Simulating the Feasibility of Using Liquid Micro-Jets for Determining Electron–Liquid Scattering Cross-Sections

Abstract

The extraction of electron–liquid phase cross-sections (surface and bulk) is proposed through the measurement of (differential) energy loss spectra for electrons scattered from a liquid micro-jet. The signature physical elements of the scattering processes on the energy loss spectra are highlighted using a Monte Carlo simulation technique, originally developed for simulating electron transport in liquids. Machine learning techniques are applied to the simulated electron energy loss spectra, to invert the data and extract the cross-sections. The extraction of the elastic cross-section for neon was determined within 9% accuracy over the energy range 1–100 eV. The extension toward the simultaneous determination of elastic and ionisation cross-sections resulted in a decrease in accuracy, now to within 18% accuracy for elastic scattering and 1% for ionisation. Additional methods are explored to enhance the accuracy of the simultaneous extraction of liquid phase cross-sections.

1. Introduction

The interaction of a low-temperature plasma with liquids is fundamental for numerous new and emerging technologies, finding applications in important domains, including environmental remediation, and the synthesis of nanomaterials and medicine [1,2,3,4,5,6,7,8,9]. The goal of high-level optimisation, of the efficacy and selectivity, of these and future generation plasma–liquid applications depends on, among other things, a detailed understanding of the underlying fundamental nanoscale physics and associated predictive modelling, underpinned by accurate and complete transport theory and cross-sections. The key driver to these applications is the role of electrons (and other radical species) at the plasma–liquid interface, but despite their central role, electron-induced transport and processes at the interface are not well understood [10].

Developing our understanding of electron transport into and within liquids is critical for enhancing the predictive power of plasma–liquid models. Fundamental to transport theory, which governs the motion of electrons into and within such environments, are complete and accurate sets of electron impact cross-sections. While there is a wealth of knowledge of electron impact cross-sections in the gas phase (e.g., databases such as LXCat [11,12]), the same is not true for the liquid environments. The scattering and transport theory of pre-solvated electrons in non-polar liquids is reaching some level of maturity (see the ab initio treatment by the authors [13,14]), however this is not the case for polar liquids. An existing Monte Carlo simulation of low-energy electrons in liquid water [15] ignored many physical processes, that are relevant to developing appropriate scattering theory. Furthermore, current models use cross-sections calculated in the gas phase, or through electron reflection measurements from amorphous ice [16]. The absence of liquid phase data represents a large knowledge gap in the literature, and experiments proposed in this manuscript aim to address, at least in part, this knowledge gap.

Here, we propose a new experimental technique that extends gas phase beam experiment methodologies [17] to include electron scattering from liquid surfaces, through the use of liquid micro-jets (LµJs). A constant replenishment of liquid, a small surface area, and a laminar flow enable the scattering of particles from a smooth and stable liquid surface, whilst operating in a vacuum environment. Until now, LµJ investigations have focused primarily on photon and photoelectron scattering [18,19,20,21,22,23,24,25,26,27,28,29,30]. In this work, we introduce electron scattering from LµJs and investigate the feasibility of developing effective electron–liquid cross-sections from the measured electron energy loss spectra (EELS). Unlike gas phase experiments, which are designed to ensure single scattering processes only, the proposed experiment is necessarily multi-scattering, with electrons being scattered from the surface and bulk atoms/molecules. The connection of this with traditional (multiple collision) swarm experiments [31,32,33,34] is thus clear.

Swarm experiments have proven to be critical in the development of accurate cross-section sets [17]. In electron swarm experiments, electrons are driven through a gaseous (or liquid) medium by an applied electric field, and macroscopic descriptors such as current, drift velocity, and diffusion coefficients are determined. Simulation techniques then evaluate the accuracy and self-consistency of the cross-section sets through comparisons of the transport coefficients. The same techniques can be used, in principle, to iteratively improve or develop cross-section sets [35,36,37,38,39,40], although the question of degeneracy in the cross-section set remains open, and other means are usually required to minimise this. Recent studies have shown promising applications of neural networks (NN), toward improving cross-section sets through swarm transport data [41,42,43,44,45,46]. Utilising a similar NN methodology, along with energy loss spectra from a LµJ experiment, we propose the generation of cross-section sets for electrons within a liquid environment. To facilitate this, a benchmarked Monte Carlo simulation technique, for simulating electron transport in liquid environments [47], has been developed and implemented for this study, with extensions to calculate EELS arising from the scattering of electrons from an LµJ.

The paper is organised as follows. In Section 2, we provide a summary of LµJs and their implementation in the proposed experiment. In Section 3, we outline the Monte Carlo (MC) simulation method utilised here and discuss the characterisation of bulk liquid and interfacial effects. The characteristic signatures of the various multiple scattering elements on the EELS are also discussed. In Section 4, we use the developed Monte Carlo software to train a neural network to predict neon’s elastic and ionisation cross-sections as a proof-of-concept. Finally, in Section 5, we provide some concluding premarks.

2. The Proposed Electron–Liquid Micro-Jet Scattering Experiment

2.1. Liquid Micro-Jets

Liquid micro-jets were first developed to overcome the limitations of probing liquids under vacuum conditions, which are often required by scattering experiments [48]. Due to the high vapour pressures found in liquids, additional treatment is required to ensure a stable surface within vacuum conditions, and hence the developments of the liquid jets. In our proposed experiment, a replenishing and sufficiently thin liquid source, exhibiting laminar flow, facilitates a smooth and stable scattering surface within a vacuum environment [49].

Siegbahn and Siegbahn [48] first utilised a replenishing $0.2\mathrm{mm}$ diameter liquid jet, and scattered electrons from that source to measure binding energies in formamide ($\mathrm{HOCN}{\mathrm{H}}_2$). In vacuo experiments were initially limited to low vapour pressure liquids, to avoid immediate freezing and/or evaporation. For high vapour pressure liquids such as water, a further reduction of exposed surface area is required to maintain a free vacuum surface [20]. To this end, significantly smaller (∼10$\mu \mathrm{m}$) liquid micro-jets were first developed to measure the velocity distributions of vapour molecules, while ensuring inter-molecular collisions were minimised [49]. Extending this, a focusing gas surrounding the nozzle was implemented to increase the length and decrease the diameter of the jet substantially [50,51,52]. Flat LµJs were also developed from the collision of two micro-jets, which results in a sheet-like scattering surface [53]. Additionally, cryogenic jet systems enable the use of super-cooled liquids with jet diameters reaching $1\mu \mathrm{m}$ [54,55]. It is clear that a wide array of configurations are achievable, and can be tailored towards the desired experiment.

Until now, applications of the aforementioned LµJ designs include an array of photoelectron spectroscopy experiments, to measure properties such as binding energies and hydrogen bonding [18,19,20,21,22,23,24,25,26,27,28,29,30]. Other LµJ experiments include mass spectrometry [56], enhanced X-ray production [57,58,59,60], X-ray emission/absorption, to probe electronic structure and molecule orientation [61,62,63], to study the molecular dynamics of evaporation [64], electrokinetic power generation [65,66], and drug delivery alternatives [67]. For further general reading, one is directed to the summary papers cited here [23,68,69].

Scattering simulations, such as the MC code developed at James Cook University, rely on the characterisation of the LµJ features. Liquid dynamics studies of LµJs found jet length stability on the order of millimetres. In one study [70], source widths of 10, 20 and $50\mu \mathrm{m}$, along with velocities ranging between 25 and $150{\mathrm{ms}}^{-1}$, were trialled for each diameter. A limited laminar flow was achieved with jet lengths between 1 and $15\mathrm{mm}$. After emergence from a circular nozzle, the jet undergoes contraction to a final cross-sectional size due to surface tension forces [71]. Typical contraction values were found to range between $90\%$ and $60\%$ [70] of the original source width, after which capillary forces break the jet into droplets through a tendency to try and reduce surface energy [71].

Within the liquid, charge build-up, and the existence of a streaming potential, due to electrokinetic charging, can hinder the accurate simulation of charged particle transport. A streaming potential is produced when a pressurised liquid is forced through the nozzle, which disrupts the electric double layer created between the liquid and the inner wall of the nozzle [72,73]. For liquid water, the magnitude of this streaming potential exceeds $60\mathrm{V}$ depending on the jet velocity and diameter [19,73]. The addition of an electrolyte has since been shown to minimise this effect [73]. Additionally, the production of electrons within an insulating medium such as water will produce a current and subsequent surface potential [68]. The magnitude of this effect will depend on the application and relevant time scales.

2.2. Crossed-Beam Electron Scattering from Liquid Micro-Jets

In our proposed experimental work, we will extend existing crossed-beam methodologies from gas to liquid environments, utilising liquid micro-jets to measure differential electron energy loss spectra (EELS), as shown schematically in Figure 1. The liquid would be propelled into a vacuum environment through a $15\mu \mathrm{m}$ circular nozzle to produce a continuous, stable, and smooth scattering surface. In this work, we assume that the jet diameter is the same as the nozzle diameter. As noted previously, this jet remains stable for several millimetres before decaying into droplets. Evaporation at the surface and the tapering of the jet’s diameter are neglected in the current work, as the characterisation of such parameters was not available, and their effects on the model are expected to be minimal. Electrons, sourced from the thermionic emission from a tungsten filament, and collimated and transported by a series of DC-potential elements, are then propelled through the vacuum normal to the jet’s direction, to undergo multiple surface and bulk scattering events before escaping (and being collected by one or more detectors) or solvating within the liquid. For the foreshadowed work, an electron source with an energy full width at half maximum (FWHM) of $0.5\mathrm{eV}$ will be used [74].

Once an electron escapes the jet, a scattered electron detector will resolve both its energy, $\epsilon$, and scattering angle, $\chi$, with a resolution of $0.8\mathrm{eV}$ (FWHM) and $1^{\circ }$, respectively [74]. Through the detection of both the energy and angular dependencies, EELS provides a high degree of insight into the underlying electron scattering cross-sections and their associated dynamics. This combination of the electron gun and scattered electron detector provides a combined energy resolution of $0.9\mathrm{eV}$. We note that, with crossed-beam experiments, some restrictions will inevitably be placed on the available detection angles. This is simply due to the physical size of the electron source, the electron detector(s), and LµJ. Additionally, while an angle-resolved EELS provides sufficient information for the derivation of differential cross-sections within the gas phase, multiple scattering effects in liquids present significant challenges when deconstructing EELS into complete (differential) cross-section sets due to the high degrees of convolution arising. The extent of this convolution is investigated, in part, in Appendix B.

2.3. Extraction of Cross-Section Sets from EELS

The derivation of cross-sections from a reflected EELS measurement is akin to the ‘Inverse Milne Problem’, which consists of deriving the reflected EELS from a ‘half-plane’ medium, of finite or infinite depth from $z=0$ to $z=L$ (where L is the half-plane depth), and extends to infinity in all other directions [75]. Outside the half-plane, a vacuum exists from which particles are propelled into the half-plane. The presence of high multiple-scattering at large L, relative to the scattering length, results in a complex combinatorial problem with a restricted set of observables, thanks to a reliance on external measurements [75].

The vast majority of treatments involve energy-independent radiative transfer [75,76] and neutron scattering [77,78]. The inclusion of energy dependencies was conceptually treated [79], with strict delta function requirements placed on both the source and detector, in order to derive total and differential cross-sections. In other studies, sufficiently thin (10–100$\mathrm{nm}$) solid films were used to derive surface and bulk inelastic mean free paths [80,81,82], assuming a knowledge of the elastic cross-section. Likewise, cross-sections for amorphous ice films were derived under a two-stream assumption, along with a variation of film thickness between 0 and $5.456\mathrm{nm}$ [16,83,84]. While the extension of these techniques towards LµJs may be feasible, other indirect approaches, such as Monte Carlo simulations, together with machine learning, could simplify the methodology significantly.

In multiple studies [41,42,43,45,46], machine learning was applied to the inversion of transport coefficients found through swarm experiments. While swarm transport coefficient measurements differ in nature to EELS, they each represent an ill-posed inversion problem associated with high degrees of multiple scattering. An extension of the current methodology towards liquid phase EELS requires a sufficiently large training dataset, which is facilitated through a highly configurable non-equilibrium Monte Carlo simulation of electron transport developed for this project. In this study, we thus seek an implicit cross-section extraction technique utilising a neural network trained through Monte Carlo simulations of an LµJ. In what follows, we outline the Monte Carlo simulation employed, and its application to the proposed LµJ measurements, before applying machine learning to this ill-posed inverse problem of determining cross-sections from LµJ-measured EELS.

3. Simulation of Electron Transport through Liquid Micro-Jets

Machine learning requires a substantial volume of training data to ensure a robust fitting process. To generate these data efficiently, a Monte Carlo simulation method was developed. Monte Carlo (MC) simulations provide flexible environments in which the manipulation of spatial and temporal parameters is both efficient and straightforward. LµJs inherently require specific and precise spatial variation in terms of the neutral density, and hence, an MC simulation is well suited for the task. The simulation technique is discussed in detail in [47], hence, here, we detail its application to LµJs, while a section of benchmark results is presented in Appendix A.

3.1. Liquid Dynamics

Relatively low-density gaseous environments involve an inherent assumption that each scattering event involves instantaneous localised interactions. In liquid environments, however, the de Broglie wavelength of the electron can be smaller than, or of the same order as, the inter-particle spacing and the mean free path, which necessitates modified coherent collision dynamics. To account for coherent scattering, a new method was recently developed which incorporated a structure factor modified cross-section [47]. Through this, the macroscopic effects of coherent scattering are realised, while retaining the efficiency of effective single particle scattering events. Therefore, the relative simplicity of modelling single particle-particle collisions is maintained, while replicating the macroscopic effects of multiple scattering. Through an integral structure factor [47,85], elastic collisions are split into three collisional processes, which are weighted by an angle-integrated structure factor, $\mathsf{\Gamma }\left(\epsilon \right)$, produced through either theoretical or independent experimental techniques.

As a result of these microscopic processes, electron scattering through liquids is accurately simulated, given an appropriate structure factor. For real systems, experimentally measured structure factors are available through X-ray or neutron scattering [86,87,88]. While its functional form will vary, the methodology remains the same. Previously [47], it was shown that these processes match the required momentum and energy transfers derived using Boltzmann’s equation [47]. While isotropic scattering was assumed, the proof of principle was extended by the present authors to show its validity for the use of differential cross-sections. In the current work, we investigate structure effects upon measured spectra, while its implementation in our machine learning model is left for future study.

3.2. Interfacial Dynamics

In addition to liquid dynamics, interfacial effects must be considered in the simulation of LµJs. A recent study investigated density-dependent inter-facial effects on electron transport [89]. Due to a change in de-localised electron energy across the interface, a density-dependent ($n_0$), and hence spatially-dependent potential $V_0(n_0(\overrightarrow{r}))$, exists, which produces an effective electric field $E_0(\tilde{r})=-\nabla V_0(\overrightarrow{r})$. A spatial density dependency was thus implemented within the MC simulation, along with the resulting potential, and hence electric field. In that study [89], it was found that a discrete change in density, and hence potential, did not significantly change the path and associated transport dynamics of such particles when compared to a realistic functional form, which in addition to computational considerations, motivated the use of a step function form for the change in density in the current work. Additionally, any variation in the potential resulting from the density change is left for future extensions of this work.

3.3. Experimental Parameters

Under realistic experimental conditions, detector resolutions and uncertainties will impact the prominence of correlations between EELS and their corresponding cross-sections. A critical factor for extracting important information is the experimental energy resolution of both the electron source and detector. In the models discussed above, an approximation to the energy spread of the electron source was incorporated by sampling a Gaussian distribution of FWHM of $0.5\mathrm{eV}$. In Figure 2, an experimentally feasible combined EELS energy resolution of $0.9\mathrm{eV}$ [74], which is incorporated into the simulations in what follows in Section 4, alongside a hypothetical resolution of $0.1\mathrm{eV}$, are shown for comparison. Distinct information loss occurs in the $0.9\mathrm{eV}$ energy-resolved spectra, for inelastic peaks with similar threshold energies, when compared to the $0.1\mathrm{eV}$ spectra. This clearly indicates that the foreshadowed experimental measurements should be conducted with as narrow an energy resolution as possible.

3.4. Liquid Micro-Jet Parameters

To develop training data, simulated electrons were scattered through a cylindrical liquid micro-jet using the benchmarked MC simulation. Each electron, with initial energy ${\epsilon }_0$, was fired along the z-axis towards a $15\mu \mathrm{m}$ jet, which was directed along the x-axis. It was assumed that the motion of the jet (∼100${\mathrm{ms}}^{-1}$) has a negligible effect on the measured EELS, as deflection angles were measured within the z-y plane and integrated over the x-axis.

Scattered electrons are detected once they reach a radial distance sufficiently far from the jet, such that the angle measured was relative to the jet’s centre in the z-y plane. The EELS was then integrated over the x-axis, such that there existed a $\left(\chi ,\epsilon \right)$ dependence, where $\epsilon$ is the electron’s final energy, before integrating over $\chi$. Each electron was simulated until it was detected, or until it lost sufficient energy, such that the probability of escape was negligible.

As a proof-of-concept for this work, we assume that the jet is gaseous neon at a liquid density of $2.13\times {10}^{28}{\mathrm{m}}^{-3}$, and at a temperature of $0\mathrm{K}$. Based on the proposed experiment [74], the energy variance of the initial electrons was sampled from a Gaussian with an FWHM of $0.5\mathrm{eV}$, centred around the initial energy ${\epsilon }_0$, while its spatial extent was assumed to be a delta function directed along the surface normal. This serves as a first approximation to the experimental work, where there will be an electron beam, with a finite diameter of $1\mathrm{mm}$, incident on the LµJ.

4. Determining Cross-Sections from Electron Energy Loss Spectra Using Machine Learning

Through a sensitivity analysis, which is presented in Appendix B, we show that there exists a highly complex and degenerate correlation between the energy loss spectra and their underlying cross-sections. It was found that, while absolute magnitudes had little effect on the EELS, the effects of both relative magnitudes and energy dependencies of the cross-section were relatively significant. Deriving cross-section sets directly from EELS, without prior knowledge, is therefore a formidable task.

In this section, we apply machine learning to this ill-posed inverse problem of determining cross-sections from LµJ-measured EELS. As a proof-of-concept, we initially consider the task of determining electron–Ne cross-sections from EELS that are calculated using our Monte Carlo simulation. In what follows, we assume gas phase isotropic scattering, due to the present limited availability of liquid phase electron scattering and differential cross-sections, the inclusion of which remains a critical step towards the determination of liquid phase electron cross-sections. We note that, given appropriate liquid phase test data, the methodology that we propose here could be used to produce effective liquid phase electron cross-sections.

4.1. Machine Learning Methodology

To obtain a solution to the “inverse EELS problem”, we apply the same general machine learning approach as proposed by Stokes et al. [41,42,43,44], for the analysis of electron swarm transport data, and utilise an artificial neural network of the form:

$$\mathbf{y}\left(\mathbf{x}\right)=\left({\mathbf{A}}_4\circ \mathrm{mish}\circ {\mathbf{A}}_3\circ \mathrm{mish}\circ {\mathbf{A}}_2\circ \mathrm{mish}\circ {\mathbf{A}}_1\right)\left(\mathbf{x}\right),$$

(1)

where ${\mathbf{A}}_n\left(\mathbf{x}\right)\equiv {\mathbf{W}}_n\mathbf{x}+{\mathbf{b}}_n$ are affine mappings defined by dense weight matrices ${\mathbf{W}}_n$ and bias vectors ${\mathbf{b}}_n$, and $\mathrm{mish}\left(x\right)=xtanh\left(ln\left(1+e^x\right)\right)$ is a nonlinear activation function [91] that is applied element-wise. The output vector, $\mathbf{y}$, contains each cross-section of interest:

$$\mathbf{y}=\left[\begin{array}{c}{\sigma }_1\left(\epsilon \right)\\ {\sigma }_2\left(\epsilon \right)\\ ⋮\end{array}\right],$$

(2)

all of which are a function of energy, $\epsilon$, which becomes an input to the neural network alongside the available EELS data:

$$\mathbf{x}=\left[\begin{array}{c}\epsilon \\ ine{I}_1\\ I_2\\ ⋮\end{array}\right],$$

(3)

where $I_1,I_2,\cdots$ are the electron intensities accumulated in each of the considered EELS energy bins. Note that we apply suitable logarithmic transformations to ensure that all inputs and outputs of the network are dimensionless and lie within $\left[-1,1\right]$. In what immediately follows, we specify that each bias vector contains 64 parameters, with the exception of ${\mathbf{b}}_4$, of which the size must match the number of cross-sections in $\mathbf{y}$. The weight matrices are sized accordingly.

In order to train the neural network, Equation (1), we require an appropriate set of example solutions to the inverse EELS problem. To ensure cross-sections provided by the network are physically plausible, we train on cross-sections from the LXCat project [11,12]. Specifically, we train with cross-sections of the form [42]:

$$\sigma \left(\epsilon \right)={\sigma }_1^{1-r}\left(\epsilon +{\epsilon }_1-{\epsilon }_1^{1-r}{\epsilon }_2^r\right){\sigma }_2^r\left(\epsilon +{\epsilon }_2-{\epsilon }_1^{1-r}{\epsilon }_2^r\right),$$

(4)

where ${\sigma }_1\left(\epsilon \right)$ and ${\sigma }_2\left(\epsilon \right)$ are a random pair of LXCat electron scattering cross-sections, sampled from the available targets, of a given type (e.g., excitation, ionisation, etc.), r is a pseudo-random number uniformly distributed between 0 and 1, and ${\epsilon }_1$ and ${\epsilon }_2$ are their respective threshold energies. Once suitable cross-sections are found for the training, corresponding LµJ EELS can be determined using our Monte Carlo simulation. In total, we consider 10,000 such training exemplars.

We implement and train the neural network, Equation (1), using the Flux.jl machine learning framework [92]. The network is initialised such that its biases are zero and its weights are uniform random numbers, as described by Glorot and Bengio [93]. Training is performed using the AdaBelief optimiser [94] with Nesterov momentum [95,96], step size $\alpha ={10}^{-3}$, exponential decay rates ${\beta }_1=0.9$ and ${\beta }_2=0.999$, and the small parameter $ϵ={10}^{-8}$. At each iteration, the optimiser is provided with a different batch of 4096 training examples, where each batch consists of 32 training cross-sections each evaluated at 128 random energies of the form $\epsilon ={10}^s$, where $s\in \left[0,2\right]$ is sampled from a continuous uniform distribution. For each batch, the optimiser adjusts the neural network weights and biases with the aim of further minimising the mean absolute error in solving the inverse EELS problem for that batch. Training is continued for 250,000 iterations, providing an equal number of potential solutions to the inverse problem. We then select every 10 for the last 100,000 cross-sections, and the quality of each of these solutions is subsequently assessed by simulating their corresponding EELS and comparing those to Ne’s EELS to find the ‘best’ regression.

4.2. Cross-Section Regression Given the EELS

We now implement and train neural networks of the form of Equation (1), to determine a selection of Ne’s cross-sections given the corresponding EELS, while assuming full knowledge of Ne’s remaining cross-sections. In total, we use eleven EELS, corresponding to initial electron energies of 100, 79.43, 63.1, 50.12, 39.81, 31.62, 25.12, 19.95, 15.85, $12.59$, and $10.0\mathrm{eV}$. In each spectra, a combined energy resolution of $0.9\mathrm{eV}$ was assumed based on the detector performance that we currently find at Flinders University. Additionally, for computational considerations, each electron was simulated until $90\%$ of its energy was lost. For the initial energies considered, no electron was simulated below $1\mathrm{eV}$, and thus we restrict the prediction domain to $\left[1\mathrm{eV},100\mathrm{eV}\right]$.

We first determine only Ne’s elastic momentum transfer cross-section (MTCS) from the considered EELS. Figure 3a shows a reasonable level of agreement between the elastic MTCS for Ne [90] and those found by the neural network, with the best regression seen to be accurate to within $9\%$. Figure 3b shows that, despite the large range in cross-sections for the 100 best regressions depicted in Figure 3a, the corresponding range of the EELS is much smaller, and in good agreement with those for Ne. This highlights the nonuniqueness of the inverse EELS problem, and suggests that there is not much room for improvement to the best cross-section fit plotted in Figure 3a unless; additional EELS were included, the energy resolution of the spectra was improved, or additional information about the unknown cross-sections was provided as input to the network.

We follow the above by now simultaneously determining both the elastic MTCS and ionisation cross-sections for Ne, using the same set of EELS. In this model, the bias vector was increased to 128, while 300,000 iterations were performed. As expected, the uncertainty in the elastic MTCS has increased here, with a larger elastic MTCS envelope found in Figure 4a, with the best regression seen to be accurate to within $18\%$. In contrast, the corresponding ionisation cross-section envelope is particularly small, with the best regression being accurate here to within $1\%$. We attribute this high accuracy to the prominent ionisation “shoulder” present in six out of eleven of the EELS. Figure 4b shows that the corresponding EELS are now close in line with those for Ne, despite having a range that is slightly larger than their counterparts in Figure 3b.

Using the MC simulation, every combination of the best 100 elastic and ionisation cross-section pairs was compared through their resulting EELS to find a further improved cross-section set. While the maximum error in the elastic MTCS fit increased to $20\%$, the average error decreased resulting in a closer fit. For ionisation, the maximum error decreased to $0.2\%$, which further emphasises the remarkable predictive capability of this model for ionisation cross-sections.

The extension of this methodology to include the prediction of excitation cross-sections, or more specifically, the simultaneous prediction of three or more cross-sections, results in a substantial decrease in accuracy within the current methodology. Thus, the simultaneous prediction of elastic, ionisation, and excitation cross-sections is left for future iterations of this method, which should include advances in either the experimental resolution, or the fitting process.

5. Conclusions

We have developed a joint Monte Carlo and machine learning solution to the inverse Milne problem, that extracts electron cross-sections based on electron energy loss spectra from a micro-jet of dense gas. Machine learning was conducted using similar techniques outlined in the literature [41,42,43,44], while a new Monte Carlo simulation of a liquid micro-jet was developed. As a proof-of-concept, we found that the neural network determined neon’s gas phase elastic momentum transfer cross-sections to within $9\%$.

The extension towards the simultaneous determination of neon’s ionisation cross-sections, in addition to elastic scattering, decreased the accuracy to within $18\%$ for elastic scattering, but was accurate within $1\%$ for the ionisation cross-section. A combinatorial search was conducted using the 100 best elastic and ionisation cross-section pairs, which resulted in an accuracy to within $20\%$ for the elastic MTCS and $0.2\%$ for ionisation. While the maximum error for neon’s elastic MTCS increased, overall, the fit was improved, with only the higher energy regime suffering in accuracy. The determination of three or more simultaneous cross-sections resulted in a substantial decrease in accuracy, and is thus left for future iterations of this method.

As expected, absolute magnitudes were difficult to determine, although the prediction of the energy dependencies showed promise. Ionisation cross-sections were remarkably well determined, while the elastic cross-section prediction saw only quite slight discrepancies. Considering a relatively well-formed energy dependence, theoretical values at the higher energies could be used to ‘fix’ each cross-section to improve the current prediction. Alternatively (or in addition), the relative flow technique used for gas phase scattering experiments to achieve absolute cross-sections could be adapted to these liquid micro-jet experiments.

The main limitation with our current technique revolves around information density. Spectra are comprised of several ‘dead zones’, in which little to no scattering information exists. Additionally, a coarse energy resolution was shown to hide important information within the spectra, especially around inelastic peaks with similar threshold energies. To increase information density, a feature extraction algorithm might be utilised for each spectrum along with an improved experimental detector resolution. In other spectral studies, Gaussian de-convolution algorithms are employed to extract peaks from spectra. Utilising a similar approach, along with an increased detector resolution, one could provide further clarity to the network, assuming an appropriate treatment of asymmetry resulting from both multiple scattering and the ionisation energy sharing profiles.

Currently, a fitting algorithm is in development which, when applied to the method outlined in this study, aims to improve the predictive capability of the neural network. Additionally, the inclusion of appropriate liquid structure factors, surface potentials and space charge effects are necessary for real-world derivations of effective electron–liquid cross-sections. Overall, through a proof-of-concept model, we have shown that utilising machine learning, along with a significant wealth of Monte Carlo training data, one can reasonably predict individual and two simultaneous electron scattering cross-sections from EELS. The extension towards predicting full, self-consistent cross-section sets first begins with improvements on the methodology to ensure an improved accuracy in determining the cross-section magnitude.