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On the basis of fluid approximation, an improved version of the model for the description of dc glow discharge plasma in the axial magnetic field was successfully developed. The model has yielded a set of analytic formulas for the physical quantities concerned from the electron and ion fluids equations and Poisson equation. The calculated results satisfy the practical boundary conditions. Results obtained from the model reveal that although the differential equations under the condition of axial magnetic field are consistent with the differential equations without considering the magnetic field, the solution of the equations is not completely consistent. The results show that the stronger the magnetic field, the greater the plasma density.

Low pressure DC glow discharge plasma technology is applied widely in many fields, like plasma etching, plasma material surface treatment, plasma electron beam source, plasma sputtering spraying, and so on [

Despite the fact that the influence of the magnetic field on the parameters of the plasma has been studied in the experiment, in theoretical research, there is still not a complete theoretical description of the influence of magnetic field on the parameters of the glow discharge. The relations between the physical quantities and some physical phenomena can not be explained strictly. Most authors use numerical methods in the theoretical study [

The present study, as a continuation of the previous work, is an attempt to improve the theoretical formulation mainly by considering the influence of axial magnetic field on the glow discharge plasma. The specific objective of this study is to obtain a set of analytic expressions that can describe the glow discharge plasma in the axial magnetic field on the whole discharge area. The final goal is to gain the relationship between the plasma parameters and the axial magnetic field. In the following presentation, we shall first discuss the aspects concerning the basic assumptions and theoretical formulation of the physical model for the glow discharge plasma in axial magnetic field. The computational results and detailed mathematical analysis shall be presented in the subsequent sections. In the final part, the above results are analyzed and corresponding conclusions are drawn.

In this study, the two-dimensional dc discharge configuration is shown in Figure

Schematic of a glow discharge with external magnetic field.

The model is based on the following assumptions:

The direction of the applied magnetic field is parallel to the direction of the electric field. The electric field is produced by the voltage at both ends of the anode and cathode. Edge effects of electric field are 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 low-temperature glow discharge plasma model, ions maintain the same temperature as neutral gas. Hence, there is no need to consider energy equation [

Generally, glow discharge current is about several milliamps. Self-magnetic field generated by glow current is much smaller than applied external magnetic field. Therefore, effect of self-magnetic field was ignored and only effect of external magnetic field on glow discharge was considered.

The basic equations employed in this theoretical formulation are the particle and momentum conservation equations coupled with one of the Maxwell equations.

In the above equations the various notations are defined as follows:

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

The plasma is assumed to be uniformly distributed in the angular direction and we obtained

From (

Equations (

We employ the variables separation method by writing

In the absence of magnetic field, using (

Calculation analyses were carried out for discharge in magnetic field at pressures ^{−3}.

Figure

Axial electron density distribution under different magnetic fields.

Axial ion density distribution under different magnetic fields.

Electric field distribution under different magnetic fields.

In the process of discharge, electrons are subjected to the interaction of electric field and magnetic field. When the electric field is parallel to the magnetic field, the electron trajectory is not affected by the magnetic field. However, when the glow plasma is formed, due to the diffusion effect of the plasma, the confinement effect of the axial magnetic field on the plasma is reflected. The radial diffusion of plasma is suppressed. And this effect not only affects the radial component of the plasma, but also affects the axial distribution of the plasma parameters and the electric field distribution in the whole region. Figures

The two-dimensional DC glow discharge model in axial magnetic field at low pressure was established. Based on a series of reasonable assumptions, the analytical solutions are obtained. It is shown that the model is consistent with the basic equations of the glow discharge without considering axial magnetic field, but the results are not the same. The results show that the magnetic field has a certain influence on the glow discharge parameters even when the magnetic field is parallel to the direction of the electric field.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (no. 11075123), the Young Scientists Fund of Nature Science Foundation of China (no. 51207171), and Fundamental Research Funds for the Central Universities (no. 2042016kf1139).

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