3.5 Excitation by a Continuous Wave
93
f (ν a , ν b ) = 1.4407 · 10
−19
ν b
ν a
ε(ν) dν = 4.3190 · 10
−9
ν b
ν a
ε(ν) dν (3.61)
where, in the second integral, the frequency unit is cm
−1 instead of Hz.
Equations (3.57–3.60) show that the quantities ν i0 B 0,i , ν i0 μ
2
i0 , and f i0 , all proportional to each other, are additive contributions to the absorption spectrum, while
B 0,i or μ
2
i0 are not. So, when spectral lines that are broadened coalesce to a single
band, with or without a structure (i.e., distinguishable peaks), their f i0 add up to
make the total f (ν a , ν b ) of the band. Of course, any interaction that causes a broadening can also alter the oscillator strengths, but the comparison between two different
situations (say, for instance, gas phase and solution) should be based on (a sum of)
the oscillator strengths. The same holds for the comparisons between theory, which
very often only determines transition frequencies and dipole moments, and experiment, which offers spectral bands with variable heights and widths depending on the
environment and other conditions, but fairly invariant oscillator strengths.
3.6 Spontaneous Emission
In 1916 Einstein showed that photon absorption and stimulated emission would never
lead to a state of equilibrium between matter and radiation and concluded that a different emission mechanism must exist [2]. Equilibrium between molecules and photons
is not the common situation in photochemical experiments and in natural (planetary
or cosmic) conditions. Most frequently, hot objects emit light that is absorbed by the
colder ones or escapes in regions where matter is rarefied. However, we can imagine a system where photons are confined (by mirror walls) and a gas of atoms or
molecules absorbs and emits light. The interactions among molecules and between
molecules and photons allow the exchange of energy among all components (note
that photons do not interact among themselves). So, if the system is thermostated,
in the long term the population N i of any molecular state |i will be proportional to
exp(−E i /K B T ). The equilibrium distribution of photons in the frequency spectrum
gives place to the spectral energy density
U ν (ν) =
8π hν
3
c 3
e
hν/K B T
− 1
−1
(3.62)
which is the famous “black body” formula proposed by Max Planck in 1900.
If we consider any two molecular states i and j, with E i < E j , there will be i → j
transitions with photon absorption and j → i transitions with stimulated emission.
Both transition rates will be proportional to B i, j U ν (ν i j ) and to the respective populations of the initial states. Since N i > N j , more photons will be absorbed than emitted,
which is the normal situation when a sample at room temperature is irradiated with
93
f (ν a , ν b ) = 1.4407 · 10
−19
ν b
ν a
ε(ν) dν = 4.3190 · 10
−9
ν b
ν a
ε(ν) dν (3.61)
where, in the second integral, the frequency unit is cm
−1 instead of Hz.
Equations (3.57–3.60) show that the quantities ν i0 B 0,i , ν i0 μ
2
i0 , and f i0 , all proportional to each other, are additive contributions to the absorption spectrum, while
B 0,i or μ
2
i0 are not. So, when spectral lines that are broadened coalesce to a single
band, with or without a structure (i.e., distinguishable peaks), their f i0 add up to
make the total f (ν a , ν b ) of the band. Of course, any interaction that causes a broadening can also alter the oscillator strengths, but the comparison between two different
situations (say, for instance, gas phase and solution) should be based on (a sum of)
the oscillator strengths. The same holds for the comparisons between theory, which
very often only determines transition frequencies and dipole moments, and experiment, which offers spectral bands with variable heights and widths depending on the
environment and other conditions, but fairly invariant oscillator strengths.
3.6 Spontaneous Emission
In 1916 Einstein showed that photon absorption and stimulated emission would never
lead to a state of equilibrium between matter and radiation and concluded that a different emission mechanism must exist [2]. Equilibrium between molecules and photons
is not the common situation in photochemical experiments and in natural (planetary
or cosmic) conditions. Most frequently, hot objects emit light that is absorbed by the
colder ones or escapes in regions where matter is rarefied. However, we can imagine a system where photons are confined (by mirror walls) and a gas of atoms or
molecules absorbs and emits light. The interactions among molecules and between
molecules and photons allow the exchange of energy among all components (note
that photons do not interact among themselves). So, if the system is thermostated,
in the long term the population N i of any molecular state |i will be proportional to
exp(−E i /K B T ). The equilibrium distribution of photons in the frequency spectrum
gives place to the spectral energy density
U ν (ν) =
8π hν
3
c 3
e
hν/K B T
− 1
−1
(3.62)
which is the famous “black body” formula proposed by Max Planck in 1900.
If we consider any two molecular states i and j, with E i < E j , there will be i → j
transitions with photon absorption and j → i transitions with stimulated emission.
Both transition rates will be proportional to B i, j U ν (ν i j ) and to the respective populations of the initial states. Since N i > N j , more photons will be absorbed than emitted,
which is the normal situation when a sample at room temperature is irradiated with