INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 38 (2005) 1577–1587
doi:10.1088/0022-3727/38/10/013
Time-dependent behaviour of electron
transport in methane–argon mixtures
Alan A Sebastian and J M Wadehra
Department of Physics and Astronomy, Wayne State University, Detroit, MI 48202, USA
Received 16 September 2004, in final form 23 February 2005
Published 6 May 2005
Online at stacks.iop.org/JPhysD/38/1577
Abstract
The Boltzmann transport equation is solved to determine the time-dependent
electron velocity distribution function in various mixtures of methane and
argon subjected to an external electric field, E. In the solution of the equation
no expansion of the distribution function is made. The concentration of
methane in the mixture is varied systematically. The distribution function is
used to calculate several time-dependent electron swarm parameters in these
gas mixtures. For each gas mixture a wide range of E/N values, from 0.1 to
1000 Td, is investigated. Steady-state values of various calculated swarm
parameters (drift velocity, Townsend ionization coefficient and characteristic
energy) agree quite well with the corresponding experimental values.
1. Introduction
Electron transport coefficients are used in the modelling
of low temperature plasma systems that have applications
in semiconductor processing, particle detectors, plasma
display modelling and plasma assisted chemical vapour
deposition. Mixtures of rare gases with hydrocarbon gases are
particularly important in these applications [1]. In the present
investigations, we have calculated accurate rates for electron
impact processes in various mixtures (with a gas density N) of
methane and argon subjected to an external constant electric
field E. Methane–argon mixtures are routinely used in gaseous
radiation detectors. Our results are based on a solution of the
time-dependent Boltzmann transport equation [2, 3], which is
solved for values of E/N ranging from 0.1 to 1000 Td (1 Td =
10−17 V cm2 ). The time-dependent behaviour of various
electron swarm parameters is obtained directly from the timedependent electron velocity distribution function (EVDF).
We present a complete set of steady-state values of the
drift velocity, Townsend ionization coefficient (α/N ) and the
characteristic energy for a wide range of values of the reduced
electric field, E/N. These steady-state results are compared
with experimental values for the argon–methane mixtures for
which such data are available. To the best of our knowledge,
these are the first systematic calculations of electron swarm
parameters in argon–methane mixtures for such a wide range
and for such large values of E/N. Previous calculations [4–9]
and experimental investigations [10–25] are either done for
100% argon or 100% methane gases or are confined to a narrow
range of E/N values for argon–methane mixtures. Also,
0022-3727/05/101577+11$30.00
© 2005 IOP Publishing Ltd
in order to explicitly show the time dependence of various
swarm parameters, we have calculated the equilibration time
for these swarm parameters. The equilibration time is defined
as the time when a particular swarm parameter reaches 99%
of its final steady-state value. The equilibration time depends
strongly on the relative concentrations of methane and argon in
the mixture. In this paper, we show values of the equilibration
times for the drift velocity, for a fixed value of E/N , as a
function of the methane concentration.
2. Method
The time-dependent behaviour of various electron swarm
parameters is determined by the EVDF that is obtained as a
solution of the Boltzmann equation. The general method [2, 3]
that we use for solving the time-dependent Boltzmann equation
is summarized here. In order to obtain the parameters that
characterize the transport of an electron beam injected in a
neutral gas mixture, which is subjected to an external electric
field E, one starts with the traditional Boltzmann’s equation:
∂f (r, v , t)
+ v · ∇f (r, v , t) + a · ∇v f (r, v , t) = R(r, v , t),
∂t
(1)
whose solution, f (r, v , t), provides the velocity distribution
function of the electrons. Here, a = −eE/m represents the
acceleration of the electrons due to the external electric field E
and the collision term R(r, v , t) represents the rate of change
in the EVDF due to all possible collision processes among
the electrons and the ambient gas particles. The collision
Printed in the UK
1577
A A Sebastian and J M Wadehra
term R(r, v , t) itself involves integrals over the distribution
function f (r, v , t), so that, in its complete form, the Boltzmann
equation is a time-dependent integrodifferential equation
whose numerical solution is usually obtained only after making
some simplifying assumptions. For our treatment of the
problem, we assume spatial homogeneity. In this special case,
the spatial gradient of the EVDF is zero and the Boltzmann
equation becomes
∂f (v , t)
+ a · ∇v f (v , t) = R(v , t).
∂t
(2)
An explicit expression for the collision term R(v , t) is [26]
Rp+ (v , t) − R − (v , t)
R(v , t) =
p
(p = elastic, inelastic, etc),
where
Rp+ (v , t)
×
N
= 2
v
2π
0
∞
vp2
dvp
(3)
π
sin ψ dψ
0
dαvp f (vp , t) σp (vp , ψ) δ(v − gp (vp , ψ))
(4)
0
represents the rate at which the electrons with initial speed vp
are scattered into a velocity space element d3 v located at v , and
R − (v , t) = Nvf (v , t)
σp (v)
(5)
p
represents the rate at which the electrons are scattered out of
the velocity space element d3 v also located at v . The energy
conserving delta function relates the initial speed vp to the final
speed v through an expression of the form v = gp (vp , ψ) [26].
Various angles are as shown in figure 1 of [3]. Thus, according
to the Boltzmann equation (2) the velocity distribution function
evolves in time, first, due to the externally applied electric field
and, second, due to the collisions among the electrons and
the ambient gas particles. The opposing nature of these two
effects leads to a final steady-state distribution function that is
independent of time.
In our approach to the solution of the Boltzmann
equation (2) we first recast the differential equation (2) into
a difference equation that is more suited for computations. If
we multiply equation (2) by a small time increment t and
add f (v , t) to it, we obtain
f (v + v , t + t) = f (v , t) + R(v , t)t
with v = at.
(6)
The EVDF described by the difference equation (6) is the
same as the distribution function described by the Boltzmann
equation (2) in the limit t → 0. Furthermore, with an
appropriate and sensible choice of v and t, equation (6)
is simpler to solve and is better suited for computations than
is equation (2). For numerical stability of the solution [3],
a suitable choice of v and t in equation (6) is such
that the time step t must be smaller than the smallest
collision time determined by the collision cross sections.
The difference equation (6), which is the starting point of
our calculations, is also the starting point of deriving the
Boltzmann equation in standard textbooks [27]. In such
a derivation, one starts with equation (6) (which essentially
1578
expresses the fact that, in the absence of collisions, a charged
particle with velocity v at time t will have, due to an electric
field E, a velocity v + v at a later time t + t), takes the
limit t → 0 and steps backwards from equation (6) to
equation (2) above. In fact, over a 100 years ago, Boltzmann
himself took this route [28] from equation (6) to equation (2)
to derive a differential equation (namely, the Boltzmann
equation) since at those times it was more natural to attempt
solutions of differential equations using techniques that were
well developed by that time. Now, in the present computer age,
one should re-think as to where the Boltzmann equation came
from in the first place rather than trying to obtain approximate
solutions of equation (2). On realizing that the Boltzmann
equation (2) originates from the difference equation (6) and that
the difference equation (6) is more suited for a computer than
the original equation (2), it becomes almost natural to work
with equation (6) and thereby numerically obtain the solution
of the Boltzmann equation. The present investigations clearly
demonstrate that this approach is computationally superior
and sensible. Using the numerical values of the distribution
function f (v , t) obtained as the solution of equation (6),
for each target gas mixture, a typical swarm parameter is
calculated to be the expectation value of a particular function
g(v ) as
g(v )f (v , t)dv
G(t) = g(v ) = .
(7)
f (v , t)dv
The steady-state value of this swarm parameter, G(t) as
t → ∞, is a function of the ratio E/N only. Note that
the distribution function, f (v, θ, t), becomes azimuthally
symmetric if the z-axis is chosen to be along −E. θ is the
angle between the velocity v and the electric field −E. Various
swarm parameters that are shown in the figures are defined and
calculated as follows [29]:
drift velocity of electrons,
vd (t) = v cos θ 2
1
ε(t) = 2 mv average energy of electrons,
N(j )vσion (j, v)
ionization rate of
Rion (t) =
j
ambient gas,
α(t)
Rion (t)
=
Townsend ionization coefficient,
N
vd (t)N
2
1 v
isotropic diffusion coefficient,
D(t) =
3 νT
vd (t)
µ(t) =
mobility of electrons,
E
eD(t)
characteristic energy of electrons.
Ec =
µ(t)
Here νT (v) and N are the total collision frequency and the total
number density, that is,
νT (v) =
N(j )vσT (j, v)
and
N=
N (j ),
j
j
where N(j ) and σT (j, v) are the number density and the total
collision cross section of the j th component in the ambient gas
mixture.
3. Numerical details
A common simplifying assumption that has frequently been
made in the solution of the Boltzmann equation is to expand
Behaviour of electron transport in methane–argon mixtures
the distribution function in an infinite series of Legendre
polynomials in θ (because of azimuthal symmetry). At low
values of E/N, it is a reasonable approximation to retain only
the first few terms in this series [5, 6]. For a given N, this twoor three-term approximation provides the velocity distribution
and the corresponding swarm parameters only for weak
external fields, and becomes greatly unreliable for larger values
of the external electric field. In principle, for larger fields one
could retain more terms in the expansion of the distribution
function but such a procedure will utilize prohibitive computer
resources and will be prone to more numerical errors since all
the derivatives in this multi-term approach will normally be
evaluated numerically. Furthermore, for particular values of
the external fields it would generally not be clear a priori,
without resorting to a separate numerical study, as to how
many terms in the expansion of the distribution function
should be retained for convergence. It is to be noted that the
difference equation (6) does not contain any errors associated
with the truncation of any series expansion of the distribution
function or errors associated with the numerical evaluation of
derivatives. Using the present procedure one can obtain
the velocity distribution function for high as well as low
values of E/N, extending from a fraction of a Townsend
to several thousands of Townsends. Especially, since the
swarm parameters take less time to reach steady state for larger
values of E/N, the present algorithm is computationally quite
economical for the larger values of E/N, where the traditional
two- or multi-term approximations break down.
In general, algorithms that utilize numerical evaluation
of derivatives for solving a partial differential equation must
satisfy the Courant condition for stability. In the context of the
Boltzmann equation, in its integrodifferential form (2), this
condition implies that the time step t would be restricted by
the condition at v , where v is the velocity step and
a is the electron acceleration as defined previously. In our
calculations, the Boltzmann equation has been expressed in a
form that does not require the evaluation of any numerical
derivative so that there is no need to impose the Courant
condition. The time step t, however, is still limited by the
largest value, νmax , of the total collision frequency νT such
that νmax t 1. To be on the cautious side in the present
calculations, we have kept the time step below its largest
allowed value by a factor of 50–100. The value of the time
step t used in these calculations depends mainly on the value
of E/N and, to a lesser extent, on the methane concentration
in the mixture. It ranges from 0.5 ps for high values of E/N
to 20 ps for low values of E/N.
The external electric field, E, defines the z-axis and in
cylindrical coordinates the distribution function f (v , t) has
azimuthal symmetry about the z-axis. Because of this symmetry, the velocity distribution function is stored in a twodimensional array, with variables vρ and vz , on a uniform
square grid. The grid size and the range of the grid were
varied to determine the convergence of results. In the final
results, we have used 151 grid points in the ρ-direction
and 301 points along the z-direction. The velocity step
v (both vρ and vz ) is chosen such that any value of the
distribution function at the array boundaries is less than 10−5
of its peak value. The acceleration of the distribution function
involves shifting the two-dimensional array along the z-axis by
an amount at. This requires a knowledge of the values of the
distribution function at locations between the grid points stored
in the velocity array and necessitates interpolation of the distribution function along the z-axis. For this interpolation we have
used a three-point Lagrangian scheme. The actual value of
the velocity step used in the calculations varied both with the
values of E/N and with values of the methane concentration
in the mixtures. The values of v ranged from 0.42 cm µs−1
for pure methane at small values of E/N to 7.6 cm µs−1 for
pure argon at large values of E/N .
The evaluation of the collision integral in equation (4) is
best performed in polar coordinates (in velocity space) using
the variables vr and vθ , which describe the vector v . The
values of the distribution function f (vr , vθ ) required for this
integration are obtained by interpolating the f (vρ , vz ) array
using a three-point Lagrangian method if v lies within the
boundaries of the f (vρ , vz ) array. On the other hand, if
v lies outside the boundaries of the f (vρ , vz ) array, then,
values of f (vr , vθ ) are obtained by extrapolation using the
logarithmic approximation, namely, f (vr + 2vr , vθ ) =
f 2 (vr +vr , vθ )/f (vr , vθ ). The number of angular steps, vθ ,
used in the evaluation of the collision integral is 50, and the
number of velocity steps, vr , is 225. Also, the velocity
step vr used in the polar integral is equal to the velocity
step v in the f (vρ , vz ) array. Evaluation of the collision
integrals is performed using the Simpson’s rule. Integration
of the velocity distribution function in equation (7) to obtain
the electron swarm parameters, at a given time, is carried out
using the trapezoidal scheme.
A step-by-step procedure for obtaining f (v , t) is:
Step 1. Start with a given distribution function at time t, which
is stored in a two-dimensional array f (vρ , vz ). It could
either be an analytical function (e.g. a Maxwellian or
a Druyvesteyan or a delta function) representing the
distribution function at the starting time t = 0 or be a
numerically generated function at some earlier time.
Step 2. Compute the collision terms R(vρ , vz ) for each value
of vρ and vz . Multiply the collision terms by t and
add to the distribution function from which they were
obtained.
Step 3. Shift the resulting array along the vz index
[f (vρ , vz ) → f (vρ , vz + v)] to obtain the new
distribution function at the later time t + t.
Step 4. Calculate various swarm parameters corresponding to
the later time t +t using the new distribution function.
Repeat from step 1 unless the swarm parameters stop
changing in time, which indicates that the steady state
has been reached.
4. Results and discussion
We present our results of several swarm parameters for various
mixtures of methane in argon and for the 100% argon and
100% methane gases. The mixtures under consideration in
this systematic study are the ones with N(methane)/N =
10%, 25%, 50% and 75%. Mixtures with other concentrations
of methane in argon were also studied but are not presented
separately. Rather, these results contribute to the surface plot
shown later in figure 10. In our calculations we have used,
for the electron–methane cross sections, the data compiled by
1579
A A Sebastian and J M Wadehra
100
-15
10
25% CH4 in Ar
10
-16
10
Argon
-17
10
Momentum Transfer
Excitation
Ionization
-18
1
-15
10
Drift Velocity (cm/µs)
Cross Section (cm2)
10
-16
10
-17
10
Methane
-18
Momentum Transfer
Excitation I
Excitation II
Attachment
Dissociation
Ionization
10
-19
10
10
1
0.1
100
10
-20
-21
1
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
Energy (eV)
Figure 1. Cross sections for electron scattering by argon and
methane. Argon: momentum transfer [33], total excitation [34]
and total ionization [35]. Methane: momentum transfer [30],
excitation I: vibrational excitation of the (ν1 + ν3 ) states [31],
excitation II: vibrational excitation of the (ν2 + ν4 ) states [31],
electron attachment [30], total dissociation [30] and total
ionization [30].
Shirai et al [30] for all but the vibrational excitation cross
sections for which we have used the data of Tawara et al
[31]. The vibrational excitation cross sections consist of
the unresolved (ν1 + ν3 ) and (ν2 + ν4 ) modes. In addition
to the momentum transfer and vibrational excitation cross
sections, we also include attachment, total dissociation and
total ionization cross sections for methane. In the case of
electron–argon, we have included the momentum transfer,
total excitation and total ionization cross sections. A critical
comparison of low energy momentum transfer cross sections
for electron scattering by argon was made recently by Buckman
and Brunger [32]. In particular, the low energy Ramsauer
minimum in the cross section occurs at an electron energy of
∼0.23 eV and the minimum value of the corresponding cross
section is ∼8.5 × 10−18 cm2 . The momentum transfer cross
section that we have used is a numerical fit to the data of Frost
and Phelps [33] and it is consistent with the critically compared
cross sections of [32]. The total excitation cross section for
electron–argon is taken from Sakai et al [34]. Finally, the total
ionization cross section of argon by electron impact is from
the work of Rapp and Englander-Golden [35]. These cross
sections are collectively shown in figure 1.
These cross sections are used in the collision term of
the Boltzmann equation (6) to obtain the time-dependent
EVDF f (v , t). This distribution function is used to calculate
various swarm parameters, such as the electron drift velocity,
1580
10% CH4 in Ar
0% CH4 in Ar (Pure Argon)
10
10
0.1
100
0.1
0.1
1
10
100
1000
E/N (Townsends)
Figure 2. Electron drift velocity as a function of E/N in 0% (pure
argon), 10% and 25% mixtures of methane with argon. Solid lines
are our present results. Experimental data for pure argon:
+, Dutton [11]; ♦, Nakamura and Kurachi [12]. 10% mixture:
, Hunter et al [15]; ×, Foreman et al [16]; , Wong et al [17].
25% mixture: ◦, de Urquijo et al [10].
ionization coefficient and characteristic energy, as a function
of time. The steady-state values of swarm parameters,
shown in figures 2–7, are our calculated data points. Note
that the solid lines, shown in these figures as the present
results, are polynomial fits to the calculated data points.
These points, some 20–30 in number depending upon the
mixture concentration and the value of E/N , are not shown
individually in the corresponding plots. However, our
calculated steady-state values of various swarm parameters are
provided in the tables.
4.1. Drift velocity
The steady-state values of the drift velocity of electrons in
various mixtures of methane and argon are shown in figures 2
and 3. For ease of comparison, all six frames in these
figures are drawn to the same scale. The calculated values
of the drift velocity, for a large range of E/N values, agree
very closely with the available experimental data. The most
recent experimental values available for drift velocity in these
mixtures are those of de Urquijo et al [10]. For comparison
purposes, we have chosen experimental data for 100% argon
from the compilation of Dutton [11] and from Nakamura
and Kurachi [12]. For 100% methane, the experimental
measurements of electron drift velocity are provided by
Behaviour of electron transport in methane–argon mixtures
100
100% Pure CH4
10
Drift Velocity (cm/µs)
1
0.1
100
75% CH4 in Ar
10
1
0.1
100
50% CH4 in Ar
10
1
0.1
0.1
1
10
100
1000
E/N (Townsends)
Figure 3. Electron drift velocity as a function of E/N in 50%
and 75% mixtures of methane with argon and in 100% pure
methane. Solid lines are our present results. Experimental data
for 50% mixture: ◦, de Urquijo et al [10]; , El-Hakeem and
Mathieson [18]; , Jean-Marie et al [19]. 75% mixture:
◦, de Urquijo et al [10]. Pure methane: , Davies et al [13];
, Hunter et al [14].
Davies et al [13] and by Hunter et al [14]. Measurements of
electron drift velocity in a mixture of 10% methane in argon
have been carried out by Hunter et al [15], by Foreman et al
[16] and by Wong et al [17]. In mixtures of 50% methane in
argon, additional measurements of the electron drift velocity
are provided by El-Hakeem and Mathieson [18] and by
Jean-Marie et al [19].
A prominent feature of the mixtures of methane with argon
is the presence of a region of negative differential conductivity
(NDC). This is the region in which the electron drift velocity
decreases for increasing values of E/N . The NDC behaviour
has been investigated by others [36, 37] and it is seen to become
less pronounced as the fraction of methane is reduced until it
disappears altogether for the case of pure argon. It is also
interesting to note that the values of the reduced electric field,
(E/N )max , at which the drift velocity reaches a local maximum
for the mixtures of methane and argon can be plotted versus
percentage of methane and is seen to fall on a straight line. If
the average electron energies that correspond to these values
of reduced electric field and methane percentage are extracted
from the data, one can note that these average energies fall very
near the regions of both the argon and methane Ramsauer–
Townsend momentum transfer cross section minima. Indeed,
this phenomenon has been observed by Wang et al [38] in
mixtures involving argon and CHF3 .
In table 1 we provide steady-state values of the
drift velocity of electrons in gas mixtures with varying
concentrations of methane in argon. Because of the presence
of a region of NDC in methane mixtures, we provide some
numerical values in table 1, in addition to those shown in
figures 2 and 3.
4.2. Townsend ionization coefficient
Steady-state values of the time-dependent density-normalized
ionization coefficient, α/N , vary with the concentrations
of methane in the mixture and with the reduced electric
field, E/N. Plots of α/N for the 100% argon and 100%
methane gases and for their mixtures are shown in figures 4
and 5, and are compared with corresponding experimental
results wherever possible. Again, for ease of comparison,
all six frames in these figures are drawn to the same
Table 1. Equilibrium values of the drift velocity of electrons in cm µs−1 , in various mixtures of methane in argon.
% Methane
E/N (Td)
0.1
1
5
10
20
30
50
100
200
300
400
500
600
700
800
900
1000
a
0
a
1.73(−1)
3.17(−1)
5.60(−1)
1.07
2.05
2.91
4.43
7.82
1.41(1)
2.05(1)
2.68(1)
3.34(1)
4.00(1)
4.68(1)
5.36(1)
6.06(1)
6.76(1)
1
2
5
10
25
50
75
100
1.67
9.58(−1)
9.32(−1)
1.20
2.06
2.91
4.51
8.30
1.54(1)
2.21(1)
2.84(1)
3.46(1)
4.04(1)
4.61(1)
5.15(1)
5.68(1)
6.18(1)
2.10
1.48
1.13
1.32
2.06
2.90
4.48
8.25
1.53(1)
2.20(1)
2.84(1)
3.44(1)
4.03(1)
4.59(1)
5.14(1)
5.66(1)
6.16(1)
2.17
2.79
1.69
1.81
2.22
2.92
4.43
8.14
1.52(1)
2.18(1)
2.81(1)
3.41(1)
3.99(1)
4.55(1)
5.09(1)
5.61(1)
6.11(1)
1.93
4.39
2.48
2.38
2.65
3.09
4.39
7.96
1.49(1)
2.14(1)
2.77(1)
3.36(1)
3.94(1)
4.49(1)
5.02(1)
5.53(1)
6.02(1)
1.19
6.88
4.31
3.59
3.53
3.75
4.50
7.52
1.41(1)
2.04(1)
2.65(1)
3.22(1)
3.77(1)
4.30(1)
4.81(1)
5.30(1)
5.77(1)
6.97(−1)
7.47
6.72
5.06
4.41
4.43
4.84
7.03
1.30(1)
1.90(1)
2.47(1)
3.01(1)
3.54(1)
4.03(1)
4.51(1)
4.97(1)
5.41(1)
4.81(−1)
6.61
8.65
6.40
5.07
4.88
5.05
6.68
1.20(1)
1.77(1)
2.31(1)
2.83(1)
3.33(1)
3.80(1)
4.25(1)
4.68(1)
5.10(1)
3.64(−1)
5.57
1.01(1)
7.63
5.75
5.26
5.21
6.41
1.12(1)
1.65(1)
2.17(1)
2.67(1)
3.14(1)
3.59(1)
4.02(1)
4.44(1)
4.83(1)
The notation a(b) means a × 10b .
1581
A A Sebastian and J M Wadehra
-15
-15
10
-20
10
-25
10
10
-20
10
-25
10
100% Pure CH4
-30
25% CH4 in Ar
-30
10
10
-35
-35
10
-40
10
-15
10
-20
10
-25
10
10
-40
10
-15
10
-20
10
α/N (cm )
2
2
α /N (cm )
-25
10
10% CH4 in Ar
-30
10
-35
10
75% CH4 in Ar
-30
10
-35
10
-40
10
-40
10
-15
10
-15
10
-20
10
-20
10
-25
10
-25
10
50% CH4 in Ar
-30
0% CH4 in Ar (Pure Argon)
-30
10
10
-35
10
-35
10
-40
10
-40
10
10
10
100
E/N (Townsends)
Figure 4. Townsend ionization coefficient as a function of E/N in
0% (pure argon), 10% and 25% mixtures of methane with argon.
Solid lines are our present results. Experimental data for pure argon:
, Lakshminarasimha and Lucas [21]; , Specht et al [22]; + ,
Abdulla et al [23]. 10% mixture: ♦, Armitage et al [20].
25% mixture: – – –, numerical fit to experimental data as found in
de Urquijo et al [10].
scale. Experimental measurements of the Townsend ionization coefficient in pure argon have been carried out by
Lakshminarasimha and Lucas [21], by Specht et al [22] and
by Abdulla et al [23]. With 10% methane in argon, the
experimental measurements of the ionization coefficient are
provided by Armitage et al [20]. For larger concentrations
of methane (25%, 50%, 75% and 100%) in the mixture,
the experimental measurements of ionization coefficient were
recently made by de Urquijo et al [10]. Note that the data
of de Urquijo et al is shown as a dashed line in figures 4 and
5. This line is derived from a fit of experimental data to an
exponential function given in [10].
The ionization coefficient for low reduced fields is
sensitive to the nature of the EVDF beginning near the
threshold of the ionization cross section and extending out
to the maximum of this cross section. Since the steady-state
EVDF is quite narrow and of low energy (below 1 eV) the
computer code must cope with the very small values of the
EVDF in the energy regime where the ionization cross section
is appreciable. Figures 4 and 5 summarize our results. Note
that the values for the Townsend ionization coefficient vary by
large orders of magnitude for the gas mixtures as the reduced
electric field is increased from below 10 Td up to 1000 Td.
Also, it may be noted that the point at which the rate of increase
1582
100
1000
E/N (Townsends)
1000
Figure 5. Townsend ionization coefficient as a function of E/N in
50% and 75% mixtures of methane with argon and in 100% pure
methane. Solid lines are our present results. Experimental
data: – – –, numerical fit to experimental data as found in de Urquijo
et al [10].
in the first Townsend ionization coefficient begins to level
off moves slowly towards higher E/N as we progress from
pure argon to pure methane. Our results are compared with
the experimental data found in several literature references;
however, the available experimental values are restricted to
the higher ranges of the reduced electric field.
Table 2 provides the steady-state values of the Townsend
ionization coefficient of electrons in various mixtures of
methane in argon. The coefficient is significantly reduced
when either the electric field (E/N ) is decreased or the
concentration of methane in argon is increased. For values
of E/N below 10 Td, the ionization coefficient is so small that
the calculated values become numerically unreliable. Several
values of the ionization coefficient given in table 2 are in
addition to those shown in figures 4 and 5.
4.3. Characteristic energy
Once the EVDF has reached steady state, it becomes easy to
calculate the average kinetic energy ε of the electron swarm
as the expectation value of 21 mv2 . On the other hand, direct
laboratory measurements of the average kinetic energy of
electrons in the swarm are not straightforward. However,
the isotropic diffusion coefficient D and the mobility µ of
the electrons, both of which are relatively easy quantities
to measure in the laboratory, are related to the average
kinetic energy of electrons. In fact, the ratio eD/µ has
Behaviour of electron transport in methane–argon mixtures
Table 2. Equilibrium values of the Townsend ionization coefficient of electrons, in cm2 , in various mixtures of methane in argon.
% Methane
E/N (Td)
10
20
30
50
100
200
300
400
500
600
700
800
900
1000
a
4.76(−23)
2.76(−20)
2.69(−19)
2.08(−18)
1.35(−17)
4.93(−17)
8.98(−17)
1.30(−16)
1.68(−16)
2.04(−16)
2.36(−16)
2.66(−16)
2.94(−16)
3.19(−16)
1
2
5
10
25
50
75
100
8.59(−21)
9.84(−20)
4.00(−19)
2.14(−18)
1.19(−17)
3.90(−17)
6.83(−17)
9.78(−17)
1.27(−16)
1.55(−16)
1.81(−16)
2.06(−16)
2.30(−16)
2.52(−16)
3.87(−22)
3.31(−20)
2.79(−19)
1.98(−18)
1.18(−17)
3.91(−17)
6.84(−17)
9.80(−17)
1.27(−16)
1.55(−16)
1.82(−16)
2.07(−16)
2.31(−16)
2.53(−16)
6.83(−24)
2.17(−20)
3.13(−19)
1.93(−18)
1.16(−17)
3.90(−17)
6.86(−17)
9.84(−17)
1.28(−16)
1.56(−16)
1.83(−16)
2.08(−16)
2.32(−16)
2.54(−16)
7.19(−26)
1.14(−20)
1.64(−19)
1.58(−18)
1.10(−17)
3.89(−17)
6.90(−17)
9.91(−17)
1.29(−16)
1.57(−16)
1.84(−16)
2.10(−16)
2.35(−16)
2.57(−16)
4.12(−30)
2.84(−23)
9.41(−21)
6.29(−19)
8.93(−18)
6.94(−17)
1.01(−16)
1.32(−16)
1.61(−16)
1.89(−16)
2.16(−16)
2.41(−16)
2.65(−16)
8.93(−18)
4.86(−33)
6.83(−27)
7.83(−23)
8.12(−20)
5.35(−18)
3.47(−17)
6.85(−17)
1.02(−16)
1.35(−16)
1.66(−16)
1.96(−16)
2.25(−16)
2.52(−16)
2.77(−16)
3.31(−42)
1.18(−30)
5.73(−25)
6.69(−21)
2.72(−18)
3.02(−17)
6.60(−17)
1.02(−16)
1.36(−16)
1.70(−16)
2.02(−16)
2.32(−16)
2.60(−16)
2.87(−16)
1.31(−42)
1.37(−33)
2.06(−27)
4.40(−22)
1.19(−18)
2.53(−17)
6.22(−17)
1.00(−16)
1.37(−16)
1.72(−16)
2.05(−16)
2.37(−16)
2.67(−16)
2.96(−16)
The notation a(b) means a × 10b .
dimensions of energy and it is directly proportional to the
average kinetic energy of electrons for some special cases [29].
These include the case when the EVDF is Maxwellian or the
case when the momentum transfer cross section is constant.
At lower values of E/N, both the longitudinal diffusion
coefficient (DL ) and the transverse diffusion coefficient (DT )
have approximately the same value as the isotropic diffusion
coefficient D. However, because of the effect of the applied
field E, the longitudinal diffusion coefficient, DL , begins to
differ considerably from the isotropic diffusion coefficient, D,
while the transverse diffusion coefficient, DT , stays close to
D for larger values of E/N. Calculations reveal that the ratio
eD/m, which is termed as the characteristic energy, follows
the same trends as the average kinetic energy of electrons when
either the electric field is varied or the concentration of methane
in argon is changed. Thus, the characteristic energy provides
a fairly good estimate of the average kinetic energy of the
electrons in the swarm and it can be directly compared with
corresponding experimental measurements.
Our calculated values of characteristic energy for a wide
range of E/N values are shown in figures 6 and 7. Again,
for easy comparison, all six frames in these two figures are
drawn to the same scale. In order to compare our calculated
results with experimental measurements, we found results
only for the 100% argon and 100% methane gases. The
characteristic energy of an electron swarm in 100% argon
was measured by Lakshminarasimha and Lucas [21] and by
Townsend [24]. In 100% methane, the electron characteristic
energy was measured by Lakshminarasimha and Lucas [21],
by Hunter et al [15] and by Millican and Walker [25]. It should
be noted that our values of eD/µ overestimate the measured
values at high reduced electric fields for the pure gases.
Numerical steady-state values of the electron characteristic
energy, for various mixtures of methane in argon, are provided
in table 3 for a wide range of values of E/N.
4.4. Time-dependent results
In our calculations, we have obtained the complete transient
behaviour of various swarm parameters from the initial
time (t = 0) to the final time when all parameters have
reached their steady-state values. For a particular value
10
1
25% CH4 in Ar
0.1
Characteristic Energy (eV)
a
0
0.01
10
1
10% CH4 in Ar
0.1
0.01
10
1
0% CH4 in Ar (Pure Argon)
0.1
0.01
0.1
1
10
100
1000
E/N (Townsends)
Figure 6. Electron swarm characteristic energy as a function of
E/N in 0% (pure argon), 10% and 25% mixtures of methane with
argon. Solid lines are our present results. Experimental data for pure
argon: +, Lakshminarasimha and Lucas [21]; ×, Townsend [24].
of E/N , the transient behaviour of a swarm parameter in
a gas mixture will be different if the initial conditions are
different. However, the final, steady-state values of various
swarm parameters will be the same, independent of the initial
conditions. As an example, figure 8 shows the time dependence
of the drift velocity and the average energy of an electron
swarm in pure methane for E/N = 5 Td. Three different
electron swarms, each with an initial Maxwellian velocity
distribution at t = 0, with average electron energies of 0.4, 1
and 5 eV are considered. It is evident from this figure that the
1583
A A Sebastian and J M Wadehra
overshoot varies with the concentration of methane in the gas
mixture, value of the applied external electric field (E/N ) and
the initial average energy of the EVDF.
The approach to steady state of all swarm parameters from
their initial Maxwellian values at t = 0 to their final values
occurs faster as the applied electric field (or E/N ) gets larger.
In order to quantify this degree of equilibration, we define the
equilibration time (τ ) of a swarm parameter as the time when
the swarm parameter reaches 99% of its final, steady-state
value. For a given value of E/N and for a fixed initial EVDF,
the value of τ for a swarm parameter varies continuously as the
concentrations of the components of a gas mixture are varied
systematically. In figure 9 we show the equilibration time, τ ,
for the drift velocity as a function of methane percentage in an
argon–methane mixture for a reduced electric field of 5 Td and
an initial average electron energy of 5 eV. We also fitted the
data to a second-order exponential decay function of the form
−c
−c
+ α2 exp
,
τ = τ0 + α1 exp
c1
c2
10
1
100% Pure CH4
Characteristic Energy (eV)
0.1
0.01
10
1
75% CH4 in Ar
0.1
0.01
10
1
50% CH4 in Ar
0.1
0.01
0.1
1
10
100
1000
E/N (Townsends)
Figure 7. Electron swarm characteristic energy as a function of
E/N in 50% and 75% mixtures of methane with argon and in 100%
pure methane. Solid lines are our present results. Experimental data
for pure methane: +, Lakshminarasimha and Lucas [21]; , Hunter
et al [15]; ◦, Millican and Walker [25].
transient behaviour of electron swarms is very different, even
though the steady-state values (10.1 cm µs−1 and 0.633 eV)
are not affected by the average energy of the initial velocity
distribution function. In particular, the drift velocity exhibits
an overshoot if the initial average energy of the electron swarm
is less than the final, steady-state value. The amount of this
where τ represents the equilibration time in microseconds and
c is the methane concentration in per cent. The remaining
coefficients are: τ0 = 0.137 µs, α1 = 1.91 µs, α2 = 0.745 µs,
c1 = 2.99 and c2 = 28.6. From this figure, it can be
seen that, among all possible mixtures of methane and argon,
equilibration is fastest for pure methane and slowest in pure
argon. This can be understood by considering the energy
loss modes available to the electron swarm in the mixture.
As the methane gas concentration is increased, the scattering
rate for vibrational energy loss channels increases, giving rise
to a larger collision term in the Boltzmann equation. This
results in a more rapid approach to equilibrium. It should
also be noted that the results shown in the figure are based
on an arbitrary value chosen for the degree of equilibration,
99% in the present case. The degree of equilibration can
be taken to be some other value close to final equilibration,
and this would have the effect of adjusting up or down the
overall curve while not changing the decay constants, c1 and c2 ,
Table 3. Equilibrium values of the characteristic energy of electrons, in eV, in various mixtures of methane in argon.
% Methane
E/N (Td)
0.1
1
5
10
20
30
50
100
200
300
400
500
600
700
800
900
1000
a
1584
0
a
1.11
3.57
7.07
7.03
7.22
7.50
7.96
8.65
9.27
9.64
9.98
1.03(1)
1.08(1)
1.12(1)
1.18(1)
1.24(1)
1.30(1)
1
2
5
10
25
50
75
100
1.67(−1)
1.56
5.01
6.63
7.21
7.47
7.81
8.24
8.75
9.19
9.64
1.01(1)
1.06(1)
1.12(1)
1.19(1)
1.26(1)
1.33(1)
1.18(−1)
1.14
4.41
6.28
7.23
7.47
7.79
8.21
8.71
9.14
9.58
1.01(1)
1.06(1)
1.12(1)
1.18(1)
1.25(1)
1.33(1)
7.82(−2)
6.90(−1)
3.67
4.96
6.81
7.33
7.72
8.12
8.59
9.02
9.47
9.94
1.05(1)
1.11(1)
1.17(1)
1.24(1)
1.31(1)
5.62(−2)
4.66(−1)
2.36
4.01
5.93
6.91
7.55
7.96
8.41
8.83
9.28
9.76
1.03(1)
1.09(1)
1.15(1)
1.21(1)
1.28(1)
4.04(−2)
2.47(−1)
1.45
2.77
4.56
5.70
6.90
7.59
7.99
8.38
8.81
9.28
9.78
1.03(1)
1.09(1)
1.15(1)
1.22(1)
3.34(−2)
1.43(−1)
9.20(−1)
1.96
3.58
4.68
6.03
7.11
7.50
7.85
8.25
8.69
9.17
9.68
1.02(1)
1.08(1)
1.14(1)
3.06(−2)
1.02(−1)
6.70(−1)
1.53
3.02
4.09
5.52
6.81
7.18
7.48
7.85
8.27
8.72
9.19
9.70
1.02(1)
1.08(1)
2.89(−2)
8.14(−2)
5.21(−1)
1.25
2.60
3.67
5.14
6.63
6.97
7.21
7.54
7.93
8.35
8.81
9.29
9.78
1.03(1)
The notation a(b) means a × 10b .
Drift Velocity (cm/µs) Average Energy (eV)
Behaviour of electron transport in methane–argon mixtures
5
Initial Energy
4
5.0 eV
1.0 eV
0.4 eV
3
2
1
16
14
12
10
8
6
4
2
0
0.00
0.05
0.10
0.15
0.20
Time (µs)
Figure 8. Time dependence of the average energy and the drift velocity of an electron swarm in 100% pure methane for E/N = 5 Td,
starting from initial Maxwellian velocity distribution functions with average electron energies of 0.4 eV, 1.0 eV and 5.0 eV.
Equilibration Time (µs)
3
2
1
0
0
10
20
30
40
50
60
70
80
90
100
Per cent CH4 in Ar
Figure 9. Equilibration time versus methane concentration: •, equilibration time for the drift velocity of electrons in a swarm with an initial
average electron energy of 5 eV for E/N = 5 Td. ——, a second-order exponential fit to the calculated equilibration times.
describing the shape of the curve. As an example, Shizgal and
McMahon [6] have calculated relaxation times for electrons
in pure argon, with E/N ranging from 0 up to 0.1 Td, for the
swarm parameters of average energy, mobility and diffusion
coefficient. Their criterion for τ is the time for relaxation of
a swarm parameter to 1/1.01 or 1/1.1 of the final steady-state
value. For E/N = 0.1 Td, using the criterion of 1/1.1, they
report calculated values of Nτ , in units of 1011 s cm−3 , for
mobility and diffusion to be 23.76 and 10.36, respectively. For
comparison, our calculated values of Nτ , in the same units, for
the same swarm parameters are 28.6 and 12.0, respectively. It
should be noted that the choice of electron–argon collision
cross sections and the initial EVDF in our work and in [6]
are different. Varying the initial energy of the electron swarm
will also affect the approach to steady state, especially since
the choice of initial average electron energy will affect the
amount of drift velocity overshoot that occurs during the time
development of the drift velocity. Since this overshoot depends
both on the initial energy and the methane percentage, the
detailed form of the equilibration time, as a function of methane
concentration, will depend on the choice of initial electron
energy.
1585
A A Sebastian and J M Wadehra
this surface plot. This is indicated by the dashed line drawn
on the contour plot of figure 10. One can also mark the
local maximum of electron drift velocity as a function of the
percentage of methane in argon. The locus of such points
is represented on the contour plot as a dotted-dashed line.
Finally, one can follow and mark the maximum of the electron
drift velocity as a function of two independent variables, E/N
and the percentage of methane in argon. The resulting path,
corresponding to the ridgeline on the three-dimensional surface
plot, is indicated on both the surface and the contour plots as a
solid line. It is interesting to observe that this line, representing
the local maxima in two dimensions, coincides with the dotteddashed line, representing the maxima of drift velocity as a
function of only the methane percentage, at low E/N (or lower
percentage of methane in argon), and then makes a transition
to coincide with the dashed line, representing the maxima of
the drift velocity as a function of E/N only, at high E/N (or
higher percentage of methane in argon).
5. Conclusions
Figure 10. Three-dimensional surface plot and a two-dimensional
contour plot of electron drift velocity versus both E/N and methane
concentration in argon.
4.5. Surface plot
In figures 2–7 and in tables 1–3 we have presented numerical
values of various electron swarm parameters only for some
selected values of E/N and for a few concentrations of
methane in argon. Our computer program, on the other hand,
is capable of calculating swarm parameters for any value,
small or large, of E/N and for any arbitrary concentration
of methane in argon. In order to show a complete picture of
our results, we have collected all of our drift velocity values
for the various percentages of methane in argon and for the
range of E/N values from 0.1 to 1000 Td in a single threedimensional surface plot and in the corresponding contour plot
in figure 10. Similar surface plots for other electron swarm
parameters are also possible but are not shown here. In order
to present the data in figure 10, some amount of interpolation
of the calculated values was performed to allow the plotting
program to smoothly represent the data. It was necessary to
make many runs of our code to accurately represent the shape
of the drift velocity surface for values of E/N below 1 Td
and for percentages of methane in argon below 5%. In this
region the drift velocity shows a significant drop in value. It
is also interesting to mark the location of the local maximum
of drift velocity as a function of the reduced electric field,
E/N (which indicates the beginning of the NDC region), on
1586
In this paper, we have investigated the time-dependent
behaviour of electron swarms in mixtures of methane in
argon that are subjected to an external static electric field E.
The domain of the values of the electric field considered
here is extensive with E/N values ranging from 0.1 to
1000 Td. The concentration of methane in the mixture is varied
systematically from 0% (pure argon) to 100% (pure methane).
For each mixture, complete time dependence of the velocity
distribution of an electron swarm from Maxwellian (at t = 0)
to steady state (t → ∞) is investigated. The EVDF is used to
obtain information about the time evolution of various electron
swarm parameters in these gas mixtures. The final steadystate values of the swarm parameters show good agreement
with the corresponding experimental results wherever such a
comparison can be made.
The present method has several advantages over
the Legendre expansion method. First, the present procedure
works for all values, both high as well as low, of E/N . Second,
in the present algorithm there is no need to evaluate any
derivatives numerically. This feature of the algorithm frees it
from the instabilities normally associated with the numerical
evaluation of the derivatives. Third, for numerical stability
of the solution of a typical partial differential equation the
time step t has to be restricted. This restricting condition
is the Courant condition, namely, at v , where a is
as defined in equation (1). In the present algorithm, since
no numerical derivatives are required, there is no need to
impose this condition. However, the time step t is still
restricted by the frequency of collisions of swarm particles
with the background gas. Finally, because of the absence of
any numerical derivatives the present algorithm is quite fast
and computationally inexpensive.
Acknowledgment
Support of this work by US Air Force Office of Scientific
Research is gratefully acknowledged.
Behaviour of electron transport in methane–argon mixtures
References
[1] Christophorou L G 1984 Electron–Molecule Interactions and
their Applications vol 2 (Orlando, Fl: Academic)
[2] Drallos P J and Wadehra J M 1988 J. Appl. Phys. 63 5601–3
[3] Drallos P J and Wadehra J M 1989 Phys. Rev. A 40 1967–75
[4] Palladino V and Sadoulet B 1975 Nucl. Instrum. Methods 128
323–35
[5] Biagi S F 1989 Nucl. Instrum. Methods A 283 716–22
[6] Demeio L and Shizgal B 1993 J. Chem. Phys. 99 7638–51
Shizgal B and McMahon D R A 1985 Phys. Rev. A 32 3669–80
[7] Krajcar-Bronic I and Kimura M 1995 J. Chem. Phys. 103
7104–13
[8] Nakamura S, Ventzek P L G and Kitamori K 1999 J. Appl.
Phys. 85 2534–9
[9] Bzenic S, Petrovic Z L, Raspopovic Z M and Makabe T 1999
Japan. J. Appl. Phys. 38 6077–83
[10] de Urquijo J, Alvarez I, Basurto E and Cisneros C 1999
J. Phys. D: Appl. Phys. 32 1646–9
[11] Dutton J 1975 J. Phys. Chem. Ref. Data 4 577–856
[12] Nakamura Y and Kurachi M 1988 J. Phys. D: Appl. Phys. 21
718–23
[13] Davies D K, Kline L E and Bies W E 1989 J. Appl. Phys. 65
3311–23
[14] Hunter S R, Carter J G and Christophorou L G 1986 J. Appl.
Phys. 60 24–35
[15] Hunter S R, Carter J G and Christophorou L G 1985 J. Appl.
Phys. 58 3001–15
[16] Foreman L, Kleban P, Schmidt L D and Davis H T 1981 Phys.
Rev. A 23 1553–7
[17] Wong L, Armitage J and Waterhouse J 1988 Nucl. Instrum.
Methods A 273 476–84
[18] El-Hakeem N and Mathieson E 1979 Nucl. Instrum. Methods
166 447–50
[19] Jean-Marie B, Lepeltier V and L’Hote D 1979 Nucl. Instrum.
Methods 159 213–19
[20] Armitage J C, Beingessner S P, Carnegie R K, Ritchie E F and
Waterhouse J 1988 Nucl. Instrum. Methods A 271 588–96
[21] Lakshminarasimha C S and Lucas J 1977 J. Phys. D: Appl.
Phys. 10 313–21
[22] Specht L T, Lawton S A and DeTemple T A 1980 J. Appl.
Phys. 51 166–70
[23] Abdulla R R, Dutton J and Williams A W 1981 Proc. 15th
ICPIG (Minsk, USSR) pp 367–8; taken from figure 1 of
Ferreira C M and Loureiro J 1983 J. Phys. D: Appl. Phys.
16 1611–21
[24] Townsend J S 1948 Motion of Slow Electrons in Gases
(London: Hutchinson)
[25] Millican P G and Walker I C 1987 J. Phys. D: Appl. Phys. 20
193–6
[26] Holstein T 1946 Phys. Rev. 70 367–84
[27] Huang K 1963 Statistical Mechanics (New York: Wiley)
pp 57–9
[28] Boltzmann L 1872 Wien. Ber. 66 275–370
Brush S G 1965 Kinetic Theory vol 2 (Oxford: Pergamon)
pp 88–175 (Engl. Transl.)
[29] Huxley L G and Crompton R W 1974 The Diffusion and Drift
of Electrons in Gases (New York: Wiley)
[30] Shirai T, Tabata T, Tawara H and Itikawa Y 2002 At. Data
Nucl. Data Tables 80 147–204
[31] Tawara H, Itikawa Y, Nishimura H, Tanaka H and Nakamura Y
1990 Collision Data Involving Hydrocarbon Molecules:
NIFS-DATA-6 Report (Nagoya, Japan) p 22
[32] Buckman S J and Brunger M J 1997 Aust. J. Phys. 50
483–509
[33] Frost L S and Phelps A V 1964 Phys. Rev. 136
A1538–45
[34] Sakai Y, Tagashira H and Sakamoto S 1977 J. Phys. D: Appl.
Phys. 10 1035–49
[35] Rapp D and Englander-Golden P 1965 J. Chem. Phys. 43
1464–79
[36] Petrović Z Lj, Crompton R W and Haddad G N 1984
Aust. J. Phys. 37 23–34
[37] Robson R E 1984 Aust. J. Phys. 37 35–44
[38] Wang Y, Christophorou L G, Olthoff J K and Verbrugge J K
1999 Chem. Phys. Lett. 304 303–8
1587