Chemistry Reference
In-Depth Information
With the known α coefficient the rising electron current density at the posi-
tion
z
within
dz
in Figure 3.32 is given by the 1D stationary balance equation for
electrons
dj
e
dz
=
α
·
j
e
=
α
·
e
·
n
e
·
v
eD
=
ν
ion
·
e
·
n
e
=
σ
ion
·
v
e
·
e
·
n
e
·
n
gas
(3.270)
where
α is the first Townsend coefficient
v
eD
is the electron drift velocity in
z
-direction
ν
ion
is the ionization frequency
σ
ion
is the electron impact ionization cross section
The total electron current density and the electron density at the position
z
is achieved
after integration of (3.270) for constant α
j
e
(
z
)
=
j
e
0
(
0
)
·
exp
(
α
·
z
)
,
(3.271)
n
e
(
z
)
=
n
e
0
(
0
)
·
exp
(
α
·
z
)
(3.272)
with
j
e
0
(
as the electron current density and the electron density at the
cathode from the external source, respectively.
In other words, starting with one electron at the cathode the number of electrons
rises on exponentially to
N
ez
with increasing path length
z
of the electron drift toward
the anode
0
)
and
n
e
0
(
0
)
N
ez
=
1
·
exp
(
α
·
z
)
(3.273)
and describes the
first electron avalanche
.
On the other hand, the produced positive ions in the gap will be accelerated in
opposite direction toward the cathode. Thereby, the ionization of neutral gas atoms
by collision with positive ions, described by the second Townsend coefficient β
in the original theory of Townsend, is neglected. The cross section of ion-impact
ionization is much lower in comparison with that of the electrons in the considered
range of kinetic energy. Impinging positive ions at the cathode surface have sufficient
high kinetic energy to produce secondary electrons, described by the
third Townsend
coefficient
γ. From the macroscopic point of view the coefficient γ describes the
number of emitted secondary electrons per impinging positive ion expressed by the
corresponding current densities at the cathode
j
es
j
+
|
cathode
.
γ
≡
(3.274)
The coefficient γ depends on the kind of gas, the kinetic energy of ions, and the
cathode material [41]. Typical values for γ in electric gas discharges are in the order
of magnitude of between 0.01 and 0.1.
