Toward a complete and comprehensive cross section database for electron scattering from NO using machine learning

Plasma medicine is a relatively new field that employs low-temperature atmospheric pressure plasmas (LTAPP), in order to induce beneficial effects in biological tissue [1, 2, 3, 4]. Key to these benefits is the formation and subsequent synergistic interactions of reactive oxygen and nitrogen species (RONS) [1]. The accurate modelling, control, and optimisation of LTAPP for plasma medicine applications is dependent on a complete and detailed understanding of all plasma-tissue interactions, of which one important subset is the interaction of electrons with RONS [2, 3, 4]. One of the most important RONS is nitric oxide (NO) [1], which has been identified as being effective in both tissue disinfection [5] and apoptosis of cancer cells [6, 7]. A precise description of the interactions between electrons and NO, in the form of electron-impact cross sections [8], is thus important for the predictive understanding of many plasma treatments. With this motivation in mind, in this investigation, we compile a comprehensive set of electron-NO cross sections that attempts to address the shortcomings of those already available in the literature.

To ensure the accuracy and self-consistency of our electron-NO cross section set, we calculate corresponding electron swarm transport coefficients for comparison with measurements in the literature. Any discrepancies observed in the swarm parameters can then be addressed through appropriate refinements of the cross section set. This inverse swarm problem of unfolding cross sections from swarm measurements has a long and successful history [9, 10, 11, 12, 13, 14, 15, 16]. Nevertheless, when the amount of available swarm data becomes limited, the inverse swarm problem becomes ill-posed and is no longer guaranteed to have a unique solution. Traditionally, this difficulty was overcome by relying on an expert in swarm analysis to rule out unphysical solutions and to select the solution that is the most physically plausible. Recently, however, Stokes et al. were able to avoid this time-consuming and possibly subjective process of manual iterative refinement by employing an artificial neural network model [17], to solve the inverse swarm problem for the biomolecule analogues tetrahydrofuran (THF, $\mathrm{C}_{4}\mathrm{H}_{8}\mathrm{O}$) [18] and $\alpha$-tetrahydrofurfuryl alcohol (THFA, $\mathrm{C_{5}H_{10}O_{2}}$) [19], with the former result being found to be of comparable quality to a conventional refinement “by hand” [20]. This application of machine learning was originally proposed by Morgan [21] in the early 1990s. Given that machine learning has in recent years been applied to problems from all fields of science, including chemical physics [22], revisiting the proposal of Morgan with a modern machine learning methodology and hardware was certainly overdue. However, most pivotal to the procedure of Stokes et al. was their use of the LXCat project [23, 24, 25], allowing for their neural network to be trained on a wealth of “real” cross sections, thus allowing their model to “learn” what constitutes a physically-plausible cross section set. In this work, we employ a similar machine learning approach to automatically and objectively refine electron-NO cross sections given independent experimental swarm data.

The remainder of this paper is structured as follows. In Sec. 2, we review relevant experimental and theoretical cross sections and assemble our initial electron-NO cross section database. In Sec. 3, we assess the accuracy and self-consistency of our initial set by simulating corresponding transport coefficients, using a multi-term Boltzmann equation solver, and comparing them to experimental measurements present in the literature. Sec. 4 details our data-driven approach to the swarm analysis of this experimental data and presents the resulting refinements to our initial cross section set. Sec. 5 assesses to what extent our refined set improves the agreement with experimental swarm data over what we proposed initially. We also use our refined set here to calculate transport coefficients for electrons in NO across a large range of density-reduced electric fields. Finally, Sec. 6 presents our conclusions and makes some suggestions for future work.

There have been three comprehensive attempts to compile cross section data for electron scattering from NO. The first, by Brunger and Buckman [26], is now quite dated and so does not figure in what follows. The second was by Professor Y. Itikawa [27], perhaps the doyen of scientists who collect and evaluate cross section data, while the most recent was from Song et al. [28]. In his compilation, Itikawa [27] reported data for the grand total cross section (TCS), elastic integral cross section (ICS), elastic momentum transfer cross section (MTCS), vibrational excitation ICS for the $0\rightarrow 1$, $0\rightarrow 2$ and $0\rightarrow 3$ quanta, a subset of the electronic-state ICS, a total ionisation cross section (TICS) and a dissociative electron attachment (DEA) cross section. Unfortunately, in many cases his recommendations did not cover a broad enough energy range, nor were all the possible open channels considered (neutral dissociation [ND] was not addressed), to fulfil the criteria of Tanaka et al. [8] and Brunger [29] that data bases for modelling and simulation studies (e.g. [30, 31, 32, 20]) must be comprehensive and complete. Song et al. [28], where possible, updated the Itikawa data base and then used that as a starting point to solve the inverse electron-swarm problem [19] in NO. In that approach, ICSs are varied, in conjunction with a two-term approximation Boltzmann equation solver [24], in order to force agreement between the simulated and measured transport coefficient data [28]. The so-determined ICS set then became their recommended cross section data base for the $e+\mathrm{NO}$ collision system. We have several reservations with the work of Song et al. [28]. Firstly, there is no a priori rationale provided by Song et al. [28] for why the two-term approximation to solving Boltzmann’s equation should be valid for all the transport coefficients they simulated over the range of $E/n_{0}$ ($E$ = applied electric field strength; $n_{0}$ = number density of the background gas [i.e. NO here] through which the electron swarm drifts and diffuses) they considered. Indeed, in Sec 3, we apply a multi-term Boltzmann solver and determine that the two-term approximation has not converged sufficiently for some of the measurements considered in their swarm analysis. Secondly, as part of their swarm analysis, Song et al. [28] use the swarm data of Takeuchi et al. [33] who determined their transport coefficients from measured distributions of electron arrival times [33]. It is known that these resulting “arrival time spectra” transport coefficients are distinct from the bulk (centre-of-mass) transport coefficients provided by many Boltzmann solvers [34, 35] and it is not clear whether Song et al. [28] have taken this into account when performing their swarm analysis. Finally, the ill-posed nature of the inverse swarm problem [18] can potentially lead to uniqueness issues in the cross sections derived. Namely, while the ICSs determined by Song et al. [28] do lead to transport coefficients that are largely consistent with the available swarm data they might not be the only set that does so. Under those circumstances the Song et al. [28] data base might not be physical. As a consequence of those concerns, we assemble our initial $e+\mathrm{NO}$ cross section data base by using extensively the work of Itikawa [27].

Brunger et al. [36] reported ICSs for 28 excited electronic states of NO, but only over the limited incident electron energy range 15–50 eV. Cartwright et al. [37] extended a subset of 9 of those cross sections, to their various threshold energies and out to 500 eV, in order to study the excited-state densities and band emissions of NO under auroral conditions. Here we extend the remaining 19 excited electronic-state ICSs to their respective threshold energies [38] and out to 500 eV, with all of these electronic-state ICSs being plotted in Fig. 1. Note that Xu et al. [39] reported some BE$f$-scaling [8] theoretical results for the $A^{2}\Sigma^{+}$, $C^{2}\Pi$ and $D^{2}\Sigma^{+}$ electronic states, and found good agreement with the ICSs of Brunger et al. [36] for the $A$- and $C$- states. For the $D$-state, however, a discrepancy was noted although that may reflect on the applicability of using the BE$f$-scaling approach for an electronic state with such a small optical oscillator strength [40]. For vibrational excitation we essentially adopt Figure 1 from Campbell et al. [41], which was constructed from the low-energy results of Josić et al. [42] and Jelisavcic et al. [43] and the higher-energy results of Mojarrabi et al. [44]. All these ICSs, for the $\nu$ = 0–1, 0–2 and 0–3 vibrational quanta, are plotted in Fig. 1. The present DEA cross sections were taken directly from Table 6 of Itikawa [27], being originally sourced from the work of Rapp and Briglia [45] (see Fig. 1). Similarly, the present TICS was taken directly from Table 5 of Itikawa [27], with the origin of the cross sections in this case being from Lindsay et al. [46] (again see Fig. 1). In the case of the elastic scattering, our initial data set is constructed below 3.38 eV by subtracting the present vibrational ICSs of Fig. 1 from the TCS measurement of Alle et al. [47]. Above 3.38 eV we use the recommended data from Table 2 of Itikawa [27]. The present elastic ICSs are also plotted in our Fig. 1. The TCSs of our initial cross section compilation were formed as follows. Below an incident electron energy of 2.6 eV we used the values of Alle et al. [47], as digitised from their Figure 1, while for $E_{0}$ above 5 eV we employed the values from Table 1 in Itikawa [27]. Between 2.6 eV and 5 eV we undertook an interpolation that ensured that the TCSs of Alle et al. [47] merged smoothly with those recommended by Itikawa [27]. The resultant TCS from this process is shown pictorially in Fig. 1. Finally, there remains the neutral dissociation cross section. There are no known measurements or calculations for the ND ICS in $e+\mathrm{NO}$ scattering, so our approach for estimating it was as follows. The sum of the individual ICSs, for all open scattering channels at some $E_{0}$, should be consistent with the TCS, at least to within the uncertainties on the TCS. In this case, we found that when we summed all our ICSs there was a quite narrow energy region ($\sim$ 5 – 20 eV) where it underestimated the TCS (to well outside the error on our TCS). We therefore assigned that residual cross section strength to be the ICS for ND, with our derived result again being plotted in Fig. 1.

Fig. 1 therefore summarises all the ICSs and the TCS that constitute our initial cross section compilation for input into our multi-term Boltzmann equation solver [18, 19], in order to determine transport coefficients that can be compared to corresponding independent results from electron-swarm experiments. The present error estimates on those ICSs and the TCS largely reflect those given in the original papers, from which our initial data base was derived. Typically, this would be $\sim\pm 25\%$ on the elastic ICS, $\sim\pm 30\%$ on the vibrational excitation ICS, $\sim\pm 7\%$ on the TICS, in the range of $\sim\pm 25\%$–$70\%$ for the various electronic-state ICS, $\sim\pm 30\%$ on the DEA ICS and $\sim\pm 80\%$ on the ND ICS that we have derived. The independent TCS data set would have an error of $\sim\pm 7\%$, a little more than cited in the relevant papers as we have included a small additional uncertainty due to the so-called “forward angle scattering effect” [48] that was previously not allowed for. It is important to remember that, when using this initial cross section compilation in conjunction with a multi-term Boltzmann equation solver, the errors on the ICSs will in principle necessarily lead to a band of allowed solutions for the transport coefficients that are to be simulated.

Another observation we should make in respect to Fig. 1 is that it is clear that the TCS we determine from adding up all our various ICSs (denoted by - - - in black and called “Final TCS”) is entirely consistent with the TCS we derived from the independent measurements available to us (denoted by — - — in red). While this is a necessary condition for a cross section compilation to be plausibly physical, as we saw in our work on THF [18] and THFA [19] it by no means a priori guarantees that the simulated transport coefficients will reproduce the available swarm data. Finally, it is also relevant to note here that all the ICSs in the present compilation are rotationally averaged, which is a direct result of the energy resolution of current-day electron spectrometers not being narrow enough to resolve the various rotational lines [26].

To assess the quality of our initial set of electron-NO cross sections, we apply a well-benchmarked multi-term solution of Boltzmann’s equation [32, 49, 50] in order to calculate swarm transport coefficients for comparison to measured values in the literature. Chronologically, electron-NO swarm measurements include drift velocities and transverse characteristic energies by Skinker et al. [51] and Bailey et al. [52], transverse characteristic energies by Townsend [53], drift velocities and attachment coefficients by Parkes et al. [54], transverse characteristic energies and ionisation coefficients by Lakshminarasimha et al. [55], transverse and longitudinal characteristic energies by Mechlińska-Drewko et al. [56], and, most recently, drift velocities and longitudinal diffusion coefficients by Takeuchi et al. [33] (for both pure NO and admixtures of NO in Ar).

When performing a Boltzmann equation swarm analysis, it is essential to precisely interpret what is being measured in each swarm experiment under consideration [34, 57]. In this regard, many of the earlier electron-NO measurements are inadequate as they are obtained using a “magnetic deflection” experiment [58] that measures a drift velocity that is distinct from the bulk (centre-of-mass) drift velocity measured in pulsed-Townsend experiments [57] and provided by our Boltzmann solver. Ultimately, we choose to calculate comparisons to the drift velocities and longitudinal diffusion coefficients of Takeuchi et al. [33], the ionisation coefficients of Lakshminarasimha et al. [55], the attachment coefficients of Parkes et al. [54], the transverse characteristic energies of Lakshminarasimha et al. [55] and Mechlińska-Drewko et al. [56], and the longitudinal characteristic energies of Mechlińska-Drewko et al. [56] and Takeuchi et al. [33] (calculated using their drift and diffusion measurements). We purposefully leave out the 421 K measurements of Parkes et al. [54], due to an observed pressure dependence in their measured attachment coefficients that they attribute to electron detachment from product ions at this temperature. We also neglect the pure NO diffusion measurements of Takeuchi et al. [33] below 11 Td, due to the pressure dependence arising from three-body attachment in this low $E/n_{0}$ regime [59, 54]. While their associated drift velocity measurements also show a pressure dependence, it is to a far lesser extent and Takeuchi et al. [33] are able to consistently estimate pressure-independent values by extrapolation to zero pressure [33]. In almost all cases considered, we interpret swarm measurements as pertaining to the centre of mass of the swarm (i.e. bulk transport coefficients). The only exception we make is for the double-shutter drift tube experiment of Takeuchi et al. [33], which measures an “arrival time spectra” drift velocity, $W_{m}$, that is distinct from the bulk drift velocity when nonconservative effects are present [34]. Fortunately, we can approximate this quantity in terms of our simulated bulk transport coefficients with [35]:

where $W$ is the bulk drift velocity, $\alpha_{\mathrm{eff}}$ is the effective first Townsend ionisation coefficient, and $D_{L}$ is the bulk longitudinal diffusion coefficient.

To perform the transport calculations, we require an appropriate quasielastic (elastic+rotation) momentum transfer cross section, which we obtain by multiplying the present elastic ICS (from Fig. 1) by the ratio of elastic MTCS to elastic ICS, each taken from the set of Itikawa [27]. When calculating the admixture transport coefficients, we use the argon cross section set present in the Biagi database [60, 61, 62]. The resulting simulated transport coefficients are plotted alongside the aforementioned experimental measurements in Figs. 2(a)–(f) for drift velocities, diffusion coefficients, ionisation coefficients, attachment coefficients, transverse characteristic energies, and longitudinal characteristic energies, respectively. Fig. 2(a) shows a qualitative agreement between our simulated drift velocities and the measurements of Takeuchi et al. [33]. In the case of pure NO, our initial cross section set results in drift velocities that are consistently too small below 2 Td and too large above 2 Td. In the admixture cases, the onset of negative differential conductivity (NDC) — the phenomenon of decreasing drift velocity with increasing $E/n_{0}$ — occurs earlier in our simulations than is shown by experiment. Fig. 2(b) indicates that our calculated longitudinal diffusion coefficients in pure NO are too large by roughly a factor of 2 on average. By comparison, the calculated admixture diffusion coefficients are much closer to the measurements and exhibit many of their qualitative features. We note that the two-term approximation is in error by up to 40% for the longitudinal diffusion coefficient measurements near 500 Td. It follows that a multi-term Boltzmann solver is also necessary here for the accurate calculation of characteristic energies. Fig. 2(c) shows ionisation coefficients calculated for our initial set that are generally larger than those measured by Lakshminarasimha et al. [55] (up to $75\%$ larger in some instances). Quantitatively, there is also room for improvement here. Fig. 2(d) shows the near absence of attachment in our simulations below $\sim 15\ \mathrm{Td}$, in stark contrast to the measurements of Parkes et al. [54]. This indicates the presence of a low-energy attachment process that is currently missing from our initial set. Fig. 2(e) shows a good quantitative agreement between our calculated transverse characteristic energies and those measured by Lakshminarasimha et al. [55]. However, the same cannot be said for the measurements of Mechlińska-Drewko et al. [56] which, while qualitatively similar, tend to be much smaller than what we calculate. Fig. 2(f) exhibits a disagreement between our calculated longitudinal characteristic energies and the pure NO measurements of Mechlińska-Drewko et al. [56] and Takeuchi et al. [33], both of which are consistently smaller than what we calculate. The agreement is much better for the mixture measurements of Takeuchi et al. [33], although the simulations are still larger than expected in the case of the 4.99% NO in Ar measurements below 3 Td.

In the previous section, we have seen clear discrepancies between the transport coefficients calculated using our initial electron-NO cross section set and those determined experimentally by a number of authors. In this section, we apply machine learning in order to refine our initial set and hopefully improve its agreement with the measured swarm data plotted in Fig. 2.

We have formed a comprehensive and self-consistent set of electron-NO cross sections, by constructing an initial set from the literature and then refining it using a multi-term Boltzmann equation analysis of the available swarm transport data. This refinement was performed automatically and objectively using a neural network model, Eq. (2), that was trained on cross sections derived from the LXCat project [23, 24, 25], to ensure that the refinements made were physically plausible. In summary, by using our initial set as a base, we obtained from the swarm data a quasielastic MTCS, a DEA cross section and a ND cross section. Compared to our initial set, we confirmed that our resulting refined cross section database was more consistent with the swarm data from which it was derived. Lastly, we used our refined set to calculate a variety of transport coefficients for electrons in NO across a large range of reduced electric fields. Notably, this revealed significant heating of the swarm above the thermal background due to our refined low-energy attachment process.

The machine learning methodology we have employed in this work has previously [18] been shown to produce cross section sets that are comparable to those refined using conventional swarm analysis, i.e. through manual iterative refinement by an expert. We thus believe that our refined set of electron-NO cross sections is of a similarly high quality. That said, we do acknowledge there is still some room for improvement in the present fit and it is thus fortunate that this machine learning approach makes it straightforward to revisit NO as new swarm data, cross section constraints, or LXCat training data becomes available. On this note, we highlight that there are presently no swarm measurements available for the effective first Townsend ionisation coefficient. We believe performing such measurements would be a worthwhile future endeavour, as they would allow one to quantify the attachment coefficient in the electropositive regime (above $\sim 75\ \mathrm{Td}$) and thus allow for the possibility of refining the DEA cross section above $\sim 4\ \mathrm{eV}$.

Given the ill-posed nature of the inverse swarm problem, we also acknowledge that our refined set carries with it some uncertainty. In this sense, an artificial neural network of the form of Eq. (2) is not ideal as it does not provide uncertainty quantification. In the future, we may address this by using an appropriate alternative machine learning model [70, 71, 72, 73, 74, 75]. We also plan to apply our data-driven swarm analysis to determine complete and self-consistent cross section sets for other molecules of biological interest, including water [76].

The data that supports the findings of this study are available within the article.