Environmental Engineering Reference
In-Depth Information
whereweusethevariable
s
and
k
0
is the absorption coefficient
in the line center. In the limiting case of an optically thick plasma,
k
0
R
D
2(
ω
ω
0
)/
ν
1, this
formula gives
1
G
(
R
)
D
k
0
)
1/2
R
7/2
.
(3.82)
(4
π
Correspondingly, in the case of the Doppler profile of the spectral line (2.138), using
anewvariable
s
0
](
mc
2
/
T
)
1/2
,wehave
a
ω
d
ω
D
π
1/2
exp(
s
2
)
ds
D
[(
ω
ω
0
)/
ω
s
2
). Substituting this into (3.80) and introducing the variable
and
k
ω
D
k
0
exp(
s
2
), we obtain
t
D
k
0
R
exp(
Z
1
Z
k
0
R
te
t
dt
p
ln(
k
0
R
/
t
)
k
0
1
s
2
G
(
R
)
D
ds
exp(
t
)
D
.
4(
π
)
3/2
R
2
4(
π
)
3/2
k
0
R
4
1
0
Replacing the upper limit by infinity in the limiting case of an optically thick plas-
ma,
k
0
R
1, and accounting for ln
k
0
R
1, we obtain
1
G
(
R
)
D
)
3/2
k
0
R
4
p
ln(
k
0
R
/
t
0
)
,
4(
π
where
t
0
is the solution of the equation
R
0
te
t
dt
p
ln(
t
/
t
0
)
D
0, that is,
t
0
D
1.52.
From this we have
1
G
(
R
)
D
)
3/2
k
0
R
4
p
ln(
k
0
R
)
0.42
.
(3.83)
4(
π
We find that propagation of resonant radiation in a gas or plasma has a specific
character. In contrast to diffusion transfer of atomic particles in gaseous matter,
radiation transfer proceeds at the wings of the spectral line for the radiative transi-
tion, and the greater the optical gas thickness, the greater the wings of the spectral
line are responsible for radiation transfer. As an example of this character of radi-
ation transfer, we consider emission of a cylindrical plasma region for the Lorenz
profile (2.136) of the spectral line. In the limit of an optically thin plasma (
k
0
d
1,
where
d
is the tube diameter), if all the photons formed leave the plasma region,
we have for the flux
j
of photons [34]
N
τ
V
S
D
N
d
4
j
D
,
τ
r
r
d
2
/4 and
S
where
V
d
are, respectively, the volume and the surface
area for a plasma region of length
L
.Inthelimitofanopticallythickplasma,
k
0
d
D
L
π
D
L
π
1, we have for the photon flux
Z
N
τ
P
(
r
)cos
θ
d
r
j
D
,
4
π
r
2
r
where
P
(
r
) is the probability that a resonant photon survives at a distance
r
from its
formation,
r
is a distance of the point of photon formation from the surface point,
