An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Peter W. Stokes, Sean P. Foster, Madalyn J. E. Casey, Daniel G. Cocks, Olmo González-Magaña, Jaime de Urquijo, Gustavo García, Michael J. Brunger, Ronald D. White
AAn improved set of electron-THFA cross sections refinedthrough a neural network-based analysis of swarm dataP W Stokes , S P Foster , M J E Casey , D GCocks , O González-Magaña , J de Urquijo , GGarcía , M J Brunger , and R D White College of Science and Engineering, James Cook University,Townsville, QLD 4811, Australia Research School of Physics, Australian National University,Canberra, ACT 0200, Australia Instituto de Ciencias Físicas, Universidad Nacional Autónoma deMéxico, 62251, Cuernavaca, Morelos, Mexico Instituto de Física Fundamental, CSIC, Serrano 113-bis, 28006Madrid, Spain College of Science and Engineering, Flinders University, BedfordPark, Adelaide, SA 5042, Australia Department of Actuarial Science and Applied Statistics, Faculty ofBusiness and Management, UCSI University, Kuala Lumpur 56000,Malaysia
E-mail: [email protected]
Abstract.
We review experimental and theoretical cross sections for electron transport in α -tetrahydrofurfuryl alcohol (THFA) and, in doing so, propose a plausible complete set. To assessthe accuracy and self-consistency of our proposed set, we use the pulsed-Townsend technique tomeasure drift velocities, longitudinal diffusion coefficients and effective Townsend first ionisationcoefficients for electron swarms in admixtures of THFA in argon, across a range of density-reducedelectric fields from 1 Td to 450 Td. These measurements are then compared to simulated valuesderived from our proposed set using a multi-term solution of Boltzmann’s equation. We observediscrepancies between the simulation and experiment, which we attempt to address by employinga neural network model that is trained to solve the inverse swarm problem of unfolding the crosssections underpinning our experimental swarm measurements. What results from our neuralnetwork-based analysis is a refined set of electron-THFA cross sections, which we confirm is ofhigher consistency with our swarm measurements than that we initially proposed. We also useour data base to calculate electron transport coefficients in pure THFA, across a range of reducedelectric fields from 0.001 Td to 10,000 Td. Keywords : swarm analysis, inverse problem, Boltzmann equation, machine learning, α -tetrahydrofurfuryl alcohol Submitted to:
J. Chem. Phys. a r X i v : . [ phy s i c s . p l a s m - ph ] J a n An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
1. Introduction
The study of non-equilibrium electron transport in biological matter underpins a diverse rangeof scientific fields and applications. Of particular interest is the medical sector, where electron-induced processes in human tissue occur in both medical imaging and therapy [1]. In theseapplications, ionising radiation liberates large numbers of low-energy secondary electrons ( ∼
30 eV )which undergo a variety of energy deposition processes in the biomolecules that constitute humantissue [2]. These thermalised electrons are known to undergo dissociative electron attachment(DEA), which has been attributed in part to the damage associated with such ionising radiation,either directly through inducing single or double strand breaks in DNA, or indirectly throughthe interactions of electron-induced radicals with DNA. Accurate kinetic simulations of electrontransport in biological matter, including a full description of the interactions with each of thevarious biomolecular constituents, are therefore required in order to fully understand radiationdamage and comprehensively inform dosimetry models.The modelling of electron transport in biomolecules has also found recent application in plasmamedicine, which is a relatively new field motivated by the synergistic interactions of low-temperatureatmospheric pressure plasmas (LTAPP) with biological tissue [3–5]. Human tissue is generallymodelled as a bulk liquid so that the system simplifies to a three-phase problem consisting of abulk gas, a gas-liquid interface, and a bulk liquid [4, 5]. While the interactions of the reactiveoxygen and nitrogen species (RONS), produced in the plasma-liquid interface, are known to inducemany of the synergistic effects [3], a predictive understanding of plasma treatments can only beobtained through a complete understanding of all of the plasma-tissue interactions, including theelectron-impact generation of radicals [2].Despite its importance, a full soft-condensed phase tissue description of electron transport iscurrently in its infancy. When applying kinetic modelling techniques, biological matter is currentlyapproximated as water vapour, despite biological media being neither water nor a gas, nor beingdecoupled entirely from electron interactions as in plasma medical device modelling, but rathera complex mixture of biological molecules in a soft-condensed phase [6–8]. As such, quantitativemodelling of electron transport through biological media requires the attainment of complete andaccurate sets of cross sections for all electron interactions with all relevant biomolecules, includingwater, in the soft-condensed phase.As low-energy electron interactions with DNA are difficult to study, focus has turned to theindividual components of DNA, in addition to their structural analogues. One component thathas received considerable attention is 2-deoxyribose, a sugar that links phosphate groups in theDNA backbone, which has well-studied surrogates including tetrahydrofuran (THF, C H O ) and α -tetrahydrofurfuryl alcohol (THFA, C H O ) [9]. Between these, THF has received the mostattention, with a number of proposed complete electron impact cross section sets present in theliterature [10–17]. In comparison, however, while individual electron-THFA cross sections areknown, a complete set is presently still lacking. In this investigation, we attempt to remedy this gapin the literature by constructing and refining a complete and self-consistent set of electron-THFAcross sections in the gas-phase, with the motivation being that such a set can be adapted to thesoft-condensed phase through appropriate modifications using pair correlations functions [18, 19].The present investigation is especially warranted given that, in comparison to THF, THFA hasbeen identified as a superior analogue for 2-deoxyribose [20, 21].Key to the derivation of our cross section set is the measurement and subsequent analysisof electron swarm transport coefficients in admixtures of THFA and argon. By comparing these n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data inverse swarm problem of unfolding cross sections from swarm measurements has a long and successful history [23–30].However, being an inverse problem, it is often the case that there is no single unique set of crosssections that is consistent with a given set of swarm measurements. This nonuniqueness poses afundamental challenge in automating swarm analysis using numerical optimisation algorithms [31–38], as while such algorithms diligently minimise the error in the associated transport coefficients,they lack the intuition about what constitutes a physically-plausible cross section set. As such,to try and ensure the most success, these iterative adjustments to the cross section set must becarefully performed by an expert that can use their prior knowledge to rule out unphysical solutions.Despite these challenges, we have recently had some success in employing machine learning modelsto solve the inverse swarm problem automatically [17, 39], an approach that was originally exploredby Morgan [40] decades earlier. By training these models on cross sections derived from the LXCatproject [41–43], they can, in a sense, “learn” what constitutes a physically-plausible cross sectionset. Recently [17], we trained an artificial neural network model in order to refine the electron-THFcross section set of de Urquijo et al . [16]. Promisingly, the set of cross sections determined by thisneural network was found to be of comparable quality to the de Urquijo et al . set that was refined“by hand”. LXCat cross sections have also been applied recently by Nam et al . [44] to train aneural network for the classification of cross sections according to their type (i.e. elastic, excitation,ionisation, or attachment).The remainder of this paper is structured as follows. In Section 2, we briefly describe our data-driven approach to solving the inverse swarm problem, including the nature of the cross sectionsand transport coefficients used to train the machine learning model. Section 3 provides a reviewof existing electron-THFA cross sections in the literature. These measured and calculated integralcross sections (ICSs) are then employed to construct a “proposed” data base for electron-THFAscattering, which is also described in this section. Note that several of the present authors have hadrecent experience in constructing data bases for electron and positron scattering problems [45–48],but that the success of this approach does depend on the volume of relevant data available andby its very construction can be highly selective. In Section 4 details of our experimental techniquefor measuring the THFA-argon gas mixture transport coefficients are provided, with the results ofthese measurements also being presented. Note also in this section that results from our Boltzmannequation analysis, using our “proposed” cross sections, are provided and compared against themeasured data. In Section 5 a refined set of THFA electron scattering cross sections, using ourmachine learning / neural network-based approach, are presented and discussed, with results fromtheir application in our Boltzmann equation analysis, for simulated transport parameters, beinggiven in Section 6. Finally, Section 7 presents our conclusions from the current investigation andgives some suggestions for future work.
2. Neural network for cross section regression
In this section, we briefly describe the architecture and application of our neural network forthe regression of electron-THFA cross sections from swarm transport data. For a more detaileddescription of this approach to inverting the swarm problem, we refer the reader to our previouspublications, Refs. [39] and [17].
An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data2.1. Neural network architecture
In this work, we perform the cross section regression by utilising neural networks of the form: y ( x ) = ( A ◦ mish ◦ A ◦ mish ◦ A ◦ mish ◦ A ) ( x ) , (1)where A n ( x ) ≡ W n x + b n are affine mappings, ◦ denotes function composition, and mish ( x ) = x tanh (ln (1 + e x )) [49] is a nonlinear activation function that is applied element-wise. The neuralnetwork, Eq. (1), is said to be fully-connected as the matrices of weights , W n , and vectors of biases , b n , are dense. The number of weights and biases is correlated with the capacity of theneural network to perform a particular nonlinear mapping from the input vector x to the outputvector y . With the exception of b , the size of which must match the output of the network, wespecify 256 biases per bias vector and size the weight matrices accordingly. Naturally, the outputof our neural network for swarm analysis contains the electron-THFA cross sections of interest: y = σ ( ε ) σ ( ε ) ... . (2)As these cross sections are functions of energy, we accordingly include the energy ε as an element ofthe input vector x . To solve the inverse swarm problem, we populate the remaining input elementswith the swarm transport coefficient measurements: x = εW W ... ( α eff /n ) ( α eff /n ) ... ( n D L ) ( n D L ) ... , (3)where W denotes the drift velocity, α eff denotes the effective Townsend first ionisation coefficient, D L denotes the longitudinal diffusion coefficient, and n is the background neutral number density.Cross section regression in this way is particularly appealing due to the versatility of neuralnetworks. So long as the energy ε remains an input to the network, we can in principle derive crosssections from any collection of experimental measurements. In fact, in forming our initial proposedset of electron-THFA cross sections in Section 3, we derive plausible vibrational and electronicexcitation cross sections in this way from the limited number of experimental measurements thatare currently available.It should lastly be noted that we normalise all inputs and outputs of the neural network byfirst taking the logarithm and then performing a linear mapping onto the domain [ − , . As aconsequence of this log-transformation, we are restricted to inputs and outputs that are positive.This poses a difficulty when predicting cross sections below threshold. In such instances, we replacecross sections equal to zero with a suitably small positive number, which we take to be − m .Consequently, if the neural network outputs a cross section less than − m , we interpret the n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data For training the neural network, Eq. (1), we use, as required, elastic, excitation, ionisation andattachment cross sections from the LXCat project [41–43, 50–68]. Specifically, we generate realisticcross sections for training by taking random pairwise geometric combinations of cross sections fromLXCat using the formula: σ ( ε ) = σ − r (cid:0) ε + ε − ε − r ε r (cid:1) σ r (cid:0) ε + ε − ε − r ε r (cid:1) , (4)where σ ( ε ) and σ ( ε ) are cross sections of a given process chosen randomly without replacement, ε and ε are their respective threshold energies, and r ∈ [0 , is a uniformly sampled mixing ratio.We apply similar geometric combinations when we wish to constrain training cross sections withinknown experimental error bars. Specifically, we ensure σ ( ε ) is itself constrained and then perturbabout it with a random σ ( ε ) and a mixing ratio r chosen small enough so as not to violate theprescribed constraints.Once cross sections have been selected for training, they must be sampled at variousenergies within the domain of interest. In this work, we are concerned with the domain ε ∈ (cid:2) − eV , eV (cid:3) , which we sample randomly within using: ε = 10 s eV , (5)where s ∈ [ − , is a uniformly distributed random number.To complete the input vector of our input/output training pair, we calculate correspondingtransport coefficients using a well-benchmarked multi-term solution of Boltzmann’s equation[22, 69, 70]. For good measure, we employ the ten-term approximation for all cross section setsused for training. Additionally, to simulate the random error present in the experimental swarmmeasurements, we multiply our simulated transport coefficients by a small amount of random noisesampled from a log-normal distribution. Specifically, we sample the natural logarithm of this noisefactor from a normal distribution with a mean of and a standard deviation of . . The neural network is implemented and trained using the
Flux.jl machine learning framework [71].Before training, we initialise the neural network biases to zero and weights to uniform randomnumbers as described by Glorot and Bengio [72]. Then, to train the network, we perform numericaloptimisation of its weights and biases so as to minimise the mean absolute error of the cross sectionsfitted by the neural network. We choose the mean absolute error measure due to its robustnessin the presence of outliers. During the optimisation we repeatedly update the weights and biasesusing the Adam optimiser [73] with step size α = 10 − , exponential decay rates β = 0 . and β = 0 . , and small parameter (cid:15) = 10 − . For each update of the neural network parameterswith the optimiser, we consider a random batch of input/output training examples. Each ofthese batches consist of random LXCat-derived cross section sets, with each set sampled at random energies using Eq. (5). In total, the training data set consists of 50,000 unique sets of crosssections. Training is continued until the transport coefficients resulting from the fitted cross sectionset best match the measured pulsed-Townsend transport coefficients that were used to perform thefit. An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 1.
Proposed THFA discrete electronic excitation cross sections, alongside theexperimental measurements [21] to which our initial proposals were fitted. See also legend infigure.
3. Electron-THFA cross section data review and initial proposals
Electronic-state scattering results for electron-THFA includes the experimental differential crosssection (DCS) data of Chiari et al . [74] and the corresponding integral cross section (ICS) dataof Duque et al . [21], the latter of which are plotted in Figure 1. These authors identify fiveRydberg electronic-state bands for THFA [20], although due to insufficient energy resolution in theexperimental apparatus these are resolved into only three separate electronic-state bands. Thesebands are reported as having threshold energies of 6.2 eV, 7.6 eV and 8.2 eV for Bands 1+2, Band3 and Bands 4+5, respectively. To construct our proposed electronic excitation cross sections forTHFA, we make use of these threshold energies alongside the ICS data of Duque et al . To interpolatethis data, as well as extrapolate to higher energies, we employ a neural network of the form of Eq.(1) to fit a plausible excitation cross section for each case. Specifically, for a given band, we inputto the network the ICS data at the four energies considered by Chiari et al . and Duque et al . (20eV, 30 eV, 40 eV and 50 eV), as well as the threshold energy for that band. The resulting neuralnetwork regression, and initial proposed ICS, for each band is plotted in Figure 1. In each case, aplausible energy location of the peak cross section value is identified automatically by the network.
Vibrational scattering data for electron-THFA consists only of the experimental DCS and ICSdata of Duque et al . [75], at the incident energies of 20 eV, 30 eV, 40 eV and 50 eV. The fourvibrational modes of THFA identified by Duque et al . are the CC stretch, CH stretch, OH stretch + combination band, and × CH stretch overtone, with respective threshold energies ofapproximately 0.12 eV, 0.33 eV, 0.5 eV and 0.7 eV. For our proposed THFA vibrational excitation n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Figure 2.
Proposed vibrational excitation cross sections, alongside the experimentalmeasurements [75–77] to which our initial proposed data were fitted. The four vibrational modeshere are CC stretch (blue), CH stretch (orange), OH stretch + combination (green), and × CH stretch (red). See also legend in figure. cross sections, we make use of this ICS data with these corresponding energy thresholds. At lowerenergies, we employ the THF data of Khakoo et al . [76], for the same vibrational modes, definedat the energies 2 eV, 3 eV, 5 eV, 10 eV, 15 eV and 20 eV. Note that here we scale the Khakoo etal . data a little, so as to match the THFA data of Duque et al . at the overlapping point of 20eV. This approach is thought to be reasonable due to THF and THFA having similar structuresand intrinsic molecular properties (e.g. dipole moment and dipole polarisability). At very lowenergies, down to the threshold in each case, we make use of the THF CC stretch data of Allan[77]. To accomplish this we need to shift Allan’s measurement to each respective THFA threshold,and also to scale so as to minimise the discontinuity with the overlapping scaled measurements ofKhakoo et al . We then subsequently performed a smoothing cubic spline interpolation [78] throughthe measurements of Duque et al ., the scaled measurements of Khakoo et al ., and the shifted andscaled measurement of Allan et al . (up to 1 eV above threshold in each case). Finally, above 50 eVwe employ the same neural network regression approach used in the previous section for electronicexcitation interpolation/extrapolation. Specifically, for each vibrational mode, we consider as inputto that neural network the threshold energy and the four measurements of Duque et al . in eachcase. The output of the neural network is then the vibrational integral cross section at points above50 eV (up to 1000 eV). We join the resulting high-energy extrapolation with the smoothing cubicspline interpolation at 50 eV by a further scaling. The final proposed vibrational excitation crosssections for THFA are plotted in Figure 2, alongside all of the experimental measurements fromwhich they are derived. Available scattering data for electron impact ionisation of THFA includes the theoretical ICS dataof Możejko and Sanche [79], the experimental ICS results of Bull et al . [80], and the theoretical
An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 3.
Proposed electron impact ionisation cross section, alongside previous experimentaland theoretical results [21, 79, 80] from which it was derived. See also legend in figure.
ICS data of Duque et al . [21]. These are each plotted in Figure 3. Both of the aforementionedtheoretical investigations use a semi-classical binary-encounter-Bethe (BEB) formalism [21, 79, 81],with Duque et al . also employing a modified IAM-SCAR approach [21]. The modification usedby Duque et al . was originally proposed in Chiari et al . [82], and allows the separation of theionisation ICS from the total “inelastic” ICS provided by the IAM-SCAR approach. The data ofBull et al . is determined from measurements of ion and electron currents using the Beer-Lambertlaw [80], and agrees quite well with the BEB results in terms of both its shape and in the position ofthe cross section maximum, while being generally lower in magnitude compared to the theoreticalresults. As the sole experimental data available, and from a group with a long history of makingreliable ionisation cross section measurements, we use the Bull et al . ionisation ICS as the basis forour proposed ionisation cross section. For energies above the maximum considered by Bull et al .(285 eV), we make use of the modified IAM-SCAR results of Duque et al . For energies below theminimum considered by Bull et al . (10 eV), we specify an ionisation threshold of 9.69 eV, as wasobtained by Dampc et al . [83] through the analysis of THFA photoelectron spectra. This resultinginitial proposed ionisation cross section can also be found plotted in Figure 3.
To our knowledge, there are currently no attachment scattering data available for THFA in theliterature. There are, however, dissociative electron attachment (DEA) cross sections availablefor some other structurally similar biomolecules, including THF [10, 15–17, 84, 85] and 3-hydroxytetrahydrofuran (3-hTHF) [84], from which we can obtain a very rough initial estimatefor that of THFA. Brunger [9] noted that the most important electronic-structure quantities fordetermining the relative magnitude of DEA cross sections between biomolecules is the dipolemoment and dipole polarisability of each molecule, with the dipole polarisability being the mostsignificant. While of course this is very much a first order approximation, it does provide at n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Figure 4.
Proposed electron attachment cross section, alongside the previous results [17, 84]from which it was derived. See also legend in figure. least some physical basis to what follows. On average, across all five of its conformers at roomtemperature [20], THFA has a large dipole polarisability of ∼ . a [20, 21], where a is the Bohrradius. For construction of our proposed electron attachment cross section for THFA, we prioritiseusing the available 3-hTHF DEA data over that for THF, as both 3-hTHF and THFA containa hydroxyl group, the presence of which has been shown experimentally to enhance DEA [84].Specifically, we choose to use the 3-hTHF DEA measurement of Aflatooni et al . [84]. Compared toTHFA, 3-hTHF has an average dipole polarisability of ∼ . a [86, 87], a value we obtain simplyby averaging the theoretically-determined values of its two most energetically stable conformersof . a and . a [86]. Accordingly, we scale up the Aflatooni et al . 3-hTHF DEAmeasurement by the ratio between the THFA and 3-hTHF dipole polarisabilities ( × . ). Thisthen completes the DEA component of our proposed attachment cross section. At energies below 0.1eV, where DEA is impossible and any attachment is inherently non-dissociative, we make use of theelectron attachment cross section for THF derived by Stokes et al . [17] using a neural network-basedanalysis of swarm transport data. Of course it is also necessary to scale this cross section ( × . ),so that the peak magnitude for DEA matches that for the scaled 3-hTHF DEA measurement usedat higher energies. Finally, we perform a smoothing cubic spline interpolation [78] over the entirerange being considered. The resulting proposed electron attachment cross section for THFA can befound plotted in Figure 4, alongside the scaled THF and 3-hTHF counterparts it was derived from. Grand total cross section (TCS) scattering data for electron-THFA includes the theoretical TCSresults from Milosavljević et al . [88], the experimental TCS data of Możejko et al . [89], thetheoretical TCS data of Zecca et al . [90], and the theoretical TCS data of Duque et al . [21]. Theseare summarised in Figure 5. The experimental data of Możejko et al . are derived by applyingthe Beer-Lambert formula [89], to the attenuated and unattenuated electron beam intensities, ina linear transmission experiment. Those data are not corrected for the forward-scattering effect0
An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 5.
Proposed grand total cross section, alongside previous experimental and theoreticaltotal cross sections [21, 88–90]. See also legend in figure. [89], and are therefore expected to be missing the significant effect of the rotational cross sectionsat lower energies. The theoretical data of Milosavljević et al . is calculated using the IAM-SCARapproach, which yields combined elastic, electronic excitation, neutral dissociation and ionisationprocesses, while lacking contributions from rotational, vibrational excitation and DEA processes.The TCS data of Zecca et al . is calculated using the IAM-SCAR+R procedure, and includes allprocesses except for vibrational excitation and DEA. Here rotational excitation is included usinga Born approximation-based method. The data of Duque et al . is also calculated using the IAM-SCAR+R procedure, but the innovation of Chiari et al . [82] is also employed in this case to separatethe ionisation channel from the rest of the “inelastic” data. The TCS data of Duque et al . thereforealso includes all processes except for vibrational excitation and DEA, but is able to individuallyresolve the elastic, rotational, discrete electronic-state excitation and ionisation cross sections. Forour proposed grand TCS for THFA, we prioritise the experimental data of Możejko et al . overthe theoretical results. However, to use this data we must first correct for the forward-scatteringeffect by increasing the magnitude of this cross section at lower energies. To determine the extentof this correction, we use the TCS data of Duque et al . as a guide, to which we add our proposedvibrational excitation and DEA cross sections to form an initial approximate grand TCS. Next, wescale the experimental data of Możejko et al . so as to best match this approximation. In particular,we scale by an energy-dependent correction of the power-law form . ε − . , which has thegreatest effect at smaller energies, while leaving the experimental data at higher energies unaffected.At energies above 370 eV, we use the same approximate grand TCS data derived from Duque etal ., but scaled down ( × . ) so as to improve continuity with the experimental data of Możejko etal . The resulting proposed grand TCS is plotted in Figure 5, alongside the various results from theliterature [21, 88–90]. n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Figure 6.
Proposed rotational integral cross section, alongside previous theoretical results[21, 90]. See also legend in figure.
Rotational scattering data for electron-THFA collisions include the theoretical ICS data of Zecca etal . [90], and the theoretical ICS data of Duque et al . [21]. These are plotted in Figure 6. Both ofthese rotational cross sections are derived using the first Born approximation (FBA), with Zecca etal . calculating the rotational excitation cross section, for J → J (cid:48) in THFA at 300 K, by weightingthe population for the J th rotational quantum number at that temperature and estimating theaverage excitation energy from the corresponding rotational constants. On the other hand, Duque et al . used the procedure of Jain [91]. All the integral rotational cross section data is given interms of a single summed rotational cross section, with the ICS data of Duque et al . calculatedfrom assuming an average rotational threshold energy of 0.74 meV. As we are relying on the TCSdata of Duque et al . to guide the form of our proposed grand TCS, for consistency we also makeuse here of the data from Duque et al . for our proposed rotational ICS, as shown in Figure 6. Elastic scattering data for electron-THFA collisions include the theoretical DCS and ICS data ofMożejko and Sanche [79], the theoretical ICS data of Milosavljević et al . [88] and the theoreticalICS data of Duque et al . [21]. These are summarised in Figure 7. The calculations of Możejkoand Sanche are performed using the independent-atom method (IAM), applied with the additivityrule (AR), and have been superseded by the calculations of Milosavljević et al . and Duque et al .,which use the IAM-SCAR procedure. That is, the IAM in conjunction with a screening correctedadditivity rule (SCAR). The ICS data of both Milosavljević et al . and Duque et al . are in goodagreement with one another, and as such we base our proposed THFA elastic ICS on both of theseresults, with some minor modifications in place for consistency with our proposed grand TCS. Theresulting proposed elastic ICS is plotted in Figure 7. Even by accounting for all of the cross sectionsproposed thus far (outlined in Sections 3.1–3.7), there still remains a discrepancy in the grand TCS2
An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 7.
Proposed elastic integral cross section, alongside previous theoretical results[21, 79, 88]. See also legend in figure.
Figure 8.
Proposed neutral dissociation cross section. See text for further details. at intermediate energies which we attribute to neutral dissociation, as discussed in the followingsection.
A challenge in obtaining complete sets of electron-biomolecule cross sections is the intractabilityof determining the neutral dissociation integral cross section from scattering experiments directly[92]. Similarly, theoretical results are rare [48]. Of course, in principle, it can be found indirectly n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Figure 9.
Proposed quasielastic momentum transfer cross section. See text for further details. by subtracting all the other scattering cross sections from the TCS, resulting in a remnant that isattributed to neutral dissociation. Proceeding with this approach results in the proposed THFAneutral dissociation cross section plotted in Figure 8, with an apparent threshold energy of . .However, as the accuracy of this remnant is predicated on the collective accuracy of all other crosssections outlined thus far, it is not anticipated to be particularly reliable. As an alternative tothis approach, swarm experiments provide an implicit way of elucidating the neutral dissociationcross section through assessment of the self-consistency of a cross section set. Indeed, the neutraldissociation cross section of THF has previously been characterised through the swarm analysis ofCasey et al . [15] and de Urquijo et al . [16], as well as recently by Stokes et al . [17] using the samemachine learning formalism described in Section 2 and employed later in Section 5 of the presentinvestigation. For the purposes of performing transport calculations, we form a proposed quasielastic(elastic+rotational) momentum transfer cross section (MTCS) by scaling our proposed quasielasticICS by the ratio of quasielastic MTCS to ICS of Duque et al . [21]. At energies below 1 eV, weutilise a power law extrapolation that is fitted to the quasielastic MTCS of Duque et al . in thisregime. Our resulting proposed quasielastic MTCS is plotted in Figure 9.
4. Pulsed-Townsend swarm measurements for assessing the self-consistency of theproposed cross section set
To assess the quality of our proposed set of electron-THFA cross sections, we perform a numberof pulsed-Townsend swarm experiments from which we obtain drift velocities, effective Townsendfirst ionisation coefficients, and longitudinal diffusion coefficients for admixtures of THFA in argon.We consider mixture ratios of 0.2%, 0.5%, 1%, 2%, and 5% THFA, across a range of reduced4
An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 10.
Measured pulsed-Townsend swarm transport coefficients (markers) of (a), thedrift velocity, W , (b), the effective Townsend first ionisation coefficients, α eff /n , and (c),the longitudinal diffusion coefficients, n D L . Simulated transport coefficients (dashed curves),derived from the proposed cross section set presented in Section 3, are also plotted for comparison. n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data − V m . Results fromthese measurements — which are tabulated in our appendix Tables A1–A5 — were derived froma pulsed-Townsend apparatus whose technique and methods of analysis of the electron avalanchewaveforms have been accounted for in detail previously [93, 94]. The pulsed Townsend method isbased on the measurement of the total displacement current due to the motion of electrons and ionswithin a parallel plate capacitor that produces a highly homogeneous field upon the application ofa highly stable and very low ripple DC voltage between 150 V to 5 kV, depending on the E/n and n conditions of the experiment. The capacitor through which the charge carriers drift andreact consists of an aluminium cathode and a non-magnetic stainless steel anode of 12 cm diametereach, separated by an accurately measured distance of 3.1 cm to within an accuracy of 0.025mm. The initial photoelectrons are generated from the cathode by the incidence of a UV laserpulse (1–2 mJ, 355 nm, 3 ns). When the dominant processes involved in the avalanche are dueto the electrons and stable, non-reacting ions, then any collisional ionisation or attachment eventsare due only to electrons. Furthermore, since the electron drift velocity is – times largerthan that of the ions, the resulting total current can readily be separated into a fast componentdue mostly to the electrons, followed by that of the slower ions. The analysis of the electroncomponent leads to the derivation of the flux electron drift velocity, W , and the density-normalisedeffective ionisation coefficient, α eff /n = ( α − η ) /n , where α and η are the ionisation and electronattachment coefficients. A very stable voltage in the range 0.2–5 kV was applied to the anode inorder to produce the highly homogeneous electric field E , according to the E/n value selectedand the gas density n in the discharge vessel. The stated purity of the commercial THFA sampleused was 99.0% (Sigma-Aldrich) and that of Ar was 99.995% (Praxair). Because of the very lowvapour pressure of THFA, namely 0.2 Torr at 293 K, the maximum pressure allowed in the dischargevessel was 0.18 Torr. This very low filling pressure value hindered the measurements of the electronswarm coefficients for pure THFA. The mere presence of the electron avalanche produces a spacecharge field which is superposed to the externally applied one. Thus care must be taken to keepthe minimum external voltage high enough so that the space charge field is smaller than 1% of thatapplied to the anode. With the present configuration, with an interelectrode distance of 3.1 cm, wecould only measure THFA mixtures with Ar successfully from 0.2% to 5% THFA. The minimumexternal voltage was 200 V. The measurements were performed at room temperature in the range293–300 K, measured with a precision of ± . , while the gas mixture pressure was monitoredwith an absolute pressure capacitance gauge ( ± . uncertainty). The displacement current dueto the electrons was measured with a very low-noise, 40 MHz amplifier with a transimpedance of V/A. The measured electron transients were analysed using the formula for the electron currentin the external circuit derived by Brambring [95]: I ( t ) = n qW L e α eff W t (cid:26) erfc (cid:20) ( W + α eff D L ) t − L √ D L t (cid:21) − e W + α eff DLDL L erfc (cid:20) ( W + α eff D L ) t + L √ D L t (cid:21)(cid:27) , (6)where L is the drift distance and erfc ( x ) = 1 − √ π (cid:82) x e − u d u is the complementary error function.Thus we have three swarm parameters to determine, namely, W , α eff and D L . The process issimplified by determining initial values of W and α eff , derived from a basic, geometrical analysis[96, 97], and inserted into a simulator to fit the whole transient, thereby obtaining D L and refinedvalues of α eff and W . Typical uncertainties for W , α eff and D L were ± , ± and ± ,respectively. Note that these transport coefficients can be related to the net ionisation frequency, R net , the bulk drift velocity, W B , and the bulk diffusion coefficient, D B,L , via [98]: R net = α eff W, (7)6 An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data W B = W + α eff D L , (8) D B,L = D L . (9)Using our proposed cross section set, we apply a well-benchmarked multi-term solution ofBoltzmann’s equation [22, 70, 99] to derive simulated pulsed-Townsend transport coefficients forcomparison to our admixture measurements. For calculating these admixture transport coefficients,we use the argon cross section set present in the Puech database [65] on LXCat [41–43]. Thesimulated transport coefficients are plotted alongside the experimental values in Figure 10. Figure10(a) compares the drift velocities and shows qualitative agreement at low-to-intermediate fields,with the proposed data set underestimating its magnitude slightly at the highest fields considered.Interestingly, while both measured and simulated drift velocities exhibit negative differentialconductivity (NDC) — drift velocity decreasing with increasing reduced electric field — there isdisagreement in the extent of NDC, with the simulated drift velocities exhibiting NDC over a muchlarger range of fields. Figure 10(b) compares the effective Townsend first ionisation coefficientsand shows that those which result from the proposed set overestimate the magnitude in theelectropositive regime, and underestimate the magnitude in the electronegative regime. That is,overall, the simulated effective Townsend first ionisation coefficients are too positive, suggestingthat an increase in the magnitude of the proposed attachment cross section is required, as wellas a decrease in magnitude of the proposed ionisation cross section. Figure 10(c) comparesthe longitudinal diffusion coefficients, and shows the poorest agreement between simulation andexperiment across all of the considered transport coefficients with the simulated diffusion coefficientsconsistently underestimating the measurements. Fortunately, the general shape of the simulateddiffusion coefficients appears to be in fair qualitative agreement with experiment.
5. Refined electron-THFA cross sections
Given the results in Figure 10, we now employ the neural network model, Eq. (1), to solve theinverse problem of mapping from our admixture swarm measurements to a selection of desiredelectron-THFA cross sections. Specifically, we choose to fit the neutral dissociation cross section,electron attachment cross section, electron impact ionisation cross section and quasielastic MTCS.By replacing those cross sections in our earlier proposed set, with those predicted by the neuralnetwork, we complete our refined set of electron-THFA cross sections.
As very little can be stated about the nature of neutral dissociation in THFA, we choose to place noexplicit constraints on the neutral dissociation cross section when training the neural network, thusforming training data, using Eq. (4), to directly combine random pairs of excitation cross sectionsfrom the LXCat project. The resulting confidence band of training examples is plotted in Figure 11,alongside the subsequent refined fit provided by the neural network, as well as the original proposedcross section for comparison. Compared to that which was proposed originally, the neural networkpredicts a neutral dissociation cross section that is substantially smaller in magnitude, peaking at . × − m versus . × − m , while also having a smaller threshold energy, with a value of3.96 eV versus 18.4 eV. Promisingly, the neural network has also resolved a plausible high-energytail — a feature that is lost when determining neural dissociation as a residual of the grand TCS. n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Figure 11.
Neural network regression results for the THFA neutral dissociation cross section,alongside the earlier proposed ICS for comparison. See also legend in figure.
Figure 12.
Neural network regression results for the THFA dissociative electron attachmentcross section, alongside the earlier proposed ICS for comparison. See also legend in figure.
Due to the current lack of electron-THFA attachment data in the literature, the attachment crosssections used for training the neural network are chosen in a similarly unconstrained fashion asthose used to refine neutral dissociation. That is, Eq. (4) is used to combine random pairs ofLXCat attachment cross sections for training, while enforcing no additional explicit cross sectionconstraints. The resulting confidence band of training examples is plotted in Figure 12, alongside the8
An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 13.
Neural network regression results for the THFA electron impact ionisation crosssection, alongside the earlier proposed ICS and the previous experimental and theoretical results[21, 79, 80] from which it was derived. See also legend in figure. refined fit provided by the neural network, as well as the original proposed attachment cross section.The neural network predicts an attachment cross section that is more uniform than that which wasproposed initially, thus resulting in some substantial differences in different energy regimes. Forexample at low energies, near − eV , the refinement is over two orders of magnitude smallerthan the initial proposal. At intermediate energies, around 3 eV, the refinement rises slightly inmagnitude while the proposal drops significantly, resulting in the refinement exceeding the proposalby almost an order of magnitude. Both the refinement and the initial proposal have a peak near6 eV, although the refined attachment cross section has a peak magnitude that is almost a factorof 4 smaller than that for the initial proposal. At higher energies, past this peak, both attachmentcross sections decay fairly rapidly in magnitude, with practically no attachment beyond 12 eV inthe refinement, compared to 10 eV for the original proposed data. Given the general agreement among the ionisation cross sections reviewed thus far, we choose toconstrain the ionisation training examples to within the vicinity of our resulting initial proposal.We make these constraints particularly stringent at higher energies, where the theoretical resultsare expected to be more accurate, and where our swarm analysis is expected to be less informative.Across all training examples we use the same threshold energy of 9.69 eV, as was done for our initialproposed ionisation cross section. Consequently, the ionisation training cross sections are formedusing a formula very similar to Eq. (4): σ ( ε ) = σ − r ( ε )1 ( ε ) σ r ( ε )2 ( ε + ε − .
69 eV) , (10)where σ ( ε ) is the ionisation cross section used for training, σ ( ε ) is our initial proposed ionisationcross section, σ ( ε ) is a randomly chosen LXCat ionisation cross section, ε is that cross section’s n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Figure 14.
Neural network regression results for the THFA quasielastic MTCS, alongside theearlier proposed MTCS for comparison. See also legend in figure. corresponding threshold energy, and we also introduce an energy-dependent mixing ratio that variesfrom . to . as the energy varies from the ionisation threshold to eV : r ( ε ) = 0 . (cid:0) ε eV (cid:1) ln (cid:0) .
69 eV10 eV (cid:1) . (11)The resulting constrained confidence band of training examples is plotted in Figure 13, alongsidethe refined fit provided by the neural network, as well as the original proposed ionisation crosssection. As was expected, the refined ionisation cross section predicted by the neural network issmaller in magnitude, peaking at . × − m compared to × − m for what we proposedinitially. Nonetheless, such a large drop in magnitude ( ∼ . times) is a little concerning giventhe reputation of Bull et al . [80] group. However, THFA is a very difficult molecule to work withexperimentally, so such a mismatch may indeed be possible in this case. As with the ionisation cross section, we expect our proposed quasielastic MTCS to be most accurateat higher energies. As such, we proceed similarly to our approach with ionisation and sample eachquasielastic MTCS for training using the following formula: σ ( ε ) = σ − r ( ε )1 ( ε ) σ r ( ε )2 ( ε ) , (12)where σ ( ε ) is the MTCS cross section used for training, σ ( ε ) is our initial proposed quasielasticMTCS, σ ( ε ) is a randomly chosen LXCat elastic cross section, and we define here the energy-dependent mixing ratio that varies from . to . to . as the energy varies from − eV to0 An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 15.
Neural network regression results for the THFA grand TCS, alongside the originalproposed TCS for comparison. See also legend in figure. to eV : r ( ε ) = 0 . −
173 ln ( ε ) ln (cid:16)
103 eV1 eV (cid:17) , − eV ≤ ε ≤ , − ln ( ε ) ln (cid:16)
103 eV1 eV (cid:17) , ≤ ε ≤ eV . (13)The resulting constrained confidence band of training examples is plotted in Figure 14, alongside therefined fit provided by the neural network, as well as the original proposed quasielastic MTCS. Therefined quasielastic MTCS provided by the neural network is essentially identical to the proposalat energies above 1 eV. Below 1 eV, however, the neural network predicts quasielastic MTCS thatis significantly smaller than the proposal, steadily decreasing in relative magnitude as energy isdecreased. The greatest difference occurs at − eV , where the refined quasielastic MTCS isroughly two orders of magnitude smaller than its counterpart from our initial proposed set. Although it is not considered explicitly, the grand TCS of each cross section set used for training isnaturally affected by the aforementioned constraints placed on the neutral dissociation, attachment,and ionisation cross sections. The resulting constrained confidence band of training examples isplotted in Figure 15, alongside the refined fit provided by the neural network, as well as the originalproposed grand TCS. The neural network predictions result in a grand TCS that is smaller thanthe initial proposal for energies above 10 eV or so, while coinciding below this energy. The greatestdifference arises at 90 eV with a reduction in magnitude of 43%, most of which being due to thereduction in the ionisation cross section (see Figure 13). n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Figure 16. (a) simulated drift velocities from both our original proposed data base and ourrefined data base, compared to corresponding results from our admixture swarm measurements.(b) corresponding percentage errors in the simulated values relative to the swarm measurements.See also legends in figures.
6. Transport coefficients of the refined electron-THFA cross section set
Using our refined set of electron-THFA cross sections, we plot revised simulated transportcoefficients in Figures 16–18 for comparison to both our admixture swarm measurements, as wellas to the transport coefficients calculated previously for our initial proposed cross section set.Respectively, the drift velocities, effective Townsend first ionisation coefficients and longitudinaldiffusion coefficients are each plotted in Figures 16(a), 17(a), and 18(a), with correspondingpercentage error differences plotted in Figures 16(b), 17(b), and 18(b). Figure 16 shows that therefined set of electron-THFA cross sections has, in general, brought the simulated drift velocities2
An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 17. (a) simulated effective Townsend first ionisation coefficients from both our originalproposed data base and our refined data base, compared to corresponding results from ouradmixture swarm measurements. (b) corresponding percentage errors in the simulated valuesrelative to the swarm measurements. See also legends in figures. closer to the results from the experimental measurements. There are some instances where themismatch has increased, but these are infrequent. The large discrepancies between simulation andexperiment remain at low fields, with the refinement having very little effect on the drift velocitiesin this regime. Figure 17 shows a substantial improvement in the effective Townsend first ionisationcoefficients after the cross-section refinement. Overall, both the relative error and the shape ofthe simulated effective Townsend first ionisation coefficients have improved, with the most benefitseen in the electropositive regime. Figure 18 shows a slight worsening in the accuracy of thesimulated longitudinal diffusion coefficients after the neural network refinement, contrary to theother transport coefficients. It should be noted, however, that the shape of the plotted longitudinaldiffusion coefficients has appeared to improve slightly with the refined cross section data set. n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Figure 18. (a) simulated longitudinal diffusion coefficients from both our original proposed database and our refined data base, compared to corresponding results from our admixture swarmmeasurements. (b) corresponding percentage errors in the simulated values relative to the swarmmeasurements. See also legends in figures.
Figure 19 shows the results of using our refined cross section set to simulate electron swarm transportcoefficients in pure gaseous THFA at 300 K, across a range of reduced electric fields from 0.001Td up to 10,000 Td. We employ our multi-term Boltzmann solver here also, but find the two-termapproximation to be fairly sufficient for all but diffusion at the highest
E/n considered, which canbe in error by up to 59% in the case of the bulk longitudinal diffusion coefficient. Figure 19(a)shows the mean energy of the electron swarm, alongside that for the background THFA vapour forcomparison. In the low-field regime, the mean electron energy is ∼
26 meV , which is substantiallylower than the thermal background of ∼
39 meV due primarily to attachment cooling. As
E/n increases, heating due to the field increases the mean energy of the swarm to eventually reach4 An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Figure 19.
Calculated mean electron energy, (a), rate coefficients, (b), drift velocities, (c), anddiffusion coefficients, (d), for electrons in pure THFA vapour at 300 K over a range of reducedelectric fields. See also the legends for further details. thermal equilibrium with the background THFA at ∼ .
175 Td . Beyond this
E/n , the meanenergy continues increasing monotonically, with its ascent occasionally slowing due to the onsetof the excitation channels (around 0.5 Td) and the ionisation channel (around 200 Td). Figure19(b) shows rate coefficients for quasielastic momentum transfer, summed excitation, attachmentand ionisation. The quasielastic momentum transfer rate coefficient remains fairly constant upuntil 30 Td, after which it increases slightly, likely due to the maximum in the magnitude of thequasielastic MTCS at ∼ .
18 eV . The summed excitation rate coefficient starts off constant at verylow fields, but begins to increase, rather early, from × − Td due to the additional opening of thevibrational excitation channels. This increase continues monotonically until reaching a maximumat ∼ , after which the summed excitation rate decreases slightly at very high fields. Theattachment rate coefficient also starts off as being constant, before increasing monotonically to apeak at ∼
350 Td , likely due to the associated peaks in the attachment cross section magnitude at ∼ . and ∼ . . As there is no appreciable attachment cross section beyond ∼
12 eV , thereis a corresponding drop in the attachment rate coefficient from this point onward. The ionisationrate coefficient is zero at low fields before increasing monotonically and becoming appreciable from n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
E/n . At low fields, the flux driftvelocity slightly exceeds the bulk, indicating that electrons are being preferentially attached at thefront of the swarm and so shifting the centre of mass towards the rear. At intermediate fields,between roughly 1 Td and 300 Td, nonconservative effects are sufficiently small such that the bulkand flux drift velocities coincide. Above 300 Td, the bulk drift velocity now exceeds the flux, dueto ionisation preferentially creating electrons at the front of the swarm. Figure 19(d) shows thebulk and flux diffusion coefficients of the swarm in both the longitudinal and transverse directionsrelative to the field. As with the drift velocities, for intermediate E/n between roughly 1 Td and300 Td, nonconservative effects are minimal and the bulk and flux diffusion coefficients coincide.The transverse diffusion coefficient differs the least between its flux and bulk counterparts. At lowfields, below 1 Td, there is only a very slight decrease in the bulk transverse diffusion compared toits flux counterpart, which we attribute to the slight preferential attachment of electrons toward thesides of the swarm. At high fields, above 300 Td, there is a substantial increase in bulk transversediffusion over the flux, which we naturally attribute to the preferential ionisation production ofelectrons toward the swarm sides. The bulk longitudinal coefficient follows a similar trend, forthese same reasons of preferential attachment and ionisation in the longitudinal direction.
7. Conclusion
We have formed a complete and self-consistent set of electron-THFA cross sections, by constructingan initially-proposed set from the literature and then refining its least-certain aspects by measuringand analysing electron swarm transport coefficients in admixtures of THFA in argon. Notably, thisswarm analysis and cross section refinement was performed automatically and objectively using aneural network model, Eq. (1), trained on cross sections from the LXCat project [41–43]. Ourneural network determined plausible cross sections for attachment and neutral dissociation, in theirentirety, from the measured swarm data, as well as cross sections for ionisation and quasielasticmomentum transfer subject to constraints given by known experimental error bars. We subsequentlyused our Boltzmann equation solver to calculate transport coefficients for this refined cross sectionset, and found an improved consistency with our experimental admixture measurements. We alsocalculated transport coefficients for electrons in pure THFA, revealing the interesting phenomenonof attachment cooling of the electron swarm below the thermal background.Given the similar methodology and swarm measurements between the present investigationand our previous successful refinement of an electron-THF cross section set [17], we believe ourrefined set of electron-THFA cross sections should be of comparable quality, if not a little better, toone hand-fitted by an expert. As there is evidently still some room for improvement, it is fortunatethat this machine learning approach makes it straightforward to revisit THFA as new swarm data,cross section constraints, or LXCat training data becomes available.A known limitation [17, 39] of the present machine learning approach to swarm analysis is thatit provides a unique solution to a problem for which multiple plausible solutions are likely to exist.In the future, we intend to address this deficiency by quantifying the uncertainty in the predictedcross sections using a suitable alternative machine learning model [100–105]. Finally, we also planto apply our machine learning approach to determine complete and self-consistent cross section setsfor other molecules of biological interest, including those for water [6].6
An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Acknowledgments
Thanks are due to A. Bustos and G. Bustos for their technical assistance. The authors gratefullyacknowledge the financial support of the Australian Research Council through the DiscoveryProjects Scheme (Grant
Data Availability Statement
The data that supports the findings of this study are available within the article.
Appendix A. Pulsed-Townsend electron swarm measurement results for variousadmixtures of THFA in argon n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Table A1.
Measured pulsed-Townsend electron swarm transport coefficients for a . admixture of THFA in argon. Estimated experimental uncertainties are ± for drift velocities, W , ± for effective Townsend first ionisation coefficients, α eff /n , and ± for longitudinaldiffusion coefficients, n D L . E/n [Td] W (cid:2) m s − (cid:3) α eff /n (cid:2) − m (cid:3) n D L (cid:2) m − s − (cid:3) An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Table A2.
Measured pulsed-Townsend electron swarm transport coefficients for a . admixture of THFA in argon. Estimated experimental uncertainties are ± for drift velocities, W , ± for effective Townsend first ionisation coefficients, α eff /n , and ± for longitudinaldiffusion coefficients, n D L . E/n [Td] W (cid:2) m s − (cid:3) α eff /n (cid:2) − m (cid:3) n D L (cid:2) m − s − (cid:3) n improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data Table A3.
Measured pulsed-Townsend electron swarm transport coefficients for a admixtureof THFA in argon. Estimated experimental uncertainties are ± for drift velocities, W , ± for effective Townsend first ionisation coefficients, α eff /n , and ± for longitudinal diffusioncoefficients, n D L . E/n [Td] W (cid:2) m s − (cid:3) α eff /n (cid:2) − m (cid:3) n D L (cid:2) m − s − (cid:3) An improved set of electron-THFA cross sections refined through a neural network-based analysis of swarm data
Table A4.
Measured pulsed-Townsend electron swarm transport coefficients for a admixtureof THFA in argon. Estimated experimental uncertainties are ± for drift velocities, W , ± for effective Townsend first ionisation coefficients, α eff /n , and ± for longitudinal diffusioncoefficients, n D L . E/n [Td] W (cid:2) m s − (cid:3) α eff /n (cid:2) − m (cid:3) n D L (cid:2) m − s − (cid:3)
10 2.55 -13.0 2.4512 2.48 -12.8 2.5714 2.49 -12.7 2.7316 2.53 -11.3 2.9318 2.59 -9.13 3.0320 2.68 -8.29 3.0623 2.83 -4.70 3.2026 2.99 -1.81 3.3530 3.23 7.66 3.4633 3.42 14.6 3.4836 3.58 22.1 3.4940 3.84 36.0 3.5545 4.16 59.6 3.6450 4.48 92.1 3.7155 4.79 132 3.7960 5.15 177 3.9870 5.79 297 3.9280 6.52 449 3.9490 7.01 556 3.89100 7.90 661 4.10120 9.08 950 4.44140 10.6 1200 4.40160 12.1 1660 4.72180 13.4 1930 5.08200 15.0 2470 4.79230 17.8 2860 5.63260 19.4 3340 5.06300 23.6330 26.0360 28.7400 32.1450 34.5
EFERENCES Table A5.
Measured pulsed-Townsend electron swarm transport coefficients for a admixtureof THFA in argon. Estimated experimental uncertainties are ± for drift velocities, W , ± for effective Townsend first ionisation coefficients, α eff /n , and ± for longitudinal diffusioncoefficients, n D L . E/n [Td] W (cid:2) m s − (cid:3) α eff /n (cid:2) − m (cid:3) n D L (cid:2) m − s − (cid:3)
26 3.89 -21.4 2.8830 4.03 -9.54 2.9933 4.16 -9.47 3.0136 4.29 -2.91 3.0940 4.55 3.40 3.2145 4.84 15.7 3.3650 5.11 30.6 3.4155 5.40 63.0 3.4060 5.73 99.1 3.3770 6.30 178 3.4880 6.88 306 3.6990 7.48 438 3.97100 8.10 582 4.10120 9.49 918 4.25140 10.9 1260 4.82160 12.4 1630 5.07180 13.5 2100 5.14200 15.3 2350 4.75230 18.1 2770 5.57260 19.8 3330 5.07330 26.5
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