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\begin{titlepage}
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        {\LARGE \textbf{Investigation of the electron energy distribution function in plasmas generated by microthrusters}} \\
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        % Your name
        \textbf{Artem Andrienko} \\
        \vspace{1.0cm}

        % University name
        \textbf{UIUC} \\
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        % Date (Year only)
        \textbf{2024} \\
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\tableofcontents \pagebreak

\section{Introduction}

To optimize the combustion of various types of fuel under conditions of non-stationary discharge, it is essential to have a clear understanding of the processes occurring in this discharge, as well as the parameters of the discharge itself. In discharges of gases such as air, \(N_2\), \(CO\), and \(CO_2\), vibrationally excited molecules form energy reservoirs that can be transferred to different degrees of freedom of the gas, dissociation, and ionization. It is evident that such characteristics and trends are very difficult to track experimentally. Moreover, in general, discharge in a supersonic air flow is a very complex object for experimental study, as the traditional probe diagnostics used in this area of experimental physics are difficult to implement and are accompanied by significant errors.

One of the most important parameters is the electron energy distribution function, knowledge of which allows for estimating and, in most cases, accurately determining various discharge parameters. By modeling the distribution function, it is possible to determine the rate constants of the reactions occurring in the non-equilibrium plasma in a supersonic flow, the electron temperature, and other characteristics.

By solving the Boltzmann equation for the electron energy distribution function, which includes various collision integrals for elastic, inelastic, and superelastic processes, the trends in the changes of the distribution function can be observed.

\pagebreak

\section{Prerequisites}

\subsection{Energy dissipation in gas discharge}

When a pulsed voltage is applied to the electrodes of a gas discharge tube, provided the electric field strength exceeds the threshold value, gas breakdown occurs in the discharge chamber. After the breakdown, the resulting plasma begins to intensively absorb the energy supplied to the discharge. In this process, electrons gain energy from the field and transfer it to the neutral gas molecules during collisions. The energy released in the gas is distributed across all degrees of freedom (vibrational, rotational, translational, electronic excitation of molecules, dissociation, etc.)

Calculating the energy distribution of electrons across various excitation channels is a self-consistent problem. On one hand, it accounts for the influence of all types of energy losses on the electron energy distribution function. On the other hand, this function is used to calculate the rate coefficients and energy losses for the same excitation channels.

The energy of the external field is transferred to the weakly ionized gas in such a way that this energy is initially acquired by the electrons, and from the electrons, it is transmitted to the gas particles. Therefore, the electron and other components of the weakly ionized gas are, from the very beginning, in nonequilibrium conditions, and the average energies of the particles associated with different components may differ. For example, the vibrational, rotational, and translational temperatures in a molecular gas depend on the method of gas excitation and the processes that lead to energy exchange between different degrees of freedom within the gas.

The specific characteristics of the electron energy balance in molecular gases are determined by energy losses during inelastic processes, which exhibit low threshold energy values and significant effective cross-sections. This applies to the excitation of rotational and vibrational states. Due to the low values of threshold energies, rotation and vibration are excited even at relatively low average electron energies. In contrast to elastic collisions, which are characterized by low energy exchange efficiency \(\delta = 2m_e/M\) (where \(m_e\) is the mass of the electron and \(M\) is the mass of the molecule), with each act of vibrational or rotational excitation, the electron loses energy equivalent to the vibrational or rotational quantum, which is usually much greater.

\begin{figure}
    \centering
    \includegraphics[width=0.5\linewidth]{figures/fig1.png}
    \caption{\label{fig:figure1}Different mechanisms of power dissipation in \(N_2\) discharge.}
\end{figure}

Figure \ref{fig:figure1} illustrates, using the example of discharge plasma in molecular nitrogen, the distribution of the total power \(\sigma E\) (\(\sigma\) is the plasma conductivity, and \(E\) is the electric field strength) supplied to the electron gas. This power is transferred to elastic losses and the excitation of rotations (1), vibrations (2), electronic states (3), and ionization (4).) From Fig. 1, it is evident that the role of elastic losses and rotational excitation is significant only at low values of the reduced electric field \(E/n\) (where \(n\) is the molecular concentration), i.e., at low electron temperatures. This is natural, as these processes are most effectively carried out at low electron energies. At electron temperatures around 1 eV, almost the entire energy contribution of the discharge is localized in the excitation of molecular vibrations. The role of electronic excitation and ionization becomes significant at higher electron temperatures (and correspondingly higher \(E/n\)), which is expected due to the high energy thresholds of these processes.

It is necessary to understand inelastic collisions in plasma and their impact on the electron distribution. The inelastic collision integral determines the rate of change of the distribution function due to electrons participating in excitation, ionization, and second-kind collisions and recombination.

For conditions of strong ionization, the form of the electron energy distribution function is mainly shaped by collisions between charged particles, primarily due to electron-electron collisions (because of the large difference in mass between electrons and ions). Therefore, electron-electron collisions establish a Maxwellian distribution of the electron energy function.

\subsection{Electronic Energy Distribution Function}
Let us list the main processes that were reflected in the Boltzmann equation for the correct description of the behavior of the electron distribution function:

\begin{itemize}
    \item the field term (electric field heats electrons),
    \item elastic collisions of electrons with atoms;
    \item inelastic collisions of electrons with atoms, including processes of excitation and de-excitation of electronic molecular levels, vibrational and rotational excitation,
    \item inter-electron collisions (electron concentration is a variable constant).
\end{itemize}

\pagebreak

For the holistic model approach full equation and reaction set should be added:

\begin{itemize}
    \item Gas heating mechanisms,
    \item Atoms and ions producing chemistry,
    \item Ionization and recombination,
    \item Dissociation into neutrals,
    \item CEX collisions,
    \item \dots
\end{itemize}

The Boltzmann kinetic equation independently describes the motion of electrons in plasma, which experience collisions from time to time both among themselves and with other components. In the general case, it can be written as [1]:

\begin{equation} \label{eq:1}
    f(\Vec{v}+ d\Vec{v}, \Vec{r} + d\Vec{r}, t + dt) - f(\Vec{v}, \Vec{r}, t) = Idt.
\end{equation}

Note that \(d\Vec{r} = \Vec{v}dt, d\Vec{v} = \frac{\Vec{F}}{m}dt\), where F is the external force acting on the electrons (in our case it's electric field), and I represents various collision integrals. Then, equation (1) could be transformed into the following form:
\begin{equation} \label{eq:2}
    \frac{df}{\partial t} + \Vec{v} \frac{\partial f}{\Vec{\partial r}} + \frac{\Vec{F}}{m} \frac{\partial f}{\Vec{\partial v}} = I.
\end{equation}
Given that the mass of electrons is much smaller than the mass of atoms, during each such collision, the absolute value of the electron velocity and, consequently, the electron energy changes only slightly, whereas the direction of motion changes in a nearly isotropic, chaotic manner. Therefore, the distribution function \(f(\Vec{v})\) is close to spherically symmetric. In connection with this, when solving the Boltzmann equation, the distribution function is expanded in terms of Legendre polynomials

\begin{equation} \label{eq:3}
    f(\Vec{v}) = \sum_{k=0}^{\infty} f_k(v)P_k(cos\theta),
\end{equation}
where \(\theta\) is the angle between velocity \(v\) and the field \(E\).
We also limit the problem to the case where the field and collision cross sections have a uniform spatial distribution. Using spherical coordinates, one can obtain:
\begin{equation} \label{eq:4}
    \frac{\partial f}{\partial t} + v cos \Theta \frac{\partial f}{\partial z} + \frac{F}{m}  \left(
    cos \Theta \frac{\partial f}{\partial v} + \frac{sin^2\Theta}{v} \frac{\partial f}{\partial cos \Theta}  \right) = I,
\end{equation}
where \(z\) is the posotion of an electron along this direction.

\begin{equation} \label{eq:5}
    \begin{split}
       \frac{1}{n} \sqrt{\frac{m \varepsilon}{2}} \frac{\partial f(\varepsilon, t)}{\partial t}
       & = \frac{\partial}{\partial t} \left( \left( \frac{1}{3} \left( \frac{\varepsilon E}{\varepsilon_0}  \right)^2 \frac{\varepsilon}{\sigma_c} + \delta k T_0 \varepsilon^2 \sigma_c\right) \frac{\partial f(\varepsilon,t)}{\partial \varepsilon} \right)
        + \frac{\partial}{\partial \varepsilon} \left( \left(  \delta \varepsilon^2 \sigma_c + 4B_0\varepsilon\sigma_{re} \right)f(\varepsilon,t) \right) - \\
       & - \sum_{n} \frac{n_m}{n} \sum_{l} \{\varepsilon\sigma_{ml}(\varepsilon) f(\varepsilon,t) - (\varepsilon+\varepsilon_{ml})\sigma_{ml}(\varepsilon+\varepsilon_{ml}) f(\varepsilon+\varepsilon_{ml},t)\} - \\
       & - \sum_{\nu} \frac{n_{\nu}}{n} \sum_{w} \{\varepsilon\sigma_{\nu w}(\varepsilon) f(\varepsilon,t) - (\varepsilon+\varepsilon_{\nu w})\sigma_{\nu w}(\varepsilon+\epsilon_{\nu w}) f(\varepsilon+\varepsilon_{\nu w},t)\} - \\
       & - \sum_{w} \frac{n_{w}}{n} \sum_{\nu} \{\varepsilon\sigma_{w \nu}(\varepsilon) f(\varepsilon,t) - (\varepsilon-\varepsilon_{w \nu})\sigma_{w \nu}(\varepsilon-\varepsilon_{w \nu}) f(\varepsilon-\varepsilon_{w \nu},t)\} + \\
       & + 2\pi e^4 \gamma_e ln\Lambda \frac{\partial}{\partial \varepsilon} \{ f(\varepsilon)\int_{0}^{\varepsilon}
       \epsilon^{1/2}f(\epsilon)d\epsilon\ + \frac{2}{3} \frac{\partial f}{\partial \varepsilon} \left(
       \int_{0}^{\varepsilon}\epsilon^{3/2}f(\epsilon)d\epsilon\ + \varepsilon^{3/2} \int_{0}^{\epsilon}f(\epsilon)d\epsilon\ \right)
    \end{split}
    \raisetag{6\normalbaselineskip}
\end{equation}
Here, \(\sigma_c\) is the transport cross-section, \(\sigma_{re}\) works for rotational excitation, \(\sigma_{ml}\)  is the cross-section for the excitation of electronic levels, \(\sigma_{vw}\) is the cross-section for the excitation of vibrational levels, and \(\sigma_{wv}\)  is the cross-section for de-excitation. The first term in the equation describes the action of the field, the second and third terms describe elastic collisions of electrons with molecules, the fourth term covers the excitation of rotational levels, the sums describe the excitation and de-excitation of electronic and vibrational levels of molecules, and the last term represents inter-electron collisions.
\subsection{Kinetic scheme}
According to \cite{1}, the non-equilibrium distribution of molecules across vibrational levels is due to an excess of vibrational energy in the molecules over the equilibrium value corresponding to the gas temperature. A large number of different processes contribute to the population of levels; however, in our problem, we consider only the lower vibrational levels (\(v\) = 0-8) of the ground state of molecular nitrogen (\(X^{1}\Sigma^{+}_g\)), where the two most likely processes are:
\begin{itemize}
    \item Vibrational excitation by electron impact (eV-exchange)
    \[ e + A(X, v_1) = e + A(X, v_2),\]
    where \(X\) is the ground electronic states of \(N_2\), and \(v_1\) and \(v_2\) are different vibrational states.
    \item Vibrational-vibrational exchange (VV)
    \[ A(X, v_1) + A(X, v_2) = A(X, v_1-1) + A(X, v_2+1)\]
\end{itemize}
In study \cite{2}, it is shown that in this case, the population of vibrational levels of molecules follows the Treanor distribution:
\begin{equation} \label{eq:6}
    \begin{split}
      & n_{\nu} = n_0\cdot \exp{(-2\beta\alpha\nu + \beta\nu^2)}, \\
      & \beta = \frac{\xi_e\hbar\omega}{T}, \alpha = \frac{\xi_e T}{2T_{\nu}} + \frac{1}{2},
    \end{split}
\end{equation}
where \(n_0\) is the population of the ground vibrational level, \( \alpha \)and \( \beta \) are related to the anharmonicity of the molecules \(\xi_e\), the vibrational temperature \(T_v\), and the translational temperature \(T\) of the gas.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{figures/fig2.png}
    \caption{\label{fig:figure2}The vibrational distribution function of nitrogen molecules in the plasma of a glow discharge at a pressure \(p\) = 2 Torr, \(T_g\) = 500K, \(N_e = 2\cdot 10^{16}\ m^{-3}\) for the first 8 vibrational levels of the ground state  : 1 – Boltzmann distribution at \(T_v\) = 5000 K, 2 – Treanor distribution at \(T_v\) = 5000 K and \(T_g\) = 500 K, 3 – calculation from \cite{2}.}
\end{figure}

The fact that during the relaxation process, deviations of the actual distribution from the Boltzmann distribution are significant only for the higher states, while they are small at lower levels (where, in general, most of the vibrational energy is concentrated due to their high occupancy), allows us to simplify the search for the actual distribution without solving the full set of equations for the level occupancies. It can be shown that the actual distribution over vibrational levels, obtained by solving the complete system of kinetic equations, differs little from the Treanor distribution for low-lying vibrational levels (\(\nu \leq 8\)), indicating the validity of applying our model. Figure \ref{fig:figure2} compares the vibrational level occupancy data for nitrogen molecules calculated using various methods.

As for the electronic levels of molecules in the discharge plasma in air, an assumption was made when solving this problem. Since the frequency of electron-neutral collisions significantly exceeds the frequencies of all other collisions, it can be assumed that the molecules are distributed among the electronic levels according to the Boltzmann distribution with a temperature equal to the effective electron temperature. Thus, for the occupancy of the electronic levels, we have:

\begin{equation} \label{eq:7}
    n_w = n_0 \cdot \exp{ \left( -\frac{E_w}{k T_{eff}^e} \right)},
\end{equation}

where \(E_w\) is the excitation threshold energy of the w-th electronic level, and \(T_{eff}^e\) is the effective electron temperature in the plasma.

\begin{equation}\label{eq:8}
    g_k \left( \varepsilon + \Delta E \right) \sigma_{kl} \left( \varepsilon + \Delta E \right) =
    g_i \varepsilon \sigma_{ik} \left( \varepsilon \right),
\end{equation}

Klein-Rosseland relations \eqref{eq:8}, expressing the principle of detailed balance for first and second-kind collision processes, were used to calculate the cross-sections of de-excitation processes. Here \(g_i, g_k\) both statistical weights of levels, and \(\Delta E\) is a threshold energy.
It is also important to note that quenching processes are thresholdless and are initiated by collisions with electrons of any energy.  Another important note is that this relation has no plasma characteristics, therefore it’s a general relation between cross-sections of first and second orders.

It should be noted that among all the electronic levels of $\text{N}_2$ only the ground state is resolved into vibrational levels, and among all vibrational levels, only the first is resolved into rotational levels, as can be observed at figure \ref{fig:figure3}. These assumptions were made for several reasons, the main one being that the most probable transitions have a probability of an order of magnitude higher than the others. The cross-sections of the processes described were taken from \cite{4}.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.4\linewidth]{figures/fig3.png}
    \caption{\label{fig:figure3}\(\text{N}_2\) excitation levels.}
\end{figure}

\pagebreak

A more detailed version of the $\text{N}_2$ processes used for calculations is presented as follows:

\begin{itemize}
    \item \textbf{Elastic collision between N\textsubscript{2} molecules and electrons:}
    \begin{align*}
        \text{N}_2 + e^- &\rightarrow \text{N}_2 + e^- \, (\text{elastic})
    \end{align*}

    \item \textbf{Rotational excitation from the ground state of N\textsubscript{2} molecule:}
    \begin{align*}
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(X^1\Sigma_g^+, J') + e^-, \, J \neq J'
    \end{align*}

    \item \textbf{Vibrational excitation from the ground state of N\textsubscript{2} molecule:}
    \begin{align*}
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(X^1\Sigma_g^+, v') + e^-, \, v = 0, \, v' = 1 \text{--} 8
    \end{align*}

    \item \textbf{Excitation of electronic levels of N\textsubscript{2} molecule:}
    \begin{align*}
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(A^3\Sigma_u^+, v') + e^-, \, v = 0, \, v' = 0 \text{--} 4 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(A^3\Sigma_u^+, v') + e^-, \, v = 0, \, v' = 5 \text{--} 10 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(B^3\Pi_g, v') + e^-, \, v = v' = 0 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(C^3\Pi_u, v') + e^-, \, v = v' = 0 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(W^3\Delta_u, v') + e^-, \, v = v' = 0 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(B'^3\Sigma_u^+, v') + e^-, \, v = v' = 0 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(A'^1\Sigma_u^-, v') + e^-, \, v = v' = 0 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(A'^1\Pi_g, v') + e^-, \, v = v' = 0 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(W'^1\Delta_u, v') + e^-, \, v = v' = 0 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(E^3\Sigma_g^+, v') + e^-, \, v = v' = 0 \\
        \text{N}_2(X^1\Sigma_g^+, v) + e^- &\rightarrow \text{N}_2(A''^1\Sigma_u^+, v') + e^-, \, v = v' = 0
    \end{align*}

    \item \textbf{Ionization of N\textsubscript{2} molecule:}
    \begin{align*}
        \text{N}_2 + e^- &\rightarrow \text{N}_2^+ + 2e^-
    \end{align*}

\end{itemize}

\pagebreak

\section{Sample results from the Master Thesis}

For example, Figure \ref{fig:figure4} shows a comparison of the results of this work with those of study \cite{7}, where the electron energy distribution function in nitrogen plasma of a glow discharge was also obtained at various values of the reduced electric field \(E/N \). A good agreement between the obtained results is observed. A couple of interesting kinks of the EEDF should be discussed here. Firstly, distinct gaps in the distribution function are clearly visible. The explanation for this is straightforward: in this energy range, electrons efficiently excite the vibrational levels of \(N_2\) molecules, leading to a reduction in electron density (an abrupt decline in the function) as energy is transferred during collisions. Additionally, the expansion of the "tail" in the electron distribution is quite evident. The explanation is simple: an increase in the electric field accelerates the electrons, resulting in a distribution tail elevation.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{figures/fig4.png}
    \caption{\label{fig:figure4}Comparing our model and Bolsig+ calculations for N2 with different field values.}
\end{figure}

\pagebreak

Figure \ref{fig:figure5} shows the dependence of the electron distribution function on the degree of plasma ionization in the N2 discharge. A strong influence of electron-electron collisions on the distribution function is clearly observed. Electron-electron collisions lead to a thermalization of the electron gas by driving the EDF towards a Maxwellian distribution. This means that the higher the degree of ionization, the closer the distribution function approaches a Maxwellian form, and the lesser the impact of vibrational excitation of heavy plasma particles on the distribution. From a graphical standpoint, this implies that in cases of high ionization, the distribution function should resemble a straight line on a logarithmic scale. This behavior is precisely reflected in the results obtained.



\begin{figure}[h!]
    \centering
    \includegraphics[width=0.6\linewidth]{figures/fig5.png}
    \caption{\label{fig:figure5}Effect of the degree of ionization on the form of the electron energy distribution function (EEDF) in \(N_2\), \(\frac{E}{N} = 50\) Td.}
\end{figure}

\pagebreak

\section{Local and Non-Local approaches}

As discussed in the previous sections, finding the electron energy distribution function (EEDF) is an important task when addressing various problems in plasma physics and different types of discharges. Overestimating or underestimating the values of the EEDF in the studied energy range may lead to incorrect evaluations of other discharge parameters, such as reaction coefficients and others.

Currently, there are several open-source codes available that allow solving the Boltzmann equation under certain approximations. Examples include \texttt{Bolsig+}\cite{3} or \texttt{Bolos}, which solve the Boltzmann equation within the so-called ``local'' approximation. However, it is crucial to emphasize that this approach has significant limitations, and very often, a meticulous analysis of discharge conditions prohibits the use of this local approximation. The question regarding the possible boundaries for the application of various approximations will be discussed further.

\section*{Local Approach}
The local approach (or local field approximation) is applied to situations where electron motion is strongly collisional, meaning that energy diffusion and collisional effects dominate over the spatial motion of electrons. In this context:
\begin{itemize}
    \item Spatial inhomogeneity becomes negligible, and the EEDF at each spatial position is determined by the homogeneous Boltzmann equation.
    \item The EEDF at each spatial position depends only on the local electric field strength.
\end{itemize}

This is the standard approach implemented in most open-source codes that solve the Boltzmann equation to obtain the electron energy distribution function.

It is, however, clear that this approximation is invalid for low-pressure discharges. Moreover, various non-local effects, such as electron heating in the sheath region of some discharges or ambipolar diffusion heating, must also be considered. The primary conditions under which this method is applicable include:
\begin{itemize}
    \item High pressures,
    \item High plasma densities,
    \item Large collision cross-sections for particle interactions.
\end{itemize}

\section*{Non-Local Approach}
Another ``extreme'' method, proposed by Bersntesin and Holstein \cite{8} and Tsendin \cite{9}, assumes the presence of significant spatial variations in the discharge system.

The non-local approach, applicable to weakly collisional plasma, can be summarized as follows:
\begin{itemize}
    \item It assumes that the nature of electron kinetics is governed by a spatially homogeneous distribution function that depends on the total energy of electrons.
    \item This spatially homogeneous EEDF is derived from the spatially averaged Boltzmann kinetic equation, which reduces to a one-dimensional ordinary differential equation, regardless of the number of spatial dimensions in the discharge problem.
\end{itemize}

If energy-consuming collisions are infrequent and the heating field is sufficiently weak, the total energy of an electron can be treated as constant during its motion across the plasma. From the perspective of timescales, one can say that the timescale of electron spatial motion is much shorter than the timescale of energy changes due to heating or collisions.

It should be mentioned that in the work \cite{10}, the application of modern open-source codes, such as \texttt{Bolsig+} and similar tools, to solving the Boltzmann equation using non-local approximations is demonstrated.

For intermediate cases, where discharge parameters fall between the two regimes discussed earlier, these approximations are no longer applicable. In such instances, the necessity for more advanced and computationally intensive methods becomes evident. Several such approaches have been proposed in works such as \cite{11} and \cite{12}. It is also worth noting the importance of these two approaches to solving the Boltzmann equation. They enable, among other things, the validation of more complex solutions obtained using Monte Carlo methods or other spectral techniques.

Next, we turn to the boundaries of applicability for the local and non-local approaches to solving the Boltzmann equation. In the work \cite{9}, a method for estimating these boundaries is proposed. This method is based on a comparison between the energy relaxation length and the characteristic length of plasma inhomogeneity.

The energy relaxation length is given by:
\begin{equation} \label{eq:9}
\lambda_\varepsilon = \lambda \sqrt{\frac{\nu_m}{\kappa \nu_m + \nu^*}},
\end{equation}
where:
\begin{itemize}
    \item $\lambda$ is the mean free path of electrons,
    \item $\nu^*$ is the total inelastic collision frequency,
    \item $\nu_m$ is the momentum transfer collision frequency,
    \item $\kappa = \frac{2m_e}{M}$ is the momentum transfer rate.
\end{itemize}

The mean free path can be estimated using the expression from \cite{13}
\begin{equation} \label{eq:10}
\lambda \approx \frac{\bar{V}}{\nu_m + \nu^*},
\end{equation}
where $\bar{V}$ is the mean electron velocity.

The study shows that the regime $\lambda_\varepsilon \gg \Lambda$ corresponds to the non-local approach, where spatial changes dominate over energy relaxation processes. On the other hand, $\lambda_\varepsilon \ll \Lambda$ corresponds to the local approach, where energy relaxation occurs faster than spatial variations.

In the following chapter, two experiments will be analyzed with results discussed in the works by \cite{6} and \cite{14}. These analyses will evaluate the applicability boundaries of the local and non-local approaches under the discharge conditions described in these experiments. Furthermore, additional considerations regarding both the electron energy distribution function and other discharge parameters will be presented.

\pagebreak

\section{VLEO plasma conditions}

\subsection{Main ideas of the experiment}

Let's consider the methodology of the work \cite{6}. This study investigates the ionization process in air-breathing plasma thrusters (ABPTs) designed for very low Earth orbit (VLEO) applications. The key focus is to model the ionization of atmospheric air using detailed plasma chemistry simulations to understand the variation in species densities and ionization degrees as a function of input electron energy. The simulation relies on solving reaction rate equations for 167 chemical reactions involving nitrogen and oxygen species at altitudes ranging from 80 to 110 km.

The reaction rate equations track the temporal evolution of species densities. The density of a species, $n_a$, is updated using:
\begin{equation} \label{eq:11}
\frac{d n_a}{d t} = \sum_b k_b(T_e, T_g) \prod_c n_{b,c},
\end{equation}
where $k_b$ is the reaction rate coefficient, dependent on electron temperature $T_e$ and gas temperature $T_g$, and $n_{b,c}$ are the reactant densities. The initial conditions for air densities were derived from the MSIS-E-90 atmospheric model, ensuring consistency with LEO altitudes. Cross-sections for electron-molecule collisions were sourced from the LXCat database, enabling accurate calculation of reaction rates based on the Boltzmann electron energy distribution.

The simulations were conducted over a typical timescale of 0.125 ms, corresponding to the time air travels through the ionization channel of the thruster at orbital speeds. Electron energy levels were carefully modulated to optimize the production of positive and negative ions. These ions were extracted using electrodes with alternating polarities, enabling charge-neutralization without the need for an external cathode. By varying the energy modes of operation, the ionization process could be controlled to achieve self-neutralization and maximize thrust efficiency.

This method provides detailed insight into how ionization parameters, such as species densities and electron energy distributions, vary with altitude and thruster conditions. The results indicate that ionization is highly sensitive to the mean electron energy and that a neutralizer-free ABPT design is feasible through careful management of ion formation and extraction processes.
\pagebreak

\subsection{Analysis of the eligibility of the local approach}

Let us return to the analysis of specific methods discussed in this work. Concerns arise regarding the use of Maxwell-Boltzmann distribution functions (equilibrium case for discharge plasma) and the open-source code \texttt{Bolsig+} for calculating reaction rate constants without validating the applicability of the local approximation under the given conditions. To address this, we conduct an evaluation.

For simplicity, we consider only the nitrogen component of the plasma. The spatial inhomogeneity length, corresponding to the size of the discharge chamber in the proposed thruster, is taken as $\Lambda = 1 \, \text{m}$. The cross-section for elastic scattering of nitrogen is approximately $\sigma_m = 10^{-19} \, \text{m}^2$, and for inelastic collisions, $\sigma^* = 10^{-20}$ to $10^{-21} \, \text{m}^2$.

We perform detailed calculations for an altitude of 80 km. The mean free path is given by:
\begin{equation} \label{eq:12}
\lambda \approx \frac{\bar{V}}{\nu_m + \nu^*} = \frac{\bar{V}}{\bar{V}(n_{\text{N}_2} \cdot \sigma_m + n_{\text{N}_2} \cdot \sigma^*)} = \frac{1}{n_{\text{N}_2} (\sigma_m + \sigma^*)},
\end{equation}
where $n_{\text{N}_2}$ is the nitrogen number density. Substituting the values for 80 km altitude, where $n_{\text{N}_2} \approx 5.7 \times 10^{20} \, \text{m}^{-3}$:
\begin{equation} \label{eq:13}
\lambda \approx \frac{1}{5.7 \times 10^{20} \cdot (10^{-19} + 10^{-21})} \approx 0.02 \, \text{m}.
\end{equation}

The energy relaxation length is calculated as:
\[
\left( \frac{\nu_m}{\kappa \nu_m + \nu^*} \right)^{1/2} \approx \left( \frac{n_{\text{N}_2} \cdot \sigma_m}{\kappa \cdot n_{\text{N}_2} \cdot \sigma_m + n_{\text{N}_2} \cdot \sigma^*} \right)^{1/2} \approx
\left( \frac{\sigma_m}{\kappa \cdot \sigma_m + \sigma^*} \right)^{1/2}.
\]
Using $\kappa = 4 \times 10^{-5}$:
\[
\left( \frac{\sigma_m}{\kappa \cdot \sigma_m + \sigma^*} \right)^{1/2} \approx \left( \frac{10^{-19}}{4 \cdot 10^{-5} \cdot 10^{-19} + 10^{-21}} \right)^{1/2} \approx 10.
\]
Thus, the energy relaxation length becomes:
\begin{equation} \label{eq:14}
\lambda_\varepsilon = \lambda \cdot \left( \frac{\nu_m}{\kappa \nu_m + \nu^*} \right)^{1/2} \approx 0.02 \, \text{m} \cdot 10 = 0.2 \, \text{m}.
\end{equation}

Comparing the energy relaxation length $\lambda_\varepsilon$ from \ref{eq:14} with the spatial inhomogeneity length $\Lambda$, it is clear that the condition for the local approximation $\lambda_\varepsilon \ll \Lambda$ is not satisfied at 80 km altitude.

Similar calculations for 90 km, 100 km, and 110 km altitudes reveal that:
- At 80–90 km, $\lambda_\varepsilon$ is on the same order as $\Lambda$, making both local and non-local approaches inapplicable. For these cases, Monte Carlo or other computationally intensive methods are required to obtain the electron energy distribution function.
- At 100–110 km, $\lambda_\varepsilon$ is an order of magnitude greater than $\Lambda$ (approximately 1 m). Thus, the non-local approach is valid, and the Boltzmann equation can be solved effectively.

\pagebreak

\subsection{A rough analysis using the local approach}

With caution and considering the limitations of the local approximation, a qualitative analysis of the properties of the electron energy distribution function (EEDF) under the plasma conditions obtained in this work is conducted. The influence of the EEDF shape on the prediction of reaction rate coefficients is also examined.

\begin{figure} [h!]
    \centering
    \includegraphics[width=0.6\linewidth]{figures/fig8.jpg}
    \caption{\label{fig:figure8} Comparison between Bolsig calculation of normalized energy distribution function and Maxwell distribution. At an altitude of \( h = 80 \, \text{km} \), the plasma and atmospheric parameters are defined as follows. The neutral density is \( n_{\text{N}_2} = 5.7 \times 10^{20} \, \text{m}^{-3} \), and the electron (plasma) density is \( n_e = 1.0 \times 10^7 \, \text{m}^{-3} \). The neutral temperature is \( T_g = 200 \, \text{K} \), and the vibrational temperature is \( T_{\text{vib}} = 300 \, \text{K} \). The mean electron energy is \( \langle E_e \rangle = 10 \, \text{eV} \).}
\end{figure}

Viewgraph \ref{fig:figure8} compares the electron energy distribution functions (EEDF) for an \texttt{N\textsubscript{2}:O\textsubscript{2}} mixture computed using \texttt{Bolsig+} versus a Maxwellian distribution with constant mean electron energy. These plasma parameters were based on Keidar's work. The results show that at low ionization degrees (e.g., $1.0 \times 10^{-13}$ in this case), the Maxwellian distribution exhibits a more populated high-energy tail, leading to an overprediction of reaction rates. This overestimation is critical for accurately evaluating ion production rates and, consequently, thrust generation.

The discrepancy in the tails can be explained by the role of electron-electron collisions, which redistribute energy among electrons. At low ionization degrees, the plasma remains in a non-equilibrium state for a longer time due to fewer electron-electron collisions. In contrast, at higher ionization degrees, frequent electron-electron collisions rapidly thermalize the distribution.

\begin{figure} [h!]
    \centering
    \includegraphics[width=0.6\linewidth]{figures/fig9.png}
    \caption{\label{fig:figure9} Normalized electron energy distribution function for N2:O2 plasma  VS electron energy.}
\end{figure}

The second figure \ref{fig:figure9} zooms into the low-energy region of the EEDF and highlights the existence of a “dip” at energies around 2--3 eV. Conditions of plasma are the same: at an altitude of \( h = 80 \, \text{km} \), the densities are: \( n_{\text{N}_2} = 5.7 \times 10^{20} \, \text{m}^{-3} \), \( n_e = 1.0 \times 10^7 \, \text{m}^{-3} \), the temperatures are \( T_g = 200 \, \text{K} \), and \( T_{\text{vib}} = 300 \, \text{K} \), the mean electron energy is \( \langle E_e \rangle = 10 \, \text{eV} \). This "kink" can be attributed to the high excitation cross-sections for vibrational levels of molecules at these energies. Electrons actively excite molecular vibrations in this range, resulting in a depletion of electrons from this energy region. While some electrons return to the region due to reverse processes, the compensation is incomplete due to the non-equilibrium nature of the plasma. For Keidar’s case, this dip is relatively small and does not significantly influence the reaction rates. However, the findings emphasize the importance of accurately knowing the true EEDF for the plasma system under study.

\pagebreak

\begin{figure} [h!]
    \centering
    \includegraphics[width=0.6\linewidth]{figures/fig10.png}
    \caption{\label{fig:figure10} The dependence of normalized electron energy distribution function $f(E)$ on electron energy E for different ionization degrees plus Maxwell distribution function to compare (\( T_{\text{exc}} = 300 \, \text{K} \).)
}
\end{figure}

The viewgraph \ref{fig:figure10} demonstrates the normalized EEDF $f(E)$ as a function of electron energy $E$ for various ionization degrees, alongside the Maxwellian distribution for comparison (at $T_{\text{exc}} = 300$ K). Plasma parameters preserved as on two previous graphs.  Key observations include:
\begin{itemize}
    \item \textbf{Collisions between charged particles}: These collisions tend to Maxwellize the EEDF, making it more closely resemble the equilibrium distribution.
    \item \textbf{Electron-ion collisions}: These collisions increase the population of high-energy electrons in the tail of the distribution, which are critical for ionization processes.
    \item \textbf{High ionization degrees}: At ionization degrees exceeding $1.0 \times 10^{-2}$, the EEDF deviates significantly from the Maxwellian distribution, especially in the high-energy tail.
\end{itemize}

In Keidar’s case, during the final stages of ionization, where the ionization degree is high, the EEDF diverges substantially from the Maxwellian approximation, which Keidar uses to calculate most reaction rates. This difference is particularly prominent in the high-energy tail of the distribution, the region crucial for ionization processes. As a result, relying on a Maxwellian distribution may lead to inaccurate predictions of ionization rates and thrust.

\pagebreak

\section{ECR plasma conditions}

\subsection{Main ideas of the experiment and selected results}

The experiment \cite{14} focuses on exploring alternative propellants to traditional xenon for use in Electron Cyclotron Resonance (ECR) gridded ion thrusters, which are crucial for electric space propulsion. Recognizing the increasing cost and scarcity of xenon and krypton, the study investigates argon, nitrogen, and air as potential substitutes. The research utilizes a 10-cm-class waveguide ECR gridded ion thruster, inspired by the $\mu10$ thruster used in JAXA’s Hayabusa missions, as a test bed for these alternative propellants.

Main ideas regarding plasma analysis in the experiment could be described as follow:

\begin{itemize}

    \item \textbf{Experimental Setup:} Conducting experiments with varying mass flow rates (75, 100, 150, and 200 $\mu$g/s) and absorbed microwave power (approximately 5 to 20 W).
    \item \textbf{Plasma Characterization:} Measuring downstream plasma properties such as electron temperature and electron number density for each propellant.

\end{itemize}

The main goals is to determine how different propellants affect plasma discharge characteristics and ion beam extraction, particularly focusing on the presence of high-current and low-current discharge modes and their correlation with visual plasma emissions.

The experimental results demonstrate variations in plasma discharge characteristics and ion beam performance among the tested propellants. For electron temperature, the experiment shows that for argon and air plasmas trends are similar to the results from other papers. At the same time, nitrogen shows unexpected behavior. Some results summarized as follows:

\begin{itemize}
    \item \textbf{Argon:}
    \begin{itemize}
        \item \textbf{Range:} 1.4 to 2.3 eV.
        \item \textbf{Trend:} $T_e$ decreases as the mass flow rate normalized by absorbed microwave power ($\dot{m}/P_{\text{abs}}$) increases.
    \end{itemize}

    \item \textbf{Nitrogen:}
    \begin{itemize}
        \item \textbf{Range:} 2.1 to 4.1 eV.
        \item \textbf{Trend:} $T_e$ increases with higher $\dot{m}/P_{\text{abs}}$, notably at a 150 $\mu$g/s flow rate.
    \end{itemize}

    \item \textbf{Air:}
    \begin{itemize}
        \item \textbf{Range:} 1.8 to 3.4 eV.
        \item \textbf{Trend:} $T_e$ decreases with increasing $\dot{m}/P_{\text{abs}}$.
    \end{itemize}
\end{itemize}

\subsection{Analysis of the results}

Let's try to analyze the result this way: the electron energy distribution function (EEDF) in ECR plasma is obtained and then used to evaluate the transport coefficients for plasmas of different gases, including argon, air, and nitrogen. The idea is that nitrogen (and air, which contains nitrogen) possesses internal degrees of freedom that allow it to store energy more effectively from collisions with electron, i.e., the energy transferred from field heating. In this sense, nitrogen is a prototypical “capacitive” gas. Its vibrational degrees of freedom form energy reservoirs that fundamentally influence the non-equilibrium properties of the plasma.

Using the energy transport equation, it is possible to determine the energy diffusion coefficient and the energy mobility parameter, which can then be used to analyze the differing behavior of nitrogen and argon concerning electron temperature, absorbed power, and mass flow rate.

To account for not only particle fluxes but also energy fluxes—affected by field heating, transfer due to collisions, and particle diffusion—we derive the energy transport equation. This equation is obtained from the Boltzmann equation in a manner analogous to the derivation of the particle transport equation, but with an additional weighting by $E^{3/2}$ (second momentum of energy) and subsequent integration. The energy transport equation can be written as follows:

\begin{equation}
\frac{\partial n_\varepsilon}{\partial t} + \frac{\partial \Gamma_\varepsilon}{\partial z} + E \Gamma = S_\varepsilon,
\label{eq:15}
\end{equation}
where:
\begin{itemize}
    \item $\frac{\partial n_\varepsilon}{\partial t}$ represents the time variation of energy density,
    \item $\frac{\partial \Gamma_\varepsilon}{\partial z}$ corresponds to the spatial variation of energy flux,
    \item $E \Gamma$ is the field heating term,
    \item $S_\varepsilon$ accounts for energy transfer due to collisions.
\end{itemize}

This equation allows us to investigate how different gases, such as nitrogen and argon, respond to variations in field heating and particle diffusion under plasma conditions. For energy density and energy flux one could obtain:
\begin{equation}
n_\varepsilon = n \int_{0}^{\infty} \varepsilon^{3/2} F_0 \, \mathrm{d}\varepsilon \equiv n \bar{\varepsilon},
\label{eq:16}
\end{equation}

\begin{equation}
\Gamma_\varepsilon = n \frac{\gamma}{3} \int_{0}^{\infty} \varepsilon^2 F_1 \, \mathrm{d}\varepsilon,
\label{eq:17}
\end{equation}

Using two-term Bolzmann equation approximation, obtain energy flux in a drift-diffusion form:

\begin{equation}
\Gamma_\varepsilon = -\mu_\varepsilon E n_\varepsilon - \frac{\partial (D_\varepsilon n_\varepsilon)}{\partial z},
\label{eq:18}
\end{equation}

Equation \ref{eq:18} contains energy mobility parameter and energy diffusion coefficients described as:

\begin{equation}
n_\varepsilon = n \int_{0}^{\infty} \varepsilon^{3/2} F_0 \, \mathrm{d}\varepsilon \equiv n \bar{\varepsilon},
\label{eq:energy_density}
\end{equation}

\begin{equation}
\Gamma_\varepsilon = n \frac{\gamma}{3} \int_{0}^{\infty} \varepsilon^2 F_1 \, \mathrm{d}\varepsilon.
\label{eq:energy_flux}
\end{equation}

In Rovey's work, the following parameters were used or implied for the \texttt{Ar/N\textsubscript{2}} plasma system:

\begin{table}[h!]
\centering
\[
\begin{array}{|l|l|}
\hline
\textbf{Parameter} & \textbf{Value} \\
\hline
\text{Neutral density (m\textsuperscript{-3})} & \text{1.0 $\times$ 10\textsuperscript{18}} \\
\hline
\text{Electron density (m\textsuperscript{-3})} & \text{1.0 $\times$ 10\textsuperscript{14}} \\
\hline
\text{Neutral temperature (K)} & \text{200} \\
\hline
\text{Excited temperature (K)} & \text{200} \\
\hline
\text{Field frequency (GHz)} & \text{2.57} \\
\hline
\end{array}
\]
\caption{Plasma parameters used in the analysis.}
\label{tb:1}
\end{table}

For argon and nitrogen plasmas, the following graphs illustrate key differences in their behavior.

\begin{figure}[h!]
    \centering
    \begin{subfigure}[b]{0.45\textwidth}
        \includegraphics[width=\textwidth]{fig11.png}
        \caption{Energy mobility versus electron mean energy for Ar and N\textsubscript{2}.}
        \label{fig:11}
    \end{subfigure}
    \quad
    \begin{subfigure}[b]{0.45\textwidth}
        \includegraphics[width=\textwidth]{fig12.png}
        \caption{Energy diffusion coefficient versus electron mean energy for Ar and N\textsubscript{2}.}
        \label{fig:12}
    \end{subfigure}
    \caption{Comparison of energy mobility and energy diffusion coefficient for Ar and N\textsubscript{2} plasmas.}
\end{figure}

Graph~\ref{fig:11} compares the energy mobility as a function of electron mean energy for argon (Ar) and nitrogen (N\textsubscript{2}). The graph indicates that the energy mobility behaves similarly for both plasmas. On the other hand, Graph~\ref{fig:12} shows the energy diffusion coefficient as a function of electron mean energy. For nitrogen plasma, the energy diffusion coefficient continuously increases with the electron mean energy. However, for argon plasma, the energy diffusion coefficient increases up to 1~eV and then starts to decline. This divergence in behavior could lead to different experimental results, as observed in Rovey’s experiments.

Just fot the sake of reminder, let as denote again: in Rovey's experiments, the electron temperature behaved differently depending on the plasma gas:
\begin{itemize}
    \item For argon, the electron temperature increases as the ratio of \((\text{mass flow rate})/(\text{absorbed power})\) decreases.
    \item For nitrogen, the electron temperature increases as the ratio of \((\text{mass flow rate})/(\text{absorbed power})\) increases.
\end{itemize}

This discrepancy might arise from the difference in absorbed power rates, which could be influenced by the divergence in energy diffusion coefficients between the two gases as we observed on ~\ref{fig:12}.

To address the importance of vibrational (and, more generally, internal) energy reservoirs in molecular plasmas, the following graph was generated under the plasma conditions outlined earlier, with a fixed electron mean energy of 1.6~eV.

\begin{figure}[h!]
    \centering
    \includegraphics[width=0.7\textwidth]{fig13.png}
    \caption{The normalized electron energy distribution functions versus electron energies for Argon and Nitrogen plasmas for the conditions presented in the table \ref{tb:1} for mean electron energy 1.6~eV.
}
    \label{fig:13}
\end{figure}

As shown on \ref{fig:13}, the kink in the electron energy distribution function (EEDF) for nitrogen plasma can be attributed to the vibrational excitation of molecular species. Additionally, the EEDF tail for nitrogen exhibits a higher population compared to argon, despite the relatively similar particle masses. This behavior can be explained by the reverse process of vibrational de-excitation. Nitrogen (N\textsubscript{2}) molecules in vibrationally excited states transfer energy to electrons via reverse collisions.

In the context of Rovey’s work, this serves as a foundation for understanding how electrons in nitrogen and argon plasmas redistribute energy acquired from high-frequency fields. The presence of more excitation levels in neutral species allows the applied electric field to convert energy into heating the neutral bulk of the plasma more efficiently. Specifically:
\begin{itemize}
    \item For argon (Ar), the available degrees of freedom include electronic excitation and translational modes. Vibrational levels are absent.
    \item For nitrogen (N\textsubscript{2}), vibrational and rotational degrees of freedom play a key role. Vibrational levels, while highly efficient for energy storage, are harder to excite at higher electron energies. In contrast, rotational levels are easily excited but store less energy.
\end{itemize}

Now, let us assume nitrogen and orxygen lacks vibrational excitation levels (e.g., these levels are ``frozen''). This was modeled in Bolsig+ by minimizing the cross-sections for vibrational excitation of molecules. The resulting EEDF is plotted below to illustrate the influence of vibrational excitation.

Figure~\ref{fig:14} illustrates the following:
\begin{itemize}
    \item The blue curve represents the EEDF for an N\textsubscript{2}:O\textsubscript{2} air mixture, with vibrational excitation of N\textsubscript{2} molecules included. The characteristic vibrational ``kink'' is clearly visible.
    \item The green curve represents the same mixture, but with vibrational excitation of N\textsubscript{2} excluded. The vibrational ``kink'' disappears, and the tail of the EEDF shows a lower population compared to the blue curve.
    \item The red curve corresponds to argon plasma, which behaves similarly to the green curve. This is expected since argon lacks vibrational degrees of freedom and primarily relies on translational and electronic excitation.
\end{itemize}

\begin{figure}[h!]
    \centering
    \includegraphics[width=0.7\textwidth]{fig14.png}
    \caption{Normalized Electron Energy Distribution Function (EEDF) vs. electron energy. The blue line corresponds to an N\textsubscript{2}:O\textsubscript{2} mixture with vibrational excitation included, the green line corresponds to the same mixture without vibrational excitation, and the red line corresponds to argon plasma.}
    \label{fig:14}
\end{figure}

\pagebreak

The reduced tail population in the green curve is due to the absence of vibrationally excited molecules that transfer energy back to electrons via superelastic collisions, which are efficient and thresholdless processes. This demonstrates the significant role of vibrational energy reservoirs in heating electrons within the plasma, in addition to energy input from the external field.

Further investigation of the internally-capacitive molecules such N\textsubscript{2} or O\textsubscript{2} and it's influence on the power extracted from the field to the discharges will be addressed in the foreseeable future. s

\pagebreak

\bibliographystyle{plain}
\bibliography{bibliography.bib}


\end{document}