Chemistry Reference
In-Depth Information
due to the radial ambipolar diffusion for the charged particles balance in the positive
column [3,50]
2.4
R
2
2.4
R
2
k
B
T
e
e
·
ν
ion
(
E
z
)
=
ν
diff
=
D
a
·
=
b
+
·
.
(3.297)
Thereby, the ionization frequency is calculated by the linear approximation of the
electron impact ionization cross section above the threshold energy
e
·
U
ion
accord-
ing to σ
ion
(
U
)
∼
e
·
(
U
−
U
ion
)
and the Maxwellian electron energy distribution
function,
2.4
R
2
∞
√
U
k
B
T
e
b
+
a
∗
·
ν
ion
=
p
·
(
U
−
U
ion
)
·
·
f
(
U
)
·
dU
=
V
diff
=
·
. (3.298)
U
ion
The derived equation
k
B
·
1
/
2
exp
e
T
e
U
ion
k
B
·
·
10
7
2
·
=
1.16
·
·
(
c
·
p
·
R
)
(3.299)
e
·
U
ion
T
e
with
c
2
combines the ratio between the mean electron energy
and the ionization threshold energy
=
(
a
·
(
U
ion
)
1
/
2
)/(
b
+
·
p
)
where
c
is a constant, depending on the kind of gas,
p
is the total pressure in Torr, and
R
is
the tube radius in cm, see Figure 3.36, [3].
The constant
c
amounts to He: 4
(
k
B
·
T
e
/
e
·
U
ion
)
with the product
(
c
·
p
·
R
)
·
10
−
3
,Ne:6
·
10
−
3
,Ar:4
·
10
−
2
,H
2
:1
·
10
−
2
,
N
2
:4
·
10
−
2
.
10
-3
10
-2
10
-1
10
0
10
4
10
4
10
3
10
3
10
-3
10
-2
10
-1
10
0
c
.
p
.
R [Torr cm]
FIGURE 3.36
Universal graph from Equation (3.299) for the estimation of the electron
temperature in the positive column in dependence on the product
(
c
·
p
·
R
)
calculated from
the tube radius
R
as well as ionization potential
U
ion
and the gas specific constant
c
.
