Calculation of the Metastable Atom Densities in Argon and Neon Abnormal Glow Discharges

In this chapter an investigation of a DC argon and neon abnormal glow discharges with metastable atom density is presented. The values of pressure are between 133.32 and 330 Pa, and the voltage range is from 250 to 400 V in the case of argon gas. In the case of neon gas, the pressure has the value of 399.92 Pa (3 Torr) and the voltage ranges from 300 to 500 V. In the frameworks, an analysis of abnormal glow discharge characteristics is carried out in the case of input data taken from the Boltzmann equation in multi term approximation (BMA), and in the case of input data obtained from BOLISG+ code. As conclusion of these differences of input data in the same gas the output results are different and it appears in the cathodic region. The spatiotemporal distributions of electron and ion densities, the potential and electric field, the mean electron energy and the metastable atom density are shown. A 1D fluid model is used to solve self-consistently the first three moments of the Boltzmann ’ s equation coupled with the Poisson ’ s equation. Our results are validated with those obtained by both recent paper and experimental results.


Introduction
The concern of the amelioration of the plasma reactor is always a domain important in the development technology, among these fields we find glow discharge. A plasma technology in a gas mixture has been studied by several authors. Ono et al. [1] have been studied oxygen-nitrogen gas mixture glow discharge plasma by intervene many chemical reactions in their model. Khomich et al. [2] have been treated the problem of the atomic deposition in the metal surface modification by nitrogen-argon mixture glow discharge in abnormal regime. Ponduri et al. [3] have been analyzed the dissociation of CO 2 by dielectric barrier glow discharge, as a consequence of utilization of CO 2 gas a lot of kinds of species intervene in the phenomena discharge. Baadj et al. have been [4] investigated Xe-Cl 2 gas mixture for the formation of XeCl* exciplex lamp by means of zerodimensional model. Li et al. [5] have been studied the plasma jet length in Ne, Ar, He and Kr in atmospheric pressure when the excimer molecule formatted from metastable state of the atomic gases, and they have been identified three modes versus of the gas flow rate, its about laminar, transition, and turbulent jet modes.
In this chapter, an research is through concerning the role of neon and argon metastable atoms in the discharge. Metastable atoms have been considered by several authors, both theoretically and experimentally. Metastable densities can experimentally be measured by optical absorption method. In theoretical means, a poise equation, including different production and loss terms is assembled to compute the metastable densities. Experimental measurements were executed, for example, for Neon gas in an RF glow discharge by Eckstein et al. [6], for Ar gas in a microwave boosted glow discharge by Uzelac and Leis [7], and for He gas as a function of discharge conditions by Browne and Dunn [8]. In the research of Smith et al. [9] purely relative absorption signals of Argon metastable atoms as a function of current and pressure were experimented. In the research of Strauss et al. [10] and Ferreira et al. [11] Argon metastable densities were experimented in a afterglow of a pulsed discharge and in the Grimm-type glow discharge, respectively, and some mechanisms for the structure of Argon metastable atoms were recommended. Argon metastable densities have also been experimented by Ferreira et al. [12], and by Ferreira and Ricard [13].
A comparison was complete with Argon metastable densities determined from a coupled-electron-metastable-atom model. Hardy and Sheldon [14] have examined Argon, Helium and Neon gases. A comparison between calculated and measured metastable densities was also realized by Kubota et al. [15] for Helium in a RF and DC glow discharge. Lymberopoulos and Economou [16] have established a combined fluid model for the electrons, Ar metastable atoms, and Ar ions in order to study the effect of metastable atoms in the discharge. In the works [17][18][19][20][21][22] rate constants of a number of collision processes in control of the demolition of metastable atoms were got by combining equilibrium equations with the experimented time-dependent variation of the metastable densities or by analyzing the reliance of the decay constants upon pressure. Den Hartog et al. [23,24] have investigated Helium gas.
Last-mentioned, Fedoseev and Sukhinin [25] have investigated the influence of metastable Ar atoms on gas discharge plasma with dust particles. Shumova et al. [26] have investigated the effect of metastable Ne atoms and dust particles in a positive column of glow discharge.
The aim of this work is to present, influence of the discharge characteristic in the case of input data obtained by BOLSIG+ code, and in the case of input data taken from the Boltzmann equation in multi term approximation (BMA). We note that both these approaches are widely used. For simulations of positive column of glow discharge, Vasilyak et al. [27] used the first approach, while Sukhinin et al. [28] used the second approach. In Section 2, the mathematical model is delineated; it comprises the boundary and initial conditions as well as the numerical method. In Section 3, the results are discussed for Ar discharge. In Section 4, the test scheme is given. In Section 5, Influence of the input data got from BOLSIG+ on the argon discharge has been shown. In Section 6, the characteristics of the Ne discharge with input data taken from BOLSIG+ are presented. Finally, the conclusion of the chapter is given in Section 7.

Discharge modeling
Our mathematical model builds on the first three moments of the Boltzmann equation. The Continuity equations and momentum transfer equations of metastable atom, electrons and positively charged ions. The energy equation is known only for electrons, and to give the effect of the electric field on the particles charged, the Poisson equation is included in the model [29,30].
The chemical reactions intervene in the discharge are indicated in the Table 1.
After that, the model in the 1D Cartesian geometry, writes by partial differential equations: Here n e , n + , n m , φ m , φ e and φ þ are number densities, densities of transport flux of the metastable atoms, electrons and ions, respectively. K m o is the rate coefficient of electron-impact excitation of ground state atoms, ε m ¼ 11:55 eV is the energy loss of excited atoms, n o is the constant background gas density, K io m is the rate coefficient of electron-impact ionization of excited atoms with the energy loss ε io À ε m À Á , K io o is the rate coefficient of electron-impact ionization of ground state atoms and ε io ¼ 15:76 eV is the energy loss of ionized atoms, K o m is the rate coefficient of deexcitation of excited atoms by electron collisions. E ¼ À∂V=∂x is the electric field strength. ε o and e are the permittivity of free space and elementary charge, respectively. K ci ¼ 8:1 Â 10 À10 cm 3 s À1 [31] is the rate coefficient of chemo-ionization processes with the energy gain ε ci ¼ 2ε m À ε io . τ m ¼ 1 μs is the metastable lifetime. ε e is the mean electron energy, φ eε is the electron energy flux. V is the electrostatic potential. P ec is the energy loss per electron due to elastic collision of electrons with the background gas [32].
Momentum transfer equations for metastable atoms, electrons, ions and electron energy [33][34][35] are: Here μ e , μ þ , D e and D + are the electron, ion for mobilities and diffusion coefficients, respectively. D eε and μ eε are the diffusivity and mobility of electron energy transport. The ion mobility has brought from Phelps and Petrović [36] and their ion diffusivity is computed agree with Einstein's relation [37]. The coefficients for electrons [38] in argon as dependences on the mean electron energy are got as of INP Greifswald for direct evaluation with the results obtained by Becker et al. [30]. D m is the metastable atom diffusivity someplace n o D m ¼ 1:7x10 18 cm À1 s À1 [19].

Boundary and initial conditions
The discharge is affected between two parallel plate electrodes and the radius of the electrode is presumed to be higher than the electrode gap and the physical characteristic distributions are approximately uniform along the radial direction. The grounded electrode has been put at x ¼ 1 cm, play the role as the anode (V anode ¼ 0). The powered electrode has been put at x ¼ 0 cm, which initiates the model discretization, play the role as the cathode (V cathode ¼ ÀV DC ).
At time t ¼ 0, the metastable atom, electron, and ion densities are supposed constant and equal to 10 3 cm À3 , and the mean electron energy equal to 1 eV.
Presuming disappearance of the metastable atom density, i.e., n m ¼ 0 has been arranged at the cathode, whilst a predominant field-driven flux shut to the cathode the expression ∂D þ n þ =∂x ¼ 0 has been used for the positive ion density ∀t > 0. At the anode, the metastable atoms and the electron density are supposed to be zero. The electron flux separating the cathode is calculated by the expression Þ∀t > 0, the mean electron energy is assumed to be 5 eV at the cathode [36] and the gas temperature is equal to 273 K in the discharge.

Numerical method
For the metastable atom and Poisson equations a finite difference method has been employed. The transport equations of the electron energy, electron and ion are also discretized spatially with the finite difference technique. In this method the exponential scheme has been employed into account [37,[39][40][41][42]. The discretization of the time terms by the right position of the finite difference technique has been used. Consequently, every discretized equation is defined as a tridiagonal matrix, which is solved by Thomas's algorithm.

Results and discussion of argon discharge
In this part, we will analysis the spatiotemporal evolution of the abnormal glow discharge in the existence of metastable atom density. The gas pressure is 133.32 Pa. The neutral species density is computed from the temperature and gas pressure with the ideal gas law. The constant value for the secondary electron yield is 0.06 [36].
The applied potential at the cathode is À400 V. The uniform subdivision of the space interval in 250 elements and a time step Δt = 10 ps, have been utilized. Figure 1 shows the temporal progression of the potential (Figure 1(a)), concentration of electrons (Figure 1(b)), concentration of ions (Figure 1(c)), concentration of metastable atoms (Figure 1(d)), the electric field (Figure 1(e)) and the mean electron energy (Figure 1(f)). We remark that the discharge is distinguished by three zones: the first one busied the time simulation amidst in 10 À11 and 9 Â 10 À6 s, the second one took the time simulation inter-time of 9 Â 10 À6 and 3 Â 10 À5 s, the final zone busied the time simulation amidst in 3 Â 10 À5 and 3.6 Â 10 À5 s.
For the first zone, we comment that the metastable atom, electrons and ion concentrations are almost identical. Therefore, the net space charge concentration is unimportant. The electric potential is distinguished by the Laplace form due to the net space charge concentration that is exist. Consequently, the electric field is seemingly constant, besides that the mean electron energy is constant.
In the second zone, we remark a pseudo emergence of the cathodic region, this is characterized by a significance of ion concentration and unimportant of the electron concentration. This is explicated by the velocity of the electron species, which speed a more than the ion species and displaced quickly starting at the cathodic region. Consequently, the amount (n e -n + ) is considerable which influence directly the electric potential as consequence the important chute that is present. Inevitably, the electric field is intense. The last earns the electron species an important energy. We comment that the metastable atom concentration is important. This discharge is sustained by the secondary emission coefficient as well as the existence of metastable atom concentration. Sooner than t = 3 Â 10 À5 s we remark a pseudo emergence of the negative glow region, where it is typified by the similar electron and ion concentrations. Consequently the net space charge concentration is slight. Hence, the electric potential and the electric field are constants. Automatically, the metastable atoms concentration is diminished. In the last zone, we remark three dissimilar regions: the anode and the plasma regions and the cathode region. The anode region is characterized by the ion concentration that is less important compared to the electron concentration. In this zone we remark that the convergence of each physical characteristics of the discharge are reached at the time 3.6 Â 10 À5 s.

Influence of the voltage and gas pressure
In this part, we will study the influence of the voltage and gas pressure on the argon discharge. So, the potential at the cathode is taking of 250 V and we will alter the gas pressure. For the influence of the applied potential on the discharge, we take the gas pressure at 133.32 Pa. bulk. Figure 2(a) shows the metastable atoms concentration plots depending on the pressure in the study state. The metastable atoms concentration augments with increasing pressure. For elevated pressure the gas density augments, which the electron diffusion coefficient turns out to be fewer and the bulk of the plasma rises which the both sheaths of the anode and cathode turn out to be small. These circumstances of the charged particle manipulate on the metastable atoms behavior in the study stated, i.e., the cathodic region is overflowing with the electron and metastable atom concentrations, which go faster the ion species in the existence of the electric field. Figure 2(b) shows the metastable atoms concentration graphs depending on the potential in the study state. For elevated potential the excitation and ionization processes increase, and the charged particle turns out to be raised in the stationary state. Consequently, the metastable atom concentration graphs become growing.
The greatest of the metastable atom concentration varies amidst in 2.47 Â 10 10 and 6.63 Â 10 11 cm À3 . We judged these results with calculated values established in the literature [9,10,12,15]. Depending on the discharge circumstances, all these value varies amidst in 2 Â 10 10 and 5 Â 10 13 cm À3 . Consequently, our calculated values something like in the exact range of the size order. We find that the values of the current densities are 0:137, 0:508 and 0:843 mA/cm 2 related to the pressures of 133:32, 250:32 and 330:32 Pa, in that order. Figure 3 represents the current-voltage properties in the stationary state. The results attained beginning of the database of BMA evaluated to those obtained by experimental method [43,44]. We find that the results get beginning of the database of the BMA are in excellent conformity judgment against to those experimental results [43,44]. Figure 4 demonstrates the comparison amidst in our results and those given by Becker et al. [30] (Figure 4(a)) ion and electron concentrations, (Figure 4(b)) electric field and electric potential, (Figure 4(c)) metastable atom concentration and (Figure 4(d)) mean electron energy. This figure substantiates the validity of our 1D code. The main dissimilarity amidst in those given by Becker et al. and our results are pointing up in the Table 2.

Validity of the model
We find that the similar discharge has been studied by Fiala et al. [45], where the hybrid model has been employed in two dimensional configurations. We find that the results got by Fiala et al., it was approximately indistinguishable to our results. In exacting, the maximum of particle concentrations is 1.1 Â 10 9 cm À3 and the electric field at the cathode is 675 V/cm for applying voltage that is equal to 126.3 V. Consequently, the hybrid model is equivalent to our fluid model in the presence of the metastable atom concentrations of these discharge circumstances. Besides our model identifier both properties of the discharge, the mean electron energy and the metastable atom concentrations.

Influence for input data of argon abnormal glow discharge
In this part, we will show the properties of the argon plasma discharge in the case of entering data computed by BOLSIG+ software [46]. We remind that the preceding results are obtained with enter data computed by multiterm estimation of the Boltzmann equation. We remind again that the preceding results are identical when are calculated exclusive of the rate coefficients K o m and K io m . The exclusion is the metastable atom concentration, which is prejudiced by these coefficients, i.e., the stepwise ionization processes is insignificant, evaluated to the both ionization of the chemo-ionization and ground state atoms processes. Consequentially, we can calculate the properties of the argon abnormal glow discharge exclusive of K o m and K io m coefficients. The secondary electron emission coefficient is 0:06. The applied voltage at the cathode is À250 V. The gas pressure is 133:32 Pa. Figure 5 shows the particle density distributions (Figure 5(a)), metastable atom concentration (Figure 5(b)) and mean electron energy (Figure 5(c)) in the stationary state. When we compare between the results given by Figures 4 and 5, we conclude clearly the influence of entering data of BOLSIG+ on the characteristics of the argon abnormal glow discharge. In exacting the cathodic region illustrated in the Figure 4(a) is totally dissimilar to the cathodic region illustrated in the Figure 5(a). We remind that the similar discharge circumstances (voltage, pressure, etc.) are used in two and multi terms approximations. But, the results are actually dissimilar. Table 3 gives the main dissimilarity between entering both data of multiterm approximation and of BOLSIG+.  Figure 6 shows the comparison amidst of the experimental results [47,48] and our results got beginning database of BOLSIG+ code for pd = 2 Torr cm. We remind that the experimental results [47,48] are given for the diameter of the electrode equal to 8 cm and the inter-electrodes spacing equal to 1 cm. We remind again that the results obtained by Ref. [47] its approximately dissimilar to the results obtained by Ref. [48] due to the experimental circumstances of the secondary electron emission coefficient. We remark that our results got from database of BOLSIG+ code are in excellent accord with the experimental results [47].

Effect of the metastable lifetime on the characteristics of argon abnormal glow discharge
We remind that the preceding computations are effected through the metastable lifetime equal to 1 μs [30], we remind that this value has been proposed by Becker et al. [30]. Figure 7 represents the effect of the metastable lifetime on the curve of the metastable atom concentration in the study state. For this reason we have used a value of metastable lifetime equal to 56 s of the theories [49] and an experimental value equal to 38 s [50]. We remark that the most of the metastable atom concentration augments from 7.76 Â 10 9 to 2.249 Â 10 11 cm À3 . We note that this effect is noticed just for metastable atom concentration and all characteristic of argon abnormal glow discharge rest unmoved in the study stated. As a consequence, the utilization of the experimental or artificial value of the metastable lifetime has an inconsequential of the abnormal glow discharge characteristics.

Characteristics of the neon discharge through entering data of the BOLSIG+
In this part, we will investigate the characteristics of the neon abnormal glow discharge through entering data of the BOLSIG+ code. The gas pressure is 3 Torr. The applied voltage is 300 V. The secondary electron emission coefficient is 0:26 [51]. Additional parameters are declared in the Appendix A. Figure 8 shows the curves of ion and electron densities (Figure 8(a)), metastable atom concentration Table 2. The major differences between our results and those given by Becker et al.

Figure 5.
Argon spatial distributions of (a) particledensities, (b) metastable atom density, and (c) mean electron energy in the stationary state for input data of BOLSIG+. (Figure 8(b)) and current density (Figure 8(c)) depending of the applied voltage in the stationary state. We note that the greatest of the metastable atom concentration is equal to 1.615 Â 10 11 cm À3 . The value of neon current density is 0:1851 mA/ cm 2 . The electric field at the cathode achieves the value of 1696:08 V/cm. We note Table 3. The major differences between input Data of BOLSIG+ and those given by multiterm approximation.

Figure 6.
Comparison between the results obtained by our calculation using database of BOLSIG+ software for pd = 2 Torr cm and those given by experimental for pd = 2 Torr cm in Argon gas.

Figure 7.
Influence of the metastable lifetime on the Argon spatial distribution of metastable atom density in the steady state.

Figure 8.
Neon spatial distributions of (a) particle densities, (b) metastable atom density, and (c) current density as a function of applied potential in the stationary state. that the most of the mean electron energy is 46:30 eV in the cathodic region. As a consequence, the fluid model by injecting the metastable atom concentrations is extremely important for investigating the abnormal glow discharge properties of several pure gases and mixed gas.

Conclusion
In order to study the effect of metastable atom density, a second order fluid electric model has been used in the case of a DC low-pressure Ar and Ne abnormal glow discharges. The Poisson equation for the potential and electric field is joined to the first three moments of the Boltzmann's conservation equations ignoring inertia of the charged particles. In the framework of the local energy approximation, the basic data employed in this chapter are calculated by Becker et al. in the case of multiterm estimation of Boltzmann equation (BMA) [30] and from BOLSIG+ software. The task of metastable atom concentration in the discharge is obvious for study into side of plasma glow discharge for several pure gases and mixed gas. We note that the abnormal glow discharge is sustained by secondary electron emission coefficient and the existence of the metastable atom concentration in this particular discharge.