Environmental Engineering Reference
In-Depth Information
respect to atom excitation by electron impact, and we find below the connection be-
tween the parameters of these processes, which proceed according to the scheme
e
C
A
0
!
e
C
A
,
(2.51)
where
A
0
and
A
denote an atom in the ground and resonantly excited states. The
connection between the parameters of direct and inverse processes is established
on the basis of the principle of detailed balance for processes (2.51). Let us take one
electron and one atom in a volume
,and
transitions between these states result from collisions with the electron. Because
this system is under equilibrium, there is a certain connection between the rate
w
0
Ω
, the atom can be found in states 0 and
0. Introducing the
interaction operator
V
which is responsible for these transitions, we have within
the framework of the perturbation theory for the transition rates
for transition 0
!
and the rate
w
0
for transition
!
2
π
„
j
2
dg
d
2
π
„
j
2
dg
0
d
w
0
D
V
0
j
,
w
0
D
V
0
j
.
ε
ε
Here
dg
0
/
d
are the statistical weights per unit energy for the corre-
sponding channels of the process. We use the definition of the cross sections of
these processes
ε
and
dg
/
d
ε
w
0
N
w
0
v
0
w
0
N
w
0
v
σ
D
v
0
D
Ω
,
σ
D
v
D
Ω
,
0
0
where
N
is the number density of particles and
v
0
and
v
are the electron
velocities for the corresponding channels (for simplicity, we consider an atom to be
motionless). For the matrix elements of the interaction operator, the time reversal
operation gives
V
0
D
D
1/
Ω
V
0
. This leads to the following relation between the cross
sections of direct and inverse processes in electron-atom collisions [88, 89]:
v
0
dg
0
d
dg
d
σ
ε
D
σ
0
v
.
(2.52)
0
ε
The statistical weights of the corresponding channels of the processes (2.51) are
d
p
0
(2
d
p
(2
dg
0
D
Ω
)
3
g
0
,
dg
D
Ω
)
3
g
,
π
„
π
„
where
g
0
and
g
are the statistical weights for given atomic states. Then, finally
formula (2.52) takes the form [88, 89]
2
g
q
v
σ
D
σ
,
(2.53)
ex
0
g
0
2
v
where
σ
D
σ
is the excitation cross section and
σ
D
σ
0
is the quenching
ex
0
q
cross section. Near the threshold [84]
A
p
ε
Δ
ε
σ
D
,
ε
Δ
ε
Δ
ε
,
(2.54)
ex
