Benchmark calculations for anisotropic scattering in kinetic models for low temperature plasma

Benchmark calculations are reported for anisotropic scattering in Boltzmann equation solvers and Monte Carlo collisional models of electron swarms in gases. The work focuses on isotropic, forward, and screened Coulomb models for angular scattering in electron-neutral collisions. The impact of scattering on electron swarm parameters is demonstrated in both conservative and non-conservative model atoms. The practical implementation of anisotropic scattering in the kinetic models is discussed.


Introduction
Modern low temperature plasma (LTP) studies are increasingly interested in quantifying the effects of various electroncollision scattering assumptions and models.Some subjects which have paid special attention to anisotropic scattering include electron scattering in liquids [1][2][3], glow discharges [4][5][6][7], capacitively coupled plasma [8], thunderstorms [9], runaway electrons [10] and strong field gradients moregenerally [11].The strong-field studies of the latter three subjects are especially concerned with the expected increasingly forward electron scattering of fast electrons, where subtle differences between even the most realistic models can form * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.dramatically hotter or cooler swarms.Studies of transport even for low field strengths are likewise concerned with the complex anisotropy of real gas cross-sections, such as that associated with the Ramsauer-Townsend minimum of Ar near 0.2 eV.
From the comments of Janssenn et al [12], Park et al [13], and Phelps and Pitchford [14], it is historically common, and sometimes even recommended [12], for researchers to partially capture anisotropic scattering effects by assuming that elastic collisions scatter isotropically according to data of the elastic momentum transfer cross-section.This is the treatment utilized by the publicly available LTP kinetic models BOLSIG+ [15], the LoKI suite (LoKI-B [16] and LoKI-MC [17]), and early versions of MultiBolt [18].This treatment is an approximation which circumvents the consideration of complicated tabulated differential scattering data or qualitative models.
At the same time, several studies have shown that a more-realistic anisotropic treatment of scattering in real gases will yield non-negligible differences in the macroscopic characteristics of an LTP.In an early study, Phelps and Pitchford demonstrated that, compared to the simplistic treatment, a complete treatment of anisotropic electron scattering in N 2 affected calculations of reaction and transport coefficients by up to 10% [14].Various approaches to anisotropic scattering have been proposed over the years (for examples, see [12]), and the proper treatment of anisotropic scattering continues to be an active area of research even today.For instance, the use of an anisotropic scattering model based on a dipole-Born approximation for rotational collisions was recently shown to significantly impact calculated electron transport coefficients in CO [19] and H 2 O [20].
Despite a wealth of literature and prior work on the topic of isotropic and anisotropic scattering, the assumption of isotropic scattering of electrons remains the default choice for most of the LTP modeling community.Monte Carlo (MC) collisional models, where the transport parameters, reaction rates, and average energy of an electron swarm are found stochastically by following an ensemble of electrons, sometimes include anisotropic scattering models [4,7,21].We also note that a few publicly available MC solvers allow for some form of anisotropic scattering [22,23].Anisotropic scattering in a Boltzmann equation (BE) solver, which solves for the same electron swarm parameters as an MC model deterministically instead, is also possible but less commonly reported [24][25][26].
One of the key factors contributing to the limited use of anisotropic scattering in LTP models is the lack of available benchmark data.At present, very limited data exists that would allow researchers to benchmark an electron kinetic model with anisotropic electron scattering.We note that some benchmark calculations for simplified energy-independent models are available from the works of Reid [24], Haddad et al [25], and Leyh et al [26].Recent works that report the effects of anisotropic scattering in real gases [13,19,20] are too complex to serve as effective benchmarks for the wider community because they require the simultaneous correct implementation of other physics, such as superelastic collisions.
This work aims to develop benchmark data for kinetic models employing anisotropic scattering to enable other researchers to benchmark their treatment of anisotropic electron scattering in electron kinetic models.To achieve this, we use the popular Reid ramp (RR) [24] and Lucas-Saelee (LS) [27] model atoms with three popular electron scattering models: isotropic scattering (ISO), forward scattering (FWD), or screened Coulomb (SC) scattering.Three independently developed electron kinetic models (MultiBolt [18,28], CAPS [29,30], and ELIOS [31,32]), are used to calculate benchmark data for these new benchmark cases.
This manuscript is organized as follows.First, in section 2, definitions relevant to scattering are introduced.The definitions of electron swarm parameters used as benchmark values and the specific expressions for ISO, FWD, and SC scattering models are also given.The codes MB, CAPS, and ELIOS, which are used to calculate the benchmark values, are also described.In section 3, the benchmark values calculated by the codes are reported.Finally, concluding remarks on the need for the ability to model anisotropic scattering are given in section 4. Additional notes on the influence of scattering on the collision operator of a BE solver are given in appendix A.

Background
In this section, the definitions of cross-sections and relevant scattering model expressions are given.The electron swarm parameters (including flux and bulk transport coefficients) reported as benchmark values are defined.The model gas cross-sections used to create the benchmark conditions are given.Finally, the codes used to calculate the benchmark values are described.

Differential and integrated cross-sections
The differential cross-section (DCS, σ(ε, Ω)) describes the probability of scattering an incident electron with kinetic energy ε = 1/2m e ∥⃗ v∥ 2 (in units of eV, m e : electron rest mass) into a solid angle element of dΩ = sin(χ)dχ dϕ.For an electron initially moving in the ẑ direction, χ = [0,π] describes the angle between the velocity vector of the scattered electron ⃗ v ′ and the z-axis, and ϕ = [0,2π] is the azimuthal angle [33].A diagram of the scattering system relevant for this work is given in figure 1.The DCS assumes isotropic azimuthal scattering such that, for this work, σ(ε, The change in electron kinetic energy following a collision is given as ε n such that the post collision electron is given as ε ′ = ε − ε n .For excitation or ionization collisions, ε n is the threshold energy for the collision (i.e. the excitation or ionization energy), and for elastic collisions, ε n /ε ≈ 2m e /M(1 − cos(χ)), where M is the mass of the neutral particle.
The total cross-section (TCS, σ(ε)) may be found from the DCS via integration over all possible scattering angles For convenience, the angular dependence of the DCS can be represented by a normalized angular scattering function I(ε, χ) The momentum transfer cross-section (MTCS, σ m (ε)), is defined as the following expression [34] Note that for elastic collisions, where m e /M << 1 and ε n is small, (4) can be approximated by the more typical σ m (ε) = 2π ´π 0 (1 − cos(χ)) σ(ε, χ) sin(χ)dχ.Here, we note that due to the integration of the DCS with the (1 − cos(χ)) term, the MTCS naturally includes some information regarding the effects of anisotropic scattering.For this reason, in a typical plasma model, isotropic scattering of elastic collisions using the MTCS results in a reasonable representation of the effects of anisotropic scattering.The simplicity of this approach is clear, but we again emphasize the importance of a more complex treatment of anisotropic scattering for both elastic and inelastic collisions as has been highlighted by previous authors [14].Often, only the MTCS is reported for elastic collisions while TCSs are given for inelastic collisions.This is the case for complete cross-section sets on the publicly available LXCat database [35].The MTCS and TCS for a certain collision are identical under the condition of isotropic electron scattering but may differ when electron scattering is anisotropic.They are always related by the following equation ( Finally, we wish to emphasize that to properly model anisotropic scattering, for a given elastic MTCS, one may either use DCS data where available or, with known MTCS data, assume some form of I(ε, χ), which can then be used to find the TCS, but in this case the MTCS must remain fixed.

Implementation of scattering
2.2.1.Anisotropic scattering in a Monte Carlo model.Monte Carlo (MC) electron kinetic models employ a stochastic approach of tracking a large number of electrons through a large number of collisions to directly predict the macroscopic properties of the electron swarm.Cross-sections are employed as input data that are compared against random numbers to determine if an electron undergoes a collision, and if so, what type of collision has occurred.When a collision occurs, two additional random numbers determine the azimuthal scattering angle, ϕ, which is uniformly distributed from 0 to 2π, and χ, which is related to a probability distribution function Generally, χ is determined by inverting the above expression such that cos(χ) = g(r), where g(r) is some function of a random number, r, uniformly distributed over [0,1].Specific examples of g(r) for specific scattering models are given in section 2.3.

Anisotropic scattering in a Boltzmann equation
model.In a BE electron kinetic model, the electron phase space F(⃗ r,⃗ v, t) is calculated via a deterministic solution to the Boltzmann equation Here, ⃗ r, ⃗ v, ⃗ a, and t are the position, velocity, and acceleration vectors and time, respectively.As is common in BE models, we employ a spatial gradient expansion to the electron density (up to second-order), and the velocity space is represented by an N ℓ truncated series of spherical harmonics, Y m ℓ (θ, ϕ) = e jmϕ P m ℓ [cos(θ)], where P m ℓ [cos(θ)] are the associated Legendre polynomials as a function of the velocity-space anisotropy [36,37].The orthogonality of the Legendre polynomial components allows the BE to be restructured into a hierarchy of the ℓ th distribution functions (i.e.Legendre polynomial coefficients) f ℓ .The lowest-order distribution, f 0 , represents the isotropic portion of the phase space (i.e. the electron energy distribution function (EEDF)).The effect of collisions on the electron phase space is accounted for by the collision operator, C [F], which in the spherical harmonics expansion of the phase space takes the form of C [f ℓ ], as follows Above, R in and R out represent the number of particles scattering in and out of a space d⃗ r • d⃗ v in a time dt respectively, and n e is the electron number density.Expressions of R in and R out for most collisions take the following form [33] Above, N is the target particle number density.
The solid angle integral in (9a) yields the same as what is defined as the 'partial cross-section' by Phelps and Pitchford [14] and employed by others [19,20].Equation (9a) can be further decomposed using the relation σ(ε, χ) = σ(ε)I(ε, χ), to yield an expression for partial angular scattering and appropriate scattering-in expression per ℓ As a result, anisotropic scattering can be treated in a BE model as long as I ℓ can be calculated.Because P 0 [cos(χ)] = 1, the evaluation of ( 10) for ℓ = 0 is identical to the normalization condition, and I 0 = 1 always.
For convenience, we also note that for a spherical harmonics-based multi-term BE model, ( 5) can be combined with (10) and can be simplified to the following

Angular scattering models
In this section, the three normalized angular scattering functions, I(ε, χ), considered in this work are defined.Expressions for the implementation of each scattering model are given for both MC and BE models.

Isotropic scattering. Isotropic scattering (ISO) means
that the electron is equally likely to scatter in any direction, trivially expressed mathematically as the following For an MC model, ISO scattering is implemented by using a uniformly distributed random number, r ∈ [0, 1], with the following expression, which determines the scattering angle of an electron following a collision For a BE model, note first that Legendre polynomials obey the orthogonality relation Above, δ mn is the Kronecker delta.By observation, 1/(4π) = 1/(4π)P 0 [cos(χ)].The orthogonality relation using x ≡ cos(χ) allows ( 10) to be evaluated exactly In this case the MTCS is identical to the TCS because When applied to all collisions, (15) above yields the collision operator seen in equation ( 17) of Stephens [18].

Forward scattering.
Forward scattering (FWD) assumes that in the event of a collision, the electron energy is decremented but the trajectory is unchanged [12].That is, χ = 0, always, resulting in the following normalized angular scattering function Above, δ(χ) is the Dirac delta function.The denominator contains sin(χ) so that I(ε, χ) obeys (3).
For FWD scattering, an MC model may employ the following relation.Note that a random number is no longer needed to determine the scattering angle cos (χ) = 1. ( Because P ℓ [cos(0)] = 1 for all ℓ, the collision term for the BE model reduces to the following, which is independent of both ℓ and ε In this case, both R in and R out will be nonzero in the collision operator for all ℓ.This differs from the ISO model, where R in is nonzero only for ℓ = 0.
The ratio between the MTCS and TCS follows In this case, the elastic TCS cannot be discerned from the MTCS because σ m (ε)/σ(ε) = 0.This means that forward scattering is nonphysical for elastic collisions in this study.

Screened Coulomb scattering.
Realistic scattering models are typically expected to be more isotropic for slow electrons but more forward for faster electrons.This behavior has historically been fulfilled by various screened Coulomb models derived from the Born approximation, particularly that of Vahedi and Surendra [5,38], and various expansions thereupon.For a detailed review of multiple screened Coulomb-like models, the work of Janssenn et al [12] and Park et al [13] is recommended.For simplicity of reporting benchmark calculations, only the screened Coulomb model of Okhrimovskyy et al [39] is chosen for consideration in this work and referred to as SC.The form of SC scattering applied here is a symmetric function of a fitting parameter ξ, typically a function of non-dimensional electron energy The use of the typical screened Coulomb potential (see [39]) yields the following screening parameter Figure 2. Normalized angular scattering of the typical SC model using (22) for fixed evaluations of ε ′ /E S .The value of I(ε, χ) for ISO scattering is given for comparison.
Above, E S is the screening energy, which is taken to be one Hartree, E S = 27.21eV.The angular dependence for SC scattering using ( 22) is given in figure 2. Note that the post-collision energy is used to evaluate the SC model here.
Evaluating the SC model for inelastic collisions using the precollision energy is less physically realistic and may significantly affect results [5,38].If the MTCS and TCS of an elastic collision are both known, ξ may be determined using (26), instead of ( 22).An MC model may apply SC scattering with the following function For a BE model, the evaluation of I ℓ for the first two terms is relatively simple It is cumbersome to evaluate this expression for an arbitrary number of higher ℓ.Higher order terms may instead be evaluated numerically at all energies via integration on a grid of χ.Evaluations for the first 10 terms of ℓ for SC scattering using (22) are given in figure 3.For higher ℓ, the value of I ℓ (ε ′ ) becomes negligible for low ε ′ /E S , which is to say that lower energy electrons do not couple into higher-order spherical harmonics terms.In all SC cases, the MTCS and TCS are related using (12).For elastic (ε n ≈ 0 eV) SC scattering, from Okhrimovskyy et al [39], this ratio obeys the following expression Figure 3. Partial angular scattering functions for the typical SC model using (22) Curves are evaluated numerically per ε ′ /E S for the first ten, ℓth order Legendre terms.
For the form of ξ given by ( 22), the value of this ratio is near 1 (nearly ISO) for small values of ε ′ /E S and approaches zero (nearly FWD) for higher ε ′ /E S .

Model atoms
In this section, the two model atoms used to conduct benchmark calculations are described.In each model atom, the benchmark conditions for comparing scattering results are defined by the combinations of scattering models used for elastic and inelastic collisions.The four cases are ISO-ISO, ISO-FWD, ISO-SC, and SC-SC.Acronyms denote the scattering model; the first acronym denotes the model given to elastic collisions and the second acronym denotes the model applied to inelastic (including ionization) collisions.That is, ISO-ISO denotes that isotropic scattering was applied to both elastic and inelastic collisions, while ISO-SC would indicate that isotropic scattering was employed for elastic collisions, while screened Coulomb scattering was employed for inelastic collisions.[24] is a conservative model atom with one constant elastic MTCS σ RR m and one linearly-increasing excitation TCS σ RR exc with an excitation threshold energy of ε n = 0.2 eV

Reid ramp model. The Reid ramp (RR) model
The mass of the neutral is 4 AMU.The temperature of the neutrals is fixed as T g = 0 K.The elastic TCS for the RR model, as is relevant for SC scattering, is given in figure 4.

2.4.2.
Lucas-Saelee model.The Lucas-Saelee (LS) model [27] is a non-conservative model atom.For this work, we assume F = 1, such that all the model atom consists of only The ratio between the mass of the electron and the neutral is m e /M = 10 −3 .The temperature of the neutrals is, again, T g = 0 K.The attachment modification of Ness and Robson [37] is not used here.The scattering models for inelastic collisions are applied to ionization.The elastic TCS relevant for SC scattering is given in figure 4.

Electron swarm parameters
For all calculations made with a certain density-reduced electric field strength E/N (in units Td = 10 −21 Vm 2 ), the benchmark calculations are given as the mean electron kinetic energy ⟨ε⟩, the electron drift velocity W F , and the densityproduct diffusion coefficients ND F T and ND F L for electron transport transverse (T) and longitudinal (L) to the electric field, respectively.These refer to the 'flux' coefficients for the drift and diffusion of individual electrons in the swarm as driven by the field [40][41][42].
When non-conservative collisions (ionization or attachment) exist, the 'bulk' transport coefficients of the electron swarm's center of mass will differ from the flux coefficients.The bulk transport coefficients W B , ND B T , and ND B L are reported for the non-conservative model gas alongside the densityreduced ionization reaction rate coefficient k iz /N.
For information on how each parameter is calculated by a particular code, refer to section 2.6 and the relevant citations.

Code description
In this section, the main features of the codes used to calculate the benchmark values are discussed.Notes are given on any approximations employed or numerical parameters.MultiBolt (MB), at the time of this writing v3.1.2,is an open source multi-term Boltzmann equation solver [18,28].Earlier implementations of MB [18] restricted the BE to ISO scattering for all collisions.The current version of MB allows the use of arbitrary scattering models as well as the popular ISO, FWD, and SC models using the collision operator equations given in appendix A. For this work, MultiBolt calculates swarm parameters using the equations given in section 2.1 of Stephens [18] for 'hydrodynamic' conditions.
A high resolution (10 000 points) linear grid of electron energies is chosen per calculation such that at least 15 orders of magnitude are captured in f 0 .Model cross-sections are evaluated functionally at all electron energies with no interpolation error.The outgoing angles of the scattered and secondary electrons from an ionization are treated as uncorrelated.
For ISO and FWD models, the expressions ( 15) and ( 19) are used for all ℓ.For SC models, the value of ( 24), (25), etc for each ℓ are solved numerically for each ε via trapezoidal integration.A linearly spaced grid of χ between 0 and π, of 500 grid-points, is used to capture the angular curvature in the integration of (10).
The MB set of benchmark calculations provides values for both N ℓ = 2 (the two-term approximation) and N ℓ = 10 terms.Note that only multi-term (N ℓ > 2) BE calculations can be expected to agree well with MC calculations.
All computation times reported in this section were completed using an Intel i7-8700 3.19 GHz processor.For the RR model atom, calculations completed within 1 and 5 min for the two-term and ten-term cases, respectively.Solutions for non-conservative swarms require more expensive iterative solutions such that, for the LS model atom, computation times were instead up to 15 and 20 min for the N ℓ = 2 and N ℓ = 10 solutions, respectively.It is worth noting that the 10 000 point grid was used here to report reliable benchmark values, but acceptable results were achieved with only 1000 points; RR and LS calculations, in this case, took no more than 5 and 10 s, respectively, to conclude.

ELIOS. The ELectron and IOn Simulation (ELIOS)
code is an MC code for the simulation of electrons under swarm conditions.Recent work with ELIOS includes benchmarking against solutions of two-term and multi-term BE solvers and simulations in atomic and molecular gases using variance reduction techniques [31,32,43].
In this work, ELIOS employs a conventional MC simulation method (without variance reduction techniques) to calculate flux and bulk transport parameters.ELIOS uses a null-collision method [44], a modified time-step technique [45], and anisotropic scattering as implemented in the manner detailed in [19].Non-conservative collisions are treated with a dynamic list of particles until a maximum number of 10 7 particles is reached (that is, similar to the method used in [41]).
Several different statistically independent electron swarms (each containing 5 × 10 4 particles) are simulated until a steady state is achieved.The total number of swarms simulated in this way is determined within the code in order to minimize the relative error in electron flux and bulk transport parameters.Typically, a simulation of 10 swarms was sufficient to achieve relative errors in transport parameters calculations below 5%.
The time at which steady state is achieved, t ss , is determined when ⟨ε⟩ deviates between time steps within less than 0.1%.After t ss , bulk transport parameters are calculated by averaging outcomes over a sampling interval of 1 ns until a final time t fin = 10 × t ss is attained.For all the simulations in this work, an energy resolution of 10 −4 eV is used for the linear interpolation of cross-sections as a function of energy.The computational time for a single simulation was found to be up to 20 m on a 2.6 GHz Intel i7 single-core processor.

CAPS.
The CUDA assisted plasma simulation (CAPS) tool is a graphics processing unit (GPU) based MC code [29,30].It features on-chip parallelism for multithreaded performance and is able to use multiple GPUs in a single node.This enables the simulation of hundreds of millions of electrons in a reasonable amount of time.Simulations were conducted on a single GPU using the Xena cluster at the center for advanced research computing (CARC) at the University of New Mexico.
For the calculations of this work, the dynamic electric field solver is disabled and the field is set as a constant in a single direction.Electron velocities and positions are computed in 3D.Binary electron-neutral collisions are determined using the null-collision method proposed by Nanbu [46,47].
For calculations in the RR model atom, a total of 500 000 electrons are simulated over 10 ns.The cross-sections are linearly interpolated to a grid of electron energies from 0 eV to 1000 eV in 0.2 eV increments.The macroscopic transport coefficients are taken as an average over the last 5 ns when the average electron energy has reached steady-state.At the end of 10 ns for the ISO-ISO case, the average number of collisions per electronper 500 ps for 1 Td, 12 Td, and 24 Td was approximately 135.14, 270.16, and 402.73, respectively.This results in a total number of several hundred million collisions for the entire electron population over the 10 ns simulation.These simulations took on the order of 5 min of real-time to complete.
For calculations in the LS model atom, the cross-sections were linearly interpolated on a grid of electron energies from 0 eV to 100 eV in 0.02 eV.In this case the electron energy reaching steady-state was found to be an insufficient criterion for the convergence of transport quantities, and the simulation was run until the average number of collisions per particle reached steady-state instead.For 10 Td, typical simulation times before reaching steady-state in this way were on the order of 10 ns, 3 ns for 20 Td, and 1 ns for 30 Td. Transport coefficients were then obtained by averaging over the time steps where the average collision count for each electron per time interval was within 0.01% of the last value in the simulation.The difference in time taken to reach steady-state for the average energy and average number of collisions is demonstrated in figure 5 for an ISO-ISO test case of E/N = 10 Td for the LS model atom.The gray-shaded region indicates the region where averaging is used to obtain the macroscopic transport coefficients.For this case, each simulation took on the order of 1 to 2 h of real-time to reach steady-state.

Benchmark calculations of swarm parameters for cold and moderate swarms
The four combinations of angular scattering models given in section 2.4 are applied to the two model atoms.For the RR model atom, swarm parameters are calculated in field strengths of 1, 12, and 24 Td by the three codes, and results are given in table 1.For the LS model atom, calculations are made for the E/N = 30 Td and reported in table 2. This much larger than typical (compared to other LS benchmarks as reported in [18]) field strength was used for this work because the influence of anisotropic scattering was found to be, as one would expect, less pronounced at lower E/N.
For increasingly-forward scattering, the magnitudes of average energy, drift velocity, and diffusion coefficients increase under hotter swarms.These increases are most dramatic in the ISO-FWD case, while both the ISO-SC and SC-SC cases are more subtle.From table 1, for the 24 Td calculations using the RR model atom, changing the scattering model from ISO-ISO to ISO-FWD increased ⟨ε⟩, W F , ND F L and ND F T by 21%, 47% , 162% and 47% respectively.Differences between ISO-SC and SC-SC results for the same reduced field strength are not usually distinguishable, and for two-term BE  (10), the multi-term BE solver results, to the right of the swarm parameter and found using δ[%] = (1 − X/Y) • 100, where X are the results of a particular code and Y are the results of MB (10) 4).The ten-term MB results agree well with the results of both ELIOS and CAPS in all models.Compared to the MC codes, the error is largest only for bulk diffusion coefficients and is within 6%.Flux and bulk drift velocities typically agree within less than 1%.

Influence of term truncation on forward and screened Coulomb scattering
Each scattering model applied to a BE solver is subject to some amount of error due solely to the two-term approximation (i.e. the use of N ℓ = 2, as opposed to N ℓ = 10).The error solely due to term-truncation can become large, most typically, while swarms are inelastic collision-dominated (as is true particularly for the RR model) and at higher field strengths [48].From the present results, it is clear that the two-term approximation can also become stressed for strong anisotropic scattering.
For the RR model atom, the term-based error between MB(2) and MB (10) results is largest for diffusion coefficients in the RR model atom for E/N = 24 Td.In this case, the termbased error in transverse diffusion coefficients increases from +14% to +25% solely due to changing scattering model from ISO-ISO to ISO-FWD.For the longitudinal diffusion coefficients, term-based error also increases from −27% to −35%, again, solely due to the use of the forward scattering model as opposed to isotropic scattering.Term-based error in this way is smaller for the LS model atom.In this case, term-based error in transverse diffusion coefficients increases only from 5% to 7% for flux coefficients and from 8% to 10% for bulk coefficients.
From both tables, the results of SC-inclusive scattering models are not as-stressed by the two-term approximation.Additional stress upon the two-term approximation depends on the strength of forward-peaking in the model.Refer to appendix B for a more in-depth examination of the interaction between the two-term approximation and anisotropic scattering models.

Benchmark calculations of distribution functions for hot swarms
Angular scattering affects not just the value of electron swarm parameters, but also the shape of the EEDFs describing the swarm from which swarm parameters are derived.This section explores these different behaviors, especially as it relates to SC scattering for high field strengths.
EEDFs are calculated using MB (using N ℓ = 10 terms), ELIOS, and CAPS in the RR model atom at 100 Td and 1000 Td for all four scattering models and depicted for comparison in both figures 6 and 7.The value of ⟨ε⟩ for these swarms varies in the range ⟨ε⟩ ∈ [1, 1.7] eV for 100 Td and ⟨ε⟩ ∈ [5,17] eV for 1000 Td.In all cases, MB and ELIOS results agree well at all ranges of ε for which ELIOS data exists.Note that MC models usually cannot fulfill content in the tails of EEDFs for many orders of decay (that is, solving for f 0 for values below 10 −7 eV −3/2 ); for this reason MB data fulfills the tails for a wider ranges of ε than ELIOS or CAPS.As was seen in previous results, ISO-FWD results are dramatically distinguishable from the other models in all cases.
From both figures, both the head and tail of the ISO-SC and SC-SC results differ to small extents in the 100 Td case, and to greater degree in the 1000 Td case.In the latter, the tail of SC-inclusive EEDFs is larger than the ISO-ISO case by two orders of magnitude for ε > 20 eV.
Similar calculations are provided instead for the LS model atom in figure 8.In this case, similarly to the 100 Td case of figure 6, the heads of the EEDFs for ISO-SC and SC-SC calculations cannot be graphically distinguished from eachother.However, from the MB results, the tails do differ.For ε > 60 eV, the tail of the ISO-SC case is larger than that of the SC-SC case by nearly one order of magnitude.This result  is unique, in this work, in distinguishing ISO-SC and SC-SC results.Content in the tail of the EEDF for SC-SC results is lower than that of the ISO-SC results despite modeling moreforward scattering due to the large size of the model atom's elastic TCS at high ε.
With these results in mind, it is relevant to kinetic modelers that SC-scattering electron swarms must be very hot to distinguish results strongly from the isotropic scattering case, and that even hotter swarms are required to distinguish effects of SC elastic scattering.

Conclusion
Benchmark calculations for anisotropic scattering in LTP kinetic models have been reported.Definitions of cross-sections were reviewed, and the details regarding the implementation of anisotropic scattering in both BE and MC models were presented.
A method for implementing any anisotropic scattering model in both BE and MC models was demonstrated.This method can be applied to both analytic expressions and tabulated data of the normalized angular scattering function I(ε, χ).In the case of the latter, the evaluations of cos(χ) and I ℓ (ε) reduce to simple lookup tables.
The influence of anisotropic scattering on swarm parameter results and its necessity in LTP models is a complicated topic.The impacts of scattering are most apparent for collisions which both dominate in a certain E/N regime and are significantly anisotropic.From the results of [19], CO anisotropic scattering is very influential in calculations for E/N < 1 Td, but negligible otherwise.In contrast, anisotropic scattering in argon [12] or N 2 [14] is thought to only have a large influence at high E/N.This can be explained by the size and anisotropy of particular collisions dominating at certain E/N.In CO, the dipole rotational excitation cross-sections are very large at low ε and also have pronounced forward scattering, but for larger ε are quickly dominated by cross-sections for other processes such as quadrupole rotational and vibrational excitation that are typically treated with isotropic scattering.From [12] and the argon elastic DCS of the BSR database [49], argon elastic scattering is strongly anisotropic for ε > 10 eV, but is nearly isotropic for lower ε, which is most critical for moderate swarms where elastic scattering dominates.
From this, it is clear that anisotropic scattering may be relevant at either low, moderate, or high E/N, depending on the gas under study.The recent work of [20] on cross-sections in H 2 O found that the best consistency to experimental swarm parameters at moderate reduced electric fields, required the treatment of anisotropic scattering as given by the dipole-Born approximation.Not all LTP studies will require the consideration of anisotropic scattering.However, for cases where treatment of anisotropic scattering is necessary, this paper presents useful benchmark data for the development of accurate anisotropic scattering models in LTP conditions.solely for isotropic scattering.In both models, BE solver results converge acceptably for N ℓ = 4 terms; for increasing termnumber, results remain similar to N ℓ = 10.Term truncation error (the difference between 2-term and 10-term results for the same scattering model) is small and graphically indistinguishable for ⟨ε⟩ and W F but becomes large for diffusion coefficients beyond 3 Td.Divergence due to scattering model is largest for the transverse diffusion coefficient.The largest term-based error observed here increases from 28% to 50% by switching from the ISO-ISO model to ISO-FWD.Error also amplifies for the longitudinal diffusion coefficient, instead from 12% to 32%.

Figure 1 .
Figure 1.Model of the scattering system for this work.The model specifically depicts an example of an electron-neutral elastic collision.

Figure 4 .
Figure 4. Elastic collision cross-sections for the model atoms.Solid lines are the elastic MTCS.Dashed lines are the elastic TCS for the case of SC scattering.

Figure 5 .
Figure 5.The time difference, in CAPS, to reach steady state for the average energy and the average number of collisions per electron.Curves are for a CAPS simulation using the LS model gas for the ISO-ISO case for E/N = 10 Td.The gray region indicates where averaging occurred to calculate transport coefficients for this particular simulation.

Figure 7 .
Figure 7.The same as figure 6, but for the RR model atom using E/N = 1000 Td instead.

Figure 8 .
Figure8.The same as figure6, but using the LS model atom instead.In this case, ISO-SC and SC-SC results can be graphically distinguished.

Figure B1 .
Figure B1.Convergence of electron swarm parameters for isotropic and anisotropic scattering per N ℓ in BE calculations in the RR model atom.Dashed lines, solid lines, and (▽) markers are 2-term, 4-term, and 10-term MB results respectively.Markers (•) are results of BOLSIG+, which solves using the two-term approximation and ISO-ISO scattering.

Table 1 .
Benchmark calculations in the RR model atom.Relative error is reported with respect to MB

Table 2 .
(10)hmark calculations in the LS model atom.The top half of the table reports the same parameters (average energy and flux transport coefficients) as table1.The ionization reaction rate coefficient and bulk transport coefficients are reported in the lower half of the table.Relative error is reported with respect to MB(10), the multi-term BE solver results, to the right of the swarm parameter and found using δ[%] = (1 − X/Y) • 100, where X are the results of a particular code and Y are the results of MB(10).Calculated electron energy distribution functions for the high field strength of E/N = 100 Td using the RR model atom.Solid and dashed lines are MB calculations (using 10 terms).Markers are MC results of ELIOS (•) and CAPS (▽).Scattering is denoted by color, with ( ) ISO-ISO, ( ) ISO-FWD, ( ) ISO-SC and ( ) SC-SC.EEDFs are given for both a linear (left) and logarithmic (right) y-axis scale.Only the SC-SC case is given using ELIOS and CAPS because, from the MC(10) results, the curves for ISO-SC and SC-SC are not graphically distinguishable.calculations, results were virtually identical.While the SC-SC model includes more anisotropic scattering than the ISO-SC model, swarm parameters for the SC-SC model are for slightly 'cooler' swarms overall due to the change in the elastic TCS necessary to implement the SC model (see figure