Formation of Electric Potential Dips and Peaks by Electron-Ion Two-Stream Instability in a Plasma Chamber with an Electron Emitter LaB₆ as the Cathode

Abstract

This paper presents a conducting-channel model aimed at elucidating the generation of high-energy particles within a plasma chamber. Initially, the chamber is charged with neutral hydrogen gas at a density of approximately ~3.3 × 10²²/m³, equivalent to 1 torr at 300 K under ideal gas conditions. A Townsend discharge (dark discharge), driven by an externally imposed electric potential (500–1000 V) across the cathode and anode, is utilized to induce partial ionization of the hydrogen gas. Once a stable conducting channel with a high conductivity is established, a low electric potential (e.g., 100–500 V) is introduced to sustain the current in the conducting channel. Our investigation then delves into the impact of a high-emissivity cathode, such as lanthanum hexaboride (LaB6), on an arc discharge. We develop a theoretical model of the conducting channel that may emerge under these conditions. As the cathode surface heats, thermionic electrons form a localized layer of negative charge density outside the cathode, leading to an electric potential dip. Our multi-fluid simulations reveal the emergence of an electron-ion two-stream instability owing to the high-density electron layer, leading to the appearance of multiple potential peaks and dips, each measuring several to tens of kV. We delineate a set of conditions conducive to the formation of these potential peaks and dips within the conducting channel. Our proposed scenario furnishes a framework for elucidating electron and ion acceleration within a weakly ionized plasma chamber.

1. Introduction

The electron-ion two-stream instability, a fundamental plasma instability, arises when electrons and ions flow past each other at a relative velocity exceeding their thermal speeds. This relative drift transfers kinetic energy to charge separation and leads to the enhancement of particle speeds and electric potential [1,2]. In addition, the electric double layers [1,2,3,4,5] can also cause similar effects. The formation of localized electrostatic structures—whether quasi-static double layers (DLs) sustained by charge separation, or dynamically evolving potential oscillations driven by beam-plasma instabilities—represents one of the most fundamental and far-reaching phenomena in plasma physics. Such structures arise in laboratory discharges, space plasmas, and astrophysical environments, and they play a decisive role in particle acceleration, energy transport, and plasma confinement. The three processes reviewed here span three distinct physical regimes: current-free expanding plasmas [1,2], current-driven glow discharges [4], and beam-plasma instabilities in bounded systems [5]. Together, they provide a cohesive picture of how electric potential is built up, sustained, and modified across a wide range of plasma conditions. Hairapetian and Stenzel [1,2] demonstrated that a two-electron-temperature plasma expanding into vacuum naturally forms a current-free double layer with a potential drop of a few to tens of eV, scaling with the hot electron temperature. Conde and León [4] showed that in a glow discharge, multiple successive double layers each carry a quantized potential drop near the argon ionization potential (~15.76 V), yielding cumulative anode potentials of 30–50 V. Kaganovich and Sydorenko [5] revealed that in a bounded beam-plasma system, two-stream instability-driven electric potential oscillations can attain saturation amplitudes exceeding those of infinite plasmas by more than an order of magnitude due to boundary-induced mode selection. Together, these results not only deepen the fundamental understanding of plasma electrodynamics but also provide direct benchmarks for models of particle acceleration in laboratory, space, and technological plasma environments.

We consider a gas plasma chamber as shown in Figure 1. The chamber is filled with hydrogen gas at a density of ~$3.3\times {10}^{22} {\mathrm{m}}^{-3}$. An arc discharge is triggered in the chamber. A cathode with high electron emissivity, such as lanthanum hexaboride (LaB₆), is used. After a stable discharge is established with a higher potential (~500–1000 V) provided by the power supply, an electric potential V of ~100 V is imposed across the cathode and anode to maintain the current I in the conducting channel. The LaB₆ cathode slab is heated and emits electrons, forming an electron layer and a potential dip around it. This potential dip can accelerate charge particles.

We note that to establish an arc discharge, a voltage of hundreds to thousands of volts is required to initialize the spark [6]. A stable arc current can then be formed with a high conductivity in the tube and maintained by a low voltage of ~100 volts with a current of a few to tens of amps [7,8]. The stable arc discharge is then maintained. The proposed theoretical model is based on the assumption that an arc discharge is maintained by an externally imposed electric potential (100–500 V) across the cathode and anode in the gas chamber. Recent progress on arc discharges has been reported [9,10,11,12,13] and a review article by Anders [14]. These studies provide a comprehensive view of plasma arcs and their interactions with metal, bridging the gap between specific, localized behaviors and broad, unifying frameworks. While the studies [9,10,11,12,13] detail complex, localized physics including chemical layer formation [9], heat transfer inaccuracies [10], tungsten tip melting [11], and energy radiation [13]. The framework by Anders [14] unites these studies and classifies different arc regimes.

It will be illustrated in this study that the applied voltage between the cathode (LaB₆) and the anode is only from 100 V to 500 V, but the potential dips and peaks formed in the chamber can be tens of kV. The proposed mechanism can be applied to particle acceleration in a weakly ionized plasma or during a gas discharge.

A key finding demonstrated in this study is that while the externally applied bias between the cathode (LaB₆) and the anode remains modest (100–500 V), the dynamically formed potential dips and peaks within the chamber can scale up to tens of kilovolts (kV). This proposed self-consistent voltage-amplification mechanism offers a novel framework for particle acceleration within weakly ionized plasmas or during intense gas discharges. Notably, analogous mechanisms involving the acceleration of ions and neutrals in rotating plasmas have been previously investigated for potential applications in controlled nuclear fusion [15,16,17]. In such fusion devices (e.g., magnetic mirrors or rotating plasma traps), achieving sufficient thermonuclear reaction rates requires ions to be accelerated to energies capable of overcoming the Coulomb barrier. By generating these high-energy particle beams through intrinsic plasma instabilities rather than relying entirely on massive external power grids, this mechanism provides a highly efficient pathway for non-thermal ion heating and acceleration in advanced fusion concepts. Furthermore, this laboratory configuration exhibits a strong phenomenological parallel to large-scale atmospheric discharge events, such as cloud-to-ground (CG) lightning driven by negatively or positively charged cloud bases. Section 5.2 presents a more detailed discussion on the lightning process.

2. Theoretical Model

In the proposed theoretical model, the plasma chamber is first imposed for demonstration with a voltage of 100 V across the anode and cathode (LaB₆). A part of the chamber is shown in Figure 1. In the previous studies [16,17], the acceleration of ions and neutrals in a rotating plasma was studied for a potential application to nuclear fusion. The chamber is filled with hydrogen molecule (H₂) gas with number density ~$3.3\times {10}^{22} {\mathrm{m}}^{-3}$, corresponding to 1 torr at 300 K for an ideal gas. In our numerical simulations, the chamber pressure with $p_0=5 \mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r}$ is also used. The heated LaB₆ slab can emit electrons. We note that the study in this paper is based on the assumption that an arc discharge with a small diameter is pre-established between the cathode and anode. It is similar to a gas-discharge lamp [7,8] except that in the proposed model, a good electron emitter, LaB₆, is used as the cathode. Electrons emitted from the LaB6 cathode surface can form a potential dip, leading to plasma instability, as discussed later.

The physical process in the present theoretical model is illustrated in the flowchart in Figure 2. The externally imposed electric potential (V) leads to the formation of partially ionized hydrogen gas. As the LaB₆ slab is heated, the emitted electrons from the LaB₆ slab form a layer of negative charge density. The net negative charge density leads to a large electrostatic potential well near the surface of LaB₆ slab with a potential dip at ${\varphi }_{dip}$. In the quasi-neutral region slightly away from the LaB₆ slab and the electron layer, the electric potential $\varphi$ increases toward the anode, and the electrons and ions are accelerated in opposite directions, which leads to the electron-ion two-stream instability to be discussed later. This instability excites strong electrostatic waves, whose amplitudes may reach tens of kilovolts (kV), resulting in multiple potential dips and peaks.

3. Simulation for the Formation of Potential Dips and Potential Peaks Generated by the Electron Layer in the Plasma Chamber

We perform multi-fluid simulations to study plasma dynamics and particle acceleration, and to investigate the formation of potential dips and peaks during an arc discharge in a plasma chamber. The initial condition of the simulation corresponds to a stable arc discharge that has already ionized the gas. We note that establishing an arc discharge requires a voltage of hundreds to thousands of volts to initiate the spark. A stable arc current can then be formed and maintained at a low voltage (e.g., 100 volts) with a current of a few to tens of amperes. The stable arc discharge is then used as the initial condition.

We set the cathode electric potential at zero volts and the anode electric potential of 100, 300, or 500 volts as the simulation boundary condition. The externally imposed potential is a time-independent boundary condition and is used for solving the one-dimensional electrostatic Poisson’s equation by using the tridiagonal matrix algorithm. It is similar to the formation of a conducting channel (return stroke) in gas-discharge lamps or in the air before the occurrence of lightning. It should be noted that the kink/sausage modes with their two-dimensional structure cannot occur in the one-dimensional approximation. In the proposed model, the emitted electrons at the cathode surface could further lead to the two-stream instability.

Figure 1 illustrates the simulation domain and setup. The left-hand side ($x=0$) is the LaB₆ slab (cathode) with ${\varphi }_c=0$, and the right-hand side ($x=x_0$) is the anode with ${\varphi }_A=100, 300 \mathrm{o}\mathrm{r} 500 \mathrm{v}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{s}$. The distance between the cathode and the anode is $x_0$. The LaB₆ slab is placed in a chamber filled with hydrogen gas, H₂. The current flows from the anode to the LaB₆ cathode. The hydrogen gas is heated and partially ionized, leading to the formation of plasma with H⁺ and e⁻.

3.1. The Governing Equations

The discharge and current flow are mostly aligned along this thin, elongated conducting tube. Hence, we simplify the key physical process into a one-dimensional problem, similar to the approach for the conducting channel associated with lightning [18]. Under this approximation, we developed a one-dimensional multi-fluid simulation code along the x-axis to study the formation of the electrostatic potential in the arc discharge. The governing equations are Poisson’s equation in electrostatics:

$${\displaystyle \frac{\partial E_x}{\partial x}}={\displaystyle \frac{e}{{\epsilon }_0}}\left(n_p-n_e\right)={\displaystyle \frac{{\rho }_c}{{\epsilon }_0}},$$

(1)

the continuity equation:

$${\displaystyle \frac{\partial n_{\alpha }}{\partial t}}= -{\displaystyle \frac{\partial }{\partial x}}(n_{\alpha }{U}_{\alpha x})+D_{0,\alpha }{\displaystyle \frac{{\partial }^2{n}_{\alpha }}{\partial x^2}},$$

(2)

the momentum equation:

$${\displaystyle \frac{\partial n_{\alpha }{U}_{\alpha x}}{\partial t}}=-{\displaystyle \frac{\partial }{\partial x}}\left(n_{\alpha }{U}_{\alpha x}^2+{\displaystyle \frac{p_{\alpha }}{m_{\alpha }}}\right)+{\displaystyle \frac{q_{\alpha }}{m_{\alpha }}}{n}_{\alpha }{E}_x-n_{\alpha }{\nu }_{\alpha n}{U}_{\alpha x},$$

(3)

and the ideal gas law:

$$p_{\alpha }=n_{\alpha }{k_{B}T}_{\alpha },$$

(4)

where e is the fundamental charge or $1.6\times {10}^{-19}$ Coulomb, ${\epsilon }_0$is vacuum permittivity, and is ${\rho }_c$ charge density. The subscript $\alpha$ indicates fluid species and can be electron (e) and proton (p), and the subscript “n” stands for “neutral”. The fluid quantities are number density n, velocity $U$, pressure p, the Boltzmann constant $k_B=1.38\times {10}^{-23}$ J/K, charge q, electric field E, collision frequency for charge particles to neutral ${\nu }_{\alpha n},$ and temperature T. The real proton to electron mass ratio is used $m_p/m_e=1836$. The diffusion of electron and ion velocities through viscosity will be included and discussed later.

In this paper, we use a three-fluid (electron, ion, and neutral) simulation. The electron and the ion are independent, with their own continuity equations and momentum equations. The quasi-neutrality is not assumed and not required.

In the simulation with the complete energy equation, we found that the arc plasma can be quickly heated to well above 1 eV. However, the plasma can be partially converted to other parts of the chamber, which lowers the plasma temperature in the conducting channel. The radiation loss also reduces the plasma temperature. The temperature of the plasma in the region where current flows through is estimated to be $\approx$6000 K [19]. The simulation initial condition at t = 0 corresponds to the time when the arc discharge is formed, and the plasma is heated to $T\approx 6000 \mathrm{K} \mathrm{b}\mathrm{y} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{y}.$ We consider that the heating of the arc plasma from electricity and the heat loss through radiation and convection reach equilibrium during the simulation and that the temperature also reaches equilibrium in the conducting channel, which likely leads to constant temperature for all species $T=T_e=T_p\approx 6000 \mathrm{K}$.

We have tried a different system length $x_0$ ranging from 2 to 50 mm, and the length can affect the stability of the simulation and the formation of electrostatic waves. In this paper, we present the results associated with the system length of $x_0=$ 2 mm, 5 mm, 8 mm, 10 mm, 20 mm, 30 mm, 40 mm, and 50 mm. In the simulations, the length x is normalized by $x_0$. The other normalization parameters are neutral number density $n_0=3.3\times {10}^{22} {\mathrm{m}}^{-3}$ for $p_0=1 \mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r}$ and $T_0=300 \mathrm{K}$, electric potential ${\varphi }_0=$1 volt, velocity $U_0=9.79\times {10}^3$ m/s (velocity of 1 eV proton) and time $t_0=x_0/U_0$. For $x_0=2 \mathrm{m}\mathrm{m}$, $t_0=2\times {10}^{-7}s$. The numerical time step and grid spacing are $∆t=t_0\times {10}^{-8}$ and $∆x=x_0/512$. The neutrals in the conducting tube are assumed to be heated from 300 K to 6000 K. With a constant pressure across the chamber, the density in the conducting channel $n_c$ will be reduced as one-twentieth of $n_0$.

The numerical time step was chosen to resolve the fastest physical time scale in the system. In the present simulations, the electron plasma frequency is on the order of ${10}^{11}$ Hz. In addition, the electron-neutral and proton-neutral collision frequencies are on the order of ${10}^{10}$ Hz and ${10}^9$ Hz, respectively. Therefore, the adopted time step, $∆t=t_0\times {10}^{-8}$, is sufficiently small to resolve both the plasma oscillation time scale and the charge-neutral collision time scales. For example, when $x_0=2 \mathrm{m}\mathrm{m}$, $U_0=9.79\times {10}^3 {\displaystyle \raisebox{1ex}{$\mathrm{m}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{s}$}\right.},$ $t_0={\displaystyle \raisebox{1ex}{$x_0$}\!\left/ \!\raisebox{-1ex}{$U_0$}\right.}=2.0\times {10}^{-7}$ s, and thus $∆t=2.0\times {10}^{-15} \mathrm{s}$, which is much smaller than $1/{\omega }_{pe}, 1/{\nu }_{en}$, and $1/{\nu }_{pn}$.

The grid spacing $\Delta x={\displaystyle \frac{x_0}{512}}$ is selected based on our numerical experience with this model. To check the effect of grid resolution, we also repeat representative simulations using 1024 grid points. The main results, including the formation of the potential dip near the cathode and the subsequent potential peaks and dips caused by the electron-ion two-stream instability, do not change significantly. Therefore, 512 grid points are used in the simulations presented in the paper to reduce the computational cost while maintaining sufficient numerical resolution.

In the present simulation study, the effects of ionization and recombination are minor over the simulation time scale of tens of nanoseconds. The hot plasma only exists in the arc discharge region. The other initial parameters at $t=0$ of our simulation are neutral density $n_n\left(x\right)=n_{c0},$ charge densities $n_p\left(x\right)=n_e\left(x\right)=n_{c0}\times 4\times {10}^{-4}$, boundary electric potential $\varphi \left(x\right)=\left(x/x_0\right){\varphi }_0$, and velocities $U_p\left(x\right)=U_e\left(x\right)=0.$ The ionization ratio is determined by using Saha’s ionization equation [20,21]. The electron collisions are considered and lead to the high temperature at $T=6000 \mathrm{K}$. Saha’s equation [20,21] gives an ionized ratio of $4\times {10}^{-4}$ for $T=6000 \mathrm{K}$. Since the ionization ratio is very small ($4\times {10}^{-4})$ and the additional ionization and recombination effects are minor over the simulation time scale, the neutral density $n_n$ and ion (electron) density${ n}_p \left(n_e\right)$ can be assumed as constant during the simulation.

The hydrogen gas is weakly ionized, with H₂ as the dominant neutral species. The proton-neutral collisional frequency [22,23,24] is given by:

$${\nu }_{pn}=1.7\times {\sigma }_{pn}{n}_n\sqrt{{\displaystyle \frac{T}{m_p}}},$$

(5)

and the electron-neutral collisional frequency is given by:

$${\nu }_{en}=2.1\times {\sigma }_{en}{n}_n\sqrt{{\displaystyle \frac{T}{m_e}}}.$$

(6)

Here, the proton-neutral collision cross-section is ${\sigma }_{pn}\approx 3\times {10}^{-19} {\mathrm{m}}^2$ and the electron-neutral collision cross-section ${\sigma }_{en}\approx 7.1\times {10}^{-20} {\mathrm{m}}^2$ [24]. Equation (6) gives that the electron-neutral collision frequency is on the order of ${10}^{10}$ Hz. The term $n_{\alpha }{\nu }_{\alpha n}{U}_{\alpha x}$ in the momentum equation comes from the charge-neutral collision effect. Since the neutral fluid is affected mainly through the proton-neutral collisional effect, the proton-neutral collision frequency ${\nu }_{pn}$ is on the order of ${10}^9$ Hz. Note here that the proton to neutral mass density ratio is $4\times {10}^{-4}$. The relation $m_p{n}_p{\nu }_{pn}=m_n{n}_n{\nu }_{np}$ gives that the neutral to proton collision frequency is ${\nu }_{np}=m_p{n}_p{\nu }_{pn}/m_n{n}_n$. As a result, the neutral-proton collision frequency is of the order of ${4\times 10}^5$ Hz due to the high neutral density compared with the charged fluid. Hence, on the nanosecond time scale, the neutral fluid can be regarded as immobile during the simulation period, since the only momentum source for the neutral fluid is neutral-ion collisions. On the other hand, the collision effect exerts a “slowing-down” or drag force on the charged fluids.

The collision frequencies between protons and neutrons, as well as between electrons and neutrals, are described in Equations (5) and (6). The electron-neutral collision frequency ${\nu }_{en}$ is on the order of ${10}^{10} \mathrm{H}\mathrm{z}.$ The proton-neutral collision frequency ${\nu }_{pn}$ is on the order of ${10}^9 \mathrm{H}\mathrm{z}.$ The background geomagnetic field is assumed to be 0.5 G, and the gyro frequency of the electron can be obtained as follows:

$$\begin{array}{c}{f}_{ce}= {\displaystyle \frac{eB}{2\pi m_e}}={\displaystyle \frac{1.6\times {10}^{-19} \mathrm{C}\times 0.5\times {10}^{-4} \mathrm{T}}{2\pi \times 9.1\times {10}^{-31} \mathrm{k}\mathrm{g}}}=1.39\times {10}^6 \mathrm{Hz}.\end{array}$$

(7)

The gyro frequency of a proton can be obtained as follows:

$$\begin{array}{c}{f}_{cp}= {\displaystyle \frac{eB}{2\pi m_p}}={\displaystyle \frac{1.6\times {10}^{-19} \mathrm{C}\times 0.5\times {10}^{-4} \mathrm{T}}{2\pi \times 9.1\times {10}^{-31} \mathrm{k}\mathrm{g}\times 1840}}=760.4 \mathrm{Hz}.\end{array}$$

(8)

The results show that the collision frequencies of electrons and protons with neutrals are much higher than the gyro frequencies of electrons and protons. Moreover, the effects of the magnetic field generated by the current in the chamber on the electron dynamics are explained in Supplementary Materials. An electron would suffer many collisions with neutrals before it could make a gyration. The gyration effect from the magnetic fields can be neglected.

In a weakly ionized gas, the electron and proton density diffusion coefficients due to thermal motion and collisions between charged and neutral fluids, to first order in the density, can be estimated as follows [25]:

$$\begin{array}{c}{D}_{0,\alpha }={\displaystyle \frac{k_{B}T}{m_{\alpha }{\nu }_{\alpha n}}}, \end{array}$$

(9a)

where $k_B=1.38\times {10}^{-23} \mathrm{J}·{\mathrm{K}}^{-1}$ is the Boltzmann constant. We obtain $D_{0,e}\approx 1230 {\mathrm{m}}^2{\mathrm{s}}^{-1}$and $D_{0,p}\approx 8.22 {\mathrm{m}}^2{\mathrm{s}}^{-1}$for $T=4000–6000 \mathrm{K}$.

On the other hand, the coefficient associated with bulk viscosity without mass density for electron and proton fluids can be estimated as follows:

$$\begin{array}{c}{\zeta }_{\alpha }={\zeta }_{\alpha ,0}{\displaystyle \frac{k_{B}T}{m_{\alpha }{\nu }_{\alpha n}}}, \end{array}$$

(9b)

where ${\zeta }_{\alpha ,0}$ is a dimensionless factor in the range of 10–100 due to the fact that the bulk viscosity is usually 10–100 times higher than the shear viscosity with ${\zeta }_{\alpha ,0}=1$. The bulk viscosity, used in the numerical simulation, is of the order of theoretical value.

At every time step, the density diffusion and velocity viscosity terms are applied to the charge density ($n_e$ and $n_p$) and to the velocity ($U_{ex}$ and $U_{px}$) via a three-point smooth scheme based on the diffusion and viscosity coefficients, respectively. The smoothing scheme plays the role of dissipation and also reduces the steepening of the fluid, similar to the dissipation term in the Burger’s equation. The three-point smooth scheme is derived from the finite difference form of the diffusion equation:

$$\begin{array}{c}{Q}_x^{\prime }={\displaystyle \frac{Q_{x-\Delta x}+\left(s_{\alpha }-2\right)Q_x+Q_{x+\Delta x}}{s_{\alpha }}}, \end{array}$$

(10)

where $s_{\alpha }=\Delta x^2/S_{\alpha }\Delta t,$ and $S_{\alpha }=D_{0, \alpha }$ for density or $S_{\alpha }={\zeta }_{\alpha }$ for velocity.

In the continuity equations, a bounded (closed) boundary condition is used, i.e., the momentum and velocity are set to zero at the boundaries. However, the emission of electrons at the surface of the LaB₆ slab can lead to an increase in the electron density just outside the LaB₆ slab. The electron emission rate due to heating of LaB₆ slab is set to 10–15 amps [26].

3.2. Two Simulation Cases Without and with Emission of Thermionic Electrons ($x_0=20 \mathrm{m}\mathrm{m},{\varphi }_A=100 \mathrm{V},p_0=1 \mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r})$

In order to examine the effect of the thermionic electron emission from LaB₆, we simulate two cases without LaB₆ and with LaB₆.

Figure 3A shows a simulation without LaB₆ slab in the cathode and a 100 V electric potential is applied across the gas chamber $(p_0=1 \mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r})$ with $x_0=20 \mathrm{m}\mathrm{m}$. For comparison, Figure 3B shows the simulation results with the same parameters as in Figure 3A, but a LaB₆ slab is applied in the cathode.

In Figure 3A, an electric potential ${\varphi }_A=100 \mathrm{V}$ is applied across the cathode ($x=0)$. Some electrons in the plasma chamber are attracted by the applied positive potential at $x=x_0$ to form a potential dip around point B. The lower potential at the cathode tends to attract ions to the left region between $x=0$ and point A. The ion density near $x=0$ is lower than the electron density near B. This may be due to a larger ion mass (lower electron mass) and hence a slower (faster) moving speed.

It is interesting to point out that the electric potential profile is similar to the potential distribution between the cathode and anode in Figure 4 of the review paper by Anders [14]. The potential drop between points B and C is the positive anode fall, while the potential drop between the origin and point A is the negative cathode fall.

In Figure 3B, a simulation with a LaB₆ slab installed in the cathode and 100 V potential is applied in the 20 mm chamber ($x_0=20 \mathrm{m}\mathrm{m}$) between the cathode and the anode. The neutral H₂ is filled with $p_0=1 \mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r}$, which is equivalent to the density of ${\displaystyle \frac{1}{20}}\times 3.3\times {10}^{22} {\mathrm{m}}^{-3}$. The thermionic electrons are emitted from the LaB₆ surface and carry a 10 A current. In this case, a potential dip near the LaB₆ surface can exceed 1000 volts. In the arc discharge, the two-stream instability occurs and leads to potential peaks of a few hundred volts. The theory of two-stream instability will be developed in the following simulation case.

In the simulation case shown in Figure 4, the density and velocity distributions in the x–t plane show clearly the presence and growth of wavy patterns, which are caused by the electron-ion two-stream instability. This case shows a clear pattern of the electrostatic waves. The potential peak can reach nearly 11.4 kV at $t=2.4 \mathrm{n}\mathrm{s}$, and the dip can reach about −0.6 kV (listed in Case 9 of Table S1). The wavy potential structure may last for a few nanoseconds and quickly fades away. The number of peaks and dips depends on the plasma density and electron velocity. These waves are nearly stationary oscillations in the plasma (electron) frame and can move in the laboratory frame.

The electron-ion two-stream instability can be solved by the linear plasma dispersion relation:

$$\begin{array}{c}1-{\left({\displaystyle \frac{{\omega }_{pp}}{\omega }}\right)}^2-{\left({\displaystyle \frac{{\omega }_{pe}}{\omega -k〈U_e〉}}\right)}^2=0, \end{array}$$

(11)

where ${\omega }_{pe}$ is the electron plasma frequency, ${\omega }_{pp}$ is the proton plasma frequency, and $k$ is the wave number [3,27,28,29]. The electron plasma frequency and the proton plasma frequency $({\omega }_{pe}$ and ${\omega }_{pp}$) can be obtained from:

$$\begin{array}{c}{\omega }_{pe}=({\displaystyle \frac{4\pi {n_{e}e}^2}{m_e}}{)}^{{\displaystyle \frac{1}{2}}}, \end{array}$$

(12)

$$\begin{array}{c}{\omega }_{pp}=({\displaystyle \frac{4\pi {n_{p}e}^2}{m_p}}{)}^{{\displaystyle \frac{1}{2}}} \end{array}$$

(13)

Here, $\omega ={\omega }_r+i{\omega }_i$ is the wave frequency. The linear dispersion relation is solved based on the electrostatic two-fluid equations for cold and homogeneous plasmas [28]. Note that the plasma frequency is on the order of ${10}^{11}$ Hz, about one order of magnitude higher than the electron-neutral collision frequency. Hence, the collision effect can be ignored when solving this dispersion relation.

The electron plasma frequency is estimated from the electron density used in the simulation. For $n_e$ approximately $1\times {10}^{19} {\mathrm{m}}^{-3}$, the electron plasma frequency is ${\omega }_{pe}$ approximately $1.78\times {10}^{11}$ rad/s. The electron-neutral collision frequency is estimated from the neutral density, temperature, and electron-neutral collision cross section, as described in Equation (6). With the parameters used in the simulation, ${\nu }_{en}$ is on the order of ${10}^{10}$ Hz. Therefore, the statement that the plasma frequency is about one order of magnitude higher than the electron-neutral collision frequency is based on these calculated characteristic frequencies.

As illustrated in Lyu [28], Equation (11) can be replaced as follows:

$$\begin{array}{c}{x}^4-2\alpha x^3+x^2\left({\alpha }^2-1-{\displaystyle \frac{m_e}{m_i}}\right)+\left(2{\displaystyle \frac{m_e}{m_i}}\alpha \right)x-{\alpha }^2{\displaystyle \frac{m_e}{m_i}}=0, \end{array}$$

(14)

where $x={\displaystyle \frac{\omega }{{\omega }_{pe}}}, and \alpha ={\displaystyle \frac{{kV}_0}{{\omega }_{pe}}}.$ There are four roots in Equation (14). Only the roots with positive growth rate (${\omega }_i>0)$ are plotted in Figure 5. As shown in Figure 5, there is a long channel tube, in which the wave structure can be analyzed by the Fourier analysis of $\omega$ and k. The solution of $\omega ={\omega }_{re}+i{\omega }_{im}$ with non-zero wave growth rate ${\omega }_{im}$ gives (a) the wave frequency ${\omega }_{re}/{\omega }_{pe}=0–0.08$ and (b) the wave number $k〈U_e〉/{\omega }_{pe}=0–1.13.$ The maximum wave growth rate ${\omega }_{im,max}/{\omega }_{pe}\approx 0.054$ occurs at $k〈U_e〉/{\omega }_{pe}\approx 1$ and ${\omega }_{re}/{\omega }_{pe}\approx 0.04$ as shown in Figure 5. The corresponding wavelength can be expressed as follows:

$$\begin{array}{c}\lambda ={\displaystyle \frac{2\pi 〈U_e〉}{{\omega }_{pe}}}. \end{array}$$

(15)

Based on the plasma parameters ($〈U_e〉\approx 2\times {10}^7$ m/s, $n_e\approx 1\times {10}^{19} {\mathrm{m}}^{-3}$, and obtain ${\omega }_{pe}=1.78\times {10}^{11} \mathrm{r}\mathrm{a}\mathrm{d}·{\mathrm{s}}^{-1}$ from Equation (12)) and the linear theory, we obtain $\lambda \approx 0.35 x_0$ and the maximum growth rate $\approx {10}^{10} {\mathrm{s}}^{-1}$ from (15). For the simulation in Figure 4, there are four potential peaks, and the corresponding wavelength is $\lambda =x_0/4=0.25 x_0$, close to the theoretical value $\lambda =0.35 x_0.$ In Figure 6a, we plot the electric potential as a function of x at different simulation times, t = 0.9, 1.0, …, 1.2 ns. It can be seen that the magnitude of the electric potential peak and dip grows with time t for all four wave peaks and dips.

The growth rate of the electric potential Peak 2 can be calculated from electron density peak values at different times. The growth rate of peaks is plotted as a function of time in Figure 6b. The growth rate of wave Peak 2 ranges from $2.4\times {10}^9 {\mathrm{s}}^{-1}$ to $8.5\times {10}^9 {\mathrm{s}}^{-1}$, which is smaller than the maximum theoretical growth rate. This difference may be caused by the non-uniform profile of $U_e$ and difference between the simulated wavelength $\left(\lambda =0.25x_0\right)$ and theoretical peak wavelength $(\lambda =0.35x_0)$.

In Figure 7, we show the simulation (Case B in Figure 3B) results at time $t=$ 0.9, 1.0, 1.1, and 1.2 ns. In this case, an electric potential dip is formed due to the high-density electron layer and the magnitude can reach one kilovolt as shown in the electric potential profiles.

Thermionic emitted electrons lead to the formation of an electron layer near the left boundary ($x\approx 0$). As a result of this electron layer, the electric potential $\varphi$ has a sharp decrease from $\varphi (x=0)$ to ${\varphi }_{dip}<0$. The electric potential $\varphi$ then increases gradually to $\varphi \left(x=x_0\right)={\varphi }_A=100$ volts at the right boundary (anode). The potential difference between ${\varphi }_{dip}$ and ${\varphi }_A$ at $x=x_0$ is large. The average electric field in the chamber is negative ($E_x<0$). The negative electric field in the chamber will accelerate electrons to high speed $U_e\approx -2.7-4.8\times {10}^7$ m/s, while ions are accelerated in the left direction with small velocity due to their large mass. The electrons and ions are accelerated in the opposite directions, which can lead to the electron-ion two-stream instability. In the electric potential profile, large amplitude electrostatic waves are formed due to this instability, and the wave amplitude may reach tens of kilovolts. The electron and ion number density and velocity profiles also show similar patterns. The wavy potential structure may last for a few ns.

In Figure 7, the peak electric potential can reach about 2000 V, the electric potential dip can reach −6.0 kV. Note that, in a weakly ionized plasma with temperature $T=$ 0.5–0.75 eV, an electron can attach to a hydrogen atom H to form a negative hydrogen particle $H^{-}$. The density of negative hydrogen ($H^{-}$) is expected to be small compared with $H^{+}$ density due to its low production rate [30]. We ignore the contribution from $H^{-}$ ions in the multi-fluid simulation. The formation of negative electric potential dips (positive electric potential peaks) can accelerate negative hydrogen ions (positive protons) toward the LaB₆ slab.

In the multi-fluid theory, unstable normal modes can arise whenever the electron and ion flowing velocities are different [28,29,31]. The total electron flow energy, integrated over the entire tube, decreases during the growth of streaming instability. The growth wave energy arises mainly from the decrease in the electron flow energy within the simulation domain. In this study, we found that the two-fluid plasma electron-ion two-stream instability can be excited. It is the relative velocity between ion flow and electron flow that provides the free energy for the electron-ion two-stream instability.

We estimate the diameter of the arc discharge by comparing a power input of 200 watts for the arc charge with the total kinetic and electric energy in the simulation domain. We calculate the total kinetic energy (electric energy) per area at $t=$ 1.8 ns of the entire simulation domain and obtain 150 $\mathrm{J}/{\mathrm{m}}^2$ (164 $\mathrm{J}/{\mathrm{m}}^2$). The energy provided by the power input at 200 watts over the period of 1.8 ns is $8.5\times {10}^{-6} \mathrm{J}$. We obtain an upper bound for the diameter $D_{cond}$ of the arc discharge:

$$\begin{array}{c} D_{cond}\le {\displaystyle \frac{2}{\sqrt{\pi }}}{\left({\displaystyle \frac{8.5\times {10}^{-6} \mathrm{J}}{314 \mathrm{J}/{\mathrm{m}}^2}}\right)}^{0.5}=0.19 \mathrm{mm}.\end{array}$$

(16)

Note that the length of the arc discharge is 2 mm, which is 10 times of $D_{cond}$. Moreover, we also simulate cases of emitted thermionic current as 15 A for three different sets of hydrogen pressure, three sets of applied voltage, and three sets of distances in the next section.

3.3. Comparisons for Simulation Results with Different Parameters

Table S1 lists 48 cases simulated with different sets of parameters, which include the neutral H₂ pressure p, the neutral density $n_n$, the electric potential ${\varphi }_A$ at the anode, and the system (tube) length $x_0$. The resulting potential peak ${\varphi }_{peak}$ and potential dip ${\varphi }_{dip}$ are also listed in Table S1.

Figure 8 shows the resulting ${\varphi }_{peak}$ and ${\varphi }_{dip}$ as a function of $x_0$ (distance between cathode and anode) under applied voltages of 100 V, 300 V, and 500 V from left to right columns. The top (bottom) panels in Figure 8 show the resulting ${\varphi }_{peak}({\varphi }_{dip})$. The red hollow circles are for the cases with 1 torr of neutral hydrogen pressure in the chamber. The black (blue) hollow circles denote the cases with 5 torr (10 torr) of neutral hydrogen (H₂) in the chamber. The resulting ${\varphi }_{peak}$ ranges from 0.1 kV to 45.0 kV, while the resulting ${|\varphi }_{dip}|$ ranges from 0.6 kV to 38.0 kV.

Table S1 and Figure 8 show that Case 47 has the largest potential peak $({\varphi }_{peak}=15.0 \mathrm{k}\mathrm{V})$. The largest potential dip is $\left|{\varphi }_{dip}\right|=15 \mathrm{k}\mathrm{V}$ in Case 47. In Case 47, $p_0=1 \mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r}, { n}_n=3.3\times {10}^{22} {\mathrm{m}}^{-3}, {\varphi }_A=500 \mathrm{V}, \mathrm{a}\mathrm{n}\mathrm{d} x_0=50 \mathrm{m}\mathrm{m}$. Figure 8 shows that the twenty-four cases with 1 torr have quasi-proportional value of ${\varphi }_{peak}$ and of ${|\varphi }_{dip}|$ with applied voltage ${\varphi }_A$.

Regarding the dependence on $x_0$ (distance between cathode and anode), a larger $x_0$ has a larger $\left|{\varphi }_{dip}\right|$ with the same filling H₂ pressure, while the dependence of ${\varphi }_{peak}{ on x}_0$ is mixing. For the cases (blue line) with $p_0=5 torr$ and long chambers ($x_0>30 \mathrm{m}\mathrm{m}$), the plasma density is high and can lead to electron density pile up and interrupting the simulation.

Figure 9 shows the simulation results of Case 47 at $t=$ 2.0 ns, 2.2 ns, 2.4 ns, and 2.6 ns. In this case, an electric potential dip is formed due to the high-density electron layer and the magnitude can reach −15.0 kV as shown in the electric potential profiles.

Thermionic emitted electrons lead to the formation of an electron layer near the left boundary ($x\approx 0$). As a result of this electron layer, the electric potential $\varphi$ has a sharp decrease from $\varphi (x=0)$ to ${\varphi }_{dip}~-15.0 \mathrm{k}\mathrm{V}$. The electric potential $\varphi$ then increases gradually to $\varphi \left(x=x_0\right)=500$ volts at the right boundary (anode). The average electric field in the chamber is negative ($E_x<0$). This negative average electric field in the chamber will accelerate electrons to high speed $U_e\approx 6.5\times {10}^7$ m/s, while ions are accelerated in the left direction with small velocity due to their large mass. The electrons and ions are accelerated in the opposite directions, which can lead to the electron-ion two-stream instability as discussed earlier. In the electric potential profile, large amplitude electrostatic waves are formed due to this instability, and the wave amplitude may reach tens of kilovolts. The electron and ion number density and velocity profiles also show similar patterns. The wavy potential structure may last for a few ns.

4. Transportation of Protons and Electrons Through High-Density Neutrals (Hydrogen Gas) with Strong Electric Fields

The formation of electron layer leads to a strong electric field, which can accelerate protons. During the acceleration, proton-neutral collisions occur, and a fraction of the protons cannot be accelerated to the peak energy as they reach the LaB₆ target. In the following, we consider the proton-neutral collision effect and use the Monte Carlo method to simulate the acceleration and collision process to obtain the energy distributions of protons reaching the LaB₆ target for possible nuclear fusion application.

In the following, the proton transportation through the high-density molecular hydrogen with a strong electric field is simulated by Geant4 10.5 (GEometry ANd Tracking) [32,33,34,35], which is a Monte Carlo toolkit used for the simulation of the transportation of particles through matter. For simplicity, we consider that the electric field in the chamber is uniform and constant in the Geant4 simulation. The schematic diagram is illustrated in Figure 10a. The uniform electric field with the strength of 5.0 × 10⁷ V/m (5.0 × 10⁶ V/m) is set with a thickness of 1 mm (10 mm), and the electric potential difference is 50 kV. The results for the cases with a total potential of 10–40 kV are similar to the 50 kV case. High-density neutrals (molecular hydrogen) are uniformly filled in the space. The picture we consider here is that a proton with an initial energy ${\epsilon }_{in}$ is injected into the hydrogen gas. The proton undergoes acceleration and collisions, and its final energy ${\epsilon }_f$ is recorded as it reaches the other side.

Figure 10b–g shows the energy distribution of outgoing protons for different incident energies ${\epsilon }_{in}$ = 0.01, 0.1, and 1 keV with acceleration layer lengths of 1 and 10 mm. The density of neutral $H_2$ is $3.3\times {10}^{23} {\mathrm{m}}^{-3}$ and the temperature is 1273 K. The number of entry protons for Monte Carlo calculations is ${10}^6~6\times {10}^6$. The mean energy $〈{\epsilon }_f〉$ and standard deviation $\sigma \left({\epsilon }_f\right)$ of the outgoing energy are also given. One can see that the mean outgoing energy is close to the electric acceleration energy 50 keV. It seems that the electric field acceleration plays a dominant role in the transportation while the energy loss from collisions is small.

The potential dip ranges from kilovolts to tens of kilovolts and can accelerate electrons. Figure 11 shows the energy distribution of incident electrons with an initial energy ${\epsilon }_{in}=$ 10 eV accelerated by the potential dip ${\varphi }_{dip}=$ 1, 5, and 30 kV with an acceleration layer length of 1 mm. The density of neutral $H_2$ is $3.3\times {10}^{23} {\mathrm{m}}^{-3}$ with a temperature of 1273 K. The number of entry electrons is ${10}^6$. The mean energy $〈{\epsilon }_f〉$ and standard deviation ${S(\epsilon }_f)$ of the final energy are given. The mean outgoing energy is close to the electric acceleration energy $e{\varphi }_{dip}=$ 1, 5, and 30 keV, where $e$ is the fundamental charge, $1.6\times {10}^{-19}$ Coulomb. The energy loss from collisions is small. The results are not sensitive to the neutral hydrogen ($H_2$) temperature for $T=$ 1000–1500 K.

5. A Brief Summary of the Conducting-Channel Model and Application

5.1. A Brief Summary of the Conducting-Channel Model

In this paper, a conducting-channel model is proposed for generating energetic particles in a plasma chamber. The chamber is initially filled with neutral hydrogen gas with density from $3.3\times {10}^{22} {\mathrm{m}}^{-3}$ to $1.65\times {10}^{23} {\mathrm{m}}^{-3}$, corresponding to 1–5 torr at 300 K for ideal gas conditions. An arc discharge is sustained by an externally applied electric potential across the cathode (slab) and anode, ranging from several hundreds to 1 kV in voltage and several amperes in current, thereby inducing a partially ionized hydrogen plasma within the chamber. As the LaB₆ slab heats up, the emitted electrons accumulate to form a layer of negative charge density locally near the cathode.

The multi-fluid simulation conducted in this study shows that the electron layer together with the imposed voltage across the plasma chamber can lead to formation of a large electrostatic potential with dips and peaks of exceeding 10 kilovolts. While our one-dimensional simulation neglects the magnetic field generated by the current flow in the arc discharge and the associated plasma instabilities, it demonstrates that the azimuthal magnetic field associated with the current filament exerts minimal effects on particle orbits. The first potential dip is formed due to the presence of the electron layer. The density diffusion and velocity viscosity in the simulation are important to create a large potential dip near $x=0$ (the cathode). As long as the first potential dip near $x=0$ is deep enough, the acceleration of ions (electrons) toward (away from) the cathode due to the large electric potential leads to the electron-ion two-stream instability. The multiple potential peaks and dips can then be formed. Electron fluid and electron particles within the chamber can be accelerated by these potential dips and peaks.

We have also carried out experiments based on the simulation of this paper. The experiment set is reported in [36]. Due to the limitation of the chamber size and to avoid changing the original plasma distributions in the chamber, the measurements of the electron density, the electric voltage of the electrons, and electron temperature are not obtained. The experiment is not completed and only measures the temperature variation of the cooling water flowing inside the flanges and chamber walls. Therefore, we cannot directly compare the difference between the simulation and the experiment.

The findings and insights derived from these simulations will be cross-referenced with experimental data in forthcoming publications, providing a comprehensive analysis of the research outcomes.

5.2. A Possible Application to the Conducting Channel Associated with Lightning

The lightning associated with the negatively (positively) charged cloud is called negative (positive) cloud-to-ground (CG) lightning. The configuration in Figure 1 is very similar to a negatively charged cloud, air, and the conducting Earth [37,38,39]. The electric potential between the charged cloud and the Earth’s ground is typically 250–400 kV [18,40]. A conducting channel is formed before the onset of lightning. The electric potential in the conducting channel associated with some plasma processes in the lightning tube can reach 50–70 MV [41]. The high electric potential along the lightning channel can lead to the presence of run-away electrons and the generation of X-rays and Gamma rays [41]. Our future work may apply the two-stream instability developed in this paper to provide a possible theory for the formulation of the high potential 50–70 MV lightning phenomena in nature.