Environmental Engineering Reference
In-Depth Information
3.3.5
Propagation of Resonant Radiation in a Dense Plasma
The above formulas allow one to analyze the character of radiation transfer in a
plasma if reabsorption processes are of importance. For an optically dense plas-
ma we have the equilibrium of absorption and emission processes at each plasma
point, and the photon flux
i
ω
is isotropic inside the plasma. One can find this flux
from the equilibrium between emitted and absorbed photons. Indeed, the rate of
absorption acts per unit volume and in a frequency range from
ω
to
ω
C
d
ω
is
, and the rate of photon emission per unit volume in this frequency range
is
N
a
ω
d
i
ω
k
ω
d
ω
ω
/
τ
. Hence, from the equality of these rates we have
a
ω
N
k
ω
τ
4
j
(0)
i
ω
D
r
D
.
ω
This gives the connection between the photon flux inside an equilibrium plasma
and outside it if the reabsorption processes create the equilibrium between atom
excitation and the photon fluxes.
We now consider the process of propagation of resonant photons in a plasma
from another standpoint, analyzing the behavior of a test emitted photon in a plas-
ma [55-57]. Let us introduce the probability
P
(
r
) that a resonant photon survives
at a distance
r
from its formation, and this distance exceeds significantly the mean
free path of the photon for the center of a spectral line 1/
k
0
. Assuming statisti-
cal character for an emission frequency, we have this probability as the product of
the probability
a
ω
d
ω
for photon emission at a given frequency and the probability
exp(
k
ω
r
) of the photon surviving, that is,
Z
P
(
r
)
D
a
ω
d
ω
exp(
k
ω
r
) .
(3.77)
Let us use this formula for the Lorentz (2.136) and Doppler (2.138) line shapes. In
the case of the Lorenz shape of spectral lines we have, introducing a new variable
s
D
(
ω
ω
0
)/
ν
,
ds
k
0
a
ω
d
ω
D
,
k
ω
D
,
π
(1
C
s
2
)
1
C
s
2
where
k
0
is the absorption coefficient for the spectral line center. Let us define
u
k
0
r
as the optical thickness for the center line frequency; according to the
problem condition,
u
D
1. Substituting this in (3.77), we obtain
Z
s
2
)
exp
s
2
1
1
π
ds
u
P
(
r
)
D
.
C
C
(1
1
1
For a dense plasma the main contribution to this integral follows from
s
1, and
for the probability
P
(
r
) for the photon to travel a given distance
r
we have for this
limit
1
p
π
P
(
r
)
D
u
,
u
1 .
(3.78)
