Environmental Engineering Reference
In-Depth Information
system results from transition of the electric field energy to electrons and then this
energy is transferred to atoms of the gas in electron-atom collisions. This leads
to a self-maintaining state of this ionized gas. Analyzing the establishment of the
electron velocity distribution function, we note that the cross section of elastic elec-
tron-atom collisions exceeds significantly the cross section of inelastic collisions.
Hence, representing the electron distribution function as a sum (3.10) of the sym-
metric and antisymmetric harmonics, we obtain the collision integral (3.12) for the
antisymmetric distribution function, and the set (3.11) of kinetic equations takes
the form
v
f
1
D
x
df
0
v
a
3
v
d
d
a
v
3
v
D
ν
v
x
f
1
(
v
),
I
ea
(
f
0
)
C
I
ee
(
f
0
) .
(3.25)
2
d
v
N
a
v
σ
is the rate of elastic electron-atom collisions,
a
Here
eE
/
m
e
,and
we include in the consideration both elastic electron-atom collisions and electron-
electron collisions.
The first equation of the set (3.25) allows one to express the electron drift velocity
through the symmetric part of the velocity distribution function as [27]
ν
D
D
Z
1
v
v
3
eE
3
m
e
d
d
2
x
f
1
d
v
D
w
e
D
,
(3.26)
v
2
ν
v
where an average is made over the spherical distribution function
f
0
for electrons.
This expression is independent of the character of establishment for the symmetric
part of the electron distribution function
f
0
.
We now examine the limit of low electron number densities, when collisions be-
tween electrons are not essential in this process. Then, taking the electron-electron
collision integral to be zero
I
ee
0 in the second equation of the set (3.25) and
using (3.18) for the electron-atom collision integral, we obtain from the second
equation of the set (3.25)
D
3
m
e
aM
ν
df
0
m
e
v
f
0
T
f
1
D
v
C
,
d
where we account for
f
0
!
0if
v
!1
. The solution of the set of equations for
f
0
and
f
1
yields
0
@
1
A
Z
v
m
e
v
0
d
v
0
m
e
u
f
0
(
v
)
D
A
exp
,
f
1
(
v
)
D
Mu
2
/3
f
0
(
v
),
T
C
Mu
2
/3
T
C
0
(3.27)
eE
/[
m
e
N
a
v
σ
(
v
)].
where
A
is the normalization constant and
u
D
eE
/(
m
e
ν
)
D
In parti
c
ular, if
ν
(
v
)
D
const
D
1/
τ
, the electron drift velocity
w
e
and the average
energy
ε
are given by
eE
m
e
eE
τ
m
e
3
2
T
M
2
w
e
w
e
D
ν
D
ε
D
C
,
(3.28)
