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ACTA PHYSICA POLONICA A No. 4 Vol. 140 (2021)

Proceedings of the Mustansiriyah International Conference on Applied Physics (MICAP-2021)

Modelling of Reduced Electric Field and

Concentration Inﬂuence on Electron Transport

Coeﬃcients of He–Ne Plasma

Maysam T. Al-Obaidi, Rafid A. Ali∗and Baidaa Hamed

Department of Physics, College of Science, Mustansiriyah University, Baghdad, Iraq

Doi: 10.12693/APhysPolA.140.299 ∗e-mail: rafidphy_1972@uomustansiriyah.edu.iq

A comprehensive theoretical investigation on the speciﬁc electron transport coeﬃcient in the He–Ne

gas mixtures state in plasma has been carried out. The performed calculations enable us to measure

the inﬂuence of the concentration of He:Ne gas mixture on plasma electronic coeﬃcients. Our approach

is based on the variation in the resistance of plasma ﬁeld placed in a varying reduced electric ﬁeld

and thermodynamic equilibrium. The eﬀects of reduced electric ﬁeld and mixture concentrations on the

electron mobility, diﬀusion coeﬃcients, and total collision frequency have been calculated by solving the

Boltzmann equation using the two-term approximation. The Monte Carlo simulation was used to solve

the Boltzmann equation. The polynomial and logistic functions were adopted through the simulation

process, and the appropriate equations have been provided, indicating the variation of plasma electronic

coeﬃcients to the variation of gas mixture concentration and reduced electric ﬁeld. The applied reduced

electric ﬁeld has been chosen to be in the limited range of 0–100 Td, and is considered for several

concentrations in the limited range of 0.1–0.7 mol. The study shows that the obtained results are in

good agreement with the results obtained using the Monte Carlo method.

topics: gas discharge plasma, electron transport coeﬃcients, electron energy distribution function

1. Introduction

Plasma is deﬁned as a “moldable” material [1] and

is one among the four elemental states of matter,

which was ﬁrst expounded by chemist Irving Lang-

muir during the 1920s [2]. The term “plasma” is

typically coined to explain a broad range of macro-

scopic neutral matters consisting of many closely

interacting and colliding free electrons and ionized

atoms or molecules, which show amalgamated be-

haviors and properties owing to the presence of

long-range Coulomb energies and forces. It should

be noted that the entire charged substances or par-

ticles ﬁeld could not be brought under the clas-

siﬁcation of plasmas. There are some conditions

or criteria to be satisﬁed to classify charged sub-

stances as plasma and to exhibit the plasma proper-

ties [3]. For instance, an ionized substance, which is

macroscopically neutral with a characteristic length

smaller than the Debye length could be considered

as plasma. The primary interactions in plasma are

electromagnetic interactions that essentially origi-

nated from the presence of an abundant number of

charged substances/particles in plasma [4, 5].

Since plasma largely contains electrically con-

ductive gaseous ionized substances, it enables the

long-range electric and magnetic ﬁelds to con-

trol the behavior and properties of matters. The

characteristics such as the existence of high conduc-

tivity and response to the electric/magnetic ﬁelds

make plasma potentially useful in a wide variety of

applications. Firstly, a suitable control is needed.

Secondly, the potential energy sources or radiations

are required to achieve the plasma. Therefore, the

lasers are routinely harnessed to produce a variety

of plasma types in the laboratory [6, 7].

Furthermore, the characteristic properties of

plasma are remarkably dependent on the interac-

tions among the substances in plasma. Notably,

the presence of “collective eﬀects” in plasma is

often found to be one of the fundamental fea-

tures that diﬀerentiate the plasma from that of

the other typical solids and ﬂuids. Furthermore,

it is because of the existence of long-range elec-

tromagnetic forces. Every individual charged sub-

stance in the plasma simultaneously interacts with

more number of other charged particles, manifest-

ing the “collective eﬀects” which are the key fac-

tors for the eﬀective physical phenomena occurring

in plasma [8, 9].

The precise collision cross-section and rate co-

eﬃcient values are crucial for determining the ki-

netic processes through modeling [1–3, 8, 9], simu-

lation [4] and diagnostics [5, 6, 8, 9] of the discharge

processes [10, 11].

299

Maysam T. Al-Obaidi et al.

It is known that the plasma at low tempera-

ture contains an adequate amount of neutral sub-

stances, where their greater internal degree of

freedom enables their active collisions and interac-

tions with electrons in the plasma. The existence

of surplus electrons and other interaction chan-

nels, such as ionization/recombination reactions for

heavy plasma substances, substantially leads to the

reformulation of electron transport equations and

mathematical expressions for various factors, such

as electron diﬀusion, electrical conductivity, ther-

mal conductivity, and viscosity coeﬃcients. Thus,

the role of inelastic collisions of electrons towards

the deduction of electron transport coeﬃcient has

become the subject of interest [12, 13].

The electron transport processes in plasma and

gaseous substances have been extensively investi-

gated over the last few decades through theoretical

and experimental studies. Most of the theoretical

investigations were devoted primarily to the analy-

sis of scalar transport features of electrons since the

rate of chemical reactions and electron energy re-

laxations are crucial and constitute the focal point

of research in plasma [14].

The ﬂuid model-based gas discharge processes

typically require the transport- and rate-coeﬃcients

as the input. The coeﬃcients are typically estimated

from the collision cross-section data by using the

electron energy Boltzmann equation (EEBE) [15],

and are largely associated with the electron en-

ergy distribution function (EEDF). In fact, the

Maxwell–Boltzmann distribution functions which

are obtained from the kinetic theory of gaseous sub-

stances provide a facile description for many basic

gas properties including the properties such as dif-

fusion, pressure, and velocity [16, 17].

In the present work, a mathematical model which

is devoted to theoretical investigation on speciﬁc

electron transport coeﬃcient of He–Ne plasma is

predicted by solving Boltzmann equation utilizing

the two-term approximation and by taking the ad-

vantages of Monte Carlo simulation method.

2. Mathematical model

In this work, the BOLSIG+program has been

employed to solve the Boltzmann equation (BE)

using the direct simulation Monte Carlo (DSMC)

method which is based on the Monte Carlo (MC)

simulation for the ﬁnite number ﬂuid-ﬂows [18].

BOLSIG+is considered a more generic and easier-

to-use tool as compared to other BE solving meth-

ods. A wide variety of fundamental processes in

gas–discharging plasma can be assessed and esti-

mated using BE which typically include the process

of electron–atom collisions, elastic collisions, elec-

tron impacted atom excitation, the transition of col-

lisions between the surrounded atoms, and the ion–

atom collisions resulting from the elastic collisions.

In the present work, the electron mobility, diﬀusion

coeﬃcients, and total collision frequency have been

calculated with varying reduced electric ﬁeld E/N,

several concentrations of helium–neon (He–Ne) gas

mixture, above the excitation temperature of 300 K.

Variations of mixture concentration can specify

the He–Ne plasma electronic transport coeﬃcients

for many applications.

2.1. The Boltzmann equation

Boltzmann introduced the probability via ex-

pounding the statistical characteristics of a non-

equilibrium thermodynamic system. Probability

has also been used in quantum physics to describe

the concept of time reversal, associating the direc-

tion of probability with the entropy exchange and

breaking the similarity between the past and the fu-

ture [19, 20] — even though the laws of mechanics

remain unchanged for time inversion [21, 22].

The Boltzmann relation gives the ratio of number

density (i.e., number per unit volume) of the sub-

stances such as atoms, ions or molecules at a certain

energy level N2to the number density in lower en-

ergy level N1. This can be written as [23]

N2

N1

=G2

G1

exp −E

kBT,(1)

where G1and G2denote the multiplicity of two en-

ergy positions, Eis the energy required to excite

the particles, kBis the Boltzmann constant, and T

is the temperature. Notably, a greater number of

particles can be excited by increasing T[18, 20].

The Boltzmann equation (BE) is

∂f

∂t =∂f

∂t force

+∂y

∂x diﬀ

+∂f

∂t coll

,(2)

where the ﬁrst term represents “force” and it can

be inﬂuenced by the external forces acting on the

particles. In turn, the second term “diﬀ” indicates

the particle diﬀusions. Finally, the term “coll” in-

dicates the forces acting between the particles in

collision [24, 25].

2.2. Equations of force, diﬀusion

and collision terms

The force term and diﬀusion term corresponding

to (2) can be written, respectively, as

∂f

∂t force

=−F(r, t)·∂f

∂p(3)

and

∂y

∂x diﬀ

=−p

m· ∇f. (4)

The equivalent form of (2) is then

∂f

∂t =p

m· ∇f+F(r, t)·∂f

∂p+∂f

∂t coll

,(5)

Here, F(r, t)represents the ﬁeld exerting on the

particles in ﬂuid, pis the momentum, and mis

the particle mass. The terms on the right side of

(5) describe the eﬀects of particle collisions, where

it will be zero if there is no collision between the

particles [26]. Regarding the “collision” term, many

300

Modelling of Reduced Electric Field and Concentration. . .

eﬀorts have been taken towards modelling the col-

lision. In this direction, the models/equations pro-

posed by Bhatnagar, Gross and Krook (known as

BGK form) [26] is one of the well-known mod-

els/equations. The modiﬁed BE as per the BGK

form is given as

∂f

∂t +p

m· ∇f+F·∂f

∂p=ν(f0−f),(6)

where νdenotes the frequency of collision and f0is

the local Maxwillian distribution function.

3. General Boltzmann equation

for a particle mixture

For a chemical mixture of particles categorized

by the indices i= 1,2,3, . . . , n, one has the

following

∂fi

∂t +pi

mi

· ∇fi+F(r, t)·∂fi

∂pi

=∂fi

∂t (7)

with fi=fi(r,pi, t)and where rrepresents the

position of particle, piand miare the momentum

and mass of species, respectively. The collision

term can be given by

∂fi

∂t coll

=(8)

n

X

j=1 ZZ d3p0dΩgij Iij (gij Ω)f0

if0

j−fifj.

It can be seen that the integration is over the

components of momentum, where f0=f0(p0

i, t),

Iij represents the diﬀerential cross-sections between

the species iand j, and dΩdenotes the solid an-

gle of collisions. The relative momentum gij can be

given by

gij =|pi−pj|=

p0

i−p0

j

,(9)

where pi,pjand p0

i,p0

Jare the i, j-th momentums

before and after collisions, respectively [27].

4. Results and discussion

Figure 1a and b represents the collisional cross-

section area of He and Ne. To solve the BE,

one needs substantial information about the colli-

sional cross-section of the gas mixture considered.

Figure 1 shows the cross-section with respect to the

electron energy of the reaction involved. Diﬀerence

in atomic structure between He and Ne leads to dif-

ferent energy levels for each gas, and consequently,

to theirs diﬀerent behavior (as can be seen in Fig. 1a

and b). The output data has been included in the

current calculations.

Figure 2a and b shows that there is a signiﬁcant

dependence of the electron energy distribution func-

tion (EEDF) on the electron mean energy (for sev-

eral values of applied electric ﬁeld E/N). The num-

ber of particles is N= 2.7×1025 m−3. The obtained

results also show that the electrons can be aﬀected

by a forward bias by increasing the applied elec-

tric ﬁeld E/N. This bias will accelerate the elec-

trons and increase energy up to a steady level of

Fig. 1. The collisional cross-section of (a) He and

(b) Ne as a function of electron energy.

Fig. 2. EEDF of (a) He and (b) Ne as a function

of the electron energy.

301

Maysam T. Al-Obaidi et al.

mean energy, which eventually increases the rate of

inelastic collisions. Above this point of the steady

energy level, the increment in E/N will be ineﬀec-

tive and the curve will fall towards the high-energy

tail. In other words, the high values of EEDF will

be at the low values of energy u. When increasing

the uvalues, the EEDF will approach the original

point due to the high energy electron resulting from

the applied forward bias. Values of EEDF for several

concentrations of He–Ne mixture are quasi logarith-

mic according to the slope of 1/(kBT). The shape

of the curves, as displayed in Fig. 2, illustrates the

Maxwell distribution for the several mixture con-

centrations.

4.1. Inﬂuence of E/N on the electron mobility

In Fig. 3, the variation in electron mobility µ/N

can be observed with respect to the increasing E/N

for several concentrations (0.1–0.6 mol) of He–Ne

mixtures, where the sharp decrease of µ/N occurs

between 0 and 40 Td. In the further range of E/N,

i.e., 40–100 Td, the mobility values become almost

steady. This can be caused by the descending elec-

tron energy as a result of ionization and excitation

processes [28].

Fig. 3. Mobility as a function of diﬀerent concen-

trations.

Fig. 4. Change in energy mobility with respect to

E/N (obtained by the BOLSIG+and the estimated

data).

4.2. Inﬂuence of the mixture concentration

on the mobility

In Table I, the inﬂuence of mixture concentration

on electrons mobility can be observed by analyzing

the Cbehavior from a low 0.1 mol to its higher

value 0.7 mol. The observed eﬀect results from the

excitation and ionization processes in relation to the

reduced electric ﬁeld E/N.

4.3. Modeling of the mobility

We have calculated the concentration values us-

ing the BOLSIG+program which bases on the de-

scription of the mobility behavior (µ/N ) depend-

ing on E/N. The obtained results are presented

in Fig. 4. Notably, the present BOLSIG+data are

found to be well-matched with the data obtained in

our present work (see Fig. 4).

The ﬁtting relationships for several concentra-

tions can be written as

µ

N=A2+A1−A2

1 + E/N

X0P,(10)

where E/N = 0–100 Td, while A1,A2,X0,P,

for a given concentration value, are constants. Poly-

nomial form has been assumed for

A1=3.32 + 0.9C+ 4.87C2×1024,(11)

A2=2.91 + 0.46C+ 2.34C2×1024,(12)

X0= 36.6+7.8C−16.22C2,(13)

P= 2.65 −0.6C+ 0.1C2,(14)

to determined the required concentration C.

4.4. Inﬂuence of E/N on the diﬀusion

coeﬃcients DN

Figure 5 shows the variation in diﬀusion coeﬃ-

cient (DN) with respect to the variation in the elec-

tric ﬁeld E/N. A slight/small increment in DN is

observed in the limited range of 0< E/N < 20 Td,

while a clear DN increment is observed in the range

of 20 Td<E/N = 20–60 Td. Next, in the range

of 60–100 Td, the DN values increase slightly.

TABLE I

Inﬂuence of the mixture concentration Con the elec-

tron mobility µ/N (E/N = 55.62 Td).

Concentration C[mol] µ/N ×1025 [1/(m V s)]

0.1 0.254

0.2 0.237

0.3 0.287

0.4 0.320

0.5 0.350

0.6 0.387

0.7 0.433

302

Modelling of Reduced Electric Field and Concentration. . .

Fig. 5. Diﬀusion coeﬃcient as a function of E/N

at diﬀerent concentration values C.

Fig. 6. Diﬀusion coeﬃcient as a function of E/N

(obtained by BOLSIG+and our estimated data.

TABLE II

Inﬂuence of mixture concentration Con diﬀusion co-

eﬃcient DN (E/N = 41.97 Td).

Concentration C[mol] DN ×1025 [1/(m s)]

0.1 0.44

0.2 0.48

0.3 0.53

0.4 0.58

0.5 0.65

0.6 0.74

0.7 0.85

The growth of DN is essentially due to the electron–

ion and electron–n collisions, which caused the drift

of the mass centre of atom (ion) and a subsequent

rerelease of electrons.

4.5. Inﬂuence of the concentration on DN

Table II shows the variation in the diﬀusion coef-

ﬁcient DN with respect to the variation of mixture

concentration Cfor a speciﬁc value of reduced elec-

tric ﬁeld E/N = 41.97 Td. This direct relationship

between DN and Cis due to the inelastic collisions.

4.6. Diﬀusion coeﬃcients modeling

The behavior of DN with respect to the change

in E/N is shown in Fig. 6. The estimated points are

obtained using the logistic function of the form

DN =B2+B1−B2

1 + E/N

r0r,(15)

where E/N = 0–100 Td, and B1,B2,r0,rare con-

stants for a given values of C. Therefore, the ﬁtting

for all concentrations is reﬂected with (15).

The parameters B1,B2,r0,rhave been obtained

utilizing polynomial function, respectively, as

B1=2.81 + 0.892C+ 0.356C2×1024,(16)

B2=9.5−0.615C+ 6.9C2×1024,(17)

r0= 73.3−34.16C−6.55C2,(18)

r= 2.36 −0.073C−0.3C2.(19)

Therefore, the concentration required in (15) is de-

termined with the use of (16)–(19).

Figure 6 shows a well-matched result between the

data obtained by the BOLSIG+program and the

estimated data with (15) for a speciﬁc concentration

value C= 0.5mol.

4.7. Inﬂuence of E/N on total collision frequency

Figure 7 shows the dependence of total collision

frequency (TCF) on E/N of the He–Ne mixture.

In the range of E/N < 20 Td, a slight increment

in the TCF is observed, whereas a sharp TCF in-

crement is observed for E/N > 20 Td. This result

is referred to the fact that the electric energy and

cross-section depend on the electron excitation and

ionization energy [28].

Table III shows the values of TCF of the He–Ne

mixture for a speciﬁc value of the reduced electric

ﬁeld. High TCF values are observed at a low mix-

ture concentration of 0.1 mol compared to a high

mixture concentration of 0.7 mol. This can be as-

cribed to the high inelastic collisions rate and the

collisional cross-section at low concentrations.

Fig. 7. Total collision frequency as a function of

E/N at diﬀerent concentrations C.

303

Maysam T. Al-Obaidi et al.

4.8. Total collision frequency (TCF) modeling

The behavior of TCF with respect to the vary-

ing values of E/N is shown in Fig. 8. The esti-

mated points of the total collision frequency νeoT

are obtained using the logistic function of the

form

νeoT

N=G2+G1−G2

1 + E/N

n0n,(20)

where E/N = (0–100) Td.

The parameters G1,G2,n0,nwhich are constants

for a given value of C, are obtained using polyno-

mial function, respectively,

G1=4.25 −3.22C−8.38 ×10−4C2×10−14,

(21)

G2=7.56 −5.2C+ 0.176C2×10−14,(22)

n0= 48.03 −2.68C−7.28C2,(23)

n= 2.47 −0.039C−0.486C2.(24)

Therefore, the concentration Crequired in (20) is

determined with the use of (21)–(24).

Fig. 8. Total collision frequency results show well-

matching between the BOLSIG+and the estimated

data under speciﬁc concentration Cof 0.5 mol.

TABLE III

Total collision frequency νeoT in the He–Ne mix-

ture under the eﬀect of the reduced electric ﬁeld

(E/N = 525.6Td).

Concentration [mol] νeoT/N ×10−13 [m3/s]

0.1 0.578

0.2 0.536

0.3 0.494

0.4 0.453

0.5 0.411

0.6 0.369

0.7 0.328

5. Conclusions

The eﬀect of induced electric ﬁeld and plasma

state in the He–Ne gas mixture concentrations on

the electron energy distribution function (EEDF)

has been investigated. The results are obtained with

the Boltzmann equation by the BOLSIG+program

supported by the direct simulation Monte Carlo

method. It is observed that increasing the elec-

tric ﬁeld and the concentration of mixture increase

the EEDF due to inelastic collisions which inﬂu-

ence the behavior of EEDF and are closely related

to the electronic coeﬃcients. The changes observed

in EEDF with respect to the electric ﬁeld indi-

cated that the performed calculations and results

obtained agree with the Maxwell equations.

The logistic and polynomial functions were ﬁt-

ted to the data. We have obtained the appropri-

ate and simpliﬁed equations to observe the behav-

ior of transport electronic coeﬃcients. The results

also showed that the data from the BOLSIG+–

Monte Carlo method were found to be well-matched

with the estimates obtained in the ﬁtting process

performed in this study. Also, the output results

turned out to be well suited to other studies [10, 27].

Overall, the obtained results suggested that the

Boltzmann equation (BE) can be one of the most

perspective tools to investigate the plasma states

and their electronic coeﬃcients from the kinetics of

electrons in feebly ionized gaseous substances.

Acknowledgments

Authors would like to express their thanks and

gratitude to the Department of physics, College of

Science, Mustansiriyah University for support and

cooperation.

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