Efficient modeling of electron kinetics under influence of externally applied electric field in magnetized weakly ionized plasma

We present a theory based on the conventional two-term (i.e. Lorentzian) approximation to the exact solution of the Boltzmann equation in non-magnetized weakly ionized plasma to efficiently obtain the electron rate and transport coefficients in a magnetized plasma for an arbitrary magnitude and direction of applied electric field E⃗ and magnetic field B⃗ . The proposed transcendental method does not require the two-term solution of the Boltzmann equation in magnetized plasma, based on which the transport parameters vary as a function of the reduced electric field E/N , reduced electron cyclotron frequency ωce/N , and angle ∠E⃗,B⃗ between E⃗ and B⃗ vectors, where N is the density of neutrals. Comparisons between the coefficients derived from BOLSIG+’s solution (obtained via the two-term expansion when B⃗≠0 ) and coefficients of the presented method are illustrated for air, a mixture of molecular hydrogen (H2) and helium (He) representing the giant gas planets of the Solar System, and pure carbon dioxide (CO2). The new approach may be used in the modeling of magnetized plasma encountered in the context of transient luminous events, e.g. sprite streamers in the atmosphere of Earth and Jupiter, in modeling the propagation of lightning’s electromagnetic pulses in Earth’s ionosphere, and in various laboratory and industrial applications of nonthermal plasmas.


Introduction
reported the first plasma fluid model for magnetized streamer discharges. The authors studied streamer propagation parallel to a magnetic field in pure CO 2 . The motivations for this study included the significant interest in the energy generation cycle of magnetohydrodynamic generators, the possible utilization of CO 2 in alternative renewable energy sources, and the use of CO 2 as an alternative to SF 6 for insulation in high-voltage transmission and distribution networks due to its lower environmental impact (e.g. Hernandez-Avila et al 2002, Seeger et al 2016, Starikovskiy et al 2021.
Transient luminous events are a set of frequently observed lightning-induced optical phenomena that were serendipitously discovered by Franz et al (1990). In particular, a transient luminous event referred to as an elve (e.g. Fukunishi et al 1996), which is the optical manifestation of the ionosphere interacting with a lightning electromagnetic pulse, occurs at lower ionospheric altitudes, i.e. ∼100 km, in the Earth's atmosphere. The Earth's geomagnetic field plays an important role in the structure of elves. For instance, Nagano et al (2003) attributed the asymmetry exhibited in the shape of elves to the Earth's geomagnetic field. A three-dimensional finite difference time domain model that accounts for the effects of electron heating on electron mobility and for the asymmetry of elves due to the geomagnetic field was reported in Marshall (2009), Marshall et al (2010). This asymmetry was not observed in studies that excluded Earth's magnetic field (Barrington-Leigh et al 2001, Veronis et al 2001, Kuo et al 2007, Liu et al 2017, Pérez-Invernón et al 2018. Following the first observation of possible transient luminous events on Jupiter reported by Giles et al (2020), Janalizadeh and Pasko (2023) developed a numerical model for the modeling of magnetized streamers in the presence of Jupiter's strong magnetic field. Similar to Starikovskiy et al (2021), streamer propagation was considered in a cylindrical coordinate system, where the magnetic field was parallel to the axis, and magnetized streamers were studied in the molecular hydrogen and helium-dominated atmosphere of Jupiter.
Following Starikovskiy et al (2021), in Janalizadeh and Pasko (2023), BOLSIG+ (Hagelaar and Pitchford 2005) was used to calculate the electron transport and rate coefficients as a function of the reduced electric field E/N, reduced electron cyclotron frequency ω ce /N, and angle ∠ ⃗ E, ⃗ B between the applied electric field ⃗ E and magnetic field ⃗ B vectors. In Janalizadeh and Pasko (2023, and references therein) it is demonstrated that in the presence of a magnetic field transport parameters of a weakly ionized plasma vary as a function of (E/N, ω ce /N, ∠ ⃗ E, ⃗ B). As done in Starikovskiy et al (2021), one may dramatically reduce the execution time of a fluid model for magnetized plasma by fitting analytical functions to lookup tables produced from the electron rate and transport coefficients calculated by BOLSIG+.
In an alternative approach, however, it is possible to use the electron rate and transport coefficients corresponding to an effective electric field E eff in non-magnetized plasma (i.e. ⃗ B = 0) to deduce plasma transport parameters for the magnetized case (i.e. ⃗ B ̸ = 0). Compared to interpolating values from 3D lookup tables corresponding to (E/N, ω ce /N, ∠ ⃗ E, ⃗ B), the proposed method requires solving a transcendental equation to obtain E eff , and subsequently using pre-computed rate and transport coefficient vectors to obtain values corresponding to E eff /N. Based on the authors' experience, 3D interpolation in a high-resolution plasma fluid model is more time-consuming than solving a transcendental equation. The authors have not compared the efficiency of the transcendental method with calculations using analytical functions fitted to 3D lookup tables (e.g. Starikovskiy et al 2021). Overall, we emphasize that the calculations realized through the fitting functions, the lookup tables, or the proposed transcendental method do not represent the most computationally expensive parts of models (i.e. streamer models (Starikovskiy et al 2021, Janalizadeh andPasko 2023)). In non-magnetized cases, we do not create lookup tables for different E and N values. Instead, we use the E/N-dependent (i.e. reduced) representations. The physicsbased simplicity of the proposed formulation can be viewed as more efficient from the same perspective.
The majority of the present work is dedicated to the development of the theoretical background quantifying E eff . Nevertheless, to accommodate readers interested mainly in the implementation aspect of the introduced method, in section 2 we describe the procedure to implement a simple and intuitive approximation to the general transcendental framework, which in later sections will be presented in detail. We conclude section 2 after demonstrating that the approximate method results in electron transport parameters in pure CO 2 plasma, that are in satisfactory agreement with BOLSIG+'s exact calculations for ⃗ B ̸ = 0.
Section 3 sets the theoretical foundation required for the introduction of the effective electric field E eff mentioned above. Here it is demonstrated that E eff , defined through the minimization of a designed error function, is in the same format of the electric field as an electron with energy ε (in electronvolts (eV)) experiences in the presence of a magnetic field (e.g. Starikovskiy et al 2021, equation (5)). Specifically, for a given magnitude of applied magnetic field (i.e. for specified ω ce /N) and for a given angle between the electric field ⃗ E and magnetic field ⃗ B applied to a weakly ionized plasma, the influence of ⃗ B on the ensemble of electrons may be interpreted in terms of an effective Hall parameter β eff , which itself varies as a function of E eff /N. Hereafter, we distinguish between the conventional energy-dependent Hall parameter β H (ε) = ω ce /ν m (ε) (where ω ce = q e B/m e , q e is the fundamental charge of an electron, m e is the mass of an electron, and ν m (ε) is the effective momentum transfer collision frequency that is a function of ε) and β eff , with the subscript 'H' used only for the former. Moreover, the dependence of quantities on the electron energy or velocity will be explicitly shown via a trailing '(ε)' or '(v)', respectively. That is, the absence of '(ε)' or '(v)' in the symbol of any plasma rate and transport coefficient implies that the corresponding coefficient has been averaged over the electron energy distribution function (EEDF). We note that the quantification of ν m (ε) for collisions of electrons with each species in a mixture requires an effective electron impact cross-section, which as mentioned in the LXCat crosssection file accompanying BOLSIG+ (Hagelaar and Pitchford 2005), equals the sum of the elastic momentum transfer crosssection and total inelastic collision cross-section. This effective cross-section is not in any way connected to the concept of E eff mentioned above.
The mathematical derivations related to section 3 may be found in appendix A, where the general transcendental method is derived. Here, we also discuss the weight function w(v) (where v = γε 1 2 in which γ = (2q e /m e ) 1 2 ) that is introduced in the definition of the error function mentioned above. Subsequently, in section 4.1 it is demonstrated that three custom weight functions investigated as part of this analysis provide similar values for β eff , and consequently we focus on one specific weight function, which results in the presented formulation lending itself to the electron rate and transport coefficients of magnetized plasma defined in e.g. Hagelaar (2016), and in some cases, directly outputted by BOLSIG+. This transition is explained in appendix B. In appendix C, further approximations justified in the case of nearly constant momentum transfer collision frequency as a function of the electron energy are introduced. This is where the simple and intuitive approximation to the general transcendental method, presented in section 2, is justified. In section 4.2 and through the comparison of some electron transport and rate coefficients with BOLSIG+ calculations corresponding to ⃗ B ̸ = 0, it is demonstrated that the proposed transcendental method, in exact and approximate form, and the magnetized plasma (i.e. ⃗ B ̸ = 0) calculations of BOLSIG+ give consistent results. The limit of nearly constant ν m (ε) is also discussed. The validity of the presented transcendental framework is demonstrated through application to three gas mixtures, i.e. air, a mixture of 88% molecular hydrogen with 12% helium representing the composition of giant gaseous planets in the Solar System, and pure carbon dioxide.

Model outline
The electron transport and rate coefficients in various gas mixtures for the non-magnetized (i.e. ⃗ B = 0) case are commonly represented as functions of the reduced applied field E/N using lookup tables or various fits with analytical functions. These are usually formulated using a combination of the solution of kinetic equations, swarm experiments, and Monte Carlo simulations, and are readily available to modelers. An example of these would be solutions corresponding to ⃗ B = 0 in CO 2 gas recently published in Starikovskiy et al (2021). The purpose of this section is to demonstrate that these ⃗ B = 0 representations can be directly used to obtain transport and rate coefficients for an arbitrary magnitude and direction of the applied magnetic field ⃗ B ̸ = 0. We note that ideas of self-consistently accounting for the electron momentum transfer collision frequency varying as a function of the applied electric field in the evaluation of the electron conductivity tensor in weakly ionized plasmas have appeared in previous publications (Pasko et al 1998, Marshall 2009, Marshall et al 2010, Kabirzadeh et al 2015, Salem et al 2016, Tonev and Velinov 2016. However, these approaches have not been rigorously justified. While a rigorous formulation and discussion of the validity of the proposed transcendental method follows in the subsequent sections, here we focus on the implementation of an approximation to the general transcendental method to illustrate and emphasize the accuracy and efficiency of the transcendental method in its simplest form (see appendix C). We note that in this case, the effective electric field E eff in the presence of a magnetic field is given by the proposed method via which resembles the expression of the electric field that an electron with energy ε experiences due to the presence of a magnetic field (e.g. Starikovskiy et al 2021, equation (5)). Whereas the Hall parameter β H (ε) = ω ce /ν m (ε) (e.g. Starikovskiy et al 2021, equation (5)), where ν m (ε)/N is the rate constant for momentum transfer due to an electron with energy ε, in equation (1 is the rate constant for momentum transfer (averaged over the EEDF). This implies that, expressed explicitly, β eff = β eff (E eff /N, ω ce /N). Nevertheless, in equation (1) the dependence of β eff on ω ce /N is suppressed, assuming that analysis is conducted for a constant applied magnetic field providing a constant value of ω ce /N. Also, the dependence of β eff on E eff /N is denoted solely by E eff to make a connection to the fixed point x of a function ϕ(x) defined as concept of a fixed point in addition to the fixed-point theorem is invoked in appendix E and the supplementary file to discuss the existence, uniqueness, and convergence of a solution E eff to non-linear equation (1).
are the components of the applied electric field ⃗ E, respectively, parallel and perpendicular to the magnetic field ⃗ B, the approximate transcendental method only requires ν m /N, which is related to the electron mobility µ e via ν m /N = (γ 2 /2)/ (µ e N) (i.e. ν m = q e /(m e µ e ) in the absence of Coulomb collisions) (Hagelaar 2016, p 17). We emphasize that this formulation employs only ν m /N (or equivalently, β eff in equation (1)) as a function of the reduced effective electric field E eff /N, i.e. ν m /N is a function of E eff /N only, and is calculated with no effect of the applied magnetic field, i.e. ⃗ B = 0. In practice, µ e can be interpreted as mobility parallel to the magnetic field µ ∥ . Furthermore, β eff introduced here is an approximation to the β eff quantity defined in section 3.
There are a number of sources available that may be used to calculate the electron mobility in various gas mixtures for the ⃗ B = 0 case. For instance, in addition to the electron mobility in air, Morrow and Lowke (1997) provide analytic functions for electron impact collision rate constants, which were used in previous modelings of streamers in air (e.g. Bourdon et al 2007, Jánský and. Additionally, Moss et al (2006) provided a MATLAB function air1.m compiled from the results of ELENDIF (Morgan and Penetrante 1990), which returns electron mobility and mean energy in addition to rate coefficients for various electron impact processes in air. This function is freely available at http://pasko. ee.psu.edu/air. There are similar MATLAB functions based on BOLSIG+ (Hagelaar and Pitchford 2005) for air (Janalizadeh and Pasko 2020), Jupiter's atmosphere (Janalizadeh and Pasko 2023), and CO 2 (provided per request from the authors).
Equation (1) is a transcendental equation for E eff and may be solved for E eff using electron mobility given by the functions mentioned above to quantify β eff . Once the value of E eff that satisfies equation (1) is obtained, the kinetics of electrons under the influence of (E/N, ω ce /N, ∠ ⃗ E, ⃗ B) are converted to an equivalent problem with (E eff /N, ⃗ B = 0). The electron rate and transport coefficients may then be calculated using the functions above, which were developed for nonmagnetized plasma. In particular, the perpendicular and Hall mobilities may be obtained via Figure 1(a) depicts the mean energy of electrons ε m in CO 2 gas under the influence of external electric and magnetic fields obtained using BOLSIG+ with accurate inclusion of an external magnetic field, while figure 1(b) depicts the same results calculated using the proposed transcendental method (i.e. equation (1)). It may be inferred that the approximate transcendental method provides results in satisfactory agreement with BOLSIG+'s exact calculations. We note that figure 1(a) is a reproduction of Starikovskiy et al (2021), figure 4(a), where the authors also use BOLSIG+ to calculate the electron mean energy. Here, we do not present the results corresponding to Starikovskiy et al (2021), figures 4(b)-(d), since in that study the respective ionization frequency ν i , electron mobility parallel to the magnetic field vector µ ∥ , and electron mobility perpendicular to the magnetic field vector µ ⊥ are presented as dimensionless quantities. Instead, in figure 2, we compare the results of BOLSIG+ and the transcendental method for ε m , ν i , µ ∥ , and µ ⊥ , as a function of ∠ ⃗ E, ⃗ B. Here, the applied electric field E = 1.5E k , where E k ≃ 80 Td (1 Td = 10 −17 V cm 2 ) is the breakdown electric field (Raizer 1991, p 137) in pure CO 2 calculated via BOLSIG+. The values of the reduced electron cyclotron frequency used for the calculations are (ω ce /N) 1 = 10 −14 , (ω ce /N) 2 = 10 −13 , and (ω ce /N) 3 = 10 −12 rad m 3 s −1 . As seen later in section 4.1, this interval of ω ce /N covers the entire range between non-magnetized (β 2 eff ≪ 1) and highly magnetized (β 2 eff ≫ 1) electrons, and consequently it is demonstrated in figure 2 that the results from the two methods are in satisfactory agreement in the entire range of magnetized CO 2 plasma. In particular, as demonstrated in equation (A.15), the error of the transcendental method grows approximately proportional to sin 4 (∠ ⃗ E, ⃗ B). Thus, the maximum discrepancy between the exact and transcendental results occurs at ∠ ⃗ E, ⃗ B = 90 • . Furthermore, for non-magnetized (fully magnetized) plasma both the exact and transcendental method trivially return E eff = E (E eff = E ∥ ). This agreement between the two methods does not necessarily hold for partially magnetized plasma, where β eff ≳ 1.
Before ending this section, we reiterate the outline used above for studies that require calculation of the electron rate and transport coefficients in a magnetized plasma. As demonstrated above, this approach reduces the problem of magnetized plasma (i.e. ⃗ B ̸ = 0) in the presence of an applied electric field ⃗ E to an equivalent problem of non-magnetized plasma (i.e. ⃗ B = 0) in the presence of an effective electric field E eff where one 1. Calculates E eff for a given (E/N, ω ce /N, ∠ ⃗ E, ⃗ B) satisfying the transcendental equation (1), and corresponding β eff (E eff /N).

Calculates electron transport and rate coefficients as if
⃗ B = 0 using this newly obtained E eff . For instance, electron mean energy ε m (E eff /N), reduced electron impact ionization frequency ν i N (E eff /N), reduced momentum transfer collision frequency νm N (E eff /N), reduced electron mobility parallel to the magnetic field µ ∥ N = (q e /m e )/ νm N (E eff /N) , and electron mobility perpendicular to the magnetic field The solution of equation (1) can be simplified if, for a given ω ce /N, β eff = (ω ce /N)/(ν m /N) can be assumed to be constant or weakly dependent on the reduced electric field E eff /N. However, we note that for typical electric fields used in applications, the β eff parameter in equation (1) can exhibit significant variations as a function of E eff /N. For example, for CO 2 gas, it changes by a factor of 5, and for air by a factor of 10. As β eff enters equation (1) in a quadratic form, these variations are important and one needs to find the solution E eff of nonlinear equation (1) to accurately solve the problem. The solutions can be simplified when β 2 eff ≪ 1 due to the high collision frequency ν m (E eff /N; N) ≳ ω ce in strong applied electric fields (e.g. Liu et al 2017) or when the orientation of the electric field with respect to the magnetic field has preferentially an E ∥ component (e.g. Pérez-Invernón et al 2018). In both cases, the effect of an external magnetic field on the system behavior can be ignored. We note that the solution flow described here follows from the case labeled as w = w 3 for β 2 H (ε) ≫ 1, where w is a weight function that will be defined in the following section.

Model formulation
Assuming a steady state and homogeneous space, in the presence of a constant electric and magnetic field, the isotropic part of the electron velocity distribution function (EVDF) f 0B in the Lorentzian approximation (e.g. Holstein 1946) where ⃗ i v is the radial unit vector in velocity space, is the solution to the differential equation (e.g. Golant et al 1980, p where the subscript B emphasizes that the EVDF is calculated for magnetic field ⃗ B ̸ = 0. The anisotropic part ⃗ f 1B is given in e.g. Loureiro and Amorim (2016, p 164) for ⃗ B ∥ẑ, whereẑ is a unit vector in the direction of the z axis. In the above equation and in a conventional cylindrical coordinate system, we have E 2 and C( f 0B ) denotes the collision term (e.g. Loureiro and Amorim 2016, pp 101-4, 110-5). The solution of equation (2) as a function of v varies with any combination Janalizadeh and Pasko 2023, appendix A2). For the remainder of this work, symbols after a semicolon represent independent external parameters on which an introduced quantity depends.
As a result of the Lorentzian, i.e. two-term expansion of the EVDF, electron impact collision rates are determined exclusively by f 0B (´∞ v=0 f 0B (v)4π v 2 dv = ne, where ne denotes electron density). On the other hand, electron transport coefficients are dependent on ⃗ f 1B . As can be seen from sources cited above (e.g. Janalizadeh and Pasko 2023, appendix A1), for a given set of (E/N, ωce/N, ∠ ⃗ E, ⃗ B), the latter is solely dependent on the derivative of the isotropic term with respect to v. Thus, in an alternative approach that does not require solving equation (2) for magnetized electrons, here we substitute f 0B with f 0 in search of the isotropic part of an EVDF in the absence of a magnetic field, i.e. ⃗ B = 0, that minimizes the residual over the v ∈ [0, ∞] interval. Before presenting the minimization process, we note that, as demonstrated in equation (2), the residual is zero for f 0B . As such, R(v) may be interpreted as a measure quantifying the difference between f 0B and f 0 = f 0 (v; E eff /N) at a given electron velocity v. We note that f 0 (v) is the solution to (e.g. Raizer 1991, p 87) 1 3 qe me (2) to obtain the above equation for a given E eff . Subsequently, we can replace the collision term C( f 0 ) in equation (3) using the definition in equation (4) to obtain Note that R(v) expressed in equation (5)  The minimization of the defined residual over v ∈ [0, ∞] may be quantified by introducing a weight function w(v) and subsequently defining an error function where we have used the definition of the inner product ⟨f, g⟩ = Dudley 1994, p 53), in which * represents the complex conjugate operator. Here, we wish to find the value of E eff that minimizes the error E. By definition, in that case we should have ∂E/∂E eff = 0. We emphasize that this minimization should generally be performed for every set of three independent external parameters used in the formulation of lookup tables (e.g. Starikovskiy et al 2021), namely, E/N, ωce/N, and ∠ ⃗ E, ⃗ B. Nevertheless, as demonstrated in appendix A, this analysis results in the general transcendental expression where 1 + β 2 eff is the factor by which the perpendicular component of the applied electric field is reduced due to the presence of a magnetic field. The quantity β eff varies only as a function of E eff /N for a given ωce/N and is given by β eff = I 1 /I 2 − 1, where I 1 and I 2 are integrals defined in equations (A.6) and (A.7), respectively. The dependence of β eff (through I 2 ) on ωce/N is only through β H (ε) since as already mentioned, f 0 corresponds to a non-magnetized EVDF such that one does not need to solve equation (2) for magnetized plasma.
Generally, E eff will be located in the [E cos(∠ ⃗ E, ⃗ B), E] interval where the upper (lower) limit of this search interval corresponds to non-magnetized, β 2 eff ≪ 1, (fully magnetized, β 2 eff ≫ 1) electrons, respectively. Therefore, once β eff is quantified, one can employ a root-finding algorithm to solve equation (7) and obtain E eff /N for a given set of input parameters (E/N, ωce/N, ∠ ⃗ E, ⃗ B). The question of the existence and uniqueness of a solution to equation (7) is addressed in appendix E. Specifically, it is demonstrated that a solution always exists, and conditions for the uniqueness of a solution and convergence of the fixed-point iteration method to find E eff are obtained such that for any given set of input parameters (E/N, ωce/N, ∠ ⃗ E, ⃗ B) one can verify these conditions. We note that the interested reader may accelerate the root-finding process by creating a two-dimensional array of β eff values varying as a function of the (E eff /N, ωce/N) pair to be subsequently used in solving the transcendental equation (7).
To numerically quantify β eff we introduce three weight functions denoted by w 1 (v), w 2 (v), w 3 (v) to demonstrate the performance of the presented transcendental method. The corresponding weight function in energy space is defined via W(ε)dε ≡ w(v)dv, where, as mentioned above, ε denotes electron energy in units of eV. The weight functions we use in this work are 1. w 1 (v) = 1, i.e. constant weight function in velocity space.
Note that since β eff ∝ I 1 /I 2 , the absolute value of the electron density or the constant γ present in the weight function do not affect any results of this analysis as the weight function is included in both I 1 and I 2 .
We emphasize that the formulation presented here is valid irrespective of the peculiarities of each weight function. As such, in choosing w 1 and w 2 we prioritize the simplicity of the weight function itself. However, this is not the case for w 3 . Specifically, w 3 has been chosen such that β eff will become proportional to the electron transport and rate coefficients directly outputted by BOLSIG+ (see appendices B and C). In other words, calculating β eff when w = w 1 or w = w 2 requires the EEDF calculated by BOLSIG+ for evaluation of the integrals I 1 and I 2 . However, when w = w 3 these integrals reduce to specific electron rate and transport coefficients that are already calculated by BOLSIG+ in the non-magnetized case (i.e. ⃗ B = 0).
Specifically, the transcendental method in the special case of w = w 3 reduces to (see appendix B) where expressions for mobilities parallel µ ∥ and perpendicular µ ⊥ to the magnetic field are given in equations (B.5) and (B.9), respectively (e.g. Hagelaar 2016, p 16). We note that whereas µ 0 ⊥ is in the same format of µ ⊥ defined in Hagelaar 2016, p 16, as opposed to µ ⊥ it is not a direct output of BOLSIG+ since the calculation of µ ⊥ by BOLSIG+ happens when ⃗ B ̸ = 0, while µ 0 ⊥ is dependent on BOLSIG+ only through BOLSIG+'s EEDF calculated for ⃗ B = 0. As such, we calculate µ 0 ⊥ manually (in MATLAB) using the EEDF output of BOLSIG+ corresponding to a defined range of E eff /N values. A given reduced gyrofrequency ωce/N quantifies β H (ε) = ωce/νm(ε) in the definition of µ 0 ⊥ through νm(ε) = Nσm(ε)v = Nσm(ε)γε 1 2 . In the calculation of µ 0 ⊥ and more generally β eff , we quantified the momentum transfer cross-section σm(ε) of each gas mixture using the LXCat set of cross-sections, which accompanied BOLSIG+. For species for which σm(ε) was not readily available in the accompanying data, we calculated σm(ε) by summing the cross-section for all inelastic processes in addition to the elastic momentum transfer cross-section. The value of σm(ε) for a mixture was obtained by weighted summation of σm(ε) of the constituent species according to their fraction of composition.

Results and discussion
4.1. Calculation of β eff for w 1 (v), w 2 (v), and w 3 (v) As inferred from section 3, whereas the determination of E eff through the transcendental equation is dependent on the applied electric and magnetic fields in addition to the angle between the two, the calculation of β eff for a given E eff /N may proceed in a standalone fashion. To quantify β eff for various weight functions, here we use BOLSIG+ to calculate the EEDF (with ⃗ B = 0) in gas mixtures considered for a wide range of E eff /N values. As already mentioned, the magnetic field enters our calculations of β eff only through the Hall parameter included in the definition of I 2 . Three gas mixtures are considered: (1) air, i.e. a mixture of 80% molecular nitrogen (N 2 ) and 20% molecular oxygen (O 2 ), (2) a mixture of 88% molecular hydrogen (H 2 ) and 12% helium (He), and (3) pure carbon dioxide (CO 2 ).
Furthermore, it is inferred from figure 3 that the difference between β eff values corresponding to various weight functions w(v) is practically insignificant. As already mentioned in section 3, w = w 3 results in the expression of β eff in terms of the electron rate and transport coefficients, which are already calculated by BOLSIG+ in the non-magnetized case (i.e. ⃗ B = 0). Therefore, for the sake of simplicity and brevity, in the remainder of this work we let w = w 3 .

Comparison of BOLSIG+ exact coefficients with present study results
As demonstrated in appendix C, when w = w 3 , the presented transcendental method for considerably magnetized electrons may be simplified even further. Specifically, if can use β eff = ωce/νm = (ωce/N)/km(E eff /N), where km = νm/N is the momentum transfer rate constant as a function of E eff /N exclusively. This is a standard rate coefficient calculated by BOLSIG+ itself such that the implementation of the transcendental method using the EEDFs calculated by BOLSIG+ may be avoided. The average ⟨ε ∂ ∂ε [σm(ε)ε 1 2 ]⟩ is a measure of the variation of momentum transfer collision frequency as a function of the electron energy. That is, σm(ε) ∝ ε − 1 2 for which the average is identically zero corresponds to a constant momentum transfer collision frequency since in that case νm(ε) = Nkm(ε) = Nσm(ε)v ∝ ε − 1 2 (γε 1 2 ) = const (e.g. Starikovskiy et al 2021). Consequently, in this section, we assume that this average is in fact negligible, and compare results with the general transcendental method with w = w 3 and β eff . We will also include the exact (no assumptions made) calculations of BOLSIG+, which have been compiled in lookup tables and subsequently interpolated for cases considered here.
Specifically, we consider two cases of ∠ ⃗ E, ⃗ B: 45 • and 90 • in addition to a large range of applied reduced fields E/N. We choose ωce/N (or equivalently ωceN 0 /N, where N 0 = 2.686 × 10 25 m −3 ) such that β eff ≃ 1 for both ∠ ⃗ E, ⃗ B considered and in the majority of the E/N range. The purpose of this choice of ωceN 0 /N is to demonstrate the performance of the transcendental method in the partially magnetized regime where, as opposed to β 2 eff ≫ 1 (β 2 eff ≪ 1), the effective electric field is not trivially E eff = E ∥ (E eff = E). In what follows, ωce/N ≃ 10 −13 rad m 3 s −1 (for all gases), which is close to the (ωce/N) 2 value in figure 3 and corresponds to partially magnetized electrons. The quantities used for comparison are the mean energy of electrons εm, the electron impact ionization frequency ν i , the mobility parallel to the magnetic field µ ∥ , and the mobility perpendicular to the magnetic field µ ⊥ . This choice of comparisons has been made to investigate the performance of the proposed transcendental method as it pertains to both the f 0 and ⃗ f 1 terms in the two-term expansion of the EVDF.
Figures 4-6 respectively depict comparisons in gas mixtures (1)-(3). All panels include β eff = (ωce/N) /km(E eff /N) after finding the solution to equation (1) to illustrate the degree of magnetization of the electrons. It is inferred from these figures that the transcendental method for w = w 3 and the approximation to this method agree to a satisfactory degree. In addition, both methods agree with the exact calculations of BOLSIG+ for the majority of the E/N range. Thus, the interested reader may initially implement the simpler approximate transcendental method presented in section 2 to evaluate and explore the method's performance for an arbitrary gas mixture.
The observed deviation of ν i results of BOLSIG+ from that of both transcendental methods at ∠ ⃗ E, ⃗ B = 90 • (i.e. E ∥ = 0) is generally considerable. On the other hand, the agreement between εm calculated for the same scenario by all methods is satisfactory. This observation emphasizes the difference in the high energy tail of the exact EEDF calculated by BOLSIG+ for ⃗ B ̸ = 0 and the EEDFs (corresponding to ⃗ B = 0) used in the transcendental methods. Specifically, both εm and ν i depend on the isotropic term of the EVDF. However, the latter involves, exclusively, the high-energy electrons represented in the tail of the EEDF (vs ε) since electron impact ionization is a collision with an energy threshold. While the abundance of these ionizing electrons controls the ionization rate constant, their exponentially lower population compared to low-energy electrons results in a negligible impact on the εm values presented. We note that the same analysis is true for rate constants of other electron impact processes with an energy threshold. The disagreement is more pronounced for electron impact collisions with a higher energy threshold.
Values corresponding to ⃗ E ∥ ⃗ B are also included in all panels. Specifically, the extremely low ionization levels when ∠ ⃗ E, ⃗ B = 90 • are demonstrated. While ∠ ⃗ E, ⃗ B = 90 • corresponds to the lowest agreement between ν i results of BOLSIG+ and the transcendental methods, the exact results of BOLSIG+ are still orders of magnitude less than scenarios in which ∠ ⃗ E, ⃗ B → 0. We note that this holds even for ∠ ⃗ E, ⃗ B as high as 45 • (see panels (b) and (d) in figures 4-6). As such, one may conclude that in a realistic scenario where ∠ ⃗ E, ⃗ B may vary in the entire range of ∠ ⃗ E, ⃗ B = 0 − 90 • , even exact ν i values corresponding to ∠ ⃗ E, ⃗ B = 90 • are so insignificant that the disagreement between BOLSIG+ and the transcendental methods has no practical significance in the framework of plasma fluid models in which these coefficients are typically employed (e.g. Starikovskiy et al 2021, Janalizadeh andPasko 2023). In other words, unless in the entire simulation domain ∠ ⃗ E, ⃗ B → 90 • and the process involves electric fields close to E k (as the threshold for significant ionization), the transcendental method provides accurate results for the ionization frequency. A similar argument may be made for the parallel mobility panel in  figure 4(g), where one observes a clear disagreement between the BOLSIG+ exact results and both transcendental methods in the low E/N region. Specifically, in this case, E ∥ = 0 such that the drift of electrons parallel to ⃗ B is absent. As for µ ⊥ depicted in figure 4(h), one can clearly infer the better performance of the exact transcendental method (i.e. w = w 3 ) compared to the approximate transcendental method described in section 2.
The results succinctly depicted in figures 4-6 may be presented in a different format. Specifically, they can be presented as two-dimensional color plots that cover the entire range of ∠ ⃗ E, ⃗ B = 0 − 90 • and ωce/N ∈ [(ωce/N) 1 , (ωce/N) 3 ] = [10 −14 , 10 −12 ] rad m 3 s −1 for a select few applied reduced fields E/N. In that case, one may initiate the transcendental method and Subsequently, a quantitative error that provides the percentage difference between BOLSIG+ and the transcendental method by normalizing it to BOLSIG+ exact values may be introduced. Due to the general satisfactory performance of the approximate transcendental method and for the sake of brevity, such figures are included in the supplementary file that accompanies this paper.
At the end of this section, we emphasize that the presented results target the regime of partially magnetized plasma. One expects a better agreement between the transcendental method and BOLSIG+ exact calculations in both cases of essentially nonmagnetized (β 2 eff ≪ 1) and fully magnetized (β 2 eff ≫ 1) plasma since in those cases E eff = E and E eff = E ∥ , respectively (see figure 2). An even more accurate transcendental method may be obtained by extending the presented model formulation to solutions of the Boltzmann equation that are more accurate than the two-term approximation.

Conclusions
We introduce a new transcendental approach to the calculation of electron transport and rate coefficients in a magnetized plasma using the theory and results of non-magnetized plasma. The obtained effective electric field results in plasma transport parameters that are in satisfactory agreement with BOLSIG+'s exact calculations in air, a mixture of 88% molecular hydrogen with 12% helium, and pure carbon dioxide. Furthermore, the effective electric field is in the same format as the electric field a single electron experiences in the presence of a magnetic field. This provides an intuitive picture, which accompanies the rigorous mathematical derivations presented here. Subsequently, a special case of the formulation is further explored to reduce calculations and use the electron rate and transport coefficients outputted by BOLSIG+. While as a result of the kinetic theory of weakly ionized and magnetized plasma the electron transport and rate coefficients are defined through a distribution function that varies with (E/N, ωce/N, ∠ ⃗ E, ⃗ B), the new method proceeds in two steps: (1) the calculation of E eff for a given (E/N, ωce/N, ∠ ⃗ E, ⃗ B) through a simple transcendental equation, and (2) the calculation of electron transport and rate coefficients in the absence of a magnetic field using E eff /N (since when ⃗ B = 0, the transport parameters become functions of E eff /N, exclusively).

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix A. Theory of general transcendental method and minimum error value
In this section we start with the definition of the error function introduced in equation (6). Using the residual R(v) given by equation (5), we have We note that based on the definition of EVDF, f 0 here is only a function of v and therefore, ∂f0 where we used the fact that ∂ ∂v Distributing the integral among the two terms above and defining I 1 and I 2 as and respectively, we arrive at (A.8) Thus, (A.9) Using the equation above, the residual corresponding to the minimum error may be expressed as .13) defining I 3 as Equivalently, where we have used 1 + β 2 eff = I 1 /I 2 . While E min depends on ∠ ⃗ E, ⃗ B implicitly through determination of β eff for any set of input parameters, one can also see the explicit dependence on ∠ ⃗ E, ⃗ B through the sin 4 (∠ ⃗ E, ⃗ B) factor. Assuming the variation of the latter as a function of ∠ ⃗ E, ⃗ B is the dominant factor (see figures 4-6 for illustrations of negligible β eff variation as a function of ∠ ⃗ E, ⃗ B), one concludes that the error of the transcendental method increases monotonically with ∠ ⃗ E, ⃗ B such that the maximum discrepancy in comparison with the exact calculations occurs at ∠ ⃗ E, ⃗ B = 90 • .

Appendix B. Simplified transcendental method when w(v) = w 3 (v)
Using w 3 (v) as defined in equation (8) and applying integration by parts to I 1 , we have (B.1) Before demonstrating that I w3 1,1 is proportional to µ ∥ , i.e. electron mobility parallel to the magnetic field (when B ̸ = 0), we show that C w3 1 (v = ∞) = 0. Let's assume there is at least a single constant k > 0 for which one can find a constant v 0 so that the and based on the inequality introduced above, Note that ne −´v 0 v=0 f 0 (v)4π v 2 dv is a bounded positive quantity and therefore, the integral above including v ℓ should be convergent. Introducing the change To be consistent with BOLSIG+'s definition of mobility (Hagelaar 2016, p 16), we now switch from v to ε to obtain I w3 In Hagelaar (2016, p 16) we have mobility parallel to the magnetic field defined as Similarly for I w3 2 we have The same arguments detailed above result in C w3 2 (v = ∞) = 0 and where µ 0 ⊥ is in the same format of perpendicular mobility Hagelaar (2016, p 16). In conclusion, for w = w 3 , the simpler transcendental expression is derived in which µ ∥ N is a function of E eff /N exclusively, whereas µ 0 ⊥ N is dependent on both E eff /N and ωce/N. Note that µ ∥ N is provided by BOLSIG+ and µ 0 ⊥ N is calculated with the EEDF corresponding to ⃗ B = 0.

Appendix E. Existence and uniqueness of a convergent solution to equation (7)
In this section we answer the questions of existence and uniqueness of a convergent solution x to equation (7), repeated here as where we have used the change of symbol E eff → x and have dropped the subscript 'eff' from β eff for the sake of brevity. By definition, x is a fixed point of the function ϕ(x), since from equation (E.1) we have x = ϕ(x). Since β(x) ∈ C[0, E], where C [a, b] is the space of all continuous functions in the interval [a, b] and a, b ∈ R, this implies ϕ(x) ∈ C[0, E] as well. It is clear from equation (E.1) that ϕ(x) ∈ [0, E], since x ⩽ E always. The previous two statements imply that the function ϕ(·) has a fixed point in the [0, E] interval (Burden and Faires 2005, p 54, theorem 2.2a), and therefore, the question of the existence of a solution to equation (E.1) is answered. In addition, if ϕ ′ (x) = dϕ/dx in (0, E) exists and so does a positive constant k < 1 such that ϕ ′ (x) ⩽ k for all x ∈ (0, E) (E.2) then the fixed point in [0, E] is unique (Burden and Faires 2005, p 54, theorem 2.2b).
The simple fixed-point iteration method can be used to solve equation (E.1) for x. The method starts with an initial approximation x 0 and then improves on this approximation using the recursive equation x n+1 = ϕ(xn) until convergence is achieved. If condition (E.2) is satisfied, then it also ensures that for any initial value x 0 , the sequence {xn} converges to the unique fixed point x (Burden and Faires 2005, p 58-59, theorem 2.3). Although checking condition (E.2) analytically might not be possible since β(x) has a highly non-linear dependence on the electron energy distribution for any x, in the supplement to this paper we use numerical results to show that convergence is indeed achieved, provided the judicious choice of the initial value x 0 = E is made (i.e. the applied electric field is used as the initial value for E eff in the fixed-point iteration method to solve equations (1) and (7) or equation (E.1) here).