Environmental Engineering Reference
In-Depth Information
these conditions, thermodynamic equilibrium is established between the atoms
and the line-center photons whose free path length is small compared with the
plasma size. Let
i
ω
be the flux of photons of frequency
inside a plasma. Then the
number of photons absorbed per unit volume per unit time in a frequency range
from
ω
,where
k
ω
is the absorption coefficient
determined by (2.148). The reduced number of absorbed photons is equal to the
corresponding number of emitted photons, which is given by
N
a
ω
d
ω
to
ω
C
d
ω
is given by
i
ω
k
ω
d
ω
ω
/
τ
.Then,
on the basis of (2.145), (2.146), and (2.148), we obtain
N
0
N
1
1
a
ω
N
k
ω
τ
r
D
ω
2
g
g
0
i
ω
D
.
(3.86)
π
2
c
2
This photon flux is isotropic and can be detected at any point in the plasma
medium that is separated from the system boundary by at least a photon mean free
path. The photon flux outside a system with a flat surface is
0
@
1
A
1
Z
π
/2
Z
π
/2
i
4
j
ω
D
i
ω
cos
d
(cos
d
(cos
D
θ
θ
)
θ
)
.
(3.87)
0
π
/2
Here
is the angle between the normal to the gas surface and the direction of pho-
ton propagation, and we have taken into account that the total photon flux outside
the system is normal to the system surface. The flux of photons of frequency
θ
ω
outside the gaseous system is
N
0
N
1
1
i
ω
D
ω
2
g
g
0
,
k
ω
L
1 .
(3.88)
2
c
2
π
If the plasma temperature is constant, we obtain (1.61) for the photon flux that
corresponds to thermodynamic equilibrium, and which is given by
exp
„
T
1
1
2
j
ω
D
ω
,
k
ω
L
1.
4
π
2
c
2
In considering radiation from a nonuniform plasma with a flat boundary and
a uniform temperature along the boundary direction, we use (3.75) for the radia-
tion flux from a flat layer of a plasma and with use of (2.148) for the absorption
coefficient, the photon flux is reduced to the form
Z
Z
N
0
N
1
1
1
u
ω
dxdu
exp
2
j
ω
D
ω
u
cos
g
g
0
d
(cos
θ
)
.
(3.89)
2
c
2
2
π
θ
0
0
D
R
0
k
ω
dx
0
and the
Here the current optical thickness of a layer is defined as
u
total optical thickness of the layer is
u
ω
D
R
0
k
ω
dx
,thatis,
du
k
ω
dx
.This
formula allows us to estimate the width of the spectral line of radiation that leaves
the plasma. The boundaries of the spectral line can be estimated from the relation
D
Z
L
u
ω
D
k
ω
dx
k
ω
L
1 .
(3.90)
0
