Environmental Engineering Reference
In-Depth Information
where, in accord with Figure 2.27, we denote the lower state by subscript
o
and the
upper state by subscript
. Equation (2.108) accounts for the fact that no transitions
occur in the absence of photons (
n
ω
D
0) and only single-photon transitions take
place. The quantity
A
does not depend on the electromagnetic field strength and is
determined only by properties of the atom.
The probability per unit time for an atomic transition with emission of a photon
can be represented in the form
1
τ
w
(
,
n
ω
!
o
,
n
ω
C
1)
D
r
C
Bn
ω
.
(2.109)
Here 1/
r
is the reciprocal lifetime of the upper state with respect to spontaneous
radiative transitions (those that proceed in the absence of the electromagnetic field)
to the lower state and
B
refers to the radiation stimulated by the external electro-
magnetic field. Both values depend only on atomic properties. The quantities
A
and
B
in relations (2.108) and (2.109) are known as the Einstein coefficients [147].
Relationships among 1/
τ
,
A
,and
B
can be obtained by an analysis of the ther-
modynamic equilibrium existing between the atoms and photons. The relation be-
tween the number densities of atoms
N
0
and
N
in the ground and excited states,
respectively, are given by the Boltzmann law (1.42):
τ
g
g
0
N
0
exp
,
„
T
N
D
where
g
0
and
g
are the statistical weights of the ground and excited states, and
the photon energy
coincides with the energy difference between the two states.
The mean number of photons in a given state is determined by the Planck distri-
bution (1.56):
„
ω
exp
1
1
„
T
n
ω
D
.
In thermodynamic equilibrium, the number of emissions per unit time must be
equal to the number of absorptions per unit time. Applying this condition to a unit
volume, we have
N
0
w
(
o
,
n
ω
!
o
,
n
ω
).
According to (2.108) and (2.109), this relation takes the form
,
n
ω
1)
D
N
w
(
,
n
ω
1
!
1
τ
C
.
N
0
An
ω
D
N
Bn
ω
(2.110)
Using the above expressions for the connection between the equilibrium number
densities of atoms and the equilibrium average number of photons in a given state,
we obtain
A
for the Einstein coefficients. We then find
the rates of the single-photon processes to be
D
g
/(
g
0
τ
)and
B
D
1/
τ
g
g
0
1
τ
n
ω
τ
w
(
o
,
n
ω
!
,
n
ω
1)
D
n
ω
,
w
(
,
n
ω
!
o
,
n
ω
1)
D
r
C
. (2.111)
τ
r
r
Note that thermodynamic equilibrium requires the presence of stimulated radia-
tion, which is described by the last term and is of fundamental importance.
