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: email@example.com
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
Plasma is deﬁned as a “moldable” material  and
is one among the four elemental states of matter,
which was ﬁrst expounded by chemist Irving Lang-
muir during the 1920s . 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 . 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  and diagnostics [5, 6, 8, 9] of the discharge
processes [10, 11].
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 .
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) ,
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 .
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 
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
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
m· ∇f. (4)
The equivalent form of (2) is then
m· ∇f+F(r, t)·∂f
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 . Regarding the “collision” term, many
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)  is one of the well-known mod-
els/equations. The modiﬁed BE as per the BGK
form is given as
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
· ∇fi+F(r, t)·∂fi
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
j=1 ZZ d3p0dΩgij Iij (gij Ω)f0
It can be seen that the integration is over the
components of momentum, where f0=f0(p0
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
where pi,pjand p0
Jare the i, j-th momentums
before and after collisions, respectively .
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
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.
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-
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
Fig. 3. Mobility as a function of diﬀerent concen-
Fig. 4. Change in energy mobility with respect to
E/N (obtained by the BOLSIG+and the estimated
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
1 + E/N
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)
P= 2.65 −0.6C+ 0.1C2,(14)
to determined the required concentration C.
4.4. Inﬂuence of E/N on the diﬀusion
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.
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)]
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.
Inﬂuence of mixture concentration Con diﬀusion co-
eﬃcient DN (E/N = 41.97 Td).
Concentration C[mol] DN ×1025 [1/(m s)]
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
1 + E/N
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)
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 .
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.
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
1 + E/N
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,
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.
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]
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.
Authors would like to express their thanks and
gratitude to the Department of physics, College of
Science, Mustansiriyah University for support and
 M.C. Kelley, The Earth’s Ionosphere:
Plasma Physics and Electrodynamics,
2nd ed., Academic Press, 2012.
 A.I. Morozov, Introduction to Plasma Dy-
namics, 1st ed., CRC Press, 2012, p. 17.
 J.A. Bittencourt, Fundamentals of Plasma
Physics, 3rd ed., Ch. 1, Springer, New York
 F.L. Braga, Appl. Math. Lett. 24, 653
 O.S. Vaulina, E.A. Linsin, E.A. Sametov,
R.A. Timirkhanov, Plasma Fusion Res. 13,
 P.K. Chu, X.P. Lu, Low Temperature
Plasma Technology: Methods and Applica-
tions, CRC Press, 2020.
 P.J. Freidberg, Plasma Physics and Fusion
Energy, Cambridge University Press, 2007,
Modelling of Reduced Electric Field and Concentration. . .
 S.N. Antipov, M.M. Vasilev, S.A. Maiarov,
O.F. Petrov, V. E. Fortov, J. Exp. Theor.
Phys. 112, 482 (2011).
 A. Piel, Plasma Physics: An Introduction
to Laboratory, Space, and Fusion Plasmas,
2nd ed., Springer, 2010, p. 4.
 V.S. Kurbanishmailov, S.A. Maiorov,
O.A. Omarov, G. Ragimkhanov, I.G. Ra-
mazanov, Bull. Dagestan State Univ.,
Series 1, Nat. Sci. 33, 7 (2018) (in
 I.R. Golyatina, A.S. Maiorov, Bull. Lebedev
Phys. Inst. 41, 285 (2014).
 Z.W. Cheng, X.M. Zhu, F.X. Liu, Y.K. Pu,
J. Plasma Sources Sci. Technol. 24, 025025
 E.A. Jawad, Ibn Al-Haitham J. Pure Appl.
Sci. 30, 38 (2017).
 V.M. Zhdanov, A.A. Stepanenko, Phys.
Proced. 71, 110 (2015).
 A. Vikharev, A. Gorbachev, D. Radishev,
J. Phys. D Appl. Phys. 52, 014001 (2018).
 M. Yitzhak, Phys. Plasmas 27, 060901
 C. Köhn, O. Chanrion, T. Neubert, Plasma
Sources Sci. Technol. 26, 015006 (2017).
 Y. Lin, F. Wang, B. Liu, J. Comput. Phys.
360, 93 (2018).
 B. Evans, S.P. Walton, Appl. Math. Model.
52, 215 (2017).
 M. Drewes, C. Weniger, S. Mendizabal,
Phys. Lett. B. 718, 1119 (2013).
 S. Yatom, A. Shlapakovski, L. Beilin,
E. Stambulchik, S. Tskhai, Y.E. Krasik,
Plasma Sources Sci. Technol. 25, 064001
 P.T. Gressman, R.M. Strain, Proc. Natl.
Acad. Sci. 107, 57445749 (2010).
 R.I. Golyatinaa, S.A. Maiorova, Plasma
Phys. Rep. 44, 453 (2018).
 G.I. Sukhinin, M.V. Salnikov, AIP Conf.
Proc. 1925, 020029 (2018).
 P.T. Gressman, R.M. Strain, J. Am. Math.
Soc. 24, 771 (2011).
 D.A. Storozhev, S.T. Surzhikov,
Teploﬁzika vysokih temperatur 53, 307
(2015) (in Russian).
 I.Y. Bychkov, S.A. Yampolskaya,
A.G. Yastremskii, Plasma Phys. Rep.
39, 374 (2013).
 W. Yang, X. Meng, X. Zhou, Q. Zhou,
Z. Dong, AIP Adv. 9, 035041 (2019).