Environmental Engineering Reference
In-Depth Information
where
J
is the atomic ionization potential, and the dispersion relation for the pho-
ton
ω
D
kc
,wehave
k
2
q
2
σ
2
g
a
g
e
g
i
σ
D
ion
.
(2.116)
rec
We now give the expression for the photoionization cross section in the limiting
cases. When atom
A
in processes (2.114) and (2.115) is a hydrogen atom in the
ground state [152, 154, 155]
exp
x
arctan
x
4
2
9
2
e
2
„
π
c
a
0
F
(
a
0
q
),
F
(
x
)
x
2
)
4
1
exp
x
σ
D
D
.
(2.117)
ion
3
2
(1
C
In the limit
x
!
0wehave
exp(4)
ω
1
8/3
0
ω
F
(0)
D
,
where
13.6 eV is the ionization potential for the hydrogen atom in
the ground state. In the limit when the photoionization process proceeds near the
threshold, this gives
„
ω
D
Ry
D
0
0
ω
8/3
0
ω
σ
D
σ
,
(2.118)
ion
where the photoionization cross section at the threshold
σ
0
and the transition fre-
quency
ω
0
for the hydrogen atom are
2
9
2
e
2
„
π
c
a
0
D
0.225
a
0
D
10
18
cm
2
σ
D
6.3
,
0
3exp(4)
m
e
e
4
„
10
16
s
1
ω
D
D
2.07
.
(2.119)
0
3
The principle of detailed balance (2.116) gives on the basis of (2.118) the follow-
ing expression for the photorecombination cross section involving a slow electron
in the ground state of the hydrogen atom:
e
2
„
3
5/3
0
5/3
0
2
c
2
q
2
σ
2
9
2
2
ω
π
ω
ω
0
)
a
0
D
σ
σ
D
D
,
rec
ion
1
c
2/3
(
ω
ω
2/3
(
ω
ω
3exp(4)
ω
ω
0
)
(2.120)
where
e
2
„
3
2
8
2
π
a
0
D
0.225
a
0
D
10
22
cm
2
.
σ
D
1.7
(2.121)
1
3exp(4)
c
In the case of a highly excited initial atomic state when a transferring electron
is described by classical laws, the photoionization cross section is given by the
Kramers formula [156]:
n
5
ω
3
D
σ
K
0
ω
σ
,
(2.122)
ion
