Environmental Engineering Reference
In-Depth Information
plasma conductivity is due to electrons because of the small electron mass. Hence,
ignoring transport of ions, for the conductivity of an ionized gas we have
1
v
v
N
e
e
2
3
m
e
3
d
d
Σ
D
,
(3.41)
2
ν
v
where
ν
is the rate of electron-atom collisions. If this rate
ν
D
1/
τ
is independent
of the collision velocity, this formula gives
N
e
e
2
τ
Σ
D
.
(3.42)
0
m
e
The same result corresponds to the tau approximation.
When electron collisions in an ionized gas occur with both atoms and ions, we
represent the rate of collisions for electrons in the form
N
a
v
σ
ea
C
N
i
v
σ
ei
,
ν
D
σ
ei
are the
diffusion cross sections for electron scattering by atoms and ions. We are guided
by a weakly ionized gas
N
a
σ
ea
and
where
N
a
and
N
i
are the atom and ion number densities and
N
i
, but because of a long-range electron-ion interac-
σ
ea
σ
ei
for thermal collisions. Therefore, collisions of electrons with ions
may be of importance for the conductivity of an ionized gas in spite of a low de-
gree of gas ionization. Using (2.39) for the diffusion cross section of electron-ion
collisions, we represent the formula for the collision rate in the form
tion,
e
4
ln
N
i
4
π
Λ
σ
ea
v
C
ν
D
N
a
,
m
e
v
3
and in the case of a relatively high ionization rate, when the collision rate is deter-
mined by electron-ion collisions, according to (3.41) the conductivity of an ionized
gasisgivenbytheSpitzerformula[35]:
2
5/2
T
3/2
e
Σ
D
.
(3.43)
3/2
m
1/2
π
e
2
ln
Λ
e
In considering an intermediate case, for definiteness we assume the diffusion
cross section of electron-atom collisions to be independent of the electron velocity.
Then we have for the conductivity of an ionized gas
Z
1
e
4
ln
e
x
xdx
1
N
i
N
a
π
Λ
T
e
σ
ea
Σ
D
Σ
Φ
(
),
D
,
Φ
(
)
D
/
x
2
,
(3.44)
ea
C
0
where the conductivity
Σ
ea
is given by (3.41) and relates to the limiting case
N
a
D
0.
It is convenient to represent the function
Φ
(
) in (3.44) in the form
Z
Z
1
1
e
x
xdx
1
e
x
xdx
x
2
Φ
(
)
D
/
x
2
D
1
C
C
0
0
h
chi(
p
)
i
,
)cos(
p
shi(
p
)cos(
p
D
1
C
)
C
(3.45)
