Environmental Engineering Reference
In-Depth Information
in a range
d
k
of the wave vector values, and the factor 2 accounts for the two inde-
pendent polarization states. From this we have the photon flux on the basis of the
dispersion relation
ω
D
kc
for photons:
j
ω
D
ω
2
d
ω
/(
π
2
c
2
). Correspondingly we
find for the absorption cross section
σ
ω
2
c
2
ω
w
(
o
,
n
ω
!
,
n
ω
1)
An
ω
j
ω
D
π
g
g
0
a
ω
τ
σ
ω
D
D
.
(2.145)
j
ω
2
r
In the same manner one can derive the expression for the stimulated cross section
σ
0
ω
as the ratio of the rate of photon formation under the action of the radiation
field
Bn
ω
to the photon flux
j
ω
.Weobtain
2
c
2
ω
Bn
ω
j
ω
D
π
a
ω
τ
σ
0
ω
D
.
(2.146)
2
r
Let us introduce the absorption coefficient
k
ω
such that the intensity of radiation
I
ω
of frequency
that propagates in a gas of atoms in direction
x
is given by
ω
dI
ω
dx
D
k
ω
I
ω
.
(2.147)
One can see that 1/
k
ω
is the mean free path of photons in a gas of atoms. According
to the definition, the absorption coefficient
k
ω
is expressed through the absorption
σ
0
ω
as
cross section
σ
ω
and the cross section of stimulated radiation
1
.
N
N
0
g
0
g
N
σ
0
ω
D
k
ω
D
N
0
σ
ω
N
0
σ
ω
(2.148)
If the distribution of atom densities is determined by the Boltzmann formu-
la (1.43), the expression for the absorption coefficient has the form
1
exp
.
„
T
k
ω
D
N
0
σ
ω
(2.149)
We now consider an important case of a transition between the ground state
and an resonantly excited state when broadening of the spectral line is determined
by collisions of these atoms with excitation transfer. The interaction operator that
governs broadening of the spectral line in this case has the form
D
D
1
D
2
3(
D
1
n
)(
D
2
n
)
R
3
U
(
R
)
.
Here
R
is the distance between colliding atoms,
n
is the unit vector directed along
R
,and
D
1
and
D
2
are the dipole moment operators for the atoms indicated. Since
the matrix element
d
is nonzero, the total cross section for collision
of these atoms that is responsible for broadening in this case, according to (2.139),
is
Djh
o
j
D
jij
d
2
/
„
v
,where
v
is the collision velocity, and the rate constant
v
σ
t
that is
responsible for broadening in the center of the spectral line is independent of the
collision velocity. Table 2.22 gives values for some resonant transitions of atoms.
σ
t
