Theoretical Calculations of the Electron Transport Parameters in CH 4-Ar and CH 4-Ne Mixtures Gases Using Monte Carlo Method Enas

The result of concentration varying of mixture methane with argon and neon gas are believed to study the change in electrons energy distribution function and then the change of the electrons transport parameters including the drift velocity, the mean energy, characteristics energy and diffusion coefficient. In the present work,a contemporary developed computer, simulation program known as Bolsig + is being used for calculating the electron transport parameters.


Introduction
The electron transport parameters of pure and mixtures gas were studied for a wide range of applied electric field.These parameters which include the drift mobility, velocity, diffusion coefficient, ionization coefficient and mean electron energy, that are described in collision cross section and the electron energy distribution function (EEDF) represented the backbone of the electron swarm conduct of pure and mixtures gas in discharge of plasma [1,2].
The solution of Boltzmann equation is generally found by utilizing the Lorenz approximation in which the initial two terms of the spherical harmonic development are considered.The numerical solution of the Boltzmann equation yields the electron energy distribution with the electric field E and gas number density N as parameters.Convenient integration of the energy distribution function yields the transport and ionizing properties of the electron swarm.
The electron transport in a gas under the have an effect of an electric field E can be simulated with the assist of a Monte Carlo method [3][4][5][6][7][8].Each electron, during its transit in the gas, performs a succession of free flights punctuated through elastic or inelastic collisions with molecules of gas defined by collision cross sections.Throughout the successive collisions for each electron, certain facts (velocity, position, and many others.) is saved to be able to calculate.
In this paper, we have studied the conduct of electrons in uniform electric fields by a Monte Carlo method.Swarm parameters are determined as a function of E/N for various rates of increase of the electric field [9].
The aim of this work is to study theoretically the electron energy distribution function and electron transport parameters in DC electric discharge processes in methane, Argon and Neon gases and their mixtures to various proportions from Monte Carlo simulation program.From this equation numerous important swarm parameters could be determined that it is as yet being utilized as a part of numerous contemporary research projects to model transport phenomena.The Boltzmann equation for electrons in an ionized gas is [10,11].……....…… (1) or "where , (

Theory
) is the electrons distribution function, a is the acceleration of charges particles and v is the velocity of charge particles.
"F j (V j ,r ,t) is the neutral species distribution function."V j is the velocity of neutral species."σ j (θ , v rj )is the differential microscopic cross section of interaction the charges particles (electron) with neutral gas species j." "d j = sin dθ d is the element solid angle, where θ and  are the polar and azimuthally angles, respectively." The electron distribution function can be written by utilizing the two-term approximation extension as follows [12]:

Transport parameters
The swarm parameters of electrons and collision cross-sections with molecules are identified with each other through the medium of the velocity distribution function of the swarm.The electron mean energy is, [13 and 14] ……………………………...…… (3) where() is expressed in electron volts.
From the connection between the drift velocity and mobility, we can compute electron mobility equation [16]: The connection between diffusion coefficient and electron energy distribution function is given by [17]

Result and Discussion
To calculate the drift velocity of electrons and the others transport parameters utilizing the Monte Carlo simulation program, knowledge of the reliance of the momentum transfer cross section on the electron energy is basis.The drift velocity does not rely on upon electron energy distribution function significantly, especially when the cross section does not fluctuate quickly with electron energy.
We present the results of several transport parameters for various mixtures of methane in argon and neon.For range of E/N values (1 Td ≤E/N≤800 Td) the diverse ratios mixtures of (CH 4 -Ar) and (CH 4 -Ne) gases are recorded in Table (1)(2)(3)(4)(5)(6)(7)(8).Tables (1 and 2) clarify the computed results for the drift velocity V d as a function of E/N , in (CH 4 -Ar) and (CH 4 -Ne) gases, respectively.Tables (3 and 4) explain the computed results for the electron mean energy, in different ratios of gas mixtures (CH 4 -Ar) and (CH 4 -Ne) gases, respectively.Tables (5 and 6) clarify the calculated results for the electron characteristics energy, in different ratios of gas mixtures (CH 4 -Ar) and (CH 4 -Ne) gases, respectively .Tables (7and 8) explain the computed results for the diffusion coefficient ,in various proportions of gas mixtures (CH 4 -Ar) and (CH 4 -Ne)gases , respectively.Figures (1)(2)(3) exhibit the cross sections for electron of methane, argon and neon as a function of electron energy.
The impact of different discharge parameters on the electron distribution function is appeared in figures 4 and 5 for (CH 4 -Ar) and (CH 4 -Ne)gases, respectively.The electron energy distribution function is strongly influenced by changing either the parameter E/N or gas mixtures.Figures (6 -9) clarify the assortment for the mean electron energy and characteristics energy vs. (E/N) in pure methane and mixture with argon and neon gas by taking into consideration various proportion mixing ratios.
Figures (10 and 11) show the diffusion coefficient for different ratios of mixtures methane with argon and neon gas.As a function of E/N in different ratios of gas mixture (CH 4 -Ar) and (CH 4 -Ne) respectively.
The drift velocity of electrons in various mixtures of (CH 4 -Ar) and (CH 4 -Ne) gases are appeared in figures12 and 13 as a function of E/N.It's necessary to note that there are measured experimentally published results that plotted with present work in the aforesaid two figures for comparison as shown in figures 14 and 15 for gases mixture (CH 4 -Ar) and (CH 4 -Ne) respectively .The results demonstrate a good agreement with the experimental values [18][19][20].

Conclusion
In this study, we have analyzed the conduct of electrons in uniform electric fields using a Monte Carlo simulation.The calculating electron energy distribution function for (CH 4 -Ar) and (CH 4 -Ne) mixtures with various concentrations has been described.
The conduct of the swarm parameters, which are drift velocity and mean kinetic electron energy rely on the proportion of the mixture components, can likely, be demonstrated by a preferential weighting of the elastic and inelastic scattering of the electrons on methane with argon and neon molecules at various estimations of E\N, additionally the results were in great concurrence with the computational work.

Table (4) The data of the mean electron Energy (eV) as a function E/N in different ratio CH 4 -Ne mixtures.
Electric field/gas density E/N (Td=10 -17 V.

2. 1 .
Boltzmann equation "The transport Boltzmann equation governing the electron distribution fundamental function; this equation can be driven simply by defining a distribution function and inspecting its time derivative." Haitham J. for Pure & Appl.Sci.Vol.30 (1) 2017

Table ( 5 )
Haitham J. for Pure & Appl.Sci.Vol.30 (1) 2017 The data of the characteristic energy of electron u ch (eV) as a function E/N in different ratio CH 4 -Ar mixtures.

Table ( 1) The data of Drift velocity V d (cm/s) of electron as a function E/N in different ratio CH 4 -Ar mixtures. Table ( 2) The data of Drift velocity V d (cm/s) of electron as a function E/N in different ratio CH 4 -Ne mixtures.
Ibn Al-Haitham J. for Pure & Appl.Sci.Vol.30(1) 2017