Issue 
Eur. Phys. J. Appl. Phys.
Volume 88, Number 3, December 2019



Article Number  30801  
Number of page(s)  6  
Section  Plasma, Discharges and Processes  
DOI  https://doi.org/10.1051/epjap/2019190224  
Published online  05 February 2020 
https://doi.org/10.1051/epjap/2019190224
Regular Article
DC radial glow discharge in axial magnetic field at low pressures
^{1}
School of Electrical and Information, ChangZhou Institute of Technology, ChangZhou 213032, P.R. China
^{2}
School of Electrical and Electronic Engineering, Shandong University of Technology, ZiBo 255049, P.R. China
^{*} email: gs59519829@126.com
Received:
8
August
2019
Received in final form:
19
October
2019
Accepted:
16
December
2019
Published online: 5 February 2020
The influence of magnetic field on DC radial glow plasma was studied by selfdesigned coaxial glow discharge device, and the influence of magnetic field on the spatial distribution of plasma density is studied. The experimental results show that the spatial density distribution of plasma from cathode to anode increases gradually in the highintensity magnetic field, and decreases gradually in the absence of magnetic field. Theoretical analysis of the above results show that the highintensity magnetic field increases the moving path of the electrons, enhances the collision efficiency between the electrons and the neutral atoms, and makes the discharge plasma density remarkably enhanced.
© EDP Sciences, 2020
1 Introduction
DC glow discharge plasma technology is applied widely in many fields, such as plasma etching, plasma material surface treatment, plasma electron beam source, plasma sputtering spraying and so on [1–3]. Many scholars have done a lot of research on the theory of the glow discharge. Most of the literatures are plate electrode structure [4–8]. Other documents are hollow cathode or abnormal glow discharge in crossed electric and magnetic fields [9–12]. In the above structure, the electric field and magnetic field are not symmetrical and the electric field distribution is not uniform [13], which makes the establishment of mathematical model more complicated.
This paper established a device of the structure of magnetron. In this model, the electric and magnetic fields are uniform and symmetrical, the electric field is perpendicular to the magnetic field. The spatial distribution of plasma density under different magnetic field conditions was measured. For better understanding of physical processes in various types of gas discharge numerical models are often applied. In this work, the fluid simulation on radial glow discharge model in axial magnetic field at low pressure is performed. Based on the analysis of the simulation results, the influence of magnetic field on the glow discharge plasma is obtained. The conclusions obtained by theoretical analysis are in agreement with the experimental results.
2 Experimental apparatus and results
The structure of magnetron glow discharge is shown in Figure 1. The distance between cathode and anode is 16 mm. The cathode is solid cylinder structure and the anode is hollow cylinder. The cylindrical height of cathode and anode is 25 mm. The inner diameter of the magnet is 20 mm and the outer diameter is 40 mm. During the experiment, the magnetic field intensity at the axial is changed by adjusting the relative position of the magnet or increasing the number of magnets. The distribution of electric and magnetic fields in glow discharge of magnetron is shown in Figure 2. The magnetic field is generated by upper and lower magnets and distributed in the gap between electrodes. The distribution of steady magnetic field in the discharge space is the axial direction. The electric field is produced by the voltage at both ends of the anode and cathode and the direction of electric field distribution is radial.
The relationship between magnetic field and glow discharge pressure is shown in Figure 3. The ignition pressure of coaxial glow discharge is 1.3 pa without applied magnetic field. With the increase of magnetic field, the glow pressure decreases gradually. When the magnetic field is 25.8 mT, the ignition pressure decreases to 0.04 pa.
In the experiment, the plasma density was measured by double probe method. In order to avoid the influence of magnetic field on the measurement results of plasma density, the relative positions of the two probes in the experiment are parallel to the magnetic lines of flux. Figure 4 shows the spatial distribution of plasma density under different magnetic fields. Under highintensity magnetic field, the plasma density increases from cathode to anode, and decreases gradually from cathode to anode in the absence of magnetic field. Figure 5 is an experimental phenomenon of glow discharge in highintensity magnetic field, and Figure 6 is an experimental phenomenon of glow discharge in a weak magnetic field. It can be seen from Figure 6 that the glow discharge near the cathode is brighter than that at the anode.
Fig. 1 Magnetron glow discharge device. 
Fig. 2 Radial glow discharge in axial magnetic field. 
Fig. 3 The relationship between ignition pressure and magnetic field. 
Fig. 4 Plasma density distribution under different magnetic fields. 
Fig. 5 Glow discharge under highintensity magnetic field. 
Fig. 6 Glow discharge under weak magnetic field. 
3 Theoretical analysis
According to Figure 2, the physical model is based on the following assumptions:

The direction of the applied magnetic field is perpendicular to the direction of the electric field. Edge effect of electric field is neglected.

Plasma formation is mainly a result of electrons collision with neutral atoms, while generation of negative ions and other factors are not considered.

It is generally considered that in lowtemperature glow discharge plasma model, ions maintain the same temperature as neutral gas. Hence, there is no need to consider energy equation [14,15].
The basic equations employed in this theoretical formulation are the particle and momentum conservation equations coupled with one of the Maxwell equations:(1) (2) (3)
In the above equations the various notations are defined as follows: ξ _{ i } is the ionization frequency; subscript α represent electron or ion; n _{ α }, u _{ α }, m _{ α } and T _{ α } are, respectively, the number density, average velocity, mass, and temperature; ν _{ αn } is the respective collision frequency of electrons and positive ions with neutral particles and they are assumed to be constants in this case; E is the electric field; B is the magnetic field; q _{ α } is the electronic charge and ϵ _{0} is the permittivity constant; k is the Boltzmann constant.
We shall first consider these equations for a steady state case in which all the terms involving time derivative vanish. Next, we neglect the inertial terms {u _{ α } ⋅ ∇} u _{ α } in equation (2), as in the present case the average flow velocity in general is much smaller than the average speed of the random thermal motions 14. Equation (2) becomes the following form(4)
According to the model assumptions (1), electric field is axial component, E _{ θ } = E _{ z } = 0 and B _{ θ } = B _{ r } = 0. Taking into account (3), conditions assumed by the model, component form of the equation (4) parallel to magnetic field is:(5)where(6)
In the direction perpendicular to the magnetic field, the equation (4) is(7)where(8) (9) (10) (11) (12) (13)
Diffusion coefficient and mobility satisfy Einstein relation:(14)
The plasma is assumed to be uniformly distributed in the angular and axial direction, we obtained(15)
Ionization frequency can be expressed as follows(16)where(17)
In the above equations, u _{ E } is the electron drift velocity along the radial direction. β _{1} is the Townsend ionization coefficient and is a function of the gas pressure p and electric field E. A and B _{ c } are constants.
Equations (1), (3), (7), (15), (16) constitute final form of glow discharge plasma physics model equation under axial magnetic field condition. In the above equations, except n _{ e }, n _{ i }, E, all other parameters can be regarded as constants in calculation.
In order to solve the above differential equations, the appropriate boundary conditions should be selected. Boundary condition for electric potential at the cathode is ϕ = 0 and at the anode ϕ = V _{0} is equal to the applied voltage. The value of electron number densities at the anode is and at the cathode n _{ e } = n _{0}; the value of ion number densities at the anode is n _{ i } = 0 and at the cathode . In the absence of a magnetic field, the mobility of electrons and ions is expressed as(18) (19)
Using (18) and (19), the collision frequency of electron and positive ion can be expressed as(20) (21)
Numerical modeling based on one dimensional extended fluid model was carried out for the voltage 600 V, the pressure range 0.01 Pa–0.1 Pa and the cathode rod radius 1 mm, anode cylinder radius is 40 mm. The physical process detailed analysis for highintensity magnetic field of 20 mT, and lowintensity magnetic field of 0 mT–0.0002 mT will be presented.
The number density profiles of electrons and ions of glow discharge at 20 mT magnetic field are shown in Figure 7. In the case of low air pressure, the electron density and ion density distribution show increasing trend. This is the same as the experimental results under the condition of 25.8 mT magnetic field in Figure 4. At the cathode, the ion density increases gradually as a result of collision ionization. Figure 8 shows the distribution of the electric field. The electric field is almost constant in the range of 0.02–0.04 m. The electric field increases gradually near the cathode. The distribution trend of electric field is consistent with that of normal glow discharge. In a flat glow discharge without magnetic field, the ion density in the cathode region is larger than the electron density, which is due to the fact that the electron diffusion rate is much higher than the ion diffusion rate.(22)
In the magnetron model, the diffusion rate of electrons is much lower than that of ions due to the influence of magnetic field, which makes the density of electrons in the cathode region larger than that of ions. According to the equation (13) (23)
Figures 9–11 are the spatial distribution of electron density, ion density and electric field at different pressures in 20 mT magnetic field. With the increasing of the pressure, the electron density, ion density and electric field increased firstly and then decreased trend in the whole discharge interval.
Figure 12 shows the electron density distribution at different weak magnetic fields. Under the condition of lowintensity magnetic field, the electron density increases with the increase of magnetic field intensity and the spatial distribution of electron density decreases gradually, which is consistent with the trend that the radial distribution of electron density is Bessel function in the absence of magnetic field.
In the discharge process, electrons are constrained by electric and magnetic fields. The equations of motion of electrons are as follows(24)
Based on the analysis of the electric field and magnetic field of Figure 1, equation (24) is expanded to scalar expression on each axis as follows.(25)
The initial condition of electron motion as follows(26) (27)
Combined equations (25), (26) and (27), we obtain electronic equation of motion(28)where(29)
The electron trajectory is from cathode to anode in magnetic field 0 mT (Fig. 13). When the magnetic field is very weak, the electron cannot form a complete cyclotron motion between the cathode and anode, as shown in Figure 14. For example, when the magnetic field of the electron is 0.0002 mT and the voltage is 500 V, the gyration radius of the electron is about 2 m, which is much larger than the distance between the anode and the cathode. According to the equation (28), under the condition of weak magnetic field, electrons have larger trajectories and the magnetic field increases the path of motion of electrons and increases the collision probability between electrons and neutral atoms, as shown in Figure 14. In this case (weak magnetic field), the density of electrons increases with the increase of magnetic field due to the increase of collision probability of electrons and neutral atoms, but the spatial density distribution of electrons is the same as that without magnetic field, as shown in Figure 12. This is because the electron does not perform complete electron cyclotron all the time, the electron can't make the plasma density accumulate in the process of ionization, and the introduction of magnetic field only increases the motion path of the electron.
Under highintensity magnetic field, the initial electrons can only move on the cathode surface and produce a thin plasma layer on the plasma positive column region. With the first thin plasma as the substrate, electrons in the plasma layer collide with neutral atoms to produce another thin plasma in the highintensity magnetic field, as shown in Figure 15. Due to impact ionization, the electron density in the plasma is larger than the initial electron density. Therefore, in the subsequent process of ionization, the increase of electron density makes the ionization stronger. Therefore, the density of the next thin plasma is higher than that of the previous one. The plasma layer is continuously produced and superimposed, and eventually develops to the anode. The plasma density in the whole space is increasing.
Fig. 7 Electron density and ion density distribution in 20 mT magnetic field. 
Fig. 8 Electric field distribution in 2 mT magnetic field. 
Fig. 9 Electron density distribution at different pressures. 
Fig. 10 Ion density distribution at different pressures. 
Fig. 11 Electric field distribution at different pressures. 
Fig. 12 Electron density distribution at different magnetic field. 
Fig. 13 Electron trajectories in the absence of magnetic field. 
Fig. 14 Electron trajectories in the weak magnetic field. 
Fig. 15 Electron trajectories in highintensity magnetic field. 
4 Conclusions
In the experiment of magnetron glow discharge, the magnetic field can reduce the glow pressure of glow discharge. At the same time, the plasma density increases gradually from cathode to anode in highintensity magnetic field. In the absence of magnetic field, the plasma density decreases gradually from cathode to anode. 1D fluid model was built for describing magnetron model at low pressure. In this work, we investigated the effect of magnetic field on the characteristic of this type of discharge. The results show the highintensity magnetic field increases the moving path of the electrons, enhances the collision efficiency between the electrons and the neutral atoms, and makes the discharge plasma density remarkably enhanced.
Acknowledgments
This work is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (19KJB140006), Changzhou Science and Technology Program of China (CJ20179061), and Changzhou Science and Technology Program of China (CJ20189024).
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Cite this article as: Shen Gao, Jianyuan Feng, Wenqi Li, Jihe Cai, DC radial glow discharge in axial magnetic field at low pressures, Eur. Phys. J. Appl. Phys. 88, 30801 (2019)
All Figures
Fig. 1 Magnetron glow discharge device. 

In the text 
Fig. 2 Radial glow discharge in axial magnetic field. 

In the text 
Fig. 3 The relationship between ignition pressure and magnetic field. 

In the text 
Fig. 4 Plasma density distribution under different magnetic fields. 

In the text 
Fig. 5 Glow discharge under highintensity magnetic field. 

In the text 
Fig. 6 Glow discharge under weak magnetic field. 

In the text 
Fig. 7 Electron density and ion density distribution in 20 mT magnetic field. 

In the text 
Fig. 8 Electric field distribution in 2 mT magnetic field. 

In the text 
Fig. 9 Electron density distribution at different pressures. 

In the text 
Fig. 10 Ion density distribution at different pressures. 

In the text 
Fig. 11 Electric field distribution at different pressures. 

In the text 
Fig. 12 Electron density distribution at different magnetic field. 

In the text 
Fig. 13 Electron trajectories in the absence of magnetic field. 

In the text 
Fig. 14 Electron trajectories in the weak magnetic field. 

In the text 
Fig. 15 Electron trajectories in highintensity magnetic field. 

In the text 
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