Journal of Physics D: Applied Physics
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Calculation of ion mobilities by means of the twotemperature displaced-distribution theory
To cite this article: P G C Almeida et al 2002 J. Phys. D: Appl. Phys. 35 1577
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INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 35 (2002) 1577–1584
PII: S0022-3727(02)35564-5
Calculation of ion mobilities by means of
the two-temperature
displaced-distribution theory
P G C Almeida1 , M S Benilov1 and G V Naidis2
1
Departamento de Fı́sica, Universidade da Madeira, Largo do Municı́pio, 9000 Funchal,
Portugal
2
Institute for High Temperatures of the Russian Academy of Sciences, Izhorskaya 13/19,
Moscow 127412, Russia
Received 5 April 2002
Published 18 June 2002
Online at stacks.iop.org/JPhysD/35/1577
Abstract
Electric-field dependent ion mobility is calculated in the framework of the
two-temperature displaced-distribution theory. The considered approach
provides a better accuracy than the conventional approach based on (the first
approximation in) the two-temperature theory and is better justified for the
case of heavy ions, while being not much more complicated than the
conventional approach. Calculated ion mobilities are in good agreement
with experimental data.
1. Introduction
Motion of ions in an applied electric field in a uniform weakly
ionized gas is characterized by the ion mobility, which, in a
general case, depends on the electric field strength. If the
gas in question is hot, in most cases there is no experimental
information in the literature and one must resort to the
kinetic theory in order to evaluate the mobility in terms of a
potential of the ion–neutral particle interaction (which in many
situations is known or may be estimated relatively easily) or
in terms of experimental data available for the same gas under
different conditions (measurements of mobilities at the room
temperature are available for many gases).
The simplest tool of the kinetic theory available in the
literature for a rapid evaluation of the mobility of ions with
account of a dependence on the local electric field is the
first approximation in the framework of the so-called twotemperature theory [1]. Such an approach is based on the
assumption that the ion distribution function is close to the
Maxwellian one, with a temperature that may be different from
the temperature of neutral particles (atoms). This assumption
implies, in particular, that the drift velocity of the ions is much
smaller than their thermal velocity. The latter assumption holds
when the mass of ions, mi , is much smaller than the mass of
atoms, ma , and is violated for mi /ma O(1).
In order to overcome this difficulty, one can assume that
the ion distribution function, being close to the Maxwellian
function with a temperature that may be different from the
0022-3727/02/131577+08$30.00
© 2002 IOP Publishing Ltd
atomic temperature, is displaced relative to the atoms, i.e.
characterized by a non-negligible mean velocity of the ions.
Such an approach may be termed the two-temperature
displaced-distribution theory. The next step is to introduce
in the (displaced) ion distribution function two different
temperatures, one in the direction along the electric field and
another in the transversal directions. The latter approach
is usually termed the three-temperature theory, and it is to
this approach that most of the attention has been paid in
the literature [1]. Obviously, the three-temperature theory is
more complex than the two-temperature displaced-distribution
theory. On the other hand, it is not quite clear whether
the introduction of different temperatures in the direction
along the field and in the transversal directions results in a
substantial enhancement of accuracy of the theory. It should
be emphasized in this connection that there are indications [1,2]
that an account of the velocity displacement plays a more
important role than the temperature anisotropy.
In [3], the two-temperature displaced-distribution description has appeared in a natural way as a part of the fluid description of ion motion with account of variable ion temperature
under conditions of non-uniform plasma, where gradients are
strong and the distribution function of the ions is far from
equilibrium with a local electric field. It should be emphasized that rates of exchange of momentum and energy between
the ion and atom fluids, calculated within this two-temperature
displaced-distribution description, take exact values in the case
when the mean ion velocity is much higher than their thermal
Printed in the UK
1577
P G C Almeida et al
velocity (i.e. the ions constitute a beam in the neutral gas).
This is essential for a correct description of ion motion both
in uniform plasmas in the case mi ma and in non-uniform
plasmas, where the ions may be accelerated up to velocities
much higher than their thermal speed.
It is of interest to apply this two-temperature displaceddistribution description to the calculation of ion mobilities in
spatially uniform plasma and to compare high-field mobilities
of ions calculated in such a way with other theoretical results
and with experimental data. It will give an idea of accuracy of
this approach as is used in the fluid description of ion motion
and may even result in a method of rapid evaluation of mobility
which, while being not much more complicated than the first
approximation in the two-temperature theory, would give more
accurate mobility values and would be better justified for the
case mi /ma O(1).
2. Theory
2.1. The model
Consider steady-state motion of ions in a uniform neutral gas
in a uniform electric field E. Equations of conservation of
momentum and energy of the ion fluid may be written in
the form
eni v i · E = ria(e) ,
eni E = r(m)
ia ,
(1)
where ni and vi are the number density and the mean (drift)
(e)
velocity of ions and r(m)
ia , ria are the rates of loss per unit
volume of, respectively, momentum and energy of the ions
due to (elastic) collisions with neutral particles.
(e)
The rates r(m)
ia and ria are expressed in terms of distribution
functions of neutral particles and ions and of the crosssection for momentum transfer between the ions and the
neutral particles. If the ionization degree is low enough,
the distribution of neutral particles is not disturbed by the
presence of ions and is described by the Maxwellian function.
The distribution of ions will be approximated by a shifted
Maxwellian function
3/2
mi (ci − vi )2
mi
fi(0) = ni
,
(2)
exp −
2kTi
2πkTi
where Ti is the temperature of the ion species. Then, one can
use results [3] and arrive at the following expressions:
r(m)
ia =
ria(e)
eni
vi ,
µi
(3)
eni
[3χk(Ti − T ) + mi vi2 ],
=
µi (mi + ma )
(4)
m i T + m a Ti
,
mi + m a
(5)
where
mia =
m i ma
,
mi + m a
µi =
η=
1578
Tia
Ti
e 3π
kTi 16
1/2
Tia =
Q(m)
ia
2kTi
πmia
,
(1,1)
Q̄ia
1/2
1
nQ̄(1,1)
ia η
χ=
Q(e)
ia
Q(m)
ia
,
2
− M 2,
3
(6)
(7)
Q̄(1,1)
ia
Q(m)
ia
Q(e)
ia
×
M=
=e
=
−M 2
∞
5 −x
x e
0
2
Q(1)
ia
∞
5 −x 2
x e
0
Q(1)
ia
2kTi
x dx,
mia
2kTia
x
mia
F (m)(2Mx)
dx,
F (e)(2Mx)
mia
vi ,
2kTia
(9)
F (m) (y) = 3
F
(e)
(8)
y cosh y − sinh y
,
y3
sinh y
.
(y) =
y
(10)
Here, T and n are the temperature and number density of
(1)
neutral particles and Q(1)
ia = Qia (g) is the velocity-dependent
cross-section for momentum transfer in collisions of ions
with neutral particles. Note that mia is the reduced mass,
is
quantity Tia may be called the reduced temperature, Q̄(1,1)
ia
the conventional average cross section for momentum transfer
evaluated at the temperature Ti , µi is the mobility of the
ions in the gas of neutral particles, η and χ are dimensionless
coefficients expressed in terms of the weighted cross-sections
(e)
Q(m)
ia and Qia . M may be termed the Mach number of the ion
motion; note that it can be written also in the form
M=
vi
,
(11)
u2a + u2i
√
√
where ui = 2kTi /mi and ua = 2kT /ma are characteristic
thermal velocities of the ions and the atoms.
Using equations (3) and (4), one can rewrite equation (1) as
vi = µi E,
3
1 ma vi2
3
.
kTi = kT +
χ 2
2
2
(12)
The first equation in equation (12) is the conventional
expression for the drift velocity. The second equation relates
the ion temperature to the temperature of neutral particles
and the ion drift velocity and represents an analog for
the considered model of the well-known Wannier equation;
note that the classical Wannier equation can be obtained from
the second equation in equation (12) by setting χ = 1. It
may be shown also that the ratio of the ion temperature to the
reduced temperature is given by the equation
2M 2
Ti
.
=1+
Tia
3χ
(13)
Now the model is complete. In order to determine the ion
drift velocity and temperature as functions of E/n and T , one
needs to solve equations in equation (12) for vi and Ti .
It should be emphasized that the form of the expressions
for the ion mobility and temperature, equation (6) and the
second equation in equation (12), has been chosen in such
a way that the structure of these equations is identical,
to the accuracy of coefficients η and χ, to that of the
respective expressions in the first approximation in the twotemperature theory (e.g. [1, equations (6-2-12) and (6-2-13)]).
(Note that what appears in place of Ti in the formulae of
Calculation of ion mobilities
2.2. Properties of coefficients η and χ
3
8
(1)
3πχ Qia (vi )
+ ···,
2 Q̄(1,1)
ia
χ=
γ1 + 4
+ ···,
3
(14)
where γ1 is a coefficient given by equation (39) of appendix.
For the particular case of a power-like dependence of the
cross-section for momentum transfer on the relative velocity,
one finds
6+n
C 2kTi n/2
,
γ1 = n. (15)
=
Q̄(1,1)
ia
2
2 mia
Thus, equation (14) in this case assumes the form
η=
3
4
π(n + 4)
2
n+4
2
n/2
χ
1.3
1.2
1.1
1
η
0.9
Coefficients η and χ, as all other quantities characterizing ion
transport, are governed by E/n and T . From the point of
view of calculations, however, it is possible and convenient
to consider formally these coefficients as functions of M and
may be
Tia . (Note that Ti in the definitions of η and of Q̄(1,1)
ia
expressed in terms of Tia by means of equation (13).) In the
particular case of a power-like dependence of the cross-section
for momentum transfer on the relative velocity, Q(1)
ia (g) =
Cg n , where C and n are constant quantities, these coefficients
are independent of Tia and depend only on M.
In the limiting case M → 0, one finds η → 1, χ → 1,
which follows from the fact that F (m) (0) = F (e) (0) = 1.
A second approximation asymptotic behaviour of the cross
(e)
sections Q(m)
ia and Qia in the opposite limiting case M → ∞
is found in appendix. (It should be emphasized that the
and Q(e)
asymptotic behaviour of Q(m)
ia
ia must be found
to a second approximation in order to determine a first
approximation asymptotic behaviour of the coefficient χ.)
Using equation (40) of appendix, one finds the following
asymptotic behaviour in the limit M → ∞:
η=
1.4
χ, η
the first approximation in the two-temperature theory is the
temperature Teff defined by an equation formally similar to
the above definition of Tia . However, if one removes T
from the definition of Teff by means of the (classical) Wannier
equation and takes into account that the temperature Tb [1],
being related to the total mean ion energy, is different from the
ion temperature Ti of the present work, then Teff will become
equal to Ti ). In other words, if coefficients η and χ are set equal
to unity, then the formulae of the present model will coincide
with the respective formulae of the first approximation in the
two-temperature theory.
1
+ ···,
((6 + n)/2)
(16)
n+4
χ=
+ ···.
3
One can see that in the particular case of a power-like
dependence of the cross-section for momentum transfer on the
relative velocity coefficients η and χ tend in the limit M → ∞
to finite values.
For an arbitrary cross-section for momentum transfer,
coefficients η and χ must be calculated by means of numerical
integration. For some model cross-sections, they may also
be found analytically [3]. For completeness, we present
here expressions for the cases when ions interact with neutral
0.8
0
2
4
6
8
10
M
Figure 1. Coefficients χ and η for the model of rigid spheres.
- - - -: limit values at large M.
particles as Maxwell molecules or rigid spheres. In accord
to the above, the coefficients η and χ are independent of Tia
in these cases and may depend only on M. For Maxwell
molecules (Q(1)
ia (g) = C/g, the model of constant collision
frequency), one gets
η = 1,
χ = 1.
(17)
As it should have been expected, this equation conforms to the
limit values given by equation (16) for n = −1.
For the model of rigid spheres (Q(1)
ia (g) = C, the model
of constant mean free path), one gets
1/2 √ 1
π
3
3χ
2
erf M
1+M −
η=
4M 2
M
8 3χ + 2M 2
1
2
+ 1+
e−M ,
(18)
2M 2
√
2
8M 2 [ π (1 + 2M 2 )erf M + 2Me−M ]
χ= √
.
3[ π (4M 4 + 4M 2 − 1)erf M + 2M(2M 2 + 1)e−M 2 ]
(19)
Graphs of these functions are shown in figure 1. Also shown
are limit √
values given by equation (16) for n = 0, which
are η = 9π/32 ≈ 0.94 and χ = 4/3 (as it should have
been expected, these values coincide with those given by
equations (16) and (19) in the limiting case M → ∞). One can
see that the coefficient η is slightly non-monotonous. While
this coefficient is (for the model of rigid spheres) not very
different from unity, coefficient χ may exceed unity by more
than 30%.
2.3. Limiting cases of low and high electric fields
If the electric field is low enough, the ion temperature is
close to the atomic temperature, Ti − T T . Using the
second equation in equation (12), one can see that the last
inequality is equivalent to vi ua , meaning that the lowfield limit corresponds to the ion drift velocity being much
smaller than thermal velocities of neutral particles, in accord to
the criterion of the low-field behaviour [1, equation (5-2-28)].
1579
P G C Almeida et al
It follows from equation (11) that M 1 in the limiting
case considered. According to the above, η → 1, χ → 1
in this limiting case and the value of the ion mobility given by
equation (6) coincides with the one evaluated by means of the
first approximation in expansion in the Sonine polynomials in
the method of Chapman–Enskog at the gas temperature.
In the opposite limiting case of high electric field, Ti T ,
or, which is equivalent, vi ua , it follows from the second
equation in equation (12) and from equation (11) that
M2 =
3χ mi
.
2 ma
(20)
(Note that in the case of heavy ions, mi ma , this equation
applies in the range Ti /T mi /ma .) One can conclude that
the Mach√number in the limiting case of high field is of the
order of mi /ma , i.e. M 1 if mi ma , M = O(1) if
mi /ma = O(1), and M 1 if mi ma .
If χ is independent of Tia and may depend only on M,
which is the case for a power-like dependence of the crosssection for momentum transfer on the relative velocity,
equation (20) may be considered as a closed equation for M.
It follows that the Mach number which can be attained by ions
in such a case is given by the root of equation (20).
In the case of intermediate electric fields, Ti − T =
O(T ), or, which is equivalent, vi = O(ua ), one finds from
equation (11) that M 1 if mi ma and M = O(1) if
mi /ma O(1).
2.4. Approximations of the model
An approximate character of the considered model stems from
the fact that the ion distribution function is replaced by the
shifted Maxwellian function (2) while calculating the rates
of momentum and energy exchange between ions and neutral
particles. The model should provide exact results in particular
cases when a difference between the exact distribution and the
shifted Maxwellian function does not affect the rates. Two
such cases can be indicated. First, this is a case of heavy ions,
mi ma , in high electric fields. In such a case vi ui ;
the ion swarm is close to a monoenergetic particle beam
and all collisions occur at approximately the same relative
velocity vi . The difference between the exact distribution
and the shifted Maxwellian function is insignificant (both
distributions are close to the delta-function), hence the model
should give exact values for the rates of momentum and
energy exchange. Indeed, employing the first equation in
equation (14) (note that M 1 in the case considered), one
arrives at the exact value of the ion mobility for the case
considered, µi = e/mia nvi Q(1)
ia (vi ). Note that the value of
the ion temperature given by the model in this particular case
need not be exact: on the one hand, it does not affect the ion
mobility; on the other hand, it is governed by the first term in
square brackets on the right-hand side of equation (4), which
is negligible relative to the second term and is not calculated
quite correctly.
Another case when the difference between the exact
distribution and the shifted Maxwellian function does not affect
the rates of momentum and energy exchange between ions and
neutral particles is the case of Maxwell molecules. A form of
the distribution function produces no effect whatsoever on the
1580
rates in this case. Indeed, substituting η = 1 into equation (6),
one arrives at the exact expression for the ion mobility for
the model of Maxwell molecules, µi = e/mia ngQ(1)
ia (g); the
classical Wannier equation, to which the second equation in
equation (12) is reduced with χ = 1, is exact for the model
of Maxwell molecules regardless of the distribution function
provided that 23 kTi is understood as the mean kinetic energy of
the ion chaotic motion [1, p 153–4].
Apart from the case of heavy ions in high electric fields,
another particular case exists in which the difference between
the exact distribution and the shifted Maxwellian function (2)
is small. This is the case of low electric field, in which both
the exact distribution and the shifted Maxwellian function are
close to the non-shifted Maxwellian function with the atomic
temperature. (Strictly speaking, this does not apply to heavy
ions since the inequality vi ua does not guarantee validity of
the inequality vi ui in this situation, however, both the exact
distribution and the shifted Maxwellian function are close to a
non-shifted delta-function in such a situation.) However, the
rate of momentum exchange in this case is governed just by
an anisotropic part of the distribution. Hence, the difference
between the exact distribution and the shifted Maxwellian
function, in spite of being infinitesimal in this particular case,
can produce a finite effect on the ion mobility. This is why the
model gives in this case the above-mentioned (approximate)
value coinciding with the first approximation in the method of
Chapman–Enskog rather than an exact value. However, the
difference between the two values is very small in most cases,
so the accuracy of the model in the limiting case of low field
is quite high.
3. Comparison with other theoretical results
Consider first the case of light ions, when M 1 regardless of
the electric field. The shift in the ion distribution function (2)
is inessential and the model considered should be equivalent to
the first approximation in the two-temperature theory. Since
η = χ = 1 at M → 0, this is indeed the case. Note
that µi in this case coincides with the value of ion mobility
evaluated by means of the first approximation in expansion in
the Sonine polynomials in the method of Chapman–Enskog at
the temperature Ti .
In [4], the high-field mobility was calculated by a moment
method for the model of rigid spheres. In accord to the above,
coefficient χ is independent of Tia for this model and is a
function of M only, hence equation (20) can be numerically
solved for M at different values of mi /ma . The solution is
illustrated by figure 2, in which the (normalized) root of this
equation is shown. The dashed lines represent the one-term
asymptotic behaviour at small and large mi /ma , which reads
mi
3 mi
M=
,
M= 2 ,
(21)
ma
2 ma
respectively.
The reduced high-field ion drift velocity for the model
of rigid spheres is given in the framework of the approach
considered by the expression
vi
f x 1/4
.
=
1/2
(1 − x)1/2
(aλ)
(22)
Calculation of ion mobilities
1/nQ(1)
ia
Here a = eE/mi is the ion acceleration, λ =
is
the mean free path between collisions of an ion with neutrals,
x = mi /(mi + ma ), and f is a function of x defined by the
equation
27πχ 1/4
f =
,
(23)
128η2
where coefficients η and χ are evaluated in terms of the highfield Mach number which is governed by equation (20) and
shown in figure 2.
In figure 3, the reduced high-field velocity calculated by
means of equation (22) is compared with accurate results
from [4] for various values of mi /ma . Also shown are the
data corresponding to the first approximation in the twotemperature theory, which are given by equation (22) with
χ = η = 1. One can see that in the case of heavy ions, the
accuracy of the present approach is better than the accuracy of
1.44
results obtained in the framework of the first approximation in
the two-temperature theory.
Note that function f = f (x) may be evaluated, to the
accuracy within 0.7%, by means of the explicit formula
f =
A=
27π
,
128
M/(mi/ma)1/2
1.28
1.24
1.2
0.1
1
mi/ma
10
Figure 2. High-field value of√the Mach number for the model of
rigid spheres normalized by mi /ma . - - - -: asymptotic behaviour
for light and heavy ions.
,
(24)
(2A − 1.7)A
,
1−A
2.3A − 2
D=
.
1−A
B=
C = A − 1,
(25)
This formula has been constructed in such a way as to give
correct values of the function f 2 and of its derivatives in the
particular cases of infinitely light ions, x = 0, and of infinitely
heavy ions, x = 1:
27π 1/2
f 2 (0) =
(26)
,
f 2 (1) = 1,
128
d(f 2 )
3
(0) =
dx
10
1.32
1/2
where
1.4
1.36
A + Bx + Cx 2
1 + Dx
27π
128
1/2
,
d(f 2 )
(1) = 0.
dx
(27)
Note that the values of the derivatives are obtained by
expanding equations (18) and (19) at M → 0 and M → ∞
and making use of equation (21).
A model of ion–atom interaction that involves inelastic
collision process is considered in [1, p 360], the total
momentum transfer cross-section being the sum of elastic
and inelastic collision cross-sections. The cross-section of
elastic process is assumed to be constant (independent on
the ion energy), while the cross-section of inelastic process
is taken equal to the same constant for relative velocities
higher than some threshold value vth and equal to zero for
velocities lower than vth . Data on the reduced high-field
ion drift velocity calculated by means of various methods
are given for different values of the reduced field strength
2
aλ/vth
at mi = ma . In figure 4, the data from [1, p 360]
are compared with results given by the present approach. One
can see that the present results are close to those given by
4
deviation (%)
0
–4
–8
–12
0
0.2
0.4
0.6
x = mi/(mi+ma)
0.8
1
Figure 3. Deviation of the high-field ion mobility for the model of
rigid spheres from results [4]. : first approximation in the
two-temperature theory. +: present work.
Figure 4. Reduced ion drift velocity vs reduced field strength.
——: first and fifth approximation in the two-temperature theory.
- - - -: present work. Points: Monte Carlo simulation.
1581
P G C Almeida et al
the Monte Carlo simulations and by the fifth approximation
in the two-temperature theory, and are more accurate than the
ones given by the first approximation in the two-temperature
theory. Note that these calculations have been performed for
mi = ma ; it should be emphasized that with an increase
of the ratio mi /ma the present two-temperature displaceddistribution approach will become still more accurate, while
accuracy of the first approximation in the two-temperature
theory will deteriorate.
It is also of interest to compare the ion energy in high
electric fields calculated by means of the present method with
results [4] obtained by means of a moment method on the
basis of the three-temperature theory, and with those provided
by the first approximation in the two-temperature theory. The
average chaotic ion energy is given in the framework of the
three-temperature theory by the formula [5]
kT
(28)
+ kT⊥ ,
2
where T and T⊥ are the ion temperatures in the direction along
the electric field and in a transversal direction, respectively. In
the present model, ε = 23 kTi and one finds in the high-field
limit
ε
1
=
.
(29)
2χ
ma vi2
In the first approximation in the two-temperature theory,
one finds in the high-field limit
ε =
1
ε
= .
2
2
ma v i
(30)
A comparison of results of the present approach with
results [4] and with those given by the first approximation in the
two-temperature theory is shown in figure 5 for the model of
rigid spheres for different ion–atom mass ratios. One can see
that both the present approach and the first approximation in
the two-temperature theory provide a relatively good accuracy
for light ions. For heavy ions, the accuracy of the first
approximation in the two-temperature theory is about 50%,
while the accuracy of the present approach is better than 15%.
4. Comparison with experimental data
In the case of ions moving in parent gases, ion mobility and
diffusion are controlled by charge exchange. The total crosssection of charge exchange may be accurately represented, for
a wide range of energies, by an expression of the form [1]
Qex = (a − b ln )2
(31)
where a and b are constants and is the relative energy of
colliding particles. The energy-dependent momentum transfer
cross-section Q(1)
ia is related to Qex as [1]
Q(1)
ia
2Qex .
(32)
Using these formulae, drift velocities have been calculated
for He+ in He, for Ne+ in Ne, for Ar+ in Ar, for Kr + in Kr, and for
Xe+ in Xe. The set of constants (a, b) in equation (31) used
in the calculations was (4.88, 0.299) for helium, (5.7, 0.37)
for neon, (7.0, 0.6) for argon, (8.2, 0.57) for krypton, and
(9.2, 0.6) for xenon, where a and b are in 10−10 m and is
in eV. Note that the data for He have been taken from [1], for
Ar from [6] and for Ne, Kr, and Xe from [7].
Results of calculations are compared with experimental
data in figures 6 and 7. Agreement between the theory and the
experiment is good.
Also shown in figure 6 are results for drift velocities in
argon obtained with the use of momentum transfer crosssections given in [9]. One can see that the effect of the variation
of the cross-section is not considerable.
5. Concluding remarks
There are different ways of application of the results
obtained. The most direct way consists in numerically solving
equation (12) for vi and Ti for given T and E/n. This can
be performed by iterations. An initial approximation may be
obtained with the use of the low-field value of the mobility
(which is given by equation (6) with η = 1 and Ti = T ).
100
H e +-He
v i (1 0 4 c m /s )
Deviation (%)
40
20
0
A r + -A r
77 K
1
0
0.2
0.4
0.6
x = mi/(mi+ma)
0.8
1
Figure 5. Deviation of the normalized high-field chaotic ion energy
for the model of rigid spheres from results [4]. ——: results [4].
: first approximation in the two-temperature theory.
+: present work.
1582
10
N e +-Ne
10
100
E/n (Td)
1000
Figure 6. Drift velocity of He+ in He (77 K and 300 K), Ne+ in Ne,
and Ar+ in Ar (both 300 K). ——: theoretical. Points:
experiment [5, 8]. - - - -: cross section from [9].
Calculation of ion mobilities
in the square brackets on the right-hand side of equation (34)
and η = 1 in the other terms, the equation obtained,
Kr+– Kr
10
1
=
vi2
vi(104 cm/s)
Xe+–Xe
1
100
1000
E/n (Td)
10000
Figure 7. Drift velocity of Kr+ in Kr and Xe + in Xe at 300 K.
——: theoretical. Points: experiment [5, 8].
The iterations converge rapidly in all the cases considered
in this paper, including at high fields (about 12 iterations
are required to achieve accuracy within 10−6 for E/n =
3 × 104 Td). A program that performs these calculations for
the above-mentioned ion–atom systems is available in [10].
In the framework of the first approximation in the twotemperature theory, an analytical formula for the ion drift
velocity for arbitrary electric fields may be obtained in the
model of rigid spheres. It is interesting to note that a similar
formula may be derived also in the framework of the approach
considered. The procedure is as follows. Writing equation (6)
for the model of rigid spheres, expressing from this equation Ti
in terms of ηvi /E, substituting the result in the second equation
in equation (12) and formally solving the obtained equation for
vi , one arrives at
1/2
aλ 2
3χkT
3χkT 2 27πχx
2
+
, (33)
−
vi =
2ma
128η2 1 − x
2ma
Equation (33) may be transformed to
1
=
vi2
η4
+
4
4vLF
η
ηHF
2
χHF 1
4
χ vHF
1/2
+
η2
,
2
2vLF
(34)
where vLF is the drift velocity evaluated in the low-field
approximation and given by the equation
vLF =
3π mi
16 kT
2kT
πmia
1/2
aλ,
1
1
+ 4
4
4vLF vHF
1/2
+
1
,
2
2vLF
(36)
will provide correct values in the liming cases of low and high
fields and may be considered as an interpolation formula for the
intermediate case. It should be emphasized that equation (36),
supplemented with equations (22), (24), and (35), represents a
closed explicit expression for the drift velocity, in contrast to,
e.g. equation (33), in which the right-hand side depends on vi
and Ti through coefficients χ and η.
Thus, equation (36), being an exact result for the model
of rigid spheres in the framework of the first approximation in
the two-temperature theory, represents a natural interpolation
formula in the framework of the present approach. Comparison
of this formula (supplemented with equations (22), (24), and
(35)) with the full (numerical) solution in the framework of
the present approach shows that its accuracy is better than one
per cent in the whole range of mass ratio and field strength
values.
The employed method, based on the two-temperature
displaced-distribution theory, allows one to rapidly evaluate
ion mobilities, including cases of a hot gas in a high
electric field, without being much more complicated than the
conventional approach based on (the first approximation in)
the two-temperature theory. It is better justified for the case
mi /ma O(1), in particular, it provides essentially better
results for the ion energy. Ion mobilities calculated in such a
way are in good agreement with experimental data and with
results of more elaborate theoretical approaches.
The fact that the application of the two-temperature
displaced-distribution description to calculation of ion
mobilities in spatially uniform plasmas has proved successful,
is important for applications of this approach in the fluid
description of ion motion under conditions of a non-uniform
plasma, where gradients are strong and the distribution
function of the ions is out of equilibrium with the local electric
field.
Acknowledgments
The work was performed within activities of the project Theory
and modelling of plasma-cathode interaction in high-pressure
arc discharges of Fundação para a Ciência e a Tecnologia.
(35)
vHF is the drift velocity evaluated in the high-field
approximation and given by equation (22), and ηHF and χHF
are values of coefficients η and χ evaluated in terms of the
high-field Mach number (i.e. the same values that are involved
in equation (22)).
The second term in the square brackets on the right-hand
side of equation (34) is dominating in the limiting case of
high fields, when coefficients η and χ tend to values ηHF
and χHF , respectively. The other terms become dominating
in the limiting case of low fields, when η and χ tend to unity.
Therefore, if one sets η = ηHF and χ = χHF in the second term
Appendix. Second approximation asymptotic
behaviour of weighted cross-sections at large Mach
numbers
We rewrite equation (9) as
∞
Q(m)
2kTia
−(x−M)2 (1)
ia
x
e
Qia
=
mia
Q(e)
0
ia


3x 2


−4Mx
−4Mx



[2Mx(1 + e
)−1+e
]
3
× 16M
dx,
4


 x (1 − e−4Mx )



4M
(37)
1583
P G C Almeida et al
At large M, dominating contributions to the integral on the
right-hand side is given by the vicinity of the point x = M.
Introducing new integration variable z = x − M, expanding
the integrand in 1/M and retaining three terms, one arrives at
∞
Q(m)
2
(1)
ia
e−z
=
Q
(v
)
i
ia
Q(e)
−M
ia


z
3M






+
3)
1
+
(γ
1




M
8










1
1
2
2
2
+ 3z γ1 + γ2 z + 3z −
+
·
·
·
dz,
×


2 M2











 M3 z
z2





1 + (γ1 + 4) + (4γ1 + γ2 + 6) 2 + · · · 
4
M
M
(38)
where
d ln Q(1)
ia (g)
(vi ),
d(ln g)
(1)
1 g 2 d2 Qia (g)
γ2 =
2 Q(1)
dg 2
ia (g)
γ1 =
(39)
.
g=vi
Replacing (to exponential accuracy) the lower integration
limit in equation (38) by −∞ and evaluating the integral,
1584
one obtains

 3M
3 3γ1 + γ2 + 2


+
+
·
·
·


√
8
16
M
Q(1)
= π
3
ia (vi ).
+
γ
+
6
M
4γ


1
2


M + ···
+
8
4
(40)
Note that the first term of the expansion on the right-hand side
of this expression describes momentum and energy exchange
between two monoenergetic particle beams and was derived
in [3].
Q(m)
ia
Q(e)
ia
References
[1] Mason E A and McDaniel E W 1988 Transport Properties of
Ions in Gases (New York: Wiley)
[2] Kumar K, Skullerud H R and Robson R E 1980 Aust. J. Phys.
33 343
[3] Benilov M S 1997 Phys. Plasmas 4 521
[4] Skullerud H R 1976 J. Phys. B: Atom. Molec. Phys. 9 535
[5] Viehland L A and Mason E A 1995 Atom. Data Nucl. Data
Tables 60 37
[6] Benilov M S and Naidis G V 1998 Phys. Rev. E 57 2230
[7] Duman E L et al 1982 Institute for Atomic Energy, Moscow
Preprint No 3532/12 (in Russian)
[8] Ellis H W, Pai R Y, McDaniel E W, Mason E A and
Viehland L A 1976 Atom. Data Nucl. Data Tables 17 177
[9] Phelps A V 1994 J. Appl. Phys. 76 747
[10] http://fisica.uma.pt/public/