Environmental Engineering Reference
In-Depth Information
where
c
is the light velocity. Therefore, if radiating atoms are distributed over ve-
locities, their radiation will be detected by a motionless receiver as being frequency
distributed. This distribution is determined by the velocity distribution function
f
(
x
) of radiating atoms.
If the distribution function over atom velocities is normalized to unity (
R
f
(
v
x
)
v
d
D
1), the frequency distribution function follows from the relation
v
x
a
ω
d
ω
D
f
(
v
x
)
d
v
x
.
Let us consider a prevalent case with a Maxwell distribution over velocities, so that
the distribution function has the form
C
exp
,
m
x
2
T
v
f
(
v
x
)
D
where
T
is the atom temperature expressed in energy units,
m
is the radiating parti-
cle mass, and
C
is the normalization constant. From this we find for the frequency
distribution function that is normalized according to condition (2.134)
mc
2
2
exp
.
1/2
1
ω
mc
2
(
ω
ω
0
)
2
a
ω
D
(2.138)
π
T
2
0
2
T
ω
0
Note that the ratio of a typical spectral line width due to Doppler broadening
Δ
ω
D
to the photon frequency
0
, that is, the ratio of the atom thermal velocity to the
light velocity, is relatively small:
ω
r
T
mc
2
Δ
ω
D
0
.
ω
For example, for helium atoms at room temperature the right-hand side of this
relation is 2.6
10
6
.
Interaction of a radiating atomwith surrounding atomic particles may determine
broadening of spectral lines in gases or plasmas. We divide this broadening into
two limiting cases. In the first case, collision of individual particles of this plasma
with a radiating atom proceeds fast, and these collisions are seldom, that is, at
each moment interaction of a radiating atom is possible with one surrounding
atomic particle only. This mechanism relates to a rare plasma. The other limiting
case corresponds to interaction of a radiating atom with many surrounding atomic
particles and relates to a dense plasma. The first case of broadening of spectral lines
is described by the impact theory, and the second case relates to the quasistatic
theory of broadening of spectral lines.
In the case of impact broadening of spectral lines, the phase of the transition ma-
trix element changes during each interaction of a radiating atom with an incident
atomic particle. Then the width of a spectral line is determined by collisions of a
radiating atomwith atomic particles of a plasma and is defined by the cross section
of these collisions. Accurately, the frequency distribution function is given by the
Lorenz formula (2.136), and in this formula we have [150] 1/
τ
D
N
h
v
σ
i
,where
N
t
