Environmental Engineering Reference
In-Depth Information
of this excited state with respect to a given transition, and so
N
/
τ
r
is the number
of emission acts per unit volume and per unit time;
a
ω
d
is the probability that
the frequency of an emitted photon lies in a given range. We assume also isotropic
character of photon emission.
In the case of an arbitrary optical thickness of this plasma layer we introduce into
this formula an additional factor, exp(
ω
R
k
ω
dx
), that an emitted photon reaches
the plasma boundary, so the above formula is generalized to the form
0
@
1
A
Z
Z
r
N
(
r
)
d
r
τ
cos
θ
k
ω
dr
0
j
ω
d
ω
D
a
ω
d
ω
r
2
exp
.
(3.74)
4
π
0
We assume uniformity of the plasma layer in the longitudinal direction, that is, all
the plasma parameters depend on a distance
x
from the plasma surface only. Tak-
ing the current optical thickness of a layer as
u
D
R
0
k
ω
dx
0
and the total optical
thickness of the layer as
u
ω
D
R
0
k
ω
dx
,thatis,
du
D
k
ω
dx
, and using the cylin-
drical symmetry
d
r
D
2
π
r
2
drd
cos
θ
, we reduce the above formula for the photon
flux to the form
Z
1
Z
u
ω
exp
.
du
N
a
ω
2
u
cos
j
ω
D
d
cos
θ
(3.75)
τ
r
k
ω
θ
0
0
0,
we use (2.149) if the number density of atoms in these states is subject to the Boltz-
mann law and use (2.145) for the absorption cross section. Then (3.75) reduces to
the form
In considering the photon emission as a result of the atomic transition
!
Z
Z
u
1
du
h
exp
1
i
1
ω
2
j
ω
D
ω
u
cos
d
cos
θ
2
π
2
c
2
θ
0
0
Z
1
h
1
exp
i
,
u
ω
cos
j
(0)
D
cos
θ
d
cos
θ
(3.76)
ω
θ
0
where the photon flux
j
(0)
under thermodynamic equilibrium according to (1.61)
ω
is given by
exp
„
T
1
1
2
ω
D
ω
j
(0)
.
4
π
2
c
2
In the limiting cases of optically thick and optically thin plasmas, (3.76) gives
j
(0)
2
j
(0)
j
ω
D
,
u
ω
1
I
j
ω
D
u
ω
,
u
ω
1.
ω
ω
