Environmental Engineering Reference
In-Depth Information
For the Doppler shape of the spectral line we introduce a new variable
u
exp
"
2
#
,
ω
ω
mc
2
2
T
0
t
D
ω
0
which gives the following expression for the probability of a resonant photon trav-
eling a distance
r
:
Z
u
e
t
dt
ln
u
t
1
1
p
π
1
P
(
r
)
D
D
u
p
ln
u
C
,
u
1 ,
(3.79)
p
π
u
C
0
where
C
0.577 is the Euler constant. Since the wings of the Doppler spectral
line drop more sharply than those of the Lorentz spectral line, the probability of
propagate large distances for the Lorentz spectral line is larger than that for the
Doppler spectral line for the same optical thickness of the layer.
Reabsorption of a resonant photon results in its formation and subsequently
in its absorption at another spatial point. We will characterize this process by the
probability
G
(
r
0
,
r
), so
G
(
r
0
,
r
) is the probability for a photon formed at point
r
to be
absorbed in a volume
d
r
0
near the point
r
0
. This probability satisfies the normaliza-
tion condition
Z
G
(
r
0
,
r
)
d
r
0
D
D
1,
where the integral is taken over all space. We obtain the expression for this proba-
bility by analogy with that in (3.73) and (3.77), and it has the form
Z
2
exp
Z
k
ω
dx
a
ω
k
ω
d
ω
G
(
r
0
,
r
)
D
.
(3.80)
π
j
r
r
0
j
4
One can use the reabsorption process in the balance equation for excited atoms in
a plasma where the reabsorption process is included along with the processes of
atom excitation and quenching by electron impact and spontaneous remission by
an excited atom. This equation has the form [58-60]
Z
N
(
r
0
)
d
r
0
Z
@
N
@
N
τ
C
1
τ
a
ω
k
ω
d
ω
t
D
N
e
N
0
k
ex
N
e
N
k
q
4
π
j
r
r
0
j
2
exp
Z
k
ω
dx
(3.81)
and is named the Biberman-Holstein equation.
In particular, for the Lorenz profile of the spectral line (2.136) and for a uniform
plasma this probability is
Z
s
)
2
exp
s
2
k
0
ds
k
0
R
G
(
R
)
D
,
4
π
R
2
(1
C
1
C
