipura [127], Tóth et al. [37], and Campeanu et al. [40]. Experimental results are from Marler and Surko [82] and Bluhme et al. [128]

Due to the large values for positronium-formation in the CCC-SCAR results, the $\sigma _\text {EL}$, shown in Fig. 20, features a first maximum not present in other theory or experiment. For energies between 30 and 80 eV, the CCC-SCAR results lie within the uncertainty of the experiment of Bluhme et al. [128]. For higher energies, the CCC-SCAR results are above the experiment of Bluhme et al. [128] and between the calculations of Billah et al. [120] and Singh and Anthony [29].

Fig. 20

Electron-loss cross section for positron scattering on carbon monoxide. Theoretical results include CCC-SCAR and Singh and Antony [115], and Ellis-Gibbings et al. [24] calculations. Experimental $\hbox {O}_{2}$ measurements are from Marler and Surko [82] and Bluhme et al. [128]

3.3.5 Total inelastic cross section

The total inelastic cross section for $\hbox {e}^+$-CO is presented in Fig. 21. As there are no experimental results for this scattering process, we present current CCC-SCAR results alongside the calculations of Singh et al. [27], Billah et al. [120], Reid and Wadehra [19] and Kothari and Joshipura [127]. For energies above 1000 eV, there is excellent agreement between all of the theoretical results, except for Billah et al. [120], which is lower than other theory. Below this energy, the CCC-SCAR results are larger than other theoretical results, except for below 70 eV, where the CCC-SCAR results are between the other theoretical calculations and the results of Billah et al. [120].

Fig. 21

Total inelastic cross section for positron scattering on carbon monoxide. Theoretical results include those of the CCC-SCAR, Singh et al. [27], Billah et al. [120], Reid and Wadehra [19] and Kothari and Joshipura [127]

3.4 $\hbox {C}_2$ scattering

3.4.1 Total cross section

The total cross section for a positron scattering on diatomic carbon is shown in Fig. 22. There is no existing experiment for high energies, however, as the IAM approach is found to be valid for the CO and $\hbox {O}_2$ molecules we expect that our calculations for C$_2$ will accurately predict the high-energy behavior. Current CCC-SCAR results are compared to the results of Singh et al. [28] and Reid and Wadehra [19]. For energies above 20 eV, good agreement is found between the CCC-SCAR results and those of Singh et al. [28]. Above 1000 eV, however, the results of Singh et al. [28] are slightly above the CCC-SCAR results. For energies between 6 and 25 eV, the CCC-SCAR results are larger than those of Singh et al. [28]. Below 6 eV, the CCC-SCAR results are instead lower, and its minimum at 5 eV is lower than that of Singh et al. [28] at 7 eV. The IAM calculations of Reid and Wadehra [19] are significantly lower than the other calculations over its entire range.

Fig. 22

Total cross section for positron scattering on diatomic carbon. Theoretical results include the CCC-SCAR and those of Singh et al. [28] and Reid and Wadehra [19]

3.4.2 Elastic cross section

The elastic cross section for this system is presented in Fig. 23. The only existing theoretical calculations are those of Reid and Wadehra [19]. As with the total cross section, these results are significantly lower than the CCC-SCAR results, except for energies below 100 eV.

Fig. 23

Elastic cross section for positron scattering on diatomic carbon. Theoretical results include those of the CCC-SCAR and Reid and Wadehra [19]

3.4.3 Total inelastic and electronic excitation cross section

For the total inelastic cross section for a positron scattering on diatomic carbon, shown in Fig. 24, the CCC-SCAR results are presented with the only other results of Reid and Wadehra [19]. Again, we find the calculations of Reid and Wadehra [19] to be lower than the CCC-SCAR results, although the differences between these results decrease with increasing incident energy. Also presented in this figure are the electronic excitation cross section which, due to there being no previous theoretical or experimental work, are presented without comparison.

Fig. 24

Total inelastic cross section for positron scattering on diatomic carbon. Theoretical results include those of the CCC-SCAR and Reid and Wadehra [19]. Also shown are the CCC-SCAR total electronic excitation cross section (Exc.) for this scattering system

3.4.4 Electron-loss, direct ionization, and positronium-formation cross sections

The positronium-formation cross section for positron scattering on diatomic carbon is shown in the bottom segment of Fig. 25. CCC-SCAR results are shown alongside the calculations of Singh and Antony [115]. Here, we find the CCC-SCAR results to be much higher than those of Singh and Antony [115] for energies below 35 eV. Above 35 eV, the CCC-SCAR results fall off substantially faster than this approach.

Fig. 25

Electron-loss (top), direct ionization (middle), and positronium-formation (bottom) cross section for positron scattering on diatomic carbon. Theoretical results include CCC-SCAR and those of Singh and Antony [115]

For the direct ionization cross section of this system, shown in the middle segment of Fig. 25, CCC-SCAR results are again shown alongside those of Singh and Antony [115]. There is very close agreement between these calculations for energies below 20 eV. Between 20 and 250 eV, the CCC-SCAR results are higher and predict a larger peak than that of Singh and Antony [115]. Above 250 eV, the CCC-SCAR results are lower than the results of Singh and Antony [115] to 5000 eV.

In the top segment of Fig. 25, we present the $\sigma _\text {EL}$ for both the CCC-SCAR and Singh and Antony [115] results for $\hbox {e}^+$-C$_2$. The CCC-SCAR calculations follow a similar shape to the calculations of Singh and Antony [115] but with a substantially higher peak at 10 eV due to the higher positronium-formation predicted by the CCC-SCAR. The second peak, resulting from the direct ionization, is also higher in the CCC-SCAR calculation. For higher energies, as with direct ionization, the CCC-SCAR results for $\sigma _\text {EL}$ are below those of Singh and Antony [115].

3.5 $\hbox {CO}_2$ scattering

3.5.1 Total cross section

In Fig. 26, we present the results for the total cross section of $\hbox {e}^+$-$\hbox {CO}_2$. Current CCC-SCAR results are presented alongside a several theoretical results [22, 23, 26, 27] and measurements for positrons [122, 124, 129, 130] and electrons [131]. As was found for CO, the current calculations are larger than the positron experiments, except below the positronium-formation energy where good agreement is found with the measurements of Zecca et al. [130]. This discrepancy is likely a result of the experiments of Sueoka and Mori [122], Kwan et al. [124], and Charlton et al. [129] having used strong guiding magnetic fields. Therefore, their results will underestimate the true result as no corrections were made for forward scattering. For energies above 400 eV, however, excellent agreement is found with the electron experiment of Garcia and Manero [131] and the theoretical calculations of Lozano et al. [23]. For energies above 2500 eV, there is also close agreement between these two calculations and those of Singh et al. [27]. Besides these agreements, large discrepancies exist between the different theoretical calculations, with all differing in magnitudes and behavior. The agreement at high energies with experiment confirms that scaling is not required for this system unlike the molecules containing H.

Fig. 26

Total cross section for positron scattering on carbon dioxide. Theoretical results include those of the CCC-SCAR, Lozano et al. [23], Singh et al. [27], Billah et al. [22], Gianturco and Mukherjee [48], and Tenfen et al. [26]. Experimental results for incident electrons are from Garcia and Manero [131] and for positrons are from Charlton et al. [129], Zecca et al. [130], Sueoka and Mori [122], Kwan et al. [124]

3.5.2 Elastic cross section

The elastic cross section for carbon dioxide is shown in Fig. 27. The current CCC-SCAR calculations are presented alongside the same theoretical calculations as the $\sigma _\text {tot}$ and the measurements of Charlton et al. [129] and Zecca et al. [130]. Excellent agreement is found between the CCC-SCAR results and the IAM-SCAR calculations of Billah et al. [22] for energies above 100 eV. Good agreement is also found with the current results and the SCOP calculations of Singh et al. [27] for energies between 20 and 80 eV. Outside of these cases, little agreement is found between the different theoretical calculations.

Fig. 27

Elastic cross section for positron scattering on carbon dioxide. Theoretical results include those of the CCC-SCAR, Lozano et al. [23], Singh et al. [27], Billah et al. [22], Gianturco and Mukherjee [48] and Tenfen et al. [26]. Experimental results for incident electrons are from Charlton et al. [129] and Zecca et al. [130]

3.5.3 Total electronic excitation cross section

In Fig. 28, the CCC-SCAR results and those of Lozano et al. [23] for the total electronic excitation cross section are shown for $\hbox {e}^+$-$\hbox {CO}_2$. Both of these calculations are in close agreement for energies above 300 eV. Below this energy, different behavior is found between the different calculations with that of Lozano et al. [23] exhibiting a much larger peak than the CCC-SCAR at a slightly lower energy.

Fig. 28

Total electronic excitation cross section for positron scattering on carbon dioxide. Theoretical results include those of the CCC-SCAR and the recommended results of Lozano et al. [23]

3.5.4 Electron-loss, direct ionization, and positronium-formation cross sections

In Fig. 29, the positronium-formation cross section is shown for $\hbox {e}^+$-$\hbox {CO}_2$. Current CCC-SCAR results are presented with the calculations of Singh and Antony [29] and the measurements of Cooke et al. [132], Murtagh et al. [133], Laricchia and Moxom [134], Griffith [118] and Kwan et al. [124] (for the lower limits). Close agreement is found between the CCC-SCAR and the experiments from threshold to 8 eV. Beyond this, however, agreement is only found again above 80 eV with the measurements of Griffith [118]. The other presented measurements and the theoretical calculations of Singh and Antony [29] predict a maximum at an energy 10 eV above that of the CCC-SCAR and a much slower drop-off in this cross section. The calculations of Lozano et al. [23] utilized an IAM-SCAR+I approach for their calculations and have similar behavior at high energies to the CCC-SCAR results, with close agreement between these calculations for energies above 50 eV. The large difference in their threshold for the results of Lozano et al. [23] and the other results is because they did not shift the threshold of their inelastic processes as was done in the CCC-SCAR approach. The discrepancies found between the current calculations and the experiments is expected to result from the IAM-SCAR approach being insufficient to model positronium-formation for $\hbox {CO}_2$.

Fig. 29

Positronium-formation cross section for positron scattering on carbon dioxide. Theoretical results include the CCC-SCAR and the calculations of Singh and Antony [29] and Lozano et al. [23]. Experimental $\hbox {CO}_{2}$ measurements are from Cooke et al. [132], Murtagh et al. [133], Laricchia and Moxom [134], Griffith [118], and the lower limits of Kwan et al. [124]

The direct ionization cross section for $\hbox {e}^+$-$\hbox {CO}_2$ is presented in Fig. 30. Current CCC-SCAR results are shown alongside the theoretical calculations of Singh and Antony [29], Lozano et al. [23], Tóth et al. [37] and Campeanu et al. [40]. The closest agreement is found between the current calculations and those of Lozano et al. [23] for energies above 100 eV, with these results slightly above the experiment of Bluhme et al. [128] for energies above 500 eV. The results of Singh and Antony [29] are in good agreement with this experiment, except for at high energies where it is larger than both the experimental and the CCC-SCAR results. For these high energies, the IAM-SCAR approach is equivalent to the IAM method, and therefore, the current high-energy results are expected to be accurate. The calculations of Tóth et al. [37] and Campeanu et al. [40] both utilize distorted-wave-based models and predict similar behavior to each other. However, this behavior is very different from the other presented theory.

Fig. 30

Direct ionization cross section for positron scattering on carbon dioxide. Theoretical results include CCC-SCAR, Singh and Antony [29], Lozano et al. [23], Tóth et al. [37] and Campeanu et al. [40]. Experimental results are from Bluhme et al. [128]

The electron-loss cross section is shown in Fig. 31 for the $\hbox {e}^+$-$\hbox {CO}_2$ scattering system. The CCC-SCAR results are shown alongside the theoretical calculations of Singh and Antony [29] and Billah et al. [22] and the experimental results of Bluhme et al. [128] and Laricchia and Moxom [134]. The CCC-SCAR results are higher than these experiments, except for energies near-threshold and at 40 eV. At higher energies, the CCC-SCAR is between the results of Billah et al. [22] and Singh and Antony [29].

Fig. 31

Electron-loss cross section for positron scattering on carbon dioxide Theoretical results are from the CCC-SCAR, Singh and Antony [29], and Billah et al. [22]. Experimental results are from Bluhme et al. [128] and Laricchia and Moxom [134]

3.5.5 Total inelastic cross section

In Fig. 32, the total inelastic cross section is shown for $\hbox {e}^+$-$\hbox {CO}_2$. CCC-SCAR results are presented alongside the calculations of Singh and Antony [29], Billah et al. [22] and Lozano et al. [23]. The results of all the theoretical calculations except for Billah et al. [22] follow a similar shape but at varying magnitudes. Between 50 and 300 eV, close agreement is found between the CCC-SCAR and the calculations of Lozano et al. [23]. For energies above 1000 eV, the CCC-SCAR is slightly higher than the other theoretical approaches.

Fig. 32

Total inelastic cross section for positron scattering on carbon dioxide. Theoretical results are from CCC-SCAR, Singh et al. [27], Billah et al. [22] and Lozano et al. [23]

3.6 $\hbox {O}_3$ scattering

3.6.1 Total cross section

The total cross section for positron scattering on ozone is presented in Fig. 33. Current CCC-SCAR results are shown alongside the only other positron results of Reid [135], which is an IAM calculation. We also present theoretical results by Bharadvaja et al. [136] and Joshipura et al. [137] and experimental results from De Pablos et al. [138] for electron scattering on ozone at high energies. Close agreement is found between the current calculations and the electron experiment for energies above 2000 eV. For $\hbox {O}_3$, scaling is therefore not required as the IAM calculation is accurate at high energies. Below this energy, the electron experimental and theoretical results are larger than the positron calculations. The CCC-SCAR results feature a local maximum at 10 eV, followed by a second maximum at 30 eV. The CCC-SCAR then smoothly decreases for energies above 100 eV, similar to the calculations of Reid [135] but at a slightly higher magnitude.

Fig. 33

Total cross section for positron scattering on ozone. Theoretical CCC-SCAR results are shown alongside the calculations of Reid [135]. Electron theoretical results are from Bharadvaja et al. [136] and Joshipura et al. [137] and electron experimental results are from De Pablos et al. [138]

3.6.2 Elastic cross section

The elastic cross section for a positron scattering on ozone is shown in Fig. 34. There are substantial differences between the current calculations and the IAM results of [135], with the CCC-SCAR almost an order of magnitude higher. The electron theoretical and experimental results are higher than the CCC-SCAR for the entire energy range but become relatively close by 5000eV.

Fig. 34

Elastic cross section for positron scattering on ozone. Theoretical CCC-SCAR results are shown alongside the calculations of Reid [135]. Electron theoretical results are from Bharadvaja et al. [136] and Patel and Joshipura [20] and electron experimental results are from De Pablos et al. [138]

3.6.3 Total electronic excitation cross section

In Fig. 35, the CCC-SCAR results for the total electronic excitation cross section for $\hbox {e}^+$-$\hbox {O}_3$ are shown alongside the calculations of Bharadvaja et al. [136] for $\hbox {e}^-$-$\hbox {O}_3$. Both of these calculations find very different behavior for this cross section. The electron calculations predict a rapid decrease in this cross section after 5 eV. The CCC-SCAR results feature a peak at 15 eV and decrease far slower than the electron calculation.

Fig. 35

Total electronic excitation cross section for positron scattering on ozone. Theoretical CCC-SCAR results are shown alongside the electron theoretical results of Bharadvaja et al. [136]

3.6.4 Electron-loss, direct ionization, and positronium-formation cross section

The positron-impact direct ionization cross section for ozone is shown in Fig. 36. As there are no existing positron measurements or calculations, the CCC-SCAR is compared to existing electron-impact theory and experiment. These include the calculations of Bharadvaja et al. [136] and Joshipura et al. [137] and the experiments of Newson et al. [139] and Siegel [140]. For energies above 1000 eV and below 30 eV, we find good agreement with the calculations of Bharadvaja et al. [136]. The electron experiments are significantly lower than the electron and positron theoretical results.

Fig. 36

Direct ionization cross section for positron scattering on ozone. Theoretical CCC-SCAR results are shown alongside the electron calculations of Bharadvaja et al. [136] and Joshipura et al. [137]. Electron experimental results are from Newson et al. [139] and Siegel [140]

As there is no existing theory or experiment for positronium-formation or electron-loss for positron scattering on ozone, we present these cross sections alongside the direct ionization cross section in Fig. 37. At high energies, the electron-loss cross section is equivalent to the direct ionization cross section, so the comparisons with electron theory and experiment will be the same here. The positronium-formation features a peak at 10 eV, followed by a smooth descent to 200 eV where it becomes minimal.

Fig. 37

CCC-SCAR calculations of electron-loss, positronium-formation, and direct ionization cross section for positron scattering on ozone

3.6.5 Total inelastic cross section

The total inelastic cross section for positron scattering on ozone is shown in Fig. 38. The CCC-SCAR results are compared against the only existing theoretical calculations of Reid [135]. The two calculations follow a similar shape above 300 eV, with the CCC-SCAR slightly lower.

Fig. 38

Total inelastic cross section for positron scattering on ozone. Theoretical CCC-SCAR results are shown alongside the calculations of Reid [135]

3.7 $\hbox {H}_2\hbox {O}$ scattering

3.7.1 Total cross section

In Fig. 39, the total cross section for positron scattering on $\hbox {H}_2\hbox {O}$ is shown. Current CCC-SCAR and CCC-SCAR+ROT results are presented alongside the theoretical results of Blanco et al. [25], Baluja et al. [141] and Sinha et al. [142]. Experimental results for positrons are from Makochekanwa et al. [143], Sueoka et al. [144], Kimura et al. [145], Loreti et al. [146] and Zecca et al. [147]. For energies above 50 eV, we also present the electron experimental results of Muñoz et al. [148]. Similar to the process taken for our $\hbox {H}_2$ results, we have scaled the H component of the current calculations to give agreement with the high energy $\sigma _{\text {tot}}$ of Muñoz et al. [148].

The rotational cross section must be accounted for to calculate the total cross section of $\hbox {H}_2\hbox {O}$ accurately. This results from this molecule being both polar and nonlinear, which results in it having an extra degree of freedom. The impact of this cross section is most notable at low energies, where the calculations of Blanco et al. [25] found it to represent almost 99% of their total cross section. In our CCC-SCAR+ROT calculations, we have summed the rotational cross sections that were obtained from the Born calculations of Blanco et al. [25] to our CCC-SCAR calculation.

For energies above 40 eV, excellent agreement is found between the current CCC-SCAR+ROT calculations and those of Sinha et al. [142]. The CCC-SCAR+ROT results are almost equivalent to the results of Blanco et al. [25] to 10 eV, lower between 30 and 150 eV, and then higher above 150 eV. Compared to the positron experiment, the current calculations are close to most of the experimental results for energies below 7 eV and above 40 eV. Between 7 and 40 eV, the current results are higher than the presented experiment.

Fig. 39

Total cross section for positron scattering on $\hbox {H}_2\hbox {O}$. Theoretical CCC-SCAR and CCC-SCAR+ROT results are shown alongside the calculations of Blanco et al. [25], Baluja et al. [141] and Sinha et al. [142]. Experimental results for positrons are from Makochekanwa et al. [143], Kimura et al. [145], Sueoka et al. [144], Loreti et al. [146] and Zecca et al. [147]. For incident electrons, the experimental results of Muñoz et al. [148] are shown

3.7.2 Elastic cross section

The elastic cross section for $\hbox {e}^+$-$\hbox {H}_2\hbox {O}$ is shown in Fig. 40. There are large discrepancies between the theoretical results for low and intermediate energies. There is also little agreement observed between the positron experiments. At 1000 eV, the CCC-SCAR and the calculations of Aouchiche et al. [32] and Sinha et al. [142] are within the error bars of the measurement of Katase et al. [149]. Above 1000 eV, these three theoretical methods are also in close agreement.

Fig. 40

Elastic cross section for positron scattering on $\hbox {H}_2\hbox {O}$. Theoretical CCC-SCAR results are shown alongside the calculations of Blanco et al. [25], Sinha et al. [142], Baluja et al. [49], Arretche et al. [43] and Aouchiche et al. [32]. Experimental results are from Tattersall et al. [150] and Loreti et al. [146] for positrons and Katase et al. [149] for electrons

3.7.3 Total electronic excitation cross section

The total electronic excitation cross section for $\hbox {e}^+$-$\hbox {H}_2\hbox {O}$ is presented in Fig. 41. CCC-SCAR results are presented alongside the theoretical results of Blanco et al. [25] and Arretche et al. [151] and the experiment of Tattersall et al. [150]. The current results are significantly higher than the previous theory by almost a factor of four. Good agreement, however, is observed with the experiment of Tattersall et al. [150] to 10 eV.

Fig. 41

Total electronic excitation cross section for positron scattering on $\hbox {H}_2\hbox {O}$. Theoretical CCC-SCAR results are shown alongside the calculations of Blanco et al. [25] and Arretche et al. [42]. Experimental results for Tattersall et al. [150] are shown for energies below the direct ionization threshold

3.7.4 Electron-loss, direct ionization, and positronium-formation cross section

The positronium-formation cross section for the $\hbox {e}^+$-$\hbox {H}_2\hbox {O}$ scattering system is shown in Fig. 42. The CCC-SCAR and Blanco et al. [25] theoretical calculations are shown alongside the experimental results of Murtagh et al. [152] and Makochekanwa et al. [143]. The current results are within the range of experiment below 7 eV and above 35 eV. Between these energies, the CCC-SCAR results are significantly higher than experiment and the calculations of Blanco et al. [25]. The calculations of Blanco et al. [25] are slightly higher than the CCC-SCAR for energies above 50 eV.

Fig. 42

Positronium-formation cross section for positron scattering on $\hbox {H}_2\hbox {O}$. Theoretical CCC-SCAR results are shown alongside those of Blanco et al. [25]. Experimental results are from Makochekanwa et al. [143] and Murtagh et al. [133]

In Fig. 43, the direct ionization cross section is shown for $\hbox {e}^+$-$\hbox {H}_2\hbox {O}$. Current CCC-SCAR results are shown alongside the calculations of Blanco et al. [25] and the Coulomb plus plane waves with full energy range (CPE) and electron-screening (ES) distorted-wave models of Tóth et al. [37]. There are no positron experimental results for direct ionization. Therefore, results are shown alongside the $\hbox {e}^-$-$\hbox {H}_2\hbox {O}$ experiments of Bolorizadeh and Rudd [153], Khare and Meath [154], Muñoz et al. [148], Straub et al. [155], and Rao et al. [156]. For energies above 150 eV, there is excellent agreement between the different theoretical methods. Below 150 eV, the current results are in closest agreement with the ES model of Tóth et al. [37] with the other theoretical results lower. There is excellent agreement with the measurements of Bolorizadeh and Rudd [153] with the CCC-SCAR results from 150 to 5000 eV.

Fig. 43

Direct ionization cross section for positron scattering on $\hbox {H}_2\hbox {O}$. Theoretical CCC-SCAR results are shown alongside those of Blanco et al. [25] and Tóth et al. [38]. Experimental results for electrons are from Bolorizadeh and Rudd [153], Khare and Meath [154], Muñoz et al. [148], Straub et al. [155] and Rao et al. [156]

The electron-loss cross section is shown in Fig. 44 for $\hbox {e}^+$-$\hbox {H}_2\hbox {O}$. The CCC-SCAR results are shown with the calculations of Blanco et al. [25] and the electron experiments that were presented for the direct ionization cross section [148, 153,154,155,156]. Below 150 eV, the CCC-SCAR results are larger than the calculations of Blanco et al. [25]. This results mainly from the larger positronium-formation cross sections found within the current calculation.

Fig. 44

Electron-loss cross section for positron scattering on $\hbox {H}_2\hbox {O}$. Theoretical CCC-SCAR results are shown alongside those of Blanco et al. [25]. Experimental results for electrons are from Bolorizadeh and Rudd [153], Khare and Meath [154], Muñoz et al. [148], Straub et al. [155] and Rao et al. [156]

3.7.5 Total inelastic cross section

The total inelastic and direct inelastic cross section are shown in Fig. 45 for positron scattering on $\hbox {H}_2\hbox {O}$. The total inelastic cross section is the sum of the direct ionization, total electron excitation, and positronium-formation cross sections. The direct inelastic cross section, on the other hand, is the sum of the total electron excitation and direct ionization cross sections. As with the total electron excitation cross section (Fig. 41), the direct inelastic results are significantly larger than the calculations of Blanco et al. [25]. For this cross section, the current CCC-SCAR calculations follow a similar shape to the results of Tattersall et al. [150] and, however, have a higher magnitude. For the total inelastic cross section, the current results are close to those of Blanco et al. [25] for energies above 100 eV. Below 100 eV, different behavior is found between both of these calculations.

Fig. 45

Total inelastic and direct inelastic cross section for positron scattering on $\hbox {H}_2\hbox {O}$. Theoretical CCC-SCAR results are shown alongside the calculations of Blanco et al. [25] and the direct inelastic experiment of Tattersall et al. [150]

3.8 $\hbox {CH}_4$ scattering

3.8.1 Total cross section

The total cross section for positron scattering on methane is shown in Fig. 46. CCC-SCAR results are shown alongside the theoretical results of Singh et al. [27], Baluja and Jain [31], Rawlins et al. [34], Zecca et al. [44], Jain and Gianturco [33] and De-Heng et al. [157]. The measurements of Charlton et al. [158], Sueoka and Mori [159], Zecca et al. [44] and Dababneh et al. [107] and the high-energy results from the recommended results of Song et al. [160] are also included. As was done for $\hbox {H}_2\hbox {O}$, the H components of the CCC-SCAR calculations have been scaled to give agreement with the high-energy electron results of Song et al. [160]. The CCC-SCAR results are close to those of Singh et al. [27] for energies between 100 and 500 eV. Above 500 eV, the CCC-SCAR follows a similar shape as the results of Singh et al. [27] but with a higher magnitude. The SMC results of Zecca et al. [44] and the model potential results of Jain and Gianturco [33] are both significantly lower than the current calculation for energies above 0.4 eV. The MBT results of Rawlins et al. [34] are higher than the current results across their calculated range and the SMC results are higher below 0.4 eV. Compared to the calculations of Baluja and Jain [31] the CCC-SCAR is larger for energies below 50 eV and above 500 eV. For the results of De-Heng et al. [157], the CCC-SCAR results are larger for energies above 500 eV and lower for energies under 500 eV. The CCC-SCAR results are in good agreement with experiment for energies below 1 eV [44] and above 80 eV [107].

Fig. 46

Total cross section for positron scattering on $\hbox {CH}_4$. Theoretical CCC-SCAR results are shown alongside the calculations of Singh et al. [27], Baluja and Jain [31], Jain and Gianturco [33], Zecca et al. [44], Rawlins et al. [34], and De-Heng et al. [157]. Experimental results for positrons are from Charlton et al. [158], Sueoka and Mori [159], Zecca et al. [44], and Dababneh et al. [107]. The recommended results of Song et al. [160] for electron scattering on methane are also given

3.8.2 Elastic cross section

The elastic cross section for $\hbox {e}^+$-$\hbox {CH}_4$ is shown in Fig. 47. CCC-SCAR results are presented alongside the calculations of Jain [30], Zecca et al. [44], and Rawlins et al. [34]. For energies below the positronium-formation threshold, the $\sigma _\text {tot}$ experimental results of Charlton et al. [158], Zecca et al. [44], and Dababneh et al. [107] are also shown. The measurements for electrons of Sakae et al. [161], which were conducted for energies up to 700 eV, have also been included. For energies above 200 eV, excellent agreement is found between the CCC-SCAR results and those of Jain [30]. Below 200 eV, the results of Jain [30] are lower than the CCC-SCAR results. The experimental results of Sakae et al. [161] have not been completed to high enough energies for the electron and positron elastic scattering to become equivalent. However, the results follow a similar shape to the current calculations, and differences decrease with increasing energy.

Fig. 47

Elastic cross section for positron scattering on $\hbox {CH}_4$. Theoretical CCC-SCAR results are shown alongside the calculations of Jain [30], Zecca et al. [44], and Rawlins et al. [34]. Experimental results for electrons are from Sakae et al. [161] and for positrons are from Charlton et al. [158], Zecca et al. [44], and Dababneh et al. [107]

3.8.3 Electron-loss, direct ionization, and positronium-formation cross section

The CCC-SCAR results for the positronium-formation cross section of $\hbox {e}^+$-$\hbox {CH}_4$ are shown in Fig. 48 alongside the calculations of Singh and Antony [29] and upper and lower limit measurements of Kauppila et al. [162]. The CCC-SCAR calculations are within the experimental limits for energies below 10 eV. The calculations of Singh and Antony [29] are smaller than the CCC-SCAR results to 30 eV. Above 30 eV, these calculations are higher than the CCC-SCAR results.

Fig. 48

Positronium-formation cross section for positron scattering on $\hbox {CH}_4$. Theoretical CCC-SCAR results are shown alongside the results of Singh and Antony [29] and the upper and lower limit measurements of Kauppila et al. [162]

Results for the direct ionization cross section are shown in Fig. 49. This cross section has had the most theoretical investigation for this molecule with CCC-SCAR results shown alongside the calculations of Singh and Antony [29], Tóth et al. [37], Campeanu et al. [41], and Fedus and Karwasz [36]. No measurements have been conducted for positron scattering for this process; therefore, we also present the recommended electron results of Song et al. [160]. For energies between 100 and 300 eV, the current CCC-SCAR calculations are lower than other theoretical results. From 300 eV to 700 eV, the CCC-SCAR results are in close agreement with the calculations of Tóth et al. [37] but are higher for energies above 700 eV. From threshold to 25 eV, there is good agreement between the CCC-SCAR and the results of Singh and Antony [29]. Between 15 and 300 eV, the different theoretical calculations are in disagreement for the magnitude of this cross section. The experimental results of Song et al. [160] are above the current CCC-SCAR results for energies above 100 eV.

Fig. 49

Direct ionization cross section for positron scattering on $\hbox {CH}_4$. The CCC-SCAR results are shown alongside the theoretical results of Singh and Antony [29], Tóth et al. [37], Campeanu et al. [41], and Fedus and Karwasz [36]. Experimental results for incident electrons are from Song et al. [160]

In Fig. 50, the electron-loss cross section for $\hbox {e}^+$-$\hbox {CH}_4$ scattering is shown. Current CCC-SCAR results are shown alongside the calculations of Singh and Antony [29] and the recommended electron results of Song et al. [160]. For energies above 50 eV, the CCC-SCAR follows a similar shape to the calculations of Singh and Antony [29] but are lower. Below 30 eV, the CCC-SCAR results are larger than those of Singh and Antony [29] and have a maximum at 15 eV, whereas the results of Singh and Antony [29] have a maximum at 25 eV.

Fig. 50

Electron-loss cross section for positron scattering on $\hbox {CH}_4$. Theoretical CCC-SCAR results are shown alongside the results of Singh and Antony [29]

3.8.4 Total inelastic and excitation cross section

The $\hbox {e}^+$-$\hbox {CH}_4$ total inelastic cross section is shown in Fig. 51. The current CCC-SCAR results are shown alongside those of Singh et al. [27]. Similar shapes are found for both calculations, but the CCC-SCAR results are significantly higher than those of Singh and Antony [29] for energies above 15 eV. Also shown in this figure is the CCC-SCAR total electronic excitation cross section, due to the absence of previous theoretical or experimental work these are presented without comparison.

Fig. 51

Total inelastic cross section for positron scattering on $\hbox {CH}_4$. Theoretical CCC-SCAR results are shown alongside the results of Singh et al. [27]. The CCC-SCAR total electronic excitation cross section (Exc.) is also shown

4 Conclusion

With the current CCC results for atomic oxygen, carbon, and hydrogen, the total, direct ionization, positronium-formation, electron-loss, total electronic excitation, total inelastic, and elastic cross sections were calculated for several simple molecules within the IAM-SCAR formalism. The current calculations modified this formalism to include the shifting of inelastic processes to the accepted threshold of each considered molecular target and the additional scaling of H atoms due to the failure of the IAM method to reproduce molecular CCC results for $\hbox {H}_2$. From comparison with one- and two-center CCC calculations for $\hbox {H}_2$, we estimate an uncertainty in the current approach of approximately 20% for all, except positronium-formation, cross sections between the positronium-formation threshold and 50 eV. The positronium-formation cross section is expected to have an error of approximately 30% to 50 eV. For energies above 100 eV, we expect this uncertainty to decrease to 10%. However, it is expected that for the non-diatomic and polar molecules this error is likely higher. Through the provision of the obtained cross sections for the wide range of considered molecules, we aim to provide results from which more advanced theoretical approaches can follow.

The current results were generally within the range of positron experiment and theory for intermediate to high energies and in agreement with electron measurements and theory for high energies. Significant differences, however, were found for the positronium-formation cross section for energies up to 30 eV, with the current approach generally considerably higher than other calculations and experiments for this energy range. Other discrepancies between theoretical results were found for the total electronic excitation cross sections of several molecules ($\hbox {O}_2$, CO, $\hbox {O}_3$, $\hbox {H}_2\hbox {O}$) for which there is a notable absence of experimental results. It is recommended that further work is undertaken for these particular cross sections.

The present work has demonstrated the advantages and limitations of using the IAM-SCAR approach to model positron collisions with molecules. The current modified IAM-SCAR approach produces accurate molecular cross sections at intermediate and high energies, especially when the underlying atomic cross sections are of high accuracy. This methodology, however, is less accurate in the low-energy regime, most notably below 1 eV, where the impact of the molecule’s structure has the most influence. Furthermore, the current approach systematically overestimates the positronium-formation cross section for energies below 30 eV. Further progress in improving the accuracy of the collision data, particularly at low projectile energies, has to come from a complete treatment of the molecular target as a whole.