Metastable Helium Atom Creation Dynamics in High-Voltage Pulsed Discharge with Transverse Magnetic Field Effects

doi:10.21203/rs.3.rs-3918067/v1

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Metastable Helium Atom Creation Dynamics in High-Voltage Pulsed Discharge with Transverse Magnetic Field Effects

https://doi.org/10.21203/rs.3.rs-3918067/v1

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published 27 May, 2024

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In this work, we investigate the effect of an external transverse weak magnetic field on the creation of metastable helium atoms and excited helium molecules in a high-voltage pulsed discharge in helium at medium pressure. A two-dimensional fluid model is used to describe a high-voltage pulsed discharge in helium in the external transverse weak magnetic field. The dynamics of discharge development in the high-voltage pulsed discharge in helium at a pressure of 30 Torr in the presence and absence of the magnetic field is studied. The effect of the external magnetic field on the behavior of the density of charged particles, metastable helium atoms, and excited helium molecules in the high-voltage pulsed helium discharge has been investigated. It is shown that in the discharge region, the density of metastable atoms decreases when a transverse magnetic field is applied, which is a consequence of an increase in the frequency of stepwise ionization.

gas-discharge

gas breakdown

pulsed discharge plasma

magnetic field

numerical simulation

Pulsed discharges in helium at medium gas pressure are widely used in electrophysical equipment [1] and in other technical devices based on dielectric barrier discharges [2]. An external magnetic field in such discharges allows to control the discharge characteristics by controlling the magnitude of the magnetic field, which gives additional relevance to such studies. Plasma behavior in the magnetic field is a special section of plasma physics, and the beginning of intensive research dates back to the time of the first experiments on plasma confinement by a magnetic field in the 1950s. [3–6]. The topic of research on the confinement of high-temperature plasma by the magnetic field nowadays is still of great interest to scholars worldwide. [7–10]. The influence of the magnetic field on low-temperature plasma of various types of gas discharges has also been studied no less actively in the past and now [1, 11–29]. In the early works [3, 11–15] it was shown that the effect of the magnetic field is equivalent to the gas pressure increase. The presence of the magnetic field leads to an increase in the brightness of the plasma column, which can be explained by the fact that the magnetic field prevents the expansion of the discharge column and reduces the diffusion losses of electrons [13]. It has been observed that the magnetic field can reduce the breakdown voltage from tens of kilovolts to several hundred volts in some gases [16]. Thus, when in independent discharge under conditions of additional pre-ionization, it is possible to realize a volumetric form of discharge at gas pressure of the order of atmospheric pressure [17, 18]. The theoretical and experimental description of the behaviour of charged particle swarms in gases in the presence of electric and magnetic fields was summarised in [19]. It has also been found [20] that the magnetic field not only leads to an increase in current density and conductivity but also leads to a decrease in the time of the arc discharge formation and the rate of expansion of the discharge channel. It was confirmed that the magnetic field can affect the energy and transport characteristics of the plasma [21]. Modern powerful computers and the development of modern numerical methods for solving differential equations provide an opportunity to implement numerical calculations of physical phenomena and processes without expensive experiment investigations. Thus, at the end of the last century, numerical models for describing gas-discharge plasma in a magnetic field appeared [23–29]. In work [23], a one-dimensional model of a low-pressure glow discharge in the transverse magnetic field was presented. The model to describe the effect of the transverse magnetic field on the distribution of charged particles in a positive column of the gas-discharge was presented in [24].

The main effect of the magnetic field on the gas-discharge plasma is that under the magnetic field, the electrons and ions under Lorentz force in gas-discharge plasma do not move in straight lines but move in helical trajectories around the magnetic field lines with radius r, called Larmor radius:

$${r_L}=m\,{\upsilon _ \bot }/q\,B$$

where ${\upsilon _ \bot }$ is the velocity component of the particle perpendicular to the direction of the magnetic field B, q is the charge of the particle, and m is the mass of the charged particle (electron or ion). The magnetic field affects the plasma transfer to the chamber wall and subsequently influences the ionization balance and the spatial distribution of the plasma. In fact, due to the decrease in plasma losses on the wall, the ionization frequency of gas-discharge plasma under a magnetic field decreases.

The study of the gas-discharge plasma in magnetic fields is currently still actively performed [25, 26, 29]. In the study of the influence of the magnetic field on gas-discharge plasma, it is of particular importance to study the processes of production of metastable atoms and excited molecules. It is also important to take these particles into account in modeling [27–30]. Metastable atoms and molecules play an important role in gas-discharge, for example, in ion generation through Penning ionization processes. Metastable atoms also play a significant role in much plasma technology, for example, as catalysts [31], and for ionization of target analytes [32]. Thus, in [27], the effect of metastable atoms in a radiofrequency discharge in argon was investigated and it was shown that at a pressure of 1 Torr, stepwise ionization accounts for a significant percentage of the total ionization rate, strongly affecting the heating of electrons and the discharge potential. Therefore, the study of the behavior of metastable particles in discharges plays an important role, especially when the magnetic field is applied to the gas-discharge gap.

In our work, the effect of a transverse magnetic field on the characteristics of a plasma of a high-voltage pulsed discharge in helium at a pressure of 30 Torr is investigated. The behavior of metastable helium atoms and excited helium molecules in the magnetic field in the high-voltage pulsed helium discharge is investigated. In addition, the spatio-temporal dynamics of the electron density and the metastable particle density in the discharge gap with and without the transverse magnetic field are also studied. It is shown that the application of the external magnetic field of small magnitude leads to an increase in the frequency of stepwise ionization in the discharge region. This in turn results in a decrease in the density of metastable particles in this region.

The object of our study in this work is a high-voltage pulsed discharge in helium at middle gas pressure. To illustrate the capabilities of the numerical model developed in our work the calculations were performed for a gas pressure of 30 Torr. A two-dimensional fluid model was used to analyze the effect of a transverse magnetic field on the characteristics of a gas-discharge. The numerical model of the gas-discharge chamber corresponded both in size and design to the real geometry. Figure 1 shows the geometry of the numerical model, which is a two-dimensional geometry that corresponds to the real geometry of the gas-discharge system. The gas-discharge chamber consists of stainless steel hemispherical electrodes with a diameter of 8 mm located at a distance of 4 mm from each other inside a quartz tube. The permanent magnets located over the gap created the magnetic field with a value of $B=0.4\,{\text{T}}$ between the electrodes.

To describe the plasma and the plasma processes in the discharge chamber, the continuity equation for particles in a drift-diffusion approximation, the drift-diffusion equation for the mean electron energy, and the Poisson equation to describe the electric field in the discharge gap are used [33].

The system of equations was supplemented by the following initial and boundary conditions:

for electron density: ${n_e}\left( {t=0,x,y} \right)={10^{15}}\,{{\text{m}}^{ - 3}}$ – initial background plasma density;
the initial mean electron energy: $\varepsilon =4\,{\text{eV}}$;
the boundary condition at the anode: $V=U\left( t \right)$. The potential at the anode walls is the same as in the experiment. The boundary condition at the cathode: $V=0$.

It is known that in a weakly ionized plasma in a not strong magnetic field (until the ions are magnetized), if the frequency of elastic collisions does not depend on the electron energy at a fixed gas pressure, then the ionization frequency in a magnetic field will only depend on the effective electric field ${{\mathbf{E}}_{ef}}$ [34–39]. From all noble gases, such an approximation is performed with satisfactory accuracy for helium. Under these conditions, the magnetic and electric fields in the kinetic equation are only in combinations ${E^2}/\left( {1+{\omega ^2}/{\nu _{ea}}^{2}} \right)$, and the ionization frequency ${v_i}$ depends only on the effective electric field ${{\mathbf{E}}_{ef}}$, since it is determined by the symmetric part of the EEDF, i.e. ${v_i}\left( {{\mathbf{E}},{\mathbf{H}}} \right)={v_{i0}}\left( {{{\mathbf{E}}_{ef}}} \right)$. Therefore, in our case to account for the effect of the magnetic field in equations electric field ${\mathbf{E}}$ is replaced by the effective electric field ${{\mathbf{E}}_{ef}}$

$${{\mathbf{E}}_{ef}}=\frac{{\mathbf{E}}}{{\sqrt {1+{\omega ^2}/{\nu _{ea}}^{2}} }},$$

where $\omega =qB/m$ is the electron cyclotron frequency in the magnetic field and ${v_{ea}}$ is the electron transport collision frequency. Eq. (2) is known as Tonks’ theorem [39]. At high values $E/N$ and low values $H/N$ it was observed better agreement between the results that follow from Tonks’ theorem and exact Monte Carlo calculations [40].

In our model, the following species are considered: helium atom in the ground state – $He\left( {{1^1}S} \right)$; two metastable atoms – $H{e^*}\left( 1 \right)\left( {{2^3}S} \right)$ and $H{e^*}\left( 2 \right)\left( {{2^1}S} \right)$; two radiative atoms – $H{e^*}\left( 3 \right)\left( {{2^3}P} \right)$ and $H{e^*}\left( 4 \right)\left( {{2^1}P} \right)$; atomic ion – $H{e^+}$; two excimer molecules – metastable $He_{2}^{*}\left( 1 \right)\left( {^{3}\sum\nolimits_{u}^{+} {} } \right)$ and radiative $He_{2}^{*}\left( 2 \right)\left( {^{1}\sum\nolimits_{u}^{+} {} } \right)$; and molecular ion: $He_{2}^{+}$. Table 1 presents the set of 56 plasma-chemical reactions used in our model. Note that the model considers collisions with not only electrons but also heavy particles, including molecules; these are the processes of ion conversion, molecular conversion, and Penning ionization.

Table 1

**Plasma-chemical reactions used in our helium model** [41–47]. ${T_e}$ **is the electron temperature in eV and** ${T_g}$ **is the gas temperature in K.**
	Reaction	Data	Process
1	$e+He \to e+He$	Cross-section	Elastic
2	$e+He \leftrightarrow e+H{e^*}\left( 1 \right)$	Cross-section	Excitation
3	$e+He \leftrightarrow e+H{e^*}\left( 2 \right)$	Cross-section	Excitation
4	$e+He \leftrightarrow e+H{e^*}\left( 3 \right)$	Cross-section	Excitation
5	$e+He \leftrightarrow e+H{e^*}\left( 4 \right)$	Cross-section	Excitation
6	$e+H{e^}\left( 1 \right) \leftrightarrow e+H{e^}\left( 2 \right)$	Cross-section	Stepwise excitation
7	$e+H{e^}\left( 1 \right) \leftrightarrow e+H{e^}\left( 3 \right)$	Cross-section	Stepwise excitation
8	$e+H{e^}\left( 1 \right) \leftrightarrow e+H{e^}\left( 4 \right)$	Cross-section	Stepwise excitation
9	$e+H{e^}\left( 2 \right) \leftrightarrow e+H{e^}\left( 3 \right)$	Cross-section	Stepwise excitation
10	$e+H{e^}\left( 2 \right) \leftrightarrow e+H{e^}\left( 4 \right)$	Cross-section	Stepwise excitation
11	$e+H{e^}\left( 3 \right) \leftrightarrow e+H{e^}\left( 4 \right)$	Cross-section	Stepwise excitation
12	$e+He \to 2e+H{e^+}$	Cross-section	Direct ionization
13	$e+H{e^*}\left( 1 \right) \to 2e+H{e^+}$	Cross-section	Stepwise ionization
14	$e+H{e^*}\left( 2 \right) \to 2e+H{e^+}$	Cross-section	Stepwise ionization
15	$e+H{e^*}\left( 3 \right) \to 2e+H{e^+}$	Cross-section	Stepwise ionization
16	$e+H{e^*}\left( 4 \right) \to 2e+H{e^+}$	Cross-section	Stepwise ionization
17	$e+He_{2}^{*}\left( 1 \right) \to 2e+He_{2}^{+}$	Cross-section	Stepwise ionization
18	$e+He_{2}^{*}\left( 2 \right) \to 2e+He_{2}^{+}$	Cross-section	Stepwise ionization
19	$e+He_{2}^{}\left( 1 \right) \to 2e+He_{2}^{}\left( 2 \right)$	Cross-section	Stepwise excitation
20	$e+He_{2}^{*}\left( 1 \right) \to e+2He$	$3.8 \times {10^{ - 15}}\,\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Metastable dissociation
21	$e+He_{2}^{+} \to He+H{e^*}\left( 3 \right)$	$3.43 \times {10^{ - 17}}{{\text{T}}_e}^{{ - 0.5}}{{\text{m}}^3}{{\text{s}}^{ - 1}}$	Diss. recomb.
22	$e+He_{2}^{+} \to He+H{e^*}\left( 2 \right)$	$1.84 \times {10^{ - 17}}{{\text{T}}_e}^{{ - 0.5}}{{\text{m}}^3}{{\text{s}}^{ - 1}}$	Diss. recomb.
23	$2e+He_{2}^{+} \to He_{2}^{*}\left( 2 \right)+e$	$3.59 \times {10^{ - 39}}{T_e}^{{{\text{-4}}{\text{.5}}}}{{\text{m}}^6}{{\text{s}}^{ - 1}}$	Recombination
24	$e+He+He_{2}^{+} \to He_{2}^{*}\left( 2 \right)+He$	$4.15 \times {10^{ - 41}}{T_e}^{{{\text{-1}}{\text{.5}}}}{{\text{m}}^6}{{\text{s}}^{ - 1}}$	Recombination
25	$H{e^}\left( 2 \right)+He \to H{e^}\left( 1 \right)+He$	$3.6 \times {10^{ - 21}}{{\text{m}}^6}{{\text{s}}^{ - 1}}$	De-excitation
26	$H{e^+}+2He \to He_{2}^{+}+He$	$3.017 \times {10^{ - 41\,}}{T_g}^{{ - 1}}{{\text{m}}^6}{{\text{s}}^{ - 1}}$	Ion conversion
27	$H{e^}\left( 1 \right)+2He \to He_{2}^{}\left( 1 \right)+He$	$3.23 \times {10^{ - 45}}\,\exp \left( { - \frac{{778}}{{{T_g}}}} \right)\,{{\text{m}}^{\text{6}}}{{\text{s}}^{{\text{-1}}}}$	Conversion into excimers
28	$H{e^}\left( 2 \right)+2He \to He_{2}^{}\left( 2 \right)+He$	$3.23 \times {10^{ - 45}}\,\exp \left( { - \frac{{778}}{{{T_g}}}} \right)\,\,{{\text{m}}^{\text{6}}}{{\text{s}}^{{\text{-1}}}}$	Conversion into excimers
29	$H{e^}\left( 3 \right)+2He \to He_{2}^{}\left( 2 \right)+He$	$3.23 \times {10^{ - 45}}\,\exp \left( { - \frac{{778}}{{{T_g}}}} \right)\,\,{{\text{m}}^{\text{6}}}{{\text{s}}^{{\text{-1}}}}$	Conversion into excimers
30	$H{e^}\left( 4 \right)+2He \to He_{2}^{}\left( 2 \right)+He$	$3.23 \times {10^{ - 45}}\,\exp \left( { - \frac{{778}}{{{T_g}}}} \right)\,\,{{\text{m}}^{\text{6}}}{{\text{s}}^{{\text{-1}}}}$	Conversion into excimers
31	$H{e^}\left( 1 \right)+H{e^}\left( 1 \right) \to H{e^+}+He+e$	$2.4 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
32	$H{e^}\left( 1 \right)+H{e^}\left( 2 \right) \to H{e^+}+He+e$	$2.4 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
33	$H{e^}\left( 1 \right)+H{e^}\left( 3 \right) \to H{e^+}+He+e$	$2.4 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
34	$H{e^}\left( 1 \right)+H{e^}\left( 4 \right) \to H{e^+}+He+e$	$2.4 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
35	$H{e^}\left( 1 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+2He+e$	$2.5 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
36	$H{e^}\left( 1 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+2He+e$	$2.5 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
37	$2H{e^*}\left( 2 \right) \to H{e^+}+He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
38	$H{e^}\left( 2 \right)+H{e^}\left( 3 \right) \to H{e^+}+He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
39	$H{e^}\left( 2 \right)+H{e^}\left( 4 \right) \to H{e^+}+He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
40	$H{e^}\left( 2 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+2He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
41	$H{e^}\left( 2 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+2He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
42	$H{e^}\left( 3 \right)+H{e^}\left( 3 \right) \to H{e^+}+He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
43	$H{e^}\left( 3 \right)+H{e^}\left( 4 \right) \to H{e^+}+He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
44	$H{e^}\left( 3 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+2He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
45	$H{e^}\left( 3 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+2He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
46	$H{e^}\left( 4 \right)+H{e^}\left( 4 \right) \to H{e^+}+He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
47	$H{e^}\left( 4 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+2He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
48	$H{e^}\left( 4 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+2He+e$	$2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
49	$He_{2}^{}\left( 1 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+3He+e$	$2.4 \times {10^{ - 16}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
50	$He_{2}^{}\left( 1 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+3He+e$	$2.4 \times {10^{ - 16}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
51	$He_{2}^{}\left( 2 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+3He+e$	$2.4 \times {10^{ - 16}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
52	$He_{2}^{}\left( 2 \right)+He_{2}^{}\left( 2 \right) \to He_{2}^{+}+2He+e$	$1.26 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}$	Penning ionization
53	$H{e^*}\left( 4 \right) \to He+hv$	$1.8 \times {10^9}\,\,{{\text{s}}^{{\text{-1}}}}$	Radiative transition
54	$H{e^}\left( 3 \right) \to H{e^}\left( 1 \right)+hv$	$1.02 \times {10^7}\,\,{{\text{s}}^{{\text{-1}}}}$	Radiative transition
55	$H{e^}\left( 4 \right) \to H{e^}\left( 2 \right)+hv$	$1.97 \times {10^6}\,\,{{\text{s}}^{{\text{-1}}}}$	Radiative transition
56	$He_{2}^{*}\left( 2 \right) \to 2He+hv$	$1 \times {10^9}\,\,{{\text{s}}^{{\text{-1}}}}$	Radiative transition

To study the effect of a magnetic field on the plasma parameters of a high-voltage pulsed discharge in helium under a pressure of 30 Torr. The applied voltage pulse to the discharge gap is shown in Fig. 2. The pulse repetition rate is 50 Hz. The magnitude of the external magnetic field is $B=0.4\,{\text{T}}$. This pressure mode is chosen to simplify the simulation process since at higher pressure it is necessary to consider all plasma-chemical processes. The incorporation of all the processes leads to model complications and necessitates large computing resources. Note that when performing calculations at low pressure, nonlocal effects need to be considered and the assumption of the Maxwell distribution function is not correct.

Figure 2 shows the applied voltage and discharge current at a pressure of 30 Torr in the case of magnetic field $B=0.4\,{\text{T}}$ and $B=0$. Figure 2 shows that in the case of the presence of a magnetic field, the breakdown time slightly decreases. It was shown that the magnetic field strongly effects the characteristics of the left side of the Paschen breakdown curve [48]. In the low-pressure mode, the breakdown voltage decreases due to an increase in the magnetic flux density, and the position of the breakdown minimum is shifted due to the magnetic field. Since the mass of a positive ion is greater than the mass of an electron, the magnetic field has a more noticeable effect on the movement of the electron than on the movement of positive ions. An empirical expression describing the variation of the secondary electron yield with the magnetic field was proposed in the work [49].

Figure 3 shows the dynamics of the electron density distribution in the discharge gap (left-hand side at $B=0$ and right-hand side at $B=0.4{\text{T}}$) at times 40 and 800 ns. From this figure, it can be seen that the presence of an external magnetic field leads to a slight delay in the breakdown time. It can be seen that in both cases, at the initial moment (t = 40 ns), the electron density is attracted to the anode. It is also seen that at the end of the voltage pulse, the electron density in the case of the presence of the magnetic field is three times greater than in the case of the absence of the magnetic field.

Figure 4 shows the metastable helium atom density distribution along the axis of symmetry between the electrodes at a pressure of 30 Torr for the cases $B=0$ and $B=0.4\,{\text{T}}$ at time of 800 ns. It can be seen from this figure that the electron density and metastable helium atom density at the end of the voltage pulse are still quite large in both cases under consideration. In the presence of the external magnetic field, the electron density is increased in the bulk plasma (Fig. 3). This is because the magnetic field leads to increased retention of electrons and ions since they no longer move in straight lines but in helical trajectories around the magnetic field lines.

From Fig. 4 it is also seen that in the near-cathode region of the discharge, there is a region where the metastable density is an order of magnitude higher than in the discharge region, which indicates a significant role of stepwise ionization in increasing the electron density (Fig. 5). As can be seen from the following Fig. 5, te total stepwise ionization is higher than the direct ionization from the ground state, which leads to a decrease the metastable density. This is also indicated by the fact that the maximum electron density is achieved at a greater distance from the cathode than the maximum metastable density. In addition, in this case, the maximum value of the density of excited atoms near the cathode increases, which is associated with a decrease in the escape of fast electrons from the cathode region and an increase in the frequency of the excitation processes.

Figure 5 shows the dynamics of the ionization rate constant of helium atoms along the symmetry axis between the electrodes at a pressure of 30 Torr in the cases $B=0$ and $B=0.4\,{\text{T}}$. In the absence of a magnetic field at low pressure, ionization is primarily due to direct ionization from the ground state, which is 2–3 orders more than stepwise ionization. From Fig. 5 it can be seen that the total stepwise ionization rate constant at the end of the 800 ns pulse remains quite large in the entire discharge region, while direct ionization is only in the narrow cathode region. Therefore, such large stepwise ionization leads to a decrease in the metastable density in the discharge region except for the narrow near-electrode region.

In this paper, the effect of an external transverse magnetic field on the behavior of metastable atoms and excimer helium molecules in a high-voltage pulsed discharge in helium at a pressure of 30 Torr is studied. The dynamics of discharge development in the absence of the magnetic field and the presence of the magnetic field are investigated. The effect of the transverse magnetic field on the charged particles density distribution, metastable helium atoms, and excimer helium molecules in the high-voltage pulsed helium discharge has been studied. It is shown that in the discharge region, the metastable density decreases when the transverse magnetic field is applied as a consequence of an increase in the frequency of stepwise ionization. It was found that the metastable density near the cathode region is several times higher than the electron density, which indicates a significant role of stepwise ionization in increasing the electron density and the manifestation of non-locality of electron kinetics in the region of an enhanced electric field near the cathode.

In conclusion, to improve the accuracy of this model, it is necessary to take into account all possible plasma-chemical processes in helium, as well as to take into account possible nonlocal effects of electron kinetics near the cathode region. Accounting for these processes is a separate task that we want to focus on in our next work.

Acknowledgements This work was partially supported by State Assignment No. FZNZ–2020–0002.

Author Contributions K.M. Rabadanov, N.A. Ashurbekov and К.О. Iminov wrote the main manuscript text. K.M. Rabadanov and G.Sh. Shahsinov carried out the main calculations. М.Z. Zakaryaeva and А.А. Murtazaeva prepared all figures. All the contents were discussed by all authors together. The manuscript was reviewed by all authors.

Competing interests The authors declare no competing interests.

Ethical Approval Not applicable.

Funding Not applicable.

Data Availability The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Metastable Helium Atom Creation Dynamics in High-Voltage Pulsed Discharge with Transverse Magnetic Field Effects

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Abstract

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1. Introduction

2. Model description

3. Results and discussion

4. Conclusion

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Version 1

	Reaction	Data	Process
1	\(e+He \to e+He\)	Cross-section	Elastic
2	\(e+He \leftrightarrow e+H{e^*}\left( 1 \right)\)	Cross-section	Excitation
3	\(e+He \leftrightarrow e+H{e^*}\left( 2 \right)\)	Cross-section	Excitation
4	\(e+He \leftrightarrow e+H{e^*}\left( 3 \right)\)	Cross-section	Excitation
5	\(e+He \leftrightarrow e+H{e^*}\left( 4 \right)\)	Cross-section	Excitation
6	\(e+H{e^}\left( 1 \right) \leftrightarrow e+H{e^}\left( 2 \right)\)	Cross-section	Stepwise excitation
7	\(e+H{e^}\left( 1 \right) \leftrightarrow e+H{e^}\left( 3 \right)\)	Cross-section	Stepwise excitation
8	\(e+H{e^}\left( 1 \right) \leftrightarrow e+H{e^}\left( 4 \right)\)	Cross-section	Stepwise excitation
9	\(e+H{e^}\left( 2 \right) \leftrightarrow e+H{e^}\left( 3 \right)\)	Cross-section	Stepwise excitation
10	\(e+H{e^}\left( 2 \right) \leftrightarrow e+H{e^}\left( 4 \right)\)	Cross-section	Stepwise excitation
11	\(e+H{e^}\left( 3 \right) \leftrightarrow e+H{e^}\left( 4 \right)\)	Cross-section	Stepwise excitation
12	\(e+He \to 2e+H{e^+}\)	Cross-section	Direct ionization
13	\(e+H{e^*}\left( 1 \right) \to 2e+H{e^+}\)	Cross-section	Stepwise ionization
14	\(e+H{e^*}\left( 2 \right) \to 2e+H{e^+}\)	Cross-section	Stepwise ionization
15	\(e+H{e^*}\left( 3 \right) \to 2e+H{e^+}\)	Cross-section	Stepwise ionization
16	\(e+H{e^*}\left( 4 \right) \to 2e+H{e^+}\)	Cross-section	Stepwise ionization
17	\(e+He_{2}^{*}\left( 1 \right) \to 2e+He_{2}^{+}\)	Cross-section	Stepwise ionization
18	\(e+He_{2}^{*}\left( 2 \right) \to 2e+He_{2}^{+}\)	Cross-section	Stepwise ionization
19	\(e+He_{2}^{}\left( 1 \right) \to 2e+He_{2}^{}\left( 2 \right)\)	Cross-section	Stepwise excitation
20	\(e+He_{2}^{*}\left( 1 \right) \to e+2He\)	\(3.8 \times {10^{ - 15}}\,\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Metastable dissociation
21	\(e+He_{2}^{+} \to He+H{e^*}\left( 3 \right)\)	\(3.43 \times {10^{ - 17}}{{\text{T}}_e}^{{ - 0.5}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\)	Diss. recomb.
22	\(e+He_{2}^{+} \to He+H{e^*}\left( 2 \right)\)	\(1.84 \times {10^{ - 17}}{{\text{T}}_e}^{{ - 0.5}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\)	Diss. recomb.
23	\(2e+He_{2}^{+} \to He_{2}^{*}\left( 2 \right)+e\)	\(3.59 \times {10^{ - 39}}{T_e}^{{{\text{-4}}{\text{.5}}}}{{\text{m}}^6}{{\text{s}}^{ - 1}}\)	Recombination
24	\(e+He+He_{2}^{+} \to He_{2}^{*}\left( 2 \right)+He\)	\(4.15 \times {10^{ - 41}}{T_e}^{{{\text{-1}}{\text{.5}}}}{{\text{m}}^6}{{\text{s}}^{ - 1}}\)	Recombination
25	\(H{e^}\left( 2 \right)+He \to H{e^}\left( 1 \right)+He\)	\(3.6 \times {10^{ - 21}}{{\text{m}}^6}{{\text{s}}^{ - 1}}\)	De-excitation
26	\(H{e^+}+2He \to He_{2}^{+}+He\)	\(3.017 \times {10^{ - 41\,}}{T_g}^{{ - 1}}{{\text{m}}^6}{{\text{s}}^{ - 1}}\)	Ion conversion
27	\(H{e^}\left( 1 \right)+2He \to He_{2}^{}\left( 1 \right)+He\)	\(3.23 \times {10^{ - 45}}\,\exp \left( { - \frac{{778}}{{{T_g}}}} \right)\,{{\text{m}}^{\text{6}}}{{\text{s}}^{{\text{-1}}}}\)	Conversion into excimers
28	\(H{e^}\left( 2 \right)+2He \to He_{2}^{}\left( 2 \right)+He\)	\(3.23 \times {10^{ - 45}}\,\exp \left( { - \frac{{778}}{{{T_g}}}} \right)\,\,{{\text{m}}^{\text{6}}}{{\text{s}}^{{\text{-1}}}}\)	Conversion into excimers
29	\(H{e^}\left( 3 \right)+2He \to He_{2}^{}\left( 2 \right)+He\)	\(3.23 \times {10^{ - 45}}\,\exp \left( { - \frac{{778}}{{{T_g}}}} \right)\,\,{{\text{m}}^{\text{6}}}{{\text{s}}^{{\text{-1}}}}\)	Conversion into excimers
30	\(H{e^}\left( 4 \right)+2He \to He_{2}^{}\left( 2 \right)+He\)	\(3.23 \times {10^{ - 45}}\,\exp \left( { - \frac{{778}}{{{T_g}}}} \right)\,\,{{\text{m}}^{\text{6}}}{{\text{s}}^{{\text{-1}}}}\)	Conversion into excimers
31	\(H{e^}\left( 1 \right)+H{e^}\left( 1 \right) \to H{e^+}+He+e\)	\(2.4 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
32	\(H{e^}\left( 1 \right)+H{e^}\left( 2 \right) \to H{e^+}+He+e\)	\(2.4 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
33	\(H{e^}\left( 1 \right)+H{e^}\left( 3 \right) \to H{e^+}+He+e\)	\(2.4 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
34	\(H{e^}\left( 1 \right)+H{e^}\left( 4 \right) \to H{e^+}+He+e\)	\(2.4 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
35	\(H{e^}\left( 1 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+2He+e\)	\(2.5 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
36	\(H{e^}\left( 1 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+2He+e\)	\(2.5 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
37	\(2H{e^*}\left( 2 \right) \to H{e^+}+He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
38	\(H{e^}\left( 2 \right)+H{e^}\left( 3 \right) \to H{e^+}+He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
39	\(H{e^}\left( 2 \right)+H{e^}\left( 4 \right) \to H{e^+}+He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
40	\(H{e^}\left( 2 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+2He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
41	\(H{e^}\left( 2 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+2He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
42	\(H{e^}\left( 3 \right)+H{e^}\left( 3 \right) \to H{e^+}+He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
43	\(H{e^}\left( 3 \right)+H{e^}\left( 4 \right) \to H{e^+}+He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
44	\(H{e^}\left( 3 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+2He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
45	\(H{e^}\left( 3 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+2He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
46	\(H{e^}\left( 4 \right)+H{e^}\left( 4 \right) \to H{e^+}+He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
47	\(H{e^}\left( 4 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+2He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
48	\(H{e^}\left( 4 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+2He+e\)	\(2.1 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
49	\(He_{2}^{}\left( 1 \right)+He_{2}^{}\left( 1 \right) \to H{e^+}+3He+e\)	\(2.4 \times {10^{ - 16}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
50	\(He_{2}^{}\left( 1 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+3He+e\)	\(2.4 \times {10^{ - 16}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
51	\(He_{2}^{}\left( 2 \right)+He_{2}^{}\left( 2 \right) \to H{e^+}+3He+e\)	\(2.4 \times {10^{ - 16}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
52	\(He_{2}^{}\left( 2 \right)+He_{2}^{}\left( 2 \right) \to He_{2}^{+}+2He+e\)	\(1.26 \times {10^{ - 15}}\,\,{{\text{m}}^{\text{3}}}{{\text{s}}^{{\text{-1}}}}\)	Penning ionization
53	\(H{e^*}\left( 4 \right) \to He+hv\)	\(1.8 \times {10^9}\,\,{{\text{s}}^{{\text{-1}}}}\)	Radiative transition
54	\(H{e^}\left( 3 \right) \to H{e^}\left( 1 \right)+hv\)	\(1.02 \times {10^7}\,\,{{\text{s}}^{{\text{-1}}}}\)	Radiative transition
55	\(H{e^}\left( 4 \right) \to H{e^}\left( 2 \right)+hv\)	\(1.97 \times {10^6}\,\,{{\text{s}}^{{\text{-1}}}}\)	Radiative transition
56	\(He_{2}^{*}\left( 2 \right) \to 2He+hv\)	\(1 \times {10^9}\,\,{{\text{s}}^{{\text{-1}}}}\)	Radiative transition