Environmental Engineering Reference
In-Depth Information
In evaluating the integral in (3.51), we take into account the threshold depen-
dence for the rate constant for atom excitation by electron impact and express it
through the rate constant for atom quenching
k
q
by a slow electron, where
k
q
is in-
dependent of the velocity of a slow electron. We have from the principle of detailed
balance (2.57) near the excitation threshold
r
ε
Δ
ε
Δ
ε
k
q
g
g
0
k
ex
D
,
where
g
0
and
g
are the statistical weights of the ground and excited states of the
atom. From this we have for the rate of atom excitation in an ionized gas for this
limiting case
2
3/2
dN
dt
D
π
T
ef
m
e
g
g
0
f
0
(
v
0
)
ν
,
(3.52)
q
where
N
a
k
q
is the rate of atom quenching by a slow electron.
In reality, excitation of atoms in a plasma is a self-consistent problem, because
atom excitation by electron impact in a plasma influences the velocity distribution
function, and according to (3.51) this part of the electron distribution function, in
turn, may determine the excitation rate. We now consider the opposite limiting
case, when all the electrons whose energy exceeds the excitation threshold expend
the energy on atom excitation. Then the distribution function for electrons decreas-
es sharply above the excitation threshold, and we take here
f
0
ν
D
q
0. Note that
although in reality the energy for this boundary condition exceeds the excitation
threshold
D
Δ
ε
,wetakeitas
f
0
(
Δ
ε
)
D
0.
We now analyze the nonstationary kinetic equation for the electron distribution
function in the case of elastic electron-atom collisions. Then we have the following
set of equations instead of (3.25) in the regime of a low number density of electrons:
@
f
0
@
a
3
v
2
@
3
f
1
)
I
ea
(
f
0
),
a
@
f
0
@
v
D
ν
v
(
v
t
C
D
f
1
.
@
v
We ignore the time dependence of the antisymmetric part of the distribution func-
tion
f
1
because establishment of the electron momentumproceeds fast, in contrast
to the energy establishment. Reducing this set of equations to the equation for the
symmetric part
f
0
of the distribution function and using (3.18) for the electron-
atom collision integral, we obtain
T
f
0
Ma
2
3
@
f
0
@
m
e
M
@
f
0
m
e
v
@
v
C
@
3
t
C
ν
C
D
0 .
(3.53)
v
2
@
v
ν
2
v
For the rate of atom excitation under the given conditions this gives
dN
dt
D
Z
dN
e
dt
D
v
@
f
0
@
2
d
4
π
v
t
T
@
f
0
Ma
2
3
m
e
M
v
f
0
@
ε
C
3
D
4
π
ν
C
.
ν
2
j
ε
D
Δ
ε
