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Journal of Engineering Science and Technology

Vol. 17, No. 2 (2022) 1550 - 1560

© School of Engineering, Taylor’s University

1550

INVESTIGATION OF CONCENTRATION INFLUENCE

ON ELECTRONIC COEFFICIENTS OF HE:NE PLASMA BY

PREDICTING A MATHEMATICAL MODEL

BAIDAA HAMED*, RAFID ABBAS ALI, MAYSAM T. AL-OBAIDI

Mustansiriyah University, Baghdad, Iraq

*Corresponding Author: baidaa800@uomustansiriyah.edu.iq,

Abstract

In this work a comprehensive investigation of specific electron transport

coefficients in plasma state He-Ne gas mixtures has been carried out. Theoretical

calculations and estimated data are presented that enable us to measure the

influence of He:Ne concentration on plasma electronic coefficients based on the

variation in a plasma field resistance situated in a varied electric field and under

a thermodynamic equilibrium. The Boltzmann equation was used to calculate

several concentrations of energy mobility and momentum frequency and varied

electric fields. Utilisation of the BOLSIG+ simulation verified the results that the

Boltzmann distribution analysis revealed. By using a simulation process,

appropriated equations which indicate the variation of plasma electronic

coefficients according to the variation of mixture concentration and reduced

electric field (E/N) have been obtained. The applied reduced electric field has

been chosen to be in the limited range of (0-100) Td, and for several

concentrations in the limited range of (0.1-0.7) mol. The between unique

information (utilizing BOLSIG+) and our estimated data. The results show a stark

resemblance involving original data (using BOLSIG+) and our estimated data,

our simulation data, where root mean square of (νe-i /N,) = 4.2×10-7, root mean

square of µƐN = 1.5×10-6 and root mean square of <

> = 9.4×10-5. This

improvement in electronic coefficients assume a significant part in the

development of energy for He:Ne laser

Keywords: Elastic and inelastic collisions, Electron-ion momentum frequency,

Energy mobility, Mean energy, Plasma electronic coefficients.

Investigation of Concentration Influence on Electronic Coefficients of . . . . 1551

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

1. Introduction

Plasma, which represents one of the four essential states of matter had been firstly

defined in the 1920s by Irving Langmuir as a ‘mouldable substance’ [1, 2]. It is

primarily comprised of a gas of atoms and molecules bearing a small number of

electrons within orbitals displaced, and free electrons [3]. Plasma is a state of matter

wherein an electrical condensation conducted upon ionized gaseous to the point that

long fields command the conduct of the matter by a range electric and attractive [4, 5].

Plasma medium possesses unbound moving negatively and positively charged

particles, therefore it is an electrically neutralised buffer with a net charge of about

nothing. Albeit these particles are unrestrained, they still have the capability of

experiencing forces to the sense that they are not “free.” An electrical current within

a magnetic field is generated by the movement of a charged also, any development

of a plasma molecule which is charged affects and is also directly affected by the

created by the other present charges. In turn, this results in varying degrees of

control collective behaviour [6, 7].

Plasma’s properties are remarkably dependent upon its particle interactions and

electron transport coefficients. Collective effects help to identify plasma’s

behaviour as being entirely unique towards that of liquids. Every charged molecule

of plasma influence instantaneously alongside a substantial amount of related

charged particles because of the vast scope of present electromagnetic powers. The

resulting effects can be represented by elastic and inelastic collisions provides a

rich variation of physical phenomena that occur in plasma [8, 9].

There has been widespread interest in plasma sciences over the past two decades

and several breakthroughs regarding the field has been made. Developing advanced

new sources of plasma was the core of this progress of new. This progress is

generally the result of advancement of current plasma sources in light of plasma

creation in electrical releases in vacuum, magnetrons and in gases with high or low

pressures and is in many ways a part of the progress made. The plasma technique

resulting from this processing is used in aerospace movement and largely within

fields of nanoscience revolving around plasma and on particular in areas where

meticulous control is necessitated. These applications involve, but are not limited

to Plasma Chemistry, lighting systems revolving around Plasma, Plasma Spray and

lighting systems [10-12].

The Boltzmann equation is one of the most powerful tools for investigating the

plasma state, from the electron kinetics in weakly ionized gases [10] to fusion [13,

14] and astrophysical plasmas. Boltzmann equation introduced into physics the idea

of probability, which was then used some years later in quantum physics [15]. The

development of models and simulation techniques for electrical discharges has

been going for more than five decades. In present paper, theoretical calculations for

several electronic coefficients in a mixture of He:Ne gas (using several gas mixture

concentrations) are presented by creating mathematical models utilizing Boltzmann

distribution function and appropriated simulation process.

In present paper, theoretical investigation of electron transport coefficients in

plasma state He-Ne gas mixtures has been introduced using BOLSIG+ simulation

method as a Boltzmann equation solver often employed in plasma modelling

community. Origin program has been utilized in a simulation process resulting

appropriated equations for theoretical calculations of electronic coefficients. Each

1552 B. Hamed et al.

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

of mean energy <

>, energy mobility µ and momentum frequency ν at several

concentrations of He-Ne gas mixture and under varied reduced electric field has

been theoretically calculated.

2. The Boltzmann Equation

The Boltzmann condition portrays the factual way of behaving of a nonequilibrium

thermodynamic framework designed in 1872 by Ludwig Boltzmann. What defines the

Boltzmann equation in modern day and particularly its uses in a wider sense are any

alteration of a perceptible amount within a system of thermodynamics for example,

particle number, charge, energy. might be utilized to decide how actual amounts change

when a fluid is in transport, such as heat energy and momentum [16, 17].

Certain behaviours trademark of liquids like consistency, heat-based

conductivity, and electric-powered conductivity can also be extracted (through

influencing the bearers of charges within a liquid like a gas). The mathematical

unknown of the equation is a function of likelihood amount within the space

revolving around particle position and momentum. This means that the equation

becomes a nonlinear integral differential equation [18].

Boltzmann equation gives the ratio of number per unit volume (the numeral

mass) of molecules, ions or atoms, denoted by N2, in a certain level of energy

comparable to the numeral mass within another lower energy level which shows as

N1, and it is given by [19]:

(1)

where, g1 and g2 represent the multiplicity of the two energy levels, in another ward

the degeneracies of the energy levels with the same energy, E is the required energy

to excite particles, K represents Boltzmann constant and T represents

thermodynamic temperature, so as T increases a greater number of particles will be

excited [20].

Boltzmann kinetic equation

The kinetic Boltzmann equation highlights the distribution function of gas

molecules f (v, r, t) in this case v becomes the velocity and r is coordinates (as

mathematic equations of time shown as t) which showcases non-equilibrium

operations within low density. The mathematical process of f helps to showcase the

mean molecules with speed between the scope of v to v, and include the coordinate

between the scope of r to r. This Boltzmann (kinetic) equation has the form only

when the distribution function x and the velocity component of v. speeds inside a

little reach from ν to ν + Δν and facilitates inside a little reach from one r to r + Δr.

In the event that the dispersion work relies just upon the direction x and the speed

part vx, then the Boltzmann (active) condition has the structure [21]:

(2)

where m is the mass of the molecule. The pace of progress of the appropriation

work is addressed by the fractional subordinate ∂f/∂t. The second term in the

situation assesses the adjustment of f because of the development of particles in

space. Third term in this situation decides the adjustment of dissemination work

that is alluding to impact of an outer power F.

Investigation of Concentration Influence on Electronic Coefficients of . . . . 1553

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

The right term represents changes in the function of distribution which is

referring to particle collision. This term is reliant on each of f as well as the idea of

interaction powers among particles and this is show by [21]:

(3)

where, f, f1, and fʹ, f1ʹ represent functions of particle distribution prior to and

following collision; v1, v2 represent the velocities of the particles before collision;

and dσ =σdΩ is the differential powerful dissipating cross area into the strong point

dΩ (depending on coordinate system of laboratory) and it depends on the law of

molecular interaction [22].

The kinetic or transport equations are generalizations of the Boltzmann equation

and showcase electron gas behaviors in crystal lattice phonons and in metals [23, 24].

3. Mean Energy

3.1. Influence of reduced electric field and concentration on mean energy

The concentration affects obviously on discharge process can be noticed in Fig. 1.

and it can be also noticed the dependence of mean energy <

> on reduced electric

field (E/N), where the increasing of mean energy according to the increasing of (E/N)

is observed also note that the mean energy values increase when moving from the

low concentration (0.1) mole to the high concentration (0.9) mole with respect to the

confined area from (0-68) Td as in Table 1, while the behavior becomes reversed in

relation to the confined region between (69 - 100) Td as in Table 2. This is because

the cross-section area is directly influenced by the reduced electric area - this impact

is high form low to high concentrations (0,1 to 0,7) mole for when E over N is

(0 - 68) Td. This behavior becomes opposite in the case of E over N= (69 - 100)Td,

wherein all of the mean energy is highly sensitive to the concentration of the gas

mixture, and as a result it will increase the rate of elastic-inelastic collisions.

Fig. 1. Mean Energy as a function of reduced electric field (E/N).

020 40 60 80 100

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Mean Energy (eV)

E/N (Td)

A1(C=0.1)

A1(C=0.2)

A1(C=0.3)

A1(C=0.4)

A1(C=0.5)

A1(C=0.6)

A1(C=0.7)

1554 B. Hamed et al.

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

Table 1. The correlation between mean

energy and Ne concentration (E/N= (0-68) Td).

E/N=45.38 Td

E/N= (0-68) Td

Concentration

(Mol.)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mean Energy

(eV)

2.63

2.665

2.71

2.75

2.81

2.87

2.95

Table 2. The relationship between mean

energy and concentration of Ne, E/N= (69-100) Td.

E/N=100 Td

E/N= (69-100) Td

Concentration

(Mol.)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mean Energy

(eV)

4.88

2.665

2.71

2.75

2.81

2.87

2.95

3.2. Mean energy modelling

For all concentrations, the fitting relationship could be obtained as shown in Fig. 1.

This relationship addresses a matching between the BOLSIG+ program values and

present model (Fig. 2) that based of utilizing strategic capacity and it can be

represented by Eq. (4):

(4)

Fig. 2. Represents percent and BOLSIG+ data

(using mathematical model) of mean energy.

Investigation of Concentration Influence on Electronic Coefficients of . . . . 1555

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

From Eq. (4) for all concentrations, parameters (A1, A2, X0, P) can be shown

through calculation of function of logistic and polynomial to determine the required

concentration calculated in Eq. (4). The equations of these parameters:

(5)

(6)

(7)

(8)

where C is concentration.

4. Energy Mobility

4.1. Influence of reduced electric field and concentration on energy mobility

The relationship between energy mobility μεN and E/N is shown in Fig. 3., where

sharp decreasing in energy mobility can be observed particularly in the limited

region between E/N= (0-50) Td. This decreasing is referred to energy losing by

ionization and excitation processes inside He:Ne mixture which is highly

influenced by reduced electric field. While in the limited region between E/N= (50-

100) Td energy mobility is almost steady.

Fig. 3. Energy mobility as a capacity of decreased electric field (E/N).

Dependences of energy mobility on gas mixture concentration can be observed,

where increasing of mobility according to the increasing of E/N can be noticed as

in Table 3, the reason for this, is the increase electron drift by increasing the

concentration with increasing the reduced electric field [10].

Table 3. The correlation between energy mobility *N and mass of Ne.

E/N =65.86Td

Concentration

(Mol.)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Energy mobility

*N×1024 (1/m/V/s)

3.74

3.97

4.23

4.53

4.89

5.32

5.84

1556 B. Hamed et al.

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

4.2. Energy mobility modeling

The following Eq. (9) illustrates the behavior of energy mobility according to the

change of E/N based of using logistic function:

(9)

where B1, B2, r0, and r are parameters, C is gas mixture concentration. This

relationship addresses a close link between the BOLSIG+ programming numbers

and percent model Fig. 4.

Fig. 4. Estimated/simulated data (using mathematical

model) of Energy mobility and calculated data.

From Eq. (9) for all concentrations, parameters (B1, B2, r0, r) can be shown

mathematically by function of numerical and mathematical to determine the

required concentration calculated in Eq. (9). The equations of these parameters are:

(10)

(11)

(12)

(13)

where C concentration.

5. e-i Momentum Frequency

5.1. Influence of reduced electric field and concentration on e-i

momentum frequency

The increasing of E/N produce sharp decreasing of (νe-i) particularly in the limited

region between (0-63) Td as shown in Fig. 5. It is clearly approved that the electron-

ion momentum frequency (νe-i) values increase when changing position from the

greater concentration (0.1) mole to the reduced concentration (0, 9) mole with

Investigation of Concentration Influence on Electronic Coefficients of . . . . 1557

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

respect to the confined area from (0-63) Td as in Table 4, while the behavior

becomes reversed in relation to the confined region between (63 - 100) Td as in

Table 5. This is because the electron-ion momentum frequency has an inverse

proportion with temperature (e-i).

Fig.5. e-i momentum frequency /N as

part of an equation of reduced electric field.

Table 4. The relationship between e-i momentum

frequency /N and concentration of Ne E/N= (0-63) Td.

E/N= 48.79 Td

E/N= (0-63) Td

Concentration

(Mol.)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

e-i momentum

frequency

/N×10-15 (m3/s)

1.53

1.51

1.48

1.45

1.42

1.39

1.35

Table 5. The relationship between e-i momentum

frequency /N and concentration of Ne E/N= (65-100) Td.

E/N=100Td

E/N= (65-100) Td

Concentration

(Mol.)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

e-i momentum

frequency

/N×10-16 (m3/s)

7.2

7.45

7.7

7.95

8.2

8.47

8.75

5.2. e-i momentum frequency /N modeling

The following Eq. (14) illustrates the behavior of e-i momentum frequency ν/N

according to the change of E/N based of using logistic function:

(14)

1558 B. Hamed et al.

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

where, , ,, are parameters, C is gas mixture concentration.

This relationship showcases a correlation between the BOLSIG+ values and

percent model (Fig. 6.):

Fig. 6. Assessed/recreated information (utilizing numerical

model) of e-i Momentum Prevalence and calculated data.

From Eq. (14) for all concentrations, parameters (, ,,) can be shown

mathematically by function of numerical and mathematical to determine the required

concentration calculated in Eq. (14). The equations of these parameters are:

(15)

(16)

(17)

(18)

Figure 6 represents the high matching between BOLSIG+ data and our estimated

data, where root mean square is in the range of (4.2×10-7). Also, Eqs. (15-18) are

used to calculate concentration in Eq. (14).

6. Conclusions

A comprehensive investigation of specific electron transport coefficients in plasma

state He-Ne gas mixtures has been carried out using the solution of Boltzmann

equation, adopting BOLSIG+ program. Through this investigation the influence of

mixture concentration on the electronic coefficients (νe-i/N, µƐN, and <

>) can be

observed. It can be also observed that by changing the reduced electric field, each

of versatile inelastic crashes and cross-area of collisions has a significant role in

discharge process.

In addition to, this investigation introduced appropriated equations of electronic

coefficients of He:Ne plasma field utilizing a simulation process for each of these

Investigation of Concentration Influence on Electronic Coefficients of . . . . 1559

Journal of Engineering Science and Technology April 2022, Vol. 17(2)

coefficients. Also simplified equations for concentration calculation have been

introduced. In general, the obtained equations through our simulation process are

shown high matching between BOLSIG+ data and our simulation data, where root

mean square of (νe-i /N,) = 4.2×10-7, root mean square of µƐN = 1.5×10-6 and root

mean square of <

> = 9.4×10-5.

Acknowledgments

Creators would want to express gratitude toward Mustansiriyah University

(www.uomustansiriyah.edu.iq) Baghdad-Iraq for its help in the current work, and

we are grateful to G. B. Ragimkhanov Dagestan State University.

Nomenclatures

E/N

Reduce electric field, 10-21 Vm2

F

External force, N

Distribution function before and after collision, (e.V)-3/2

Multiplicity of two energy level

K

Boltzmann constant, J/K

m

Particle mass, kg

Density of charge, m-3

Number of densities, m-3

T

Thermodynamic temperature, K

v

Velocity of electron, m/s2

Greek Symbols

Differential effective scattering

Solid angle

∂f/∂t

Rate of the change of the distribution rate of the change of the

distribution function

<

>

Mean energy, e.V

µ

Mobility, m2 /vs.

Momentum frequency, m3/s

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