Environmental Engineering Reference
In-Depth Information
processes involving excited atoms do not violate the Maxwell energy distribution of
electrons with electron temperature
T
e
.
Formula (3.66) reflects the character of the equilibrium for resonantly excited
atoms. Note that thermodynamic equilibrium is ascertained by comparison of the
lifetime of excited atoms
in a gas with the typical time (
N
e
k
q
)
1
for atomic
quenching (not excitation!), and establishment of thermodynamic equilibrium for
excited atoms requires the criterion
τ
N
e
k
q
τ
1 .
(3.67)
3.3.2
Stepwise Ionization of Atoms
In analyzing the three body process (2.70) of electron-ion recombination, we con-
sider this process as electron capture in a bound state with an ionization poten-
tial of the order of the thermal electron energy
T
e
, and subsequently the atom
state varies as a result of successive collisions with electrons until transition to
the ground state. We now consider the stepwise ionization of atoms that results
from a series of reverse transitions until the atom is ionized under the action of
collisions with electrons.
The mean electron energy in a gas discharge plasma is considerably lower than
the atomic ionization potential
T
e
J
,
(3.68)
and stepwise ionization of atoms by electron impact can take place only at a high
number density of electrons, so there are no competing channels for transitions
between excited states. The stepwise ionization process is the inverse process with
respect to three body recombination of electrons and ions. In inverse processes,
atoms undergo the same transformations but in opposite directions. As a result,
on the basis of the principle of detailed balance, we obtain the connection between
the rate constant for stepwise ionization
k
st
and the three body electron-ion recom-
bination coefficient
according to (2.71). This gives the rate constant for stepwise
ionization of atoms in a plasma in the limiting case when one can ignore atom
radiation [50]:
α
m
e
T
e
2
exp
exp
3/2
m
e
e
10
„
D
N
e
g
e
g
i
g
a
J
T
e
2
C
g
i
g
a
J
T
e
k
st
D
,
(3.69)
π
„
2
3
T
e
where
C
2 is a numerical factor, and is identical for various atoms.
Let us compare the rate constant for stepwise ionization (3.69) and the rate con-
stant for direct ionization in a single collision, where we use the Thomson for-
mula (2.60) for the ionization cross section, and because of criterion (3.68), the
threshold cross section is used. The rate constant for direct ionization within the
