Environmental Engineering Reference
In-Depth Information
framework of the Thomson model is
Z
exp
2
1/2
1/2
2
ε
J
T
e
m
e
k
ion
D
σ
ion
d
ε
1/2
T
3/2
π
e
8
1/2
exp
e
4
J
2
π
T
e
m
e
J
T
e
D
,
(3.70)
where
is the energy of the incident electron. Taking the atomic ionization poten-
tial
J
of the order of the atomic value
ε
m
e
e
4
/
2
,weobtainfromthis
ε
D
„
0
T
e
ε
7/2
k
ion
k
st
1 ,
(3.71)
0
because of criterion (3.68), that is, stepwise ionization is a more effective process
in a low-temperature plasma than direct atom ionization by electron impact. Note
that (3.69) for the rate constant for stepwise ionization holds true if the population
of all the intermediate excited atom states corresponds to the Boltzmann formu-
la (1.43). Usually this is not fulfilled, and (3.69) may be used as an estimate.
To analyze the character of stepwise ionization, we consider this process if the
first stage of excitation of the atom ground state results in transition to a resonant-
ly excited state. Then according to the definition of the ionization rate constant
we have
N
0
k
st
k
st
N
,where
N
0
and
N
are the number densities of atoms in
the ground and resonantly excited states, respectively, and
k
st
and
k
st
are the rate
constants for stepwise atom ionization if it starts from the ground and resonantly
excited atom states, respectively. We use (3.69) for the rate constant for stepwise
ionization from the resonantly excited state and (3.66) for the number density of
resonantly excited atoms. Then on the basis of the above definition of the ionization
rate constant we have
D
)
exp
m
e
e
10
„
4
g
i
g
a
1
J
T
k
st
,
(3.72)
3
T
e
1
C
1/(
N
e
k
q
τ
where
g
i
and
g
a
are the statistical weights for the ion and atom in the ground
state,
J
is the ionization potential of an atom in the ground state, and
τ
is the
effective lifetime of the first resonantly excited state. In the limit
N
e
k
q
1 (3.72)
is converted into (3.69). This shows that additional channels for decay of excited
states along with transitions in collisions with electrons lead to a decrease of the
ionization rate constant.
τ
3.3.3
Excited Atoms in a Helium Plasma
If excited atoms partake only in processes of excitation and quenching by electron
impact, their number density is connected with the number density of atoms in
the ground state by the Boltzmann formula (1.43). If the lifetime of excited atoms
is determined also by other processes, the number density of excited atoms is less
