Electron Thermalization in Molecular Gases
Takeo Nishigori
Kibi Kokusai University, Takahashi-shi, Okayama, 716-8508 Japan
Abstract. Thermalization of hot electrons in molecular gases is described by a Boltzmann equation, with the Boltzmann
operator consisting of a Fokker-Plank operator for elastic processes and a difference operator for inelastic ones. The
eigenvalue approach of Shizgal and co-workers for a Fokker-Plank equation is extended to solve this Boltzmann
equation. The inelastic interaction is much stronger than the elastic one, and two well-separated time scales are involved
in the relaxation processes. This makes the analysis difficult, and the convergence of the eigenmode expansion is very
slow. It is shown, however, that a high precision calculation involving a few hundred eigenmodes shows convergence.
1. INTRODUCTION
Thermalization of hot electrons in molecular gases, as well as in rare gases, is of interest in understanding
collisional energy loss of electrons and various transport phenomena in ionized gases [1-8]. In the case of rare
gases, the Boltzmann equation reduces to a Fokker-Planck equation, and a successful eigenvalue approach has been
developed by Shizgal and co-workers [1, 3]. Nishigori has found that the memory-function approach is useful to
complement the eigenvalue approach, which converges slowly at short times [4-6]. In the case of molecular gases,
inelastic processes are important, and one uses a Boltzmann equation with a nonlocal difference operator describing
the inelastic collisions [4, 7, 8]. Demeio and Shizgal applied the eigenvalue approach to a molecular gas by using
the Fokker-Planck equation in an approximate way [8]. In a previous paper [4], the memory-function method was
applied on the basis of the Boltzmann equation taking correct account of the inelastic processes. The result is
correct at short times and has the correct asymptotic limit, but the validity in the intermediate time region has not yet
confirmed.
The aim of the present paper is to extend the eigenvalue approach of Shizgal and co-workers to take correct
account of the inelastic collisions; we therefore consider the eigenvalue problem of the Boltzmann operator instead
of the Fokker-Planck one. The inelastic interaction is stronger than the elastic one by a factor of molecule-electron
mass ratio, M / m ≥ 7, 000 . The electron therefore loses the initial energy very rapidly to reach the inelastic
threshold energy, below which electron thermalizes slowly via elastic collisions. These two well-separated time
scales involved in the relaxation processes make the analysis difficult, and the convergence of the eigenmode
expansion is very slow. Present paper shows that a few hundred eigenmodes are required to obtain the convergence.
Formulation is given in Sec. 2 to obtain a symmetric matrix representation of the Boltzmann operator. Speed
polynomials [9] are used. Numerical results are given in Sec. 3 to examine the convergence of the eigenmode
expansion. Reid’s ramp model [10] is used as in the previous paper [4] for the inelastic collision cross section.
Concluding remarks are given in the final section.
2. FORMALISM
Consider the isotropic component f 0 (v, t ) of the velocity distribution function f (v , t ) of monenergetic hot
electrons dilutely dispersed in a spatially homogeneous molecular gas without any external electric field. We use
the dimensionless variables
x = v / vb ; vb = 2kTb / m
and
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
888
(1)
t ' = t / τ ; τ = 2 M / nmσ 0 vb ,
(2)
where Tb is the bath temperature, m the electron mass, M the molecular mass, n the number density, and
σ0
Defining the distribution function φ ( x, t ') by the relation
some convenient hard-sphere cross section.
f (v , t )d v = 4π v f 0 (v, t )dv = φ ( x, t ')dx , one has the Boltzmann equation [4, 7, 8],
2
∂
φ ( x, t ') = Bφ ( x, t '),
∂t '
(3)
φ ( x, 0) = δ ( x − x0 )
(4)
subject to the initial condition
with initial speed x0 . The operator B consists of a Fokker-Planck operator P for elastic collisions and a nonlocal
difference operator Q for inelastic ones;
B = P +Q;
(5)
∂
∂
[2( x 2 − 1)σ ( x) + xσ ( x) ],
∂x
∂x
ξ x2
ξ x2
2M
Q=
x ∑ { σ ij ( x 2 )[ n j i 2 D− − ni ] + σ ij ( x+2 )[ ni D+ − n j i +2 ] },
ξ j x−
ξj x
m i< j
P=
(6)
(7)
where
x± = x 2 ± ε ij / kTb , ε ij = ε j − ε i > 0,
(8)
ni is the population density of the molecular internal state i with energy ε i and degeneracy ξi , σ ij ( x ) is the
2
inelastic collision cross section for the electron energy x 2 , and the displacement operators D± are defined by
D± f ( x) = f ( x± ) D± f ( x) = f ( x± )
(9)
for any function f ( x) .
The equilibrium distribution is a Maxwellian φ ( x, ∞) ∝ M ( x) = x 2 exp(− x 2 ) . It is convenient to introduce a
distribution function g ( x, t ') by
φ ( x, t ') = M ( x) g ( x, t ').
(10)
An equation for g ( x, t ') follows by noting the relation
B M ( x) = M ( x)B † .
(11)
Here the adjoint operator B = P + Q is given by
†
†
†
P † = [−2( x 2 − 1)σ ( x) +
∂
∂
xσ ( x)] ,
∂x
∂x
(12)
and
Q† =
ξ x2
2M
x ∑ { σ ij ( x 2 )ni [D− − 1] + σ ij ( x+2 )n j i +2 [D+ − 1] },
m i< j
ξj x
(13)
where use has been made of the relation
D± † =
x
D∓ .
x∓
(14)
The Boltzmann equation (3) and the initial condition (4) are rewritten for the distribution function g ( x, t ') as
∂
1
g ( x, t ') = B † g ( x, t '); g ( x, 0) =
δ ( x − x0 ).
M ( x)
∂t '
(15)
It should be noted that a polynomial representation of the adjoint operator B † is symmetric with respect to the
weight function M ( x) ;
889
†
Bnm
=
∫
∞
0
M (x )Pn (x )B †Pm (x )dx =
=
∞
∫
0
∫
∞
0
Pm (x )BM (x )Pn (x )dx
†
M (x )Pm (x )B † Pn (x )dx = Bmn
,
(16)
where {Pn } is a set of orthonormal polynomials. All the eigenvalues of the matrix B† are therefore real, and this is
the reason that we have introduced the distribution function g ( x, t ') .
The present paper is concerned with the energy relaxation
∞
∞
0
0
H (t ') = E (t ') / E∞ = ∫ h( x)φ ( x, t ')dx = ∫ h( x) M ( x) g ( x, t ')dx;
h( x ) =
2 2
x .
3
(17)
(18)
The Boltzmann equation gives a formal solution
†
g ( x, t ') = et ' B g ( x, 0).
Introducing the eigenvalues −ωλ and eigenfunctions ψ λ as
(19)
B †ψ λ ( x) = −ωλψ λ ( x),
(20)
g ( x, 0) = ∑ aλψ λ ( x),
(21)
g ( x, t ') = ∑ aλψ λ ( x)e −ωλ t ' .
(22)
and expanding the initial distribution as
λ
one has
λ
Using the relation (11), one can prove the orthogonality
∫
∞
M ( x)ψ µ ( x)ψ λ ( x)dx = δ µ , λ ,
0
(23)
which yields
∞
aλ = ∫ M ( x)ψ λ ( x) g ( x, 0)dx = ψ λ ( x0 )
0
(24)
for the coefficients in the expansion (21). The distribution function (22) is thus evaluated in terms of the
eigenvalues −ωλ and the eigenfunctions ψ λ ( x ) .
In the polynomial representation, the eigenfunctions can be evaluated by the expansion
ψ λ ( x) = ∑ψ nλ Pn ( x).
(25)
n
The solution of the eigenvalue problem of the matrix B† ,
B† ψ λ = −ωλ ψ λ ;
ψ λ = (ψ 0 ,ψ 1 ,...) ,
λ
λ
T
(26)
(27)
gives the eigenvalues −ωλ and the coefficients ψ n in the eigenfunction (25). The distribution function φ ( x, t ') is
then given by Eqs. (10) and (22), and the energy relaxation is evaluated by
(28)
H (t ') = ∑ h λ e−ωλ t ' ;
λ
λ
∞
hλ = ψ λ ( x0 ) ∫ h( x) M ( x)ψ λ ( x)dx.
0
(29)
3. NUMERICAL RESULTS
To numerically examine the present formalism, we have assumed a hard sphere elastic collision, σ ( x) = 1.0 , and
Reid’s ramp model [10]
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TABLE 1. Convergence of Eigenvalues with Increasing Matrix Dimension N.
ω1
ω10
ω20
ω40
N
101
5.31892
116.702
399.82
6042.2
201
5.31133
116.517
399.28
1456.2
301
5.31052
116.516
399.22
1455.5
401
5.31033
116.507
399.19
1455.3
501
5.31027
116.505
399.18
1455.3
2
10( x kTb − 0.2) / σ 0 ,
0,
σ 12 ( x 2 ) = 
x 2 kTb ≥ 0.2eV
otherwise
ω60
ω80
121960.
3122.0
3095.8
3095.4
3095.2
572460.
11592
5244.0
5243.4
5243.2
(30)
for inelastic collisions with a He gas ( M = 4 amu, σ 0 = 6 A2 , i, j = 1,2). The threshold energy is 0.2 eV, which is
5.3 times the thermal average energy E∞ = (3 / 2) kTb . The initial electron energy was chosen to be sixteen times the
thermal average energy, i.e., (2 / 3) x02 = 16 .
As the basis set of polynomials, we have used the speed polynomials [9] {Bn ( x)} , which are orthogonal with
respect to the steady distribution M ( x) , and hence expected to lead to an efficient calculation [11]. The matrix
elements of the elastic collision P† can be evaluated analytically. Those of the inelastic collision Q† are written
explicitly in the present model as
∞
∞
2M
2M
†
Qnm
xn1σ 12 ( x+2 ) Bm ( x+ )dx − ∫ M ( x+ ) Bn ( x)
xn1σ 12 ( x+2 ) Bm ( x)dx
= ∫ M ( x + ) Bn ( x)
0
0
m
m
(31)
∞
∞
2M
2M
2
2
+ ∫ M ( x+ ) Bn ( x+ )
xn1σ 12 ( x+ ) Bm ( x) dx − ∫ M ( x) Bn ( x)
xn1σ 12 ( x ) Bm ( x)dx.
0
0
m
m
It is seen that the pair of the first and third terms on the right-hand side, and each of the second and fourth terms are
symmetric with respect to the interchange of n and m . The integrals in Eq. (31) were evaluated by the quadrature
based on the speed polynomials [9]; 1,000 quadrature points have been employed with a sufficiently small weight of
7.58 ×10−156 at the largest point x = 18.977 .
Eigenvalues have been calculated by the standard Jacobi routine. Table 1 shows the convergence of some
selected eigenvalues with increasing matrix dimension. The matrix of dimension 401 shows the convergence of first
81 eigenvalues in four significant figures. The convergence is very slow in comparison with the case without
inelastic collisions, where a matrix of dimension less than 100 is enough to get convergence [3].
Figure 1 illustrates the results for the energy relaxation at short times. Convergence of the eigenmode expansion
is slow in this short-time region, and as many as 401 eigenmodes are required to get convergence. Two distinctive
relaxation modes are clearly seen in this figure. Electrons lose energy very quickly by the inelastic collisions, and
the average energy H (t ') decreases from the initial value of 16.0 to the inelastic threshold value of 5.3 at
t ' = 0.00023 . Below this threshold, the electrons lose energy slowly due to much weaker elastic interactions.
The results over much longer time scale are shown in Fig. 2. The eigenvalue expansion converges rapidly at the
long time region, and even the 101 mode calculation is seen to give a reasonable result. The energy approaches very
slowly to the equilibrium value of 1.0. The relaxation time t '1.1 is found to be 0.1233 in this result, where the
relaxation time is defined [1, 3, 7] as the time when the average energy reaches within 10% of the equilibrium
energy, i.e., H (t '1.1 ) = 1.1 . The relaxation time is about 500 times longer than the time required for the initial energy
to reach the inelastic threshold energy via inelastic collisions.
Also, we show in Figure 3 the slow convergence of the eigenmode expansion at small times. Each dotted line
was obtained by truncating the eigenmode expansion (28) at λ = 100, 160, 180, 190, 200, or 300 in the converged
501 mode calculation. The initial values are 5.287, 6.083, 7.509, 12.45, 15.37, and 16.00, respectively; i.e., nearly
300 modes are necessary to obtain the correct initial value. This is highly in contrast to the memory-function
approach, which converges rapidly at short times; the result involving only two exponential modes (see Eq. (11) in
Ref. 4) has been shown by the broken line, and that involving ten modes by the solid line. Even the simple two
exponential mode approximation describes accurately the early energy relaxation due to the inelastic collisions. The
unified theory proposed in previous papers [12] for rare gases will therefore be useful also for the molecular gases.
The time evolution of the distribution function φ ( x, t ') is determined from Eqs. (10) and (22). Figure 4 displays
891
Energy H(t')
16
101 modes
14
201
12
301
401
10
501
8
6
4
2
0
0.000 0.002 0.004 0.006 0.008 0.010
time t'
FIGURE 1. Convergence of energy with increasing number of eigenmodes at small times.
Energy H(t')
16
101 modes
14
201
12
301
10
401
501
8
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
time t'
FIGURE 2. Convergence of energy with increasing number of eigenmodes in a longer time scale.
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the distribution function at the initial time and at t ' = 0.00008 . The initial delta function at x = x0 , corresponding to
energy E (0) / E∞ = 16 , is approximated in the present analytical polynomial expansion by an oscillating function.
Energy H(t')
16
14
101, …, 301 eigenmodes
12
2 modes in MF method
10
10 modes in MF method
8
6
4
2
0.00 0.05 0.10 0.15 0.20 0.25 0.30
*10-3
time t'
FIGURE 3. The eigenmode expansion converges slowly at small times. The memory function (MF) method is very efficient
and even two exponential modes give a good description of the initial relaxation due to the inelastic collisions.
20
t’ =0
t’ =0.00008
phi(x,t')
10
0
-10
-20
8
9
10 11 12 13 14 15 16
energy
FIGURE 4. Time evolution of the distribution function. The initial distribution is a delta function at energy 16.0. At t’
=0.00008 the average energy decreases to 7.283.
893
8
6
phi(x,t')
4
2
0
-2
-4
-6
-8
4
5
6
7
8
9
10 11 12
energy
FIGURE 5. The distribution function at t’ =0.0002 with average energy of 5.439 slightly above the inelastic threshold energy.
t’ =
0.01
1.0
0.04
0.08
phi(x,t')
0.8
0.12
0.16
0.6
0.6
and 1.0
0.4
0.2
0.0
0
1
2
3
4
5
6
7
8
energy
FIGURE 6. Time evolution of the distribution function when only the elastic collisions contribute to the thermalization. The
average energies are 2.238, 1.464, 1.210, 1.105 and 1.053 at t’ =0.01, 0.04, 0.08, 0.12 and 0.16, respectively. The distribution at
t’ =0.6 and 1.0 is the equilibrium Maxwellian at 290K.
894
At t ' = 0.00008 , the average electron energy decreases to 7.283. The distribution function shows a very small peak
at energy 16, and another peak at energy 10.6; difference between the peaks is about 5.4, closely matches the
inelastic threshold energy of 5.3, which is the energy lost by one inelastic collision; this may be interpreted as the
Lewis effect [12, 7]. There must be other small peaks at the lower energies, but they are not evident because of the
unphysical oscillations in the present analytical approach (cf. a finite difference approach in Ref. 7).
Figure 5 illustrates the distribution with the average energy of 5.439 slightly above the inelastic threshold energy.
The peak at energy around 10.6 due to inelastic collisions almost disappears, and another peak appears at energy
nearly equal to the average energy. This peak moves toward the lower energy side as the distribution evolves in
time as seen in Fig. 6. A thermal peak is gradually formed, and an equilibrium Maxwellian distribution is
established at about t’=0.6.
CONCLUDING REMARKS
The eigenvalue approach of Shizgal for electron thermalization has been extended to include the inelastic
processes in molecular gases. The speed polynomials have been employed to obtain a matrix representation of the
Boltzmann operator. In this polynomial representation, it has been found that a few hundred eigenmodes have to be
included to get a convergence of the energy relaxation. The quadrature discretization method developed by Shizgal
and co-workers [1, 3, 5, 11] will be useful to improve this slow convergence.
In the memory-function approach, only two exponential modes are enough to describe the initial rapid energy
relaxation, while nearly 300 exponential modes are necessary in the eigenvalue approach. Note that both the two
approaches are based on an identical matrix [5]. The unified theory proposed in the previous papers [5] applies the
memory-function approach at short times, and will thus be useful to avoid calculating many higher modes in the
eigenvalue approach.
Applications to realistic molecular gases such as CH4 [7, 8] and H2 [2, 13] are necessary to fully understand the
usefulness of the present formalism.
REFERENCES
1. Shizgal, B. et al., Radiat. Phys. Chem. 34, 35-50 (1989).
2. Okigaki, S. et al., ”Electron Thermalization Processes in Gaseous Mixtures” in Rarefied Gas Dynamics 18, edited by B. D.
Shizgal and D. P. Weaver, Prog. Astronautics and Aeronautics 159, 97-102; J. Chem. Phys. 96, 8324-8329 (1992).
3. Shizgal, B. and McMahon, D. R. A., Phys. Rev. A 32, 3669-3680 (1985).
4. Nishigori, T., “Short-Time Expansion for the Analysis of Electron Transport in Molecular Gases” in Rarefied Gas Dynamics
22, edited by T. J. Bartel and M. A. Gallis, AIP Conference Proceedings 585, American Institute of Physics, New York, 2001,
pp. 94-100.
5. Nishigori, T. and Nagata, K., “Unified Theory of the Eigenvalue and Memory-Function Approaches in Electron
Thermalization in Gases -II” in Rarefied Gas Dynamics 21, edited by R. Brun et al., Editions Cepadues, Toulouse, 1999, vol.
1, pp. 39-46.
6. Nishigori, T. and Shizgal, B., J. Chem. Phys. 89, 3275-3278 (1988).
7. Kowari, K., Demeio, L., and Shizgal, B., J. Chem. Phys. 97, 2061-2074 (1992).
8. Demeio, L. and Shizgal, B., J. Chem. Phys. 99, 7638-7651 (1993).
9. Shizgal, B. and Blackmore, R., J. Comput. Phys. 55, 313-327 (1984).
10. Reid, I. D., Aust. J. Phys. 32, 231-254 (1979).
11. Leung, K, Shizgal, B. and Chen, H., J. Math. Chem. 24, 291-319 (1998).
12. Lewis, H, W., Phys. Rev. 125, 937-940 (1962).
13. Kowari, K., Phys. Rev. A 53, 853 (1996).
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