The LisbOn KInetics Boltzmann solver

LoKI-B describes the electron kinetics of a plasma with electron density n_e, excited from a general mixture of atomic/molecular gases, by applying an electric field along the z-axis with the form

$$\begin{eqnarray}&&\vec{E}(t)=-{E}_{p}\cos \left(\omega t\right){\vec{e}}_{z},\end{eqnarray} \tag{ 1 }$$

where the oscillation frequency ω is either set to zero (in the DC case) or taken much larger than the typical collision frequency ν_c (in the HF case). Each gas k of the mixture has mass M_k and particle density N_k, such that the total gas density is $N=p/({k}_{B}{T}_{g})\equiv {\sum }_{k}{N}_{k}$, with p the total gas pressure and T_g the gas temperature (k_B is the Boltzmann constant), assumed equal for all k-gas components. Moreover, each gas k is composed by several electronic levels k_i with particle density ${N}_{{k}_{i}}$, such that the k-gas density is ${N}_{k}\equiv {\sum }_{i}{N}_{{k}_{i}}$. We also introduce the following normalised densities: the gas fraction in the mixture χ_k ≡ N_k/N; the population of level k_i, relative to the k-gas density, ${\xi }_{{k}_{i}}\equiv {N}_{{k}_{i}}/{N}_{k};$ and the reduced density of level k_i, relative to the total gas density, ${\delta }_{{k}_{i}}\equiv {N}_{{k}_{i}}/N$, such that ${\delta }_{{k}_{i}}={\xi }_{{k}_{i}}{\chi }_{k}$. Note that each k_i-level can include several vibrational sub-levels ${k}_{{i}_{v}}$, each one composed by several rotational sub-levels ${k}_{{i}_{{v}_{J}}}$. The definition of the normalised densities for each one of these sub-levels follows the same ontology as for the k_i levels, and therefore can be obtained by a straightforward generalisation of ${\xi }_{{k}_{i}}$ and ${\delta }_{{k}_{i}}$.

In the classical two-term approximation, the electron distribution function can be written assuming a separation between the Legendre expansion of the energy distribution and the space-time profile of the electron density n_e(z, t) as follows

$$\begin{eqnarray}\begin{array}{rcl}F(z,u,t) & \simeq & \left\{{f}^{0}(u)+\left[{\vec{f}}^{1}(u)\cdot \displaystyle \frac{\vec{v}}{v}\right]\cos \omega t\right\}{n}_{e}(z,t)\\ & = & \left[{f}^{0}(u)+{f}^{1}(u)\cos \theta \cos \omega t\right]{n}_{e}(z,t),\end{array}\end{eqnarray} \tag{ 2 }$$

where $u={m}_{e}{v}^{2}/(2e)$ is the electron kinetic energy in eV (with m_e and e the electron mass and charge, respectively); f⁰(u) ≡ f(u) is the isotropic part of the electron distribution function, identified with the EEDF in the framework of the two-term approximation, and satisfying the normalisation condition ${\int }_{0}^{\infty }f(u)\sqrt{u}{du}=1;$ and f¹(u) is the (real part of the) first-anisotropy of the electron distribution function.

Under the previous conditions, the EBE including a non-conservative collisional operator can be written as

$$\begin{eqnarray}&&\sqrt{\displaystyle \frac{{m}_{e}}{2{eu}}}\displaystyle \frac{\langle {\nu }_{\mathrm{eff}}\rangle }{N}{uf}(u)+\displaystyle \frac{1}{N}\sqrt{\displaystyle \frac{{m}_{e}}{2e}}\displaystyle \frac{{dG}(u)}{{du}}=S(u)\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&{f}^{1}(u)=-\displaystyle \frac{\zeta \left(E/N\right)}{{{\rm{\Omega }}}_{\mathrm{PT}}(u)}\displaystyle \frac{{df}(u)}{{du}},\end{eqnarray} \tag{ 3b }$$

or

$$\begin{eqnarray}&&\displaystyle \frac{1}{3}\displaystyle \frac{{\alpha }_{\mathrm{eff}}}{N}{{uf}}^{1}(u)+\displaystyle \frac{1}{N}\sqrt{\displaystyle \frac{{m}_{e}}{2e}}\displaystyle \frac{{dG}(u)}{{du}}=S(u)\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{f}^{1}(u)=-\displaystyle \frac{\left({\alpha }_{\mathrm{eff}}/N\right)}{{{\rm{\Omega }}}_{\mathrm{SST}}(u)}f(u)-\displaystyle \frac{\left(E/N\right)}{{{\rm{\Omega }}}_{\mathrm{SST}}(u)}\displaystyle \frac{{df}(u)}{{du}},\end{eqnarray} \tag{ 4b }$$

where (3a) and (4a) or (3b) and (4b) correspond to the isotropic and anisotropic components of the EBE, respectively, in the following approximations: the space-homogeneous equations (3a)–(3b) assume an exponential temporal growth of the electron density with a growth constant $\langle {\nu }_{\mathrm{eff}}\rangle \equiv \langle {\nu }_{\mathrm{ion}}\rangle -\langle {\nu }_{\mathrm{att}}\rangle \equiv {\sum }_{k,i}{\delta }_{{k}_{i}}\left[\langle {\nu }_{{k}_{i},\mathrm{ion}}\rangle -\langle {\nu }_{{k}_{i},\mathrm{att}}\rangle \right]$, corresponding to the electron net creation frequency for the system under study (with $\langle {\nu }_{\mathrm{ion}}\rangle$ and $\langle {\nu }_{\mathrm{att}}\rangle$ the corresponding electron mean frequencies for ionisation and attachment, respectively, and where the sums extend over all levels of all gases in the mixture); the time-constant equations (4a)–(4b) assume an exponential spatial growth of the electron density in the direction opposite to that of the applied electric field, with a growth constant ${\alpha }_{\mathrm{eff}}\equiv \alpha -\eta \equiv {\sum }_{k,i}{\delta }_{{k}_{i}}\left({\alpha }_{{k}_{i}}-{\eta }_{{k}_{i}}\right)$, corresponding to the effective electron Townsend coefficient (with α and η the corresponding first and second Townsend coefficients for ionisation and attachment, respectively). Note that the inclusion of growth models for the electron density is necessary if one intends a space and time independent description, yet including non-conservative collisional mechanisms such as electron ionisation and attachment. The temporal growth model was used to simulate Pulsed Townsend (PT) discharges by Tagashira et al [65], using the formulation of Thomas [66] for the one dimensional continuity equation of electrons under the action of an uniform electric field. The spatial growth model is usually adopted when simulating steady-state Townsend (SST) discharges maintained with a DC electric field [65].

Equations (3a)–(3b) can be used in either the DC or the HF cases, whereas equations (4a)–(4b) are for the DC case only. In these equations, the quantity E/N is the reduced electric field, where $E\equiv {E}_{p}/\zeta$ (with $\zeta =1\,{\rm{or}}\,\sqrt{2}$ for the DC or the HF cases, respectively); the quantities Ω_PT and Ω_SST have the dimensions of a cross section, being defined as

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{PT}}(u)\equiv {{\rm{\Omega }}}_{c}(u)+\displaystyle \frac{\left[{m}_{e}/(2{eu})\right]{\left(\omega /N\right)}^{2}}{{{\rm{\Omega }}}_{c}(u)}\end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{\mathrm{SST}}(u)\equiv {\sigma }_{c}(u),\end{eqnarray} \tag{ 5b }$$

where ${{\rm{\Omega }}}_{c}(u)\equiv {\sigma }_{c}(u)+\sqrt{{m}_{e}/(2{eu})}\left(\langle {\nu }_{\mathrm{eff}}\rangle /N\right)$, with ${\sigma }_{c}(u)\,\equiv {\sum }_{k}{\chi }_{k}{\sigma }_{k,c}$ the electron-neutral total momentum-transfer cross section, for the system under study, and σ_k,c the corresponding cross section for gas k; $G\equiv {G}_{E}+{G}_{\mathrm{el}}+{G}_{\mathrm{CAR}}+{G}_{{ee}}$ is the upflux function that contains power gain/loss contributions due to, in order, the applied electric-field, the elastic collisions, the rotational collisions (in the continuous approximation for rotations, CAR, with a Chapman-Cowling correction due to the gas temperature [57, 67–69], written here for the case of homonuclear diatomic molecules), and the electron-electron collisions [68], respectively

$$\begin{eqnarray}&&{G}_{E}(u)=N\sqrt{\displaystyle \frac{2e}{{m}_{e}}}\displaystyle \frac{u}{3}\displaystyle \frac{1}{\zeta }\displaystyle \frac{E}{N}{f}^{1}(u)\end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray}&&{G}_{\mathrm{el}}(u)=-\displaystyle \sum _{k}2{\gamma }_{k}{\nu }_{k,c}^{\mathrm{el}}(u){u}^{3/2}\left[f(u)+\displaystyle \frac{{k}_{B}{T}_{g}}{e}\displaystyle \frac{{df}(u)}{{du}}\right]\end{eqnarray} \tag{ 6b }$$

$$\begin{eqnarray}&&{G}_{\mathrm{CAR}}(u)=-\displaystyle \sum _{k}4{B}_{k}{\nu }_{0,k}{u}^{1/2}\left[f(u)+\displaystyle \frac{{k}_{B}{T}_{g}}{e}\displaystyle \frac{{df}(u)}{{du}}\right]\end{eqnarray} \tag{ 6c }$$

$$\begin{eqnarray}&&{G}_{{ee}}(u)=-2{\nu }_{{ee}}(u){u}^{3/2}\left[I(u)f(u)+J(u)\displaystyle \frac{{df}(u)}{{du}}\right],\end{eqnarray} \tag{ 6d }$$

where ${\gamma }_{k}\equiv {m}_{e}/{M}_{k},{\nu }_{k,c}^{\mathrm{el}}\equiv {N}_{k}\sqrt{2{eu}/{m}_{e}}\,{\sigma }_{k,c}^{\mathrm{el}}$ is the electron-neutral elastic momentum-transfer collision frequency for the gas k (${\sigma }_{k,c}^{\mathrm{el}}$ being the corresponding cross section), ${\nu }_{0,k}\,\equiv {N}_{k}\sqrt{2{eu}/{m}_{e}}\,{\sigma }_{0,k}$ is the electron-neutral collision frequency for the rotational excitation of gas k (${\sigma }_{0,k}\equiv 8\pi {Q}_{k}^{2}{a}_{0}^{2}/15$ being the corresponding cross section, with a₀ the Bohr radius), B_k and Q_k are the rotational constant and the quadrupole-moment constant (in units of ${{ea}}_{0}^{2}$) of the gas k, respectively, I(u) and J(u) are the relevant Spitzer integrals [39, 70], and ${\nu }_{{ee}}\,\equiv 4\pi {\left[{e}^{2}/\left(4\pi {\varepsilon }_{0}{m}_{e}\right)\right]}^{2}\mathrm{ln}{{\rm{\Lambda }}}_{c}/{v}^{3}{n}_{e}$ is the electron-electron collision frequency (with ε₀ the vacuum permittivity, $\mathrm{ln}{{\rm{\Lambda }}}_{c}$ the Coulomb logarithm (${{\rm{\Lambda }}}_{c}\equiv 12\pi {n}_{e}{\lambda }_{D}^{3}$) and λ_D the Debye length); $S\equiv {\sum }_{i,j\gt i}\,{S}_{i,j}+{\sum }_{i}{S}_{i,\mathrm{ion}}+{\sum }_{i}{S}_{i,\mathrm{att}}$ is the discrete collision operator that contains power loss/gain contributions due to inelastic/superelastic mechanisms, ionisation and attachment, respectively

$$\begin{eqnarray}\begin{array}{rcl}{S}_{i,j}(u) & = & {\delta }_{i}\left[(u+{V}_{i,j}){\sigma }_{i,j}(u+{V}_{i,j})f(u+{V}_{i,j})-u{\sigma }_{i,j}(u)f(u)\right]\\ & & +{\delta }_{j}\displaystyle \frac{{g}_{i}}{{g}_{j}}\left[u{\sigma }_{i,j}(u)f(u-{V}_{i,j})\right.\\ & & \left.-(u+{V}_{i,j}){\sigma }_{i,j}(u+{V}_{i,j})f(u)\right]\end{array}\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}&&\begin{array}{l}{S}_{i,\mathrm{ion}}(u)={\delta }_{i}\left\{{\displaystyle \int }_{u+{V}_{i,\mathrm{ion}}}^{2u+{V}_{i,\mathrm{ion}}}u^{\prime} {\sigma }_{i,\mathrm{ion}}^{\sec }(u^{\prime} ,u^{\prime} -{V}_{i,\mathrm{ion}}-u)f(u^{\prime} ){du}^{\prime} \right.\\ \quad +\,\left.{\displaystyle \int }_{2u+{V}_{i,\mathrm{ion}}}^{\infty }u^{\prime} {\sigma }_{i,\mathrm{ion}}^{\sec }(u^{\prime} ,u)f(u^{\prime} ){du}^{\prime} -u{\sigma }_{i,\mathrm{ion}}(u)f(u)\right\}\end{array}\end{eqnarray} \tag{ 7b }$$

$$\begin{eqnarray}&&{S}_{i,\mathrm{att}}(u)=-{\delta }_{i}u{\sigma }_{i,\mathrm{att}}(u)f(u).\end{eqnarray} \tag{ 7c }$$

In (7a)–(7c), the levels i, j and the reduced densities δ_i, δ_j now refer to any level for any gas in the system; σ_i,j is the electron-collision cross section for the $i\to j$ excitation with energy threshold V_i,j and statistical weights g_i and g_j, respectively (note that the third and fourth superelastic terms in (7a) were written resorting to the Klein-Rosseland relation [71]); ${\sigma }_{i,\mathrm{att}}(u)$ is the electron-collision cross section for the attachment of level i; ${\sigma }_{i,\mathrm{ion}}^{\sec }(u^{\prime} ,u)$ is the single differential cross section (SDCS) for the ionisation of level i with energy threshold ${V}_{i,\mathrm{ion}};$ and

$$\begin{eqnarray}&&{\sigma }_{i,\mathrm{ion}}(u)={\int }_{0}^{(u-{V}_{i,\mathrm{ion}})/2}{\sigma }_{i,\mathrm{ion}}^{\sec }(u,u^{\prime} ){du}^{\prime} \end{eqnarray} \tag{ 8 }$$

is the integral cross section for the ionisation of level i. The SDCS takes into account the distribution of the energy available after an ionisation event, caused by a primary electron with energy $u^{\prime}$, yielding a secondary electron born in the event with energy u and a scattered electron with energy $u^{\prime} -{V}_{i,\mathrm{ion}}-u$.

Note that the effect of rotational inelastic/superelastic mechanisms, in the electron kinetics of homonuclear diatomic molecules, can be considered by adopting either the CAR approach of (6c) or the discrete approach of the collision operator (7a), yielding similar results provided that the rotational constants Q_k and B_k are coherently matched to the rotational cross section set σ_i,j [57].

LoKI-B allows the user to choose between different descriptions of the ionisation mechanism, namely: (i) taking ionisation as a conservative mechanism, described by the inelastic part of the collisional operator (7a), in which case the first terms in the left-hand side of (3a) and (4a) should be set to zero; (ii) taking ionisation as a non-conservative mechanism, described by the collisional operator (7b) with a SDCS, and using the EBE given by (3a)–(4b); (iii) taking ionisation as a non-conservative mechanism, and prescribing no-sharing of energy between the secondary electron, introduced at u = 0, and the scattered electron, introduced at energy $u^{\prime} -{V}_{i,\mathrm{ion}}$. In this case the SDCS for the secondary electrons writes ${\sigma }_{i,\mathrm{ion}}^{\sec }(u^{\prime} ,u)={\sigma }_{i,\mathrm{ion}}(u^{\prime} )\delta (u)$ and the ionisation operator becomes

$$\begin{eqnarray}\begin{array}{rcl}{S}_{i,\mathrm{ion}}(u) & = & {\delta }_{i}\left[\Space{0ex}{3ex}{0ex}(u+{V}_{i,\mathrm{ion}}){\sigma }_{i,\mathrm{ion}}(u+{V}_{i,\mathrm{ion}})f(u+{V}_{i,\mathrm{ion}})\right.\\ & & -u{\sigma }_{i,\mathrm{ion}}(u)f(u)+\left.\sqrt{\displaystyle \frac{{m}_{e}}{2e}}\displaystyle \frac{\langle {\nu }_{i,\mathrm{ion}}\rangle }{N}\delta (u)\right],\end{array}\end{eqnarray} \tag{ 9 }$$

corresponding to the inelastic part of (7a) plus an extra term that describes the introduction of new electrons at zero energy; (iv) taking ionisation as a non-conservative mechanism, and prescribing equal-energy-sharing between the secondary and the scattered electrons, introduced at energy $(u^{\prime} -{V}_{i,\mathrm{ion}})/2$. In this case the SDCS for the secondary electrons is ${\sigma }_{i,\mathrm{ion}}^{\sec }(u^{\prime} ,u)={\sigma }_{i,\mathrm{ion}}(u^{\prime} )\delta [u-(u^{\prime} -{V}_{i,\mathrm{ion}})/2]$ and the ionisation operator becomes

$$\begin{eqnarray}\begin{array}{rcl}{S}_{i,\mathrm{ion}}(u) & = & {\delta }_{i}\left[4(2u+{V}_{i,\mathrm{ion}}){\sigma }_{i,\mathrm{ion}}(2u+{V}_{i,\mathrm{ion}})f(2u+{V}_{i,\mathrm{ion}})\right.\\ & & -\left.u{\sigma }_{i,\mathrm{ion}}(u)f(u)\right],\end{array}\end{eqnarray} \tag{ 10 }$$

where the first term of ${S}_{i,\mathrm{ion}}(u){du}$ describes the entrance of both the secondary and the scattered electrons at a total energy between $2u$ and $2u+2{du}$, produced by a primary electron with energy $u^{\prime} =2u+{V}_{i,\mathrm{ion}}$, whereas the second term describes the exit of primary electrons with energy u after ionizing collisions.

Note that the current version of LoKI-B does not include any external sources of electrons, such as electron beam injection or photoionization. However, the model adopted in the code can be easily upgraded as to consider these effects: (i) on the right-hand side of the EBE, with the addition, to the source function S, of an energy-resolved gain-term; (ii) on the left-hand side of the EBE, with the addition of the corresponding energy-integral term, describing the growth of the electron density in a stationary situation. The inclusion of these external sources of electrons, as well as the handling of the non-conservative electron-collision mechanisms (particularly the attachment), must be carefully monitored in all cases, since they can be responsible for the development of significant anisotropies, in which case the two-term expansion of the EBE adopted here is no longer valid.

LoKI-B adopts specific models to define the total momentum-transfer cross section, σ_c, and the SDCS for the ionisation of level $i,{\sigma }_{i,\mathrm{ion}}$.

The momentum-transfer cross section, for an electron-neutral collision described by a differential (angle-resolved) cross section σ(u, θ), is generally defined as [72]

$$\begin{eqnarray}&&{\sigma }_{c}(u)\equiv \frac{1}{2}{\int }_{0}^{\pi }\sigma (u,\theta )(1-\cos \theta )\sin \theta d\theta .\end{eqnarray} \tag{ 11 }$$

For a gas k, the electron-neutral momentum-transfer cross section σ_k,c(u) is constructed by weighting the contributions of the different types of collisional mechanisms to give

$$\begin{eqnarray}&&{\sigma }_{k,c}(u)=\displaystyle \sum _{i}{\xi }_{{k}_{i}}{\sigma }_{{k}_{i},c}^{\mathrm{el}}(u)+\displaystyle \sum _{i,j\gt i}\left[{\xi }_{{k}_{i}}{\sigma }_{{k}_{(i,j)},c}(u)+{\xi }_{{k}_{j}}{\sigma }_{{k}_{(j,i)},c}(u)\right],\end{eqnarray} \tag{ 12 }$$

where ${\sigma }_{{k}_{i},c}^{\mathrm{el}},{\sigma }_{{k}_{(i,j)},c}$ and ${\sigma }_{{k}_{(j,i)},c}$ are the electron-neutral momentum-transfer cross sections for the elastic scattering of level k_i, the inelastic excitation ${k}_{i}\to {k}_{j}$, and the superelastic de-excitation ${k}_{j}\to {k}_{i}$, respectively. The latter cross sections, whose magnitudes are usually much smaller than those of ${\sigma }_{{k}_{i},c}^{\mathrm{el}}$, are often taken assuming an isotropic scattering (also due to the lack of data), in which case one can identify the momentum-transfer cross section with the integral cross section (integrated over all scattering angles), i.e. ${\sigma }_{{k}_{(i,j)},c}\simeq {\sigma }_{{k}_{(i,j)}}$. Also, the elastic momentum-transfer cross section is usually identified with that of the highly-populated ground-state of the gas. Therefore

$$\begin{eqnarray}&&{\sigma }_{k,c}(u)\simeq {\sigma }_{k,c}^{\mathrm{el}}(u)+\displaystyle \sum _{i,j\gt i}\left[{\xi }_{{k}_{i}}{\sigma }_{{k}_{(i,j)}}(u)+{\xi }_{{k}_{j}}{\sigma }_{{k}_{(j,i)}}(u)\right].\end{eqnarray} \tag{ 13 }$$

The solution of the EBE (3a)–(4b) requires information on both the elastic and the total momentum-transfer cross sections, but the way to obtain it depends on the momentum-transfer data available for each gas. If the elastic momentum-transfer cross section is available, equation (13) can be used directly to deduce the corresponding total σ_k,c, knowing the electron-scattering inelastic/superelastic cross sections for the transitions considered together with the relevant populations, to be self-consistently determined (if possible) within a kinetic model. Conversely, if the data available is for a total momentum-transfer cross section, which in databases is usually termed as effective ${\sigma }_{k,c}^{\mathrm{eff}}$ (see section 3.3), then the elastic momentum-transfer cross section can be deduced from ${\sigma }_{k,c}^{\mathrm{el}}={\sigma }_{k,c}^{\mathrm{eff}}-{\sum }_{i,j\gt i}\left[{\xi }_{{k}_{i}}^{\mathrm{prescribed}}{\sigma }_{{k}_{(i,j)}}+{\xi }_{{k}_{j}}^{\mathrm{prescribed}}{\sigma }_{{k}_{(j,i)}}\right]$, where the populations ${\xi }_{{k}_{i}}^{\mathrm{prescribed}}$ and ${\xi }_{{k}_{j}}^{\mathrm{prescribed}}$ correspond to some prescribed distribution, coherent with the measured/estimated data of ${\sigma }_{k,c}^{\mathrm{eff}}$, and allowing to obtain a unique result (density independent) for the elastic cross section. By default, LoKI-B adopts a Boltzmann distribution at the gas temperature for ${\xi }_{{k}_{i}}^{\mathrm{prescribed}}$, but the user can modify this choice. The previous comments support the recommendation of replacing, whenever possible, the data for effective momentum-transfer cross sections with that for elastic momentum-transfer cross sections.

The implementation of the ionisation operator (7b) requires information on the SDCS, discriminating its values as a function of the primary and the secondary electron energies. In general, the available experimental data for SDCS is scarce and often measured for only a few energy values of the primary electron. Here, we have adopted the comprehensive set of measurements of Opal et al [73], for the double differential cross section of different gases, resolved on both the energy and the angle of the secondary electron. This pioneering work proposes differential ionization cross sections with a very reliable shape [74], which can be normalized from available experimental data for the corresponding integral cross sections (see (8)). Indeed, after integration over the angles, and by comparing the energy distributions of the secondary electrons for different energies of the primary electrons, these authors proposed a fitting expression for the SDCS

$$\begin{eqnarray}&&{\sigma }_{i,\mathrm{ion}}^{\sec }(u^{\prime} ,u)=\displaystyle \frac{{\sigma }_{i,\mathrm{ion}}(u^{\prime} )}{w\arctan \left[\left(u^{\prime} -{V}_{i,\mathrm{ion}}\right)/(2w)\right]}\displaystyle \frac{1}{1+{\left(u/w\right)}^{\beta }},\end{eqnarray} \tag{ 14 }$$

where β 2 and w is a parameter close to the ionisation threshold V_i,ion, which is set for each gas according to the original estimates [73]. Note that (14) satisfies exactly the expression (8) for the integral cross section ${\sigma }_{i,\mathrm{ion}}$, and therefore it is used in LoKI-B to describe the energy distribution of the electrons involved in ionisation events, using available data (see section 3.3) and mitigating possible normalisation errors. Note also that (14) is not the sole fitting expression proposed for the SDCS in the literature, a possible alternative being given in [75].

LoKI-B solves the EBE (3a)–(4b) given: the working conditions E/N (or T_e, when choosing a generalised Maxwellian as EEDF, see section 3.5), p, T_g, ω/2π and n_e, along with: the χ_k fractions of the various k-gases in the mixture; the distributions of populations ${\xi }_{{k}_{i}},{\xi }_{{k}_{{i}_{v}}}$ and ${\xi }_{{k}_{{i}_{{v}_{J}}}}$ of electronic levels k_i, vibrational levels ${k}_{{i}_{v}}$ and rotational levels ${k}_{{i}_{{v}_{J}}}$, respectively (if applicable); and the sets of cross sections ${\sigma }_{{k}_{i},c}^{\mathrm{el}}$ (or ${\sigma }_{k,c}^{\mathrm{eff}}$, see section 3.2), σ_i,j, ${\sigma }_{i,{\rm{att}}}$ and ${\sigma }_{i,\mathrm{ion}}$, where here i, j stand for any level of any gas in the system. Note that T_g is needed to calculate the elastic and the CAR operators (6b) and (6c), whereas p is required to calculate the total gas density in the particular case of HF excitations (for obtaining the reduced frequency ω/N) and/or if electron-electron collisions are considered in the calculations (for obtaining the ionisation degree n_e/N).

Input parameters are set from a text input-file, in a user-friendly flexible way, e.g. allowing to (i) prepare simulations for either a sole value or a range of values of the applied reduced electric field or electron temperature; (ii) choose among different ionisation models (conservative, no-sharing or equal-energy-sharing) and electron-density growth models (temporal or spatial); (iii) define the distributions of populations directly in the input file, from user-defined text files, or via functions for typical distributions at user-defined temperatures (e.g. Boltzmann or Treanor); (iv) list the filename(s) with the set(s) of cross sections to adopt in the simulations; (v) provide details about the energy grid and the solution algorithm to adopt in the simulations (see section 3.4). The code uses SI units for all physical quantities, except the energies that are expressed in eV (electron-volt) and the reduced fields that are expressed in Td (Townsend; $1\,\mathrm{Td}={10}^{-21}\,{\mathrm{Vm}}^{2}$).

By default, LoKI-B uses electron scattering cross sections obtained from the LXCat open-access website [38]. The cross sections can be assembled into a single file, or given from multiple files. As input, LoKI-B accepts also extra sets of cross sections, which are not used in the calculation of the EEDF, but remain available for integration over the calculated EEDF, to obtain electron macroscopic parameters with interest for various purposes (e.g. global models, spectroscopy analysis, actinometry diagnostics, etc.). The choice of the input data is the responsibility of the users, meaning that the selection of any cross section set (obtained or not from LXCat) must be carefully validated and evaluated according to the specific needs and the conditions of interest for the simulations.

LXCat is organised to provide 'data required for modeling low temperature plasmas' [38] in the most effective way. In the case of electron-neutral scattering cross sections, LXCat provides easy access to complete sets of cross sections, defined as those giving a good description of all electron energy and momentum losses [72], yielding electron swarm parameters in agreement with available measured data (within experimental uncertainties), when used in a two-term Boltzmann solver [72, 76]. Each complete set assembles cross sections for electron collisions with different neutral targets, usually associated with the electronic ground-state of each gas. To date, on LXCat these data are tagged with the name of the corresponding gas (e.g. Ar, N₂, O₂, CO₂, ⋯) and are grouped under the category Ground states, where one can find cross sections for the collisions of electrons with the electronic/vibrational/rotational ground-states of a neutral gas. For example, the datasets N₂ may contain (i) the elastic/effective momentum-transfer cross section for N₂; (ii) excitation cross sections for different electronic levels (including the ionisation level), from the electronic ground-state N₂(X); (iii) excitation cross sections for different vibrational levels N₂(X,v), from the vibrational ground-state N₂(X, v = 0); (iv) a global rotational excitation cross section for N₂(X).

This practical organisation of LXCat can create some difficulties when a detailed discrimination and handling of data is intended. The reason is of technical nature, and it relates with the fact that, to date, electron collisions with the Ground state of a gas are described as a collection of mechanisms e + gas name -> ⋯, with no possibility to specify the internal structure of the electronic ground-state of the gas. To circumvent this limitation, LoKI-B checks the free-field COMMENT, of the LXCat datasets, to look for details on the full structure of reactants and products, for each electron-impact reaction. This workaround is the practical solution to prepare any cross section datafile obtained from LXCat for use in LoKI-B, and it is already implemented in the database IST-Lisbon. For example, in IST-Lisbon the v = 0 to v = 1 vibrational excitation within ground-state N₂(X) is described by LXCat metadata as e + N₂ -> e + N₂(v=0---v=1), but is discriminated in the COMMENT field as e + N₂(X,v=0) -> e + N₂(X,v=1). Note also that, in IST-Lisbon, the cross sections for excitations from vibrational levels N₂(X, v > 0) and from rotational levels N₂(X, v = 0, J), can be found in the datasets termed N₂-vib and N₂-rot, respectively.

The EBE is numerically solved in an uniform energy grid ${u}_{n}=(n-1/2){\rm{\Delta }}u$, defined in the energy interval [0, u_max] with $n=1,2,\ \cdots { \mathcal N }$ cells of fixed step-size ${\rm{\Delta }}u={u}_{\max }/{ \mathcal N }$, each cell n being limited by nodes $n-1/2$ and n + 1/2. The user can define the energy grid in two ways: (i) by setting the maximum energy u_max and the number of cells ${ \mathcal N };$ (ii) by prescribing a number of decades in the fall of the EEDF, between f₁ and ${f}_{{ \mathcal N }}$, as to ensure an automatic adjustment of the energy grid. Typical values for the decade-fall are within 10 to 25, considering not only the use of double precision, but also the order of magnitude of the various quantities (kinetic energy, cross sections) multiplying the EEDF in the calculations. The user is responsible for adequately defining the energy grid, namely by noticing that small energy steps are required to resolve any relevant structure in the cross sections, such as near-threshold variations and resonances, as well as situations involving the presence of a Ramsauer minimum.

Equations (3a)–(4b) are written for each grid-cell, where the isotropic and anisotropic functions f and f¹ are defined, with the corresponding upfluxes G_x (x = E, el, CAR, ee) and cross sections σ_y (y = k, c; k_(i,j)) being set on the grid-nodes. The cross sections are interpolated linearly from the data provided by the user, setting to zero all the values up to the corresponding energy threshold, i.e. ${\sigma }_{{y}_{n}}=0$ for n ≤ INT $\left({V}_{i,j}/{\rm{\Delta }}u\right)$.

The EBE is discretised adopting the finite differences scheme presented in [9]. The result is a system of algebraic equations that can be written under matrix form, with tridiagonal elements coming from the left-hand sides of (3a) and (4a), and some sparse off-tridiagonal elements due to the electron-neutral collisional operator in the right-hand sides. In particular, equations (3a)–(4b) are composed by terms proportional to f, df/du and dG/du whose discretised forms can be generally summarised as follows

$$\begin{eqnarray}&&{\left[a(u)f(u)\right]}_{n}={a}_{n}{f}_{n}\end{eqnarray} \tag{ 15a }$$

$$\begin{eqnarray}&&{\left[a^{\prime} (u)f(u\pm {V}_{i,j})\right]}_{n}=a{{\prime} }_{n}{f}_{n\pm \mathrm{INT}({V}_{i},j/{\rm{\Delta }}u)}\end{eqnarray} \tag{ 15b }$$

$$\begin{eqnarray}&&{\left[b(u)\displaystyle \frac{{df}(u)}{{du}}\right]}_{n}={b}_{n}\displaystyle \frac{{f}_{n+1}-{f}_{n-1}}{2{\rm{\Delta }}u}\end{eqnarray} \tag{ 15c }$$

$$\begin{eqnarray}&&\,\begin{array}{r}\begin{array}{l}{\left.\displaystyle \frac{{dG}_{x}(u)}{du}\right|}_{n;x=E,{\rm{e}}{\rm{l}},{\rm{C}}{\rm{A}}{\rm{R}}}={\left\{\displaystyle \frac{d}{du}\left[{g}_{x}(u)\left(f(u)+{c}_{x}\displaystyle \frac{df(u)}{du}\right)\right]\right\}}_{n}\\ \,=\,\left[\displaystyle \frac{{g}_{{x}_{n-1/2}}}{{\rm{\Delta }}u}\left(\displaystyle \frac{{c}_{x}}{{\rm{\Delta }}u}-\displaystyle \frac{1}{2}\right)\right]{f}_{n-1}\\ \,\,-\,\left[\displaystyle \frac{{g}_{{x}_{n+1/2}}}{{\rm{\Delta }}u}\left(\displaystyle \frac{{c}_{x}}{{\rm{\Delta }}u}-\displaystyle \frac{1}{2}\right)+\displaystyle \frac{{g}_{{x}_{n-1/2}}}{{\rm{\Delta }}u}\left(\displaystyle \frac{{c}_{x}}{{\rm{\Delta }}u}+\displaystyle \frac{1}{2}\right)\right]{f}_{n}\\ \,\,+\,\left[\displaystyle \frac{{g}_{{x}_{n+1/2}}}{{\rm{\Delta }}u}\left(\displaystyle \frac{{c}_{x}}{{\rm{\Delta }}u}+\displaystyle \frac{1}{2}\right)\right]{f}_{n+1}\end{array}\end{array}\end{eqnarray} \tag{ 15d }$$

$$\begin{eqnarray}\begin{array}{rcl}{\left.\displaystyle \frac{{{dG}}_{{ee}}(u)}{{du}}\right|}_{n} & = & {\left\{\displaystyle \frac{d}{{du}}\left[{ \mathcal I }(u)f(u)+{ \mathcal J }(u)\displaystyle \frac{{df}(u)}{{du}}\right]\right\}}_{n}\\ & = & {A}_{n-1}{f}_{n-1}-\left({A}_{n}+{B}_{n}\right){f}_{n}+{B}_{n+1}{f}_{n+1},\end{array}\end{eqnarray} \tag{ 15e }$$

where $a(u),a^{\prime} (u)$ and b(u) are generic functions of u; c_x is a constant; ${g}_{x}(u),{ \mathcal I }(u)$ and ${ \mathcal J }(u)$ can be obtained straightforwardly from (6a)–(6d); and the coefficients A_n and B_n are linear functions of the EEDF given by

$$\begin{eqnarray}&&{A}_{n}=-\displaystyle \frac{{{ \mathcal I }}_{n+1/2}}{2{\rm{\Delta }}u}+\displaystyle \frac{{{ \mathcal J }}_{n+1/2}}{{\left({\rm{\Delta }}u\right)}^{2}}\equiv \displaystyle \sum _{m=1}^{{ \mathcal N }}{a}_{n,m}{f}_{m}\end{eqnarray} \tag{ 16a }$$

$$\begin{eqnarray}&&{B}_{n}=\displaystyle \frac{{{ \mathcal I }}_{n-1/2}}{2{\rm{\Delta }}u}+\displaystyle \frac{{{ \mathcal J }}_{n-1/2}}{{\left({\rm{\Delta }}u\right)}^{2}}\equiv \displaystyle \sum _{m=1}^{{ \mathcal N }}{b}_{n,m}{f}_{m}.\end{eqnarray} \tag{ 16b }$$

The previous equations are subjected to the following conditions [9]: (i) the flux boundary conditions G(0) = G(u_max) = 0, which imply ${g}_{{x}_{1/2}}={g}_{{x}_{{ \mathcal N }+1/2}}=0$ and ${A}_{0}\,={A}_{{ \mathcal N }}={B}_{1}={B}_{{ \mathcal N }+1}=0;$ (ii) the energy conservation condition for dG_ee/du, which yields a_n,m = b_m,n; and (iii) the imposition of a Maxwellian EEDF for dG_ee/du = 0, which is ensured by setting ${a}_{n,m}={a}_{m-1,n+1}$.

After discretisation, the isotropic part of the EBE can be written as ${\sum }_{m=1}^{{ \mathcal N }}{C}_{n,m}(f){f}_{m}=0$, which is solved coupled to the normalisation condition ${\sum }_{m=1}^{{ \mathcal N }}{f}_{m}\sqrt{{u}_{m}}{\rm{\Delta }}u=1$. The matrix C(f) is nonlinear due to the presence of the following terms Firconsecutive iterations; (ii) a Newton-Raphson-based approachst, the temporal/spatial growth-term of the electron density, associated with non-conservative mechanisms, involves either the effective ionisation rate coefficient $\langle {\nu }_{\mathrm{eff}}\rangle /N$ or the effective Townsend reduced coefficient α_eff/N, both being functions of the EEDF (see section 3.5). Second, the expression of the electron-electron upflux G_ee involves linear functions of the EEDF, as confirmed by (6d) and (16a)–(16b).

In the absence of nonlinear effects (e.g., for electropositive gases, with the ionization taken as a conservative mechanism and no electron-electron collisions), the EBE is solved adopting a very effective direct solution method. Conversely, when the matrix C(f) contains nonlinear terms, the EBE must be solved using some iterative method. In the case of weak nonlinear effects (e.g., low electronegativity and low ionization degree), the solution can be obtained by using two nested iterative algorithms, based on successive direct solutions. The algorithms adopt (i) a mixing of solutions to converge over $\langle {\nu }_{\mathrm{eff}}\rangle /N$ or α_eff/N, looking for relative differences of the EEDF below a user-prescribed value (typically 10⁻⁹), between two consecutive iterations; (ii) a Newton-Raphson-based approach that converges over the EEDF, looking for relative variations below a user-prescribed value (typically 10⁻⁹), between two consecutive iterations. Final convergence checks also the ratio of the power 'dissipated' in electron-electron collisions to the power gained from the applied electric field Θ_ee/Θ_E, looking for values of this ratio below ${10}^{-9}\mbox{--}{10}^{-10}$. In the case of strong nonlinear effects, the user is recommended to choose a relaxation method for solving the EBE. Here, the default convergence criterion demands relative variations of the EEDF below a user-prescribed value (typically 10⁻⁹), between two consecutive iterations.

On output, LoKI-B calculates the isotropic and the anisotropic parts of the electron distribution function, in addition to various electron macroscopic parameters obtained by integration over the EEDF, namely: (i) swarm parameters (transport parameters and rate coefficients), which can be used to adjust complete sets of electron-scattering cross sections [47, 76] or as input parameters in macroscopic (fluid/global) plasma models [1, 77, 78]; (ii) power-transfer terms [1, 70], providing the distribution of power among the different collisional channels and controlling the calculation errors.

The rate coefficients for excitation/de-excitation mechanisms, between levels i and j > i, are given by [1]

$$\begin{eqnarray}&&{C}_{i,j}=\sqrt{\displaystyle \frac{2e}{{m}_{e}}}{\int }_{0}^{\infty }u{\sigma }_{i,j}(u)f(u){du}\end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray}&&{C}_{j,i}=\sqrt{\displaystyle \frac{2e}{{m}_{e}}}{\int }_{0}^{\infty }\displaystyle \frac{{g}_{i}}{{g}_{j}}(u+{V}_{i,j}){\sigma }_{i,j}(u+{V}_{i,j})f(u){du}.\end{eqnarray} \tag{ 17b }$$

Note that (17a)–(17b) can be applied not only to the cross sections adopted for solving the EBE, but also to the extra set of cross sections defined by the user as additional input data of LoKI-B (see section 3.3).

The ionisation and the attachment rate coefficients are defined similarly to (17a) as

$$\begin{eqnarray}&&{C}_{i,\mathrm{ion}}\equiv \displaystyle \frac{\langle {\nu }_{i,\mathrm{ion}}\rangle }{N}=\sqrt{\displaystyle \frac{2e}{{m}_{e}}}{\int }_{0}^{\infty }u{\sigma }_{i,\mathrm{ion}}(u)f(u){du}\end{eqnarray} \tag{ 18a }$$

$$\begin{eqnarray}&&{C}_{i,\mathrm{att}}\equiv \displaystyle \frac{\langle {\nu }_{i,\mathrm{att}}\rangle }{N}=\sqrt{\displaystyle \frac{2e}{{m}_{e}}}{\int }_{0}^{\infty }u{\sigma }_{i,\mathrm{att}}(u)f(u){du}.\end{eqnarray} \tag{ 18b }$$

For simulations involving the time-growth of the electron density, equations (18a)–(18b) are solved within each iteration on the convergence parameter $\langle {\nu }_{\mathrm{eff}}\rangle /N$ (see section 3.4).

The reduced transverse free-diffusion coefficient and the reduced mobility are calculated from [1, 70]

$$\begin{eqnarray}&&{D}_{e}N=\displaystyle \frac{1}{3}\sqrt{\displaystyle \frac{2e}{{m}_{e}}}{\int }_{0}^{\infty }\displaystyle \frac{u}{{{\rm{\Omega }}}_{x}(u)}f(u){du}\end{eqnarray} \tag{ 19a }$$

$$\begin{eqnarray}&&{\mu }_{e}N=-\displaystyle \frac{1}{3}\sqrt{\displaystyle \frac{2e}{{m}_{e}}}{\int }_{0}^{\infty }\displaystyle \frac{u}{{{\rm{\Omega }}}_{x}(u)}\displaystyle \frac{{df}(u)}{{du}}{du}.\end{eqnarray} \tag{ 19b }$$

In (19a), x = c, SST for temporal or spatial growth of the electron density, respectively. Equation (19b) corresponds either to the DC mobility, at ω = 0 and $x=\mathrm{PT},\ \mathrm{SST}$ for temporal or spatial growth of the electron density, respectively; or to the real part of the HF mobility, at ω ν_c and x = PT. Equations (3b) and (4b) can be integrated to define the electron mean drift velocity ${v}_{d}\equiv \sqrt{2e/{m}_{e}}{\int }_{0}^{\infty }(u/3){f}^{1}(u){du}$, yielding

$$\begin{eqnarray}&&{v}_{d}={\mu }_{e}E,\,{\rm{for}}\,{\rm{a}}\,{\rm{PT}}\,{\rm{situation}}\end{eqnarray} \tag{ 20a }$$

$$\begin{eqnarray}\begin{array}{rcl}{v}_{d} & = & {\mu }_{e}E-{D}_{e}\displaystyle \frac{1}{{n}_{e}}\displaystyle \frac{{{dn}}_{e}}{{dz}}\\ & = & {\mu }_{e}E-{D}_{e}{\alpha }_{\mathrm{eff}},\,{\rm{for}}\,{\rm{a}}\,{\rm{SST}}\,{\rm{situation}},\end{array}\end{eqnarray} \tag{ 20b }$$

and equation (4a) can be integrated to define the effective Townsend coefficient

$$\begin{eqnarray}&&{\alpha }_{\mathrm{eff}}=\displaystyle \frac{\langle {\nu }_{\mathrm{ion}}\rangle -\langle {\nu }_{\mathrm{att}}\rangle }{{v}_{d}}=\displaystyle \frac{\langle {\nu }_{\mathrm{eff}}\rangle }{{v}_{d}}.\end{eqnarray} \tag{ 21 }$$

For DC simulations involving the spatial-growth of the electron density, equations (20b) and (21) can be combined to yield

$$\begin{eqnarray}&&{D}_{e}{\alpha }_{\mathrm{eff}}^{2}-{\mu }_{e}E{\alpha }_{\mathrm{eff}}+\langle {\nu }_{\mathrm{eff}}\rangle =0,\end{eqnarray} \tag{ 22 }$$

and in this case (22) is solved within each iteration on the convergence parameter α_eff (see section 3.4).

The balance equation for the electron power-density, per electron at unit gas density, is obtained by multiplying (3a) or (4a) by $u\sqrt{2e/{m}_{e}}$ and integrating it over all energies. The result writes (in eV s⁻¹ m³)

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{growth}}}{N}+\displaystyle \frac{{{\rm{\Theta }}}_{E}}{N}+\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{coll}}^{\mathrm{gain}}}{N}+\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{coll}}^{\mathrm{loss}}}{N}=0,\end{eqnarray} \tag{ 23 }$$

where the different terms represent, in order, the variation of the power density due to non-conservative mechanisms inducing a time/space net growth of the electron density, the power density gained from the applied electric field, and the power density gained/lost in collisional events, respectively

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{growth}}}{N}=\left\{\begin{array}{l}-\tfrac{\langle {\nu }_{\mathrm{eff}}\rangle }{N}\varepsilon ,\,{\rm{for}}\,{\rm{a}}\,{\rm{PT}}\,{\rm{situation}}\\ -\tfrac{{\alpha }_{\mathrm{eff}}}{N}\left({\mu }_{\varepsilon }E-{D}_{\varepsilon }{\alpha }_{\mathrm{eff}}\right),\,{\rm{for}}\,{\rm{a}}\,{\rm{SST}}\,{\rm{situation}}\end{array}\right.\end{eqnarray} \tag{ 24a }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{E}}{N}=\displaystyle \frac{{v}_{d}}{\zeta }\displaystyle \frac{E}{N}\end{eqnarray} \tag{ 24b }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{x=\mathrm{el},\mathrm{CAR}}^{\mathrm{gain}}}{N}=-{\int }_{0}^{\infty }{g}_{x}(u)c\displaystyle \frac{{df}(u)}{{du}}{du}\end{eqnarray} \tag{ 24c }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{{ee}}^{\mathrm{gain}}}{N}=-{\int }_{0}^{\infty }{ \mathcal J }(u)\displaystyle \frac{{df}(u)}{{du}}{du}\end{eqnarray} \tag{ 24d }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{\sup }^{\mathrm{gain}}}{N}=\displaystyle \sum _{i,j\gt i}{\delta }_{j}{C}_{j,i}{V}_{i,j}\end{eqnarray} \tag{ 24e }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{x=\mathrm{el},\mathrm{CAR}}^{\mathrm{loss}}}{N}=-{\int }_{0}^{\infty }{g}_{x}(u)f(u){du}\end{eqnarray} \tag{ 24f }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{{ee}}^{\mathrm{loss}}}{N}=-{\int }_{0}^{\infty }{ \mathcal I }(u)f(u){du}\end{eqnarray} \tag{ 24g }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{inel}}^{\mathrm{loss}}}{N}=-\displaystyle \sum _{i,j\gt i}{\delta }_{i}{C}_{i,j}{V}_{i,j}.\end{eqnarray} \tag{ 24h }$$

In these equations, the labels 'inel' and 'sup' refer to inelastic and superelastic collisions, respectively, associated with rotational, vibrational, electronic excitation/de-excitation and ionisation/attachment mechanisms. In (24a), $\varepsilon \equiv {\int }_{0}^{\infty }{u}^{3/2}f(u){du}$ is the electron mean energy, and D_ε and μ_ε are the diffusion coefficient and the mobility for the electron-energy flux, respectively, defined by expressions similar to (19a) and (19b) where the integrand function is further multiplied by u.

The parameters defined by (17a)–(24h) can be calculated adopting the distribution function f(u), as obtained from the solution to the EBE, or some prescribed EEDF, such as a generalised Maxwellian [79–81]

$$\begin{eqnarray}&&f(u)={\left(\displaystyle \frac{2}{3{k}_{B}{T}_{e}}\right)}^{3/2}\displaystyle \frac{{\rm{\Gamma }}{\left(\tfrac{5}{2s}\right)}^{3/2}}{{\rm{\Gamma }}{\left(\tfrac{3}{2s}\right)}^{5/2}}\exp \left[-{\left(\displaystyle \frac{{\rm{\Gamma }}\left(\tfrac{5}{2s}\right)}{{\rm{\Gamma }}\left(\tfrac{3}{2s}\right)}\displaystyle \frac{2{eu}}{3{k}_{B}{T}_{e}}\right)}^{s}\,\right],\end{eqnarray} \tag{ 25 }$$

where Γ represents the gamma-function, T_e is the electron kinetic temperature such that $({k}_{B}{T}_{e})/e\equiv (2/3)\varepsilon$, and s is a positive real number that allows shaping the EEDF. In particular, (25) yields a Maxwellian EEDF and a Druyvesteyn EEDF for s = 1 and s = 2, respectively.

The two-term approximation adopted in LoKI-B to solve the EBE is valid in a situation of small anisotropies [1], corresponding to electron–neutral mean-free-paths much smaller than any characteristic dimension of the plasma container, and an energy gain from the electric field, between collisions, much smaller than the electron kinetic energy. For a homogeneous scenario like the one described here, the latter condition is the most relevant and can be approximately written as (see (20a) and (24b))

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{E}\sim {\mu }_{e}{E}^{2}\ll \varepsilon {\nu }_{c},\end{eqnarray} \tag{ 26 }$$

which yields, for a Maxwellian EEDF at 1 eV temperature and a typical reduced collision frequency ${\nu }_{c}/N\sim {\sigma }_{c}\langle v\rangle \sim {10}^{-19}\,\times {10}^{6}={10}^{-13}$ m³ s⁻¹, $E/N\ll \sqrt{{m}_{e}\varepsilon /e}({\nu }_{c}/N)\sim 300$ Td. This is a conservative limit for the reduced electric-field, often exceeded in calculations because the two-term approximation yields fairly good results still. However, users are advised to check the EEDF results when taking E/N values much above the estimated validity limit, namely by comparing the solution obtained for the isotropic and the anisotropic parts of the electron distribution function. Also, HF calculations are valid only if the oscillation frequency of the electric field is much larger than the typical frequency for the electron energy relaxation, which can be approximately written as (see (6b))

$$\begin{eqnarray}&&\omega \gg \gamma N({\nu }_{c}/N),\end{eqnarray} \tag{ 27 }$$

corresponding to ω/(2π) 20 MHz–20 kHz for ν_c/N ~ 10⁻¹³ m³ s${}^{-1},\gamma \sim 5\times {10}^{-5},{T}_{g}=300$ K and p ~ 10⁵ Pa–130 Pa. Again, users are advised to check the validity of the HF approximation for the particular working conditions considered.

LoKI-B contains several control features upon the input data and the calculation results, and in some cases it returns warning messages for the benefit of users. In particular: (i) the code controls the maximum value u_max of the energy grid adopted, comparing it with the cutoff energy u_cutoff of the electron-neutral elastic momentum-transfer cross section. When u_cutoff < u_max a warning message is returned, and if the user accepts to proceed with the calculations these are carried out by setting to zero the values of any cross section from its corresponding u_cutoff and up to u_max. LoKI-B does not extrapolate cross sections beyond the cutoff energies defined in the input data-files, again because it adopts an ontology that separates the tool and the data. The user is recommended to always careful examine the cross sections being used and, if needed, he/she should adopt adequate laws to extrapolate them, chosen according to the specific processes considered (e.g. dipole allowed or dipole forbidden electron excitation, ionization, etc); (ii) the code returns a warning message if the calculation of the elastic momentum-transfer cross section from the corresponding effective produces negative values for ${\sigma }_{k,c}^{\mathrm{el}}$ (see section 3.2), which are then set to zero. In this case, users are advised to check the cross section data adopted; (iii) the code controls the quality of the Boltzmann-matrix inversion, by monitoring the error upon the fractional electron power-balance. The latter is obtained from (23), by normalizing this equation with respect to a reference power-density Θ_ref, defined for each calculation as the sum of all power-gain terms (24b), (24c), (24e). The relative error $\left[{{\rm{\Theta }}}_{E}+{{\rm{\Theta }}}_{\mathrm{coll}}^{\mathrm{gain}}+{{\rm{\Theta }}}_{\mathrm{coll}}^{\mathrm{loss}}+{{\rm{\Theta }}}_{\mathrm{growth}}\right]/{{\rm{\Theta }}}_{\mathrm{ref}}$ (with typical value ~10⁻¹⁰) degrades for very-low E/Ns, due to the smallness of the power-density values for all the power-transfer channels, and for very-high E/Ns, if the u_max grid-limit adopted in the calculations is not large enough to prevent ill-conditioned EBE-matrices.

An obvious verification of LoKI-B corresponds to the solution of the EBE for a model gas where the sole electron-scattering mechanisms are elastic collisions with the neutrals. The problem can be solved analytically, yielding the well-known Margenau EEDF [82]

$$\begin{eqnarray}&&f(u)\propto \exp \left[-{\int }_{0}^{u}\displaystyle \frac{{du}^{\prime} }{\tfrac{e}{3\gamma {m}_{e}}{\left(\tfrac{E/N}{{\nu }_{c}^{\mathrm{el}}/N}\right)}^{2}+\tfrac{{k}_{B}{T}_{g}}{e}}\right],\end{eqnarray} \tag{ 28 }$$

and for a constant momentum-transfer rate coefficient the result is a Maxwellian EEDF at temperature

$$\begin{eqnarray}&&{T}_{e,\mathrm{Mxw}}\equiv {T}_{g}+\displaystyle \frac{1}{3{k}_{B}\gamma {m}_{e}}{\left(\displaystyle \frac{{eE}/N}{{\nu }_{c}^{\mathrm{el}}/N}\right)}^{2}.\end{eqnarray} \tag{ 29 }$$

For this verification test we consider a model gas with ${\nu }_{c}^{\mathrm{el}}/N={10}^{-12}$ m³ s⁻¹ and M = 40 amu, running LoKI-B for E/N values between 0 and 50 Td, adjusting the maximum value of the energy grid as to ensure a fall of 15 decades for the EEDF (see section 3.4). Figure 1 shows the EEDF and the ratio of the anisotropic-to-isotropic components of the electron distribution function, at E/N = 10 Td (corresponding to ${k}_{B}{T}_{e}/e=4.53$ eV) and ${ \mathcal N }=1000$ cells. As expected, the fractional anisotropic component of the electron distribution function increases with the electron energy, yet remaining below the unit.

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To evaluate quantitatively the deviation of the EEDF with respect to a Maxwellian distribution function, f_Mxw, one can follow different approaches. The simplest is to monitor the differences of the electron kinetic temperatures obtained from the curve in figure 1 and calculated from (29), ${\rm{\Delta }}{T}_{e}/{T}_{e,\mathrm{Mxw}}\,\equiv | {T}_{e}-{T}_{e,\mathrm{Mxw}}| /{T}_{e,\mathrm{Mxw}}$. One can also compare directly the numerical and the analytical EEDFs, as shown in figure 2, which presents the relative error between the distribution functions and the mean value of this error over the entire kinetic-energy scale, $\overline{{\rm{\Delta }}f/{f}_{\mathrm{Mxw}}}\equiv \overline{| {f}_{\mathrm{numerical}}(u)-{f}_{\mathrm{Mxw}}(u)| /{f}_{\mathrm{Mxw}}(u)}$. Note that the steady increase in the relative error of f(u), as the kinetic energy increases, is due to the division by an exponentially decreasing function, as was confirmed by calculating the corresponding absolute errors. The relative errors for T_e and f(u) are plotted in figure 3, as a function of E/N at constant ${ \mathcal N }=1000$ cells, revealing small standard deviations σ around average values $\langle .\rangle$ between $5\times {10}^{-4}$ and 10⁻³ respectively: $\langle {\rm{\Delta }}{T}_{e}/{T}_{e,\mathrm{Mxw}}\rangle \,\pm {\sigma }_{{T}_{e}}$ = $5.33\times {10}^{-4}\pm {10}^{-6}$ and $\langle \overline{{\rm{\Delta }}f/{f}_{\mathrm{Mxw}}}\rangle \pm {\sigma }_{f}=1.57\,\times {10}^{-3}\pm 8\times {10}^{-6}$.

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The small error-dispersion observed in figure 3 is preserved for other values of ${ \mathcal N }$ but, as expected, it is possible to decrease the relative error by increasing the number of grid-cells. Figure 4 shows the evolution of the average relative error for the electron temperature and the EEDF, as a function of the number of grid-cells, and from the data in this log-log representation one can estimate the order of the algorithm adopted in the calculations. Indeed, the slope of the trend line with the squares is approximately 2, which is coherent with the use of a second-order-accurate discretization scheme in solving the EBE, whereas the slope of the trend line with the triangles is roughly 1.6, thus revealing that the evaluation of the electron temperature is related not only with the discretization scheme adopted for the EBE, but also with the integration routine used to obtain macroscopic values from the EEDF (first-order mid-point quadrature rule).

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A second verification check of LoKI-B analyses the behavior of the electron-electron collision operator (6d) for an increasing influence of these collisions, namely due to an increase in the electron density. For this verification check we consider a Reid-type model gas [43] with mass M = 4 amu, temperature T_g = 300 K and the following electron-scattering cross sections: constant elastic momentum-transfer cross section ${\sigma }_{c}^{\mathrm{el}}=6\times {10}^{-20}$ m²; ramp inelastic cross section with threshold 0.2 eV and slope 10⁻¹⁹ m² eV⁻¹. Figure 5 shows the EEDF calculated at E/N = 10 Td and ${ \mathcal N }=1000$ cells, for electron densities varying between 10¹⁷ and 10²² m⁻³, corresponding to ionisation degrees ${n}_{e}/N={10}^{-6}-{10}^{-1}$ at p = 414 Pa and T_g = 300 K. As expected, the EEDF approaches a Maxwellian distribution function with the increase in the ionisation degree, which confirms the correct behavior for the operator G_ee.

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The development of LoKI-B involves also its benchmarking against numerical calculations obtained with other codes, namely the very popular BOLSIG+ [27]. For these comparisons we have adopted the same Reid model gas as before [43], except for the value of the gas temperature taken equal to 0 K, according to the definition settings of this model gas. The calculations consider E/N values between 1 and 100 Td, adopting an energy grid with ${ \mathcal N }=1000$ cells, adjusted as to ensure a 13 decade-fall for the EEDF.

Figures 6(a)–(b) shows, as a function of E/N, the electron reduced mobility, as the example of an electron macroscopic parameter calculated with LoKI-B and BOLSIG+, and the relative difference between the predictions of the codes for ${\mu }_{e}N,{D}_{e}N$ and ε. In general, the results obtained with LoKI-B and BOLSIG+ are in good agreement, within errors of ${10}^{-2} \% -1 \%$ ($\langle {\rm{\Delta }}X/\langle X \rangle\rangle \pm {\sigma }_{X}=2.20\times {10}^{-3}\pm 3\times {10}^{-6}$, $4\times {10}^{-4}\pm 3\times {10}^{-4},2\times {10}^{-3}\pm 2\times {10}^{-3}$, for $X={\mu }_{e}N,{D}_{e}N,\varepsilon$, respectively).

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As mentioned previously (see section 3.5), swarm parameters can be used to adjust complete sets of electron-scattering cross sections, and this procedure of code/data validation was also adopted in the development of LoKI-B, by comparing calculation results with available measurements for these parameters.

As an example for an atomic gas, figures 7(a)–(b) shows, as a function of E/N, calculations and measurements of the electron reduced mobility and the electron characteristic energy ${u}_{k}\equiv {D}_{e}/{\mu }_{e}$ of argon, obtained for T_g 300 K, 77 K. LoKI-B calculations used the Ar complete set of cross sections, published in the IST-Lisbon database of LXCat [83]. The results in this figure reproduce the typical validation agreement between calculations and measurements, within the experimental uncertainty.

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Argon was used also to analyse the impact, on the EEDF and the electron swarm parameters, of the different descriptions available for the ionisation (see section 3.1): using a SDCS given by (14), according to Opal's model [73]; prescribing specific distributions of the energy available after each ionisation, between the secondary and the scattered electrons, namely by assuming no-sharing of energy with the secondary electron or equal-energy-sharing between both electrons; taking ionisation as a conservative mechanism, with no production of secondary electron. Figures 8(a)–(b) plots the EEDF at E/N = 300 Td and the first reduced Townsend coefficient as a function of E/N, for an argon plasma. LoKI-B calculations used the IST-Lisbon cross section set, and adopted the four descriptions mentioned above for the ionisation mechanism, assuming a spatial-growth of the electron density when considering the production of secondary electrons due to ionisation events.

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An observation of figure 8(a) shows the following: (i) for all non-conservative descriptions, regardless of the specific energy-sharing-model adopted (SDCS, no-sharing or equal-energy-sharing), the introduction of secondary electrons at low energies is responsible for an enhancement of the body of the EEDF and a simultaneous depletion of its tail, as to yield a normalised distribution; (ii) the description no-sharing, given by (9), produces the highest enhancement of the EEDF, in the region close to u = 0. This description is partly identical to the treatment of electron-impact ionisations as a conservative mechanism, the difference being associated with the introduction of secondary electrons at zero energy. In this case the scattered electrons have the largest possible energy $u^{\prime} -{V}_{\mathrm{ion}}$, causing the EEDF tail to be slightly above all other solutions for non-conservative descriptions; (iii) both SDCS and equal-energy-sharing descriptions induce a larger distribution of energy between the secondary and the scattered electrons, leading to the corresponding decrease of the EEDF in the region close to u = 0.

The changes observed in the EEDF have consequences upon the swarm parameters, namely the first Townsend coefficient. Figure 8(b) shows that, as expected for high E/N, the calculated values of α/N are smaller (and in better agreement with the available measurements), when electron-impact ionisations are described as a non-conservative mechanism.

The procedure of validation of a complete set of electron-scattering cross sections using swarm parameters was applied also to molecular gases. For example, figure 9 shows, as a function of E/N, calculations and measurements of the electron reduced mobility of nitrogen, obtained for T_g 300 K. LoKI-B calculations used the N₂ complete set of electron-scattering cross sections, published in the IST-Lisbon database of LXCat [106]. The N₂ cross sections describe the elastic momentum-transfer due to electron collisions with the ground-state N₂(X); the excitation of electronic levels N₂(A${}^{3}{{\rm{\Sigma }}}_{u}^{+}$, ${{\rm{B}}}^{3}{{\rm{\Pi }}}_{g}$, ${{\rm{W}}}^{3}{{\rm{\Delta }}}_{u}$, ${{\rm{B}}}^{3}{{\rm{\Sigma }}}_{u}^{-}$, ${{\rm{a}}}^{1}{{\rm{\Sigma }}}_{u}^{-}$, ${{\rm{a}}}^{1}{{\rm{\Pi }}}_{g}$, ${{\rm{w}}}^{1}{{\rm{\Delta }}}_{u}$, ${{\rm{C}}}^{3}{{\rm{\Pi }}}_{u}$, ${{\rm{E}}}^{3}{{\rm{\Sigma }}}_{g}^{+}$, ${{\rm{a}}}^{1}{{\rm{\Sigma }}}_{g}^{+}$, higher-singlets) from the ground-state N₂(X); the excitation of vibrational levels N₂(X, v = 1–10) from the vibrational ground-state N₂(X, v = 0); and the ionisation of N₂(X). For simulations at low E/N values, the dataset N₂ is complemented with the description of the rotational excitation/de-excitation of the ground-state N₂(X, v = 0) by electron impact, i.e. e + N₂(X, v = 0, J) $\leftrightarrow$ e + N₂(X, v = 0, J+2). As mentioned, this description can adopt either the continuous (CAR) approach, using the G_CAR rotational operator (6c), or a discrete approach that considers a set of cross sections for rotational transitions involving the levels N₂(X, v = 0, J = 0–32), which corresponds to the N₂-rot dataset published also in the IST-Lisbon database. The results in figure 9 reveal a good agreement between calculations and measurements, within the experimental uncertainty, also validating the equivalence between the continuous and the discrete descriptions of rotational excitation/de-excitation mechanisms for nitrogen.

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Nitrogen plasmas can be used also to demonstrate the distribution of the electron energy between various gain/loss channels. Figures 10(a)–(b) plots, as a function of E/N, the fractional power-transfer Θ_x/Θ_ref due to the different phenomena considered in the EBE (x = E, el, inel, sup, growth), and the largest power-densities per electron at unit gas density Θ_x/N ($x=E,\sup -\mathrm{rotational}$), respectively. LoKI-B calculations used the same data and rotational descriptions as before. Figure 10(a) discriminates between electron power-gain mechanisms (positive values) and electron power-loss mechanisms (negative) (see (24a)–(24h)), and in the case of inelastic/superelastic electron collisions it further discriminates the various collisional channels, i.e., rotational, vibrational, electronic excitation/de-excitation, and ionisation. As indicated, the results in this figure are normalised to the sum of the different power-gain terms, ${{\rm{\Theta }}}_{\mathrm{ref}}$ (see sections 3.5 and 3.6). Here, ${{\rm{\Theta }}}_{\mathrm{ref}}\simeq {{\rm{\Theta }}}_{\sup -\mathrm{rotational}}$ for E/N < 0.1 Td (when adopting the CAR approach) or E/N < 1 Td (when using the discrete approach for rotations), and above these field values ${{\rm{\Theta }}}_{\mathrm{ref}}\simeq {{\rm{\Theta }}}_{E}$, as can be confirmed from figure 10(b) or by checking, for each E/N, which mechanism gives ${{\rm{\Theta }}}_{x}/{{\rm{\Theta }}}_{\mathrm{ref}}=1$ in figure 10(a). An observation of figure 10(b) reveals that CAR yields a slightly smaller power-transfer to the rotational channel, which results in the relative differences depicted in figure 10(a) between CAR and the discrete approach for rotations, in the range 0.1 Td $\lesssim \,E/N\lesssim 10$ Td. As expected, for reduced electric fields above 1 Td the electron power-losses are dominated by the vibrational and the electronic collisional channels.

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One of the main features of LoKI-B is the flexibility of operation with mixtures of atomic/molecular gases. This section presents results of the electron kinetics for nitrogen-oxygen and air-argon mixtures, including an example of validation of swarm parameters, the analysis of the effect of vibrational distribution functions (VDF) upon the EEDF, and the influence, on the EEDF and the electron power-transfer, of admixing dry-air to argon plasmas.

Figure 11 shows, as a function of E/N, calculations and measurements of the electron reduced mobility for dry-air (80% N₂–20% O₂), obtained for T_g 300 K. LoKI-B calculations used data published in the IST-Lisbon database of LXCat [106, 119], namely the N₂ and the O₂ complete sets of electron-scattering cross sections, complemented by the datasets N₂-rot and O₂-rot for the (discrete) description of rotational transitions. The information on the electron cross sections contained in N₂ and N₂-rot was already given in section 4.3. The O₂ cross sections include the elastic momentum-transfer due to electron collisions with the ground-state O₂(X); the excitation, from the ground-state O₂(X), of electronic levels O₂(a${}^{1}{{\rm{\Delta }}}_{g}$, ${{\rm{b}}}^{1}{{\rm{\Sigma }}}_{g}^{+}$), O₂(A${}^{3}{{\rm{\Sigma }}}_{u}^{+}$, ${{\rm{C}}}^{3}{{\rm{\Delta }}}_{u}$, ${{\rm{c}}}^{1}{{\rm{\Sigma }}}_{u}^{-}$) bound-states and leading to dissociation (e + O₂(X) $\to$ e + O(³P) + O(³P)) and continuum Herzberg bands, O₂(B${}^{3}{{\rm{\Sigma }}}_{u}^{-}$) leading to dissociation (e + O₂(X) $\to$ e + O(³P) + O(¹D)) and continuum Schumann-Runge bands, and O₂(9.97eV, 14.7 eV); the excitation of vibrational levels O₂(X, v = 1-4) from the vibrational ground-state O₂(X, v = 0); the electron dissociative attachment of O₂(X); and the ionisation of O₂(X). The O2-rot cross sections describe the rotational transitions by electron impact involving the levels O₂(X, v = 0, J = 1, 3, 5, ⋯, 31), i.e. e + O₂(X, v = 0, J) $\leftrightarrow$ e + O₂(X, v = 0, J+2). The results in figure 11 reveal good agreement between calculations and measurements, within the experimental uncertainty. For comparison, figure 11 shows also the results of μN in the absence of rotational transitions. Note that the inclusion of these mechanisms is essential to obtain a good agreement with the experiment, at low E/N. In particular, not considering these effects for N₂ (O₂) leads to an underestimation in the computed values of μN by 39% (8%), at 0.1 Td.

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To analyse the effect of the vibrational transitions upon the EEDF, we have considered the VDFs plotted in figure 12, corresponding to (i) a Treanor-Gordiets distribution [120, 121] at T_g = 300 K and T_vib = 4000 K in nitrogen

$$\begin{eqnarray}{\xi }_{v}=\left\{\begin{array}{ll}\exp \left\{-v\left[\tfrac{{E}_{1}-{E}_{0}}{{k}_{B}{T}_{\mathrm{vib}}}-(v-1)\tfrac{{\rm{\Delta }}E}{{k}_{B}{T}_{g}}\right]\right\} & v\leqslant {v}^{\star }\\ {\xi }_{{v}^{\star }}\tfrac{{v}^{\star }}{v} & v\geqslant {v}^{\star }\end{array}\right.,\end{eqnarray} \tag{ 30 }$$

with ${v}^{\star }=1/2\left[1+({E}_{1}-{E}_{0})/{\rm{\Delta }}E+{T}_{g}/{T}_{\mathrm{vib}}\right]$, where E_v (v = 0, 1) is the energy of vibrational levels 0, 1, and Δ E is the anharmonicity factor in eV; (ii) results recently calculated/measured in an oxygen ICP [122], which are representative of such distributions in O₂ regardless of the specific discharge conditions [123–125].

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Figure 13 shows EEDFs calculated with LoKI-B at E/N = 30 Td, considering different N₂-O₂ mixtures with the VDFs plotted in figure 12. The results confirm the relevance of electron-vibrational excitation/de-excitation mechanisms in nitrogen, signalled by the sharp decrease in the body of the EEDF due to a vibrational barrier around 2 eV, which smooths down for increasing percentages of oxygen in the mixture (see the insert in this figure). Simulation results (not shown) confirm also that in oxygen, and for the working conditions considered, electron-vibrational mechanisms have little influence on the EEDF. As expected, the effect of vibrational excitations can be enhanced by setting to zero the populations of the excited vibrational levels, thus removing their influence as energy reservoirs for electron-impact superelastic collisions. The effect can be observed in figure 13, by comparing the EEDFs obtained for air plasmas, either considering the VDFs of figure 12 or by setting to zero the population of all vibrational levels other than the ground-states N₂(X, v = 0) and O₂(X, v = 0).

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Figures 14(a)–(b) analyses the influence, on the EEDF (calculated at 30 Td) and the electron power-transfer, of admixing to argon plasmas a small percentage (below 5%) of dry-air (80% N₂ – 20% O₂, considering the VDFs plotted in figure 12). As expected, the addition of air increases the relevance of the power transferred to vibrational transitions (with the corresponding decrease in the relative power transferred to elastic collisions and to electronic excitations, see figure 14(b)), introducing a more distinctive separation between the EEDF regions below and above ~2 eV (see figure 14(a)), as consequence of the electron-impact vibrational excitations and de-excitations, respectively.

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The response of the electrons in a gas-mixture subjected to an electric-field pulse is a current hot topic in LTPs, mainly due to applications related with environmental remediation, synthesis of value-added chemicals and new fuels, surface treatment and medicine [126–129].

In this context, LoKI-B was used to study the evolution in time of the electron kinetics, when an electric-field pulse with a duration of 10 μs is applied to a 80% N₂–20% O₂ gas mixture (dry-air). The simulations assume a time-dependent reduced electric-field (see figure 15) given by

$$\begin{eqnarray}&&\displaystyle \frac{E(t)}{N}=\displaystyle \frac{{E}_{0}}{N}\sqrt{\displaystyle \frac{t}{{t}_{0}}}{e}^{-t/{t}_{0}},\end{eqnarray} \tag{ 31 }$$

with E₀/N = 100 Td and t₀ = 1 μs, taking as initial condition for the EEDF the zero-field solution of the EBE (which corresponds to a Maxwellian distribution at the gas temperature T_g = 300 K). Moreover, because LoKI-B handles the populations of excited states in a user-friendly way, the study considers the evolution of the EEDF in the presence of Boltzmann distribution functions, at T_g = 300 K, for the vibrational states N₂(X, v) and O₂(X, v) and for the rotational states N₂(X, v = 0, J) and O₂(X, v = 0, J), which are assumed not to change within the 10 μs time-interval of the simulations. The calculations were done following the same strategy adopted in several models [126–129], which consists in solving the time-independent EBE for a range of E/N to produce, as a function of either the reduced electric-field or the electron mean-energy, a table of rate coefficients that can be interpolated at different times during the execution of the model. Here, this quasi-static approximation was implemented by resolving the pulse rise into 500 E/N values, corresponding to time-steps below Δ t = 0.1 μs.

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For these working conditions, figures 16(a)–(c) shows the evolution in time of the EEDF, calculated with LoKI-B at E/N = 0, 20, 43 Td, respectively, corresponding to the values signalled with arrows (I), (II) and (III) in figure 15. Note that figure 16(a) represents the initial condition for the EEDF; note also that the EEDFs in figures 16(b)–(c) exhibit the vibrational barrier usually found in nitrogen plasmas, whose influence in the losses of the electron energy is mitigated with the increase in the electric-field intensity.

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The EEDFs calculated at different times can be used to obtain the time variation of pertinent electron macroscopic parameters, such as rate coefficients and power densities. For the same conditions as before, figures 17(a)–(b) represents, as a function of time, the ionisation and the attachment rate coefficients, and the fractional power transferred from the electric field to the different electron-collision channels. Note the electronegative feature of the gas mixture, for which η/N > α/N during the whole duration of the pulse; note also that, as expected, the major power-transfer channels are due to rotational mechanisms (at low E/N) and vibrational mechanisms (at high E/N, namely around the electric-field peak).

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The results presented here intend to demonstrate the flexibility of LoKI-B to exploit some relevant physical aspects related with the electron kinetics in LTPs. However, for what concerns the evolution in time of the EEDF, e.g. in reactive gas mixtures under the action of an electric-field pulse, a more cautious approached should be considered, involving the solution of the time-dependent EBE (instead of using a quasi-static approximation, based on the solution of its time-independent form), eventually coupled to a time-dependent description of the kinetics of heavy-species, namely if the time-scales considered are above ~100 μs. Work is in progress to upgrade LoKI-B with a time-dependent algorithm for monitoring the temporal evolution of the EEDF.