Environmental Engineering Reference
In-Depth Information
2.4.6
Broadening of Spectral Lines
Radiation due to a certain radiative transition is monochromatic, that is, the width
of frequencies of emitted photons is small compared with the frequency of this
radiative transition. Nevertheless this narrow spectrum is of importance for reab-
sorption processes, and therefore we analyze now the mechanisms of broadening
of spectral lines [45, 164]. Let us introduce the distribution function of emitted pho-
tons
a
ω
,so
a
ω
d
ω
is the probability that the frequency of the emitted photon lies
in a range between
. Because the probability is normalized to 1, the
frequency distribution function of photons
a
ω
satisfies the relation
Z
a
ω
d
ω
and
ω
C
d
ω
ω
D
1 ,
(2.134)
and the distribution function of emitted photons is given by
a
ω
D
δ
(
ω
ω
0
) ,
(2.135)
where
0
is the frequency of an emitted photon. This means that the spectral line
is narrow, that is, the line width
ω
Δ
ω
is small compared with the frequency:
Δ
ω
ω
0
.
We first find the frequency distribution function
a
ω
due to a finite lifetime
of transition states. Note that the amplitude of radiative transitions is expressed
through the matrix element of the dipole moment operator between transition
states, and we analyze the time dependence of the matrix element. Since the sta-
tionary wave function of state
k
contains a time factor exp(
), where
E
k
is
the energy of this state, the matrix element for transition between stationary states
i
and
k
includes the time factor exp(
iE
k
t
/
„
i
ω
0
t
), where
ω
D
(
E
i
E
k
)/
„
. If we account
0
for a finite lifetime
τ
r
of an upper state and represent the wave function of this state
as exp(
r
), the time dependence of the matrix element for transition
between two states has the form
c
(
t
)
iE
i
t
/
„
t
/2
τ
r
). Hence, the frequency
dependence for a matrix element, the Furie component from the function
c
(
t
), is
given as
c
ω
j
D
(
i
ω
0
t
t
/2
τ
i
(
ω
ω
0
)
C
1/2
τ
j
, and therefore the frequency distribution function
of emitted photons is
1
1
2
a
ω
Dj
c
ω
j
D
πτ
2
,
(2.136)
2
(
ω
ω
0
)
2
C
ν
ν
D
where
) is the spectral line width. Above we accounted for the normal-
ization of the frequency distribution function according to condition (2.134). A fre-
quency distribution function of this shape is named a Lorenz distribution function.
We now consider another mechanism of broadening of spectral lines due to mo-
tion of emitting atoms. Indeed, if an emitting particle moves with respect to a
receiver with velocity
1/(2
τ
0
, according to the
Doppler effect it is perceived by the receiver as having frequency
x
and emits a photon of frequency
ω
v
0
1
c
,
C
v
x
ω
D
ω
(2.137)
