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MODULE 14 (701)
The Absorption of Light by Atoms and Molecules
The interaction of light with matter is one of the most
fundamental and important of natural phenomena.
Our existence is completely dependent on the food generated by
photosynthesis (absorption of sunlight by green plants, e.g.).
Photosynthesis is one of the multitude of processes that cannot
proceed under thermal conditions but require the input of light.
It is one of the phenomena encompassed in the study of the
Photosciences.
In all photoprocesses the absorption of a photon is the primary
initiating event, and we start by studying the interaction.
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The Fermi Golden Rule (FGR) governs the rate of a transition
between a pair of states situated in a radiation field.
2
k f i  2 V fi r ( E fi )
r(Efi) is the density of final (f) states at the transition frequency
and the modulus term is the perturbation that drives the
transition.
[the convention is to write the upper state first and the direction of the arrow indicates
the direction of the transition]
r(Efi) is relatively straightforward – a count of the total number of
states at energy Efi that can be accessed by the perturbation.
The Vfi quantity requires some further development.
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In the FGR derivation the perturbation was Hfi which naturally
(through the hamiltonian) leads to an energy term, Vfi.
When we are considering the interaction of light with matter an
energy term has little relevance.
We need to relate the perturbation influence to something that we
can either measure, or calculate with a reasonable degree of
accuracy.
Light is a form of electromagnetic radiation (EMR) that is
classically described by orthogonal oscillating electric and
magnetic fields.
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The frequency of the oscillation determines the color of the light
and the amount of energy in the photon.
In the classical picture light was thought of as a wave,
characterized by a wavelength and a period, permeating space
between the source and the observer.
Einstein argued that light was composed of bundles, or particles,
localized in a small volume of space, traveling with velocity, c.
These photons can be thought of as electromagnetic
disturbances, or impulses, that continue in straight line motion
unless scattered or annihilated by interactions with matter.
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The annihilation process is absorption.
The photon is completely eliminated and its energy becomes part
of the total energy of the absorbing entity.
As we saw during our perturbation treatment, a basic requirement
for an effective interaction is an energy correspondence between
the photon and a pair of energy levels of the absorber.
The absorbers that we deal with in the Photosciences are usually
molecules, but sometimes atoms.
Generally speaking, both follow the same rules, but molecules
have vibrations and rotations to add to the complications.
In this Module we shall be referring to molecules but keep in mind
that the atomic situation is quite similar.
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All absorbers have bound electrons in common.
The presence of electrons in their appropriate orbitals gives rise to
chemical bonding and molecular stability.
In a given quantum state the electronic charge distribution is a
fixed quantity, independent of time.
If that quantum state is subjected to an electrical field, the charge
distribution will become polarized (a perturbation).
If the field is oscillating at the right frequency, the perturbation
can promote the transition to another quantum state.
If this happens then the absorber has undergone an electric
dipole transition and the photon has been annihilated.
The frequencies of ultraviolet and visible light are appropriate for
promoting electric dipole transitions in atoms and molecules.
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Consider the combination of a pair of states and a photon
1  h  2
The double-headed arrow is used because the principle of
microscopic reversibility requires that every process must have its
inverse.
Let us first consider state 1
If it is an eigenstate a measurement of energy provides only E1.
The state can be described as
1   1 ( x)e  iE1t /
where 1 is the time-independent wave function.
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Then
11*  1 *( x)e
 iE1t /
1 ( x)e
iE1t /
 1 *  x 1 ( x)
and the probability density is time-independent.
Thus eigenstate 1 is a stationary state of the system.
The probability density and the corresponding charge distribution
are independent of time, even though the wavefunctions
themselves fluctuate in time.
Exactly the same conclusion applies to eigenstate 2
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Now consider a transition in progress in a system that is a mix of
the two eigenstates and a photon of specified frequency.
Measurements of the energy (??) would lead to one of the
eigenvalues E1 or E2, and the wavefunction describing the system
would be a linear combination of the eigenstate functions.
  c1 1e iE1t /  c2 2e iE2t /
 *   c1*c1 1* 1  c2*c2 2* 2
 c2*c1 2* 1 ei ( E2  E1 )t /
 c1*c2 1* 2 e  i ( E2  E1 ) t /
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 *   c1*c1 1* 1  c2*c2 2* 2
 c c  1 e
*
2 1
*
2
i ( E2  E1 ) t /
 c1*c2 1* 2 e  i ( E2  E1 ) t /
In the first two terms in the equation the time-dependence has
cancelled, as earlier.
These terms represent the (stationary) charge distributions of the
pure eigenstates.
The last two terms contain complex exponentials that oscillate in
time at some frequency, call it .
Thus the total charge distribution of the superposition contains
contributions from the unperturbed states plus two oscillatory
terms that arise from the mixing.
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The Euler relationships
E2  E1 E2  E1


2
h
The time-dependent changes of the electronic charge distribution
that are set up during a transition are oscillations of the electric
dipole moment (m),
This is given by
m  er
and is a vector formed by the product of the electronic charge (a
scalar) with the expectation value of its displacement vector from
the center of mass of the nucleons
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We need to find an expression that will provide the amplitude of
the oscillating electric dipole moment of the system when it is in
the mixture of states (during the transition).
i.e. calculate the expectation value of the dipole moment operator
mˆ  er
This is the form of the operator for a one electron, one nucleon
(hydrogenic) atom. For a polyatomic system the product of all the
charges with their position vectors need to be summed.
We designate our two pure eigenstates as initial (i) and final (f),
and set both coefficients equal to unity.
mˆ   f mˆ  f   i mˆ  i 
e
i ( Ei  E f ) t /
 i mˆ  f  e
 i ( Ei  E f ) t /
 f mˆ  i
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The first two integrals are zero for reasons of parity (i.e. the pure
states have no oscillating dipoles).
It is the last two terms that describe the oscillations in the
required expectation value.
It is the magnitude of the integrals that gives the amplitude of the
oscillations ( = (Ei –Ef)/h)).
Thus the amplitude of the oscillations of the electric dipole
moment during an electric dipole transition between the states is
proportional to a matrix element, defined by
ˆ i d  f mˆ fi i
m fi   m
*
f
This matrix element is named the transition dipole moment.
i
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The value of the transition dipole moment depends on both states
involved in the transition,
During the radiative event the system is described by a mixture of
both i and f
Thus when our absorber undergoes a transition induced by a
i
photon, it is the perturbation of the electric dipole of the absorber
by the photon that mixes the states
The matrix element for the mixing appears in the FGR.
Thus when the wavefunction is a mixture of wavefunctions of two
non-degenerate quantum states then the probability density of
the system is no longer independent of time.
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The wavefunction contains terms that oscillate in time with a
specific frequency, governed by the energy difference.
It is the transition dipole moment that governs the amplitude of
the electric dipole moment during the transition.
Moreover, if an electronic system is in a state that is of higher
energy than another state, then a transition to the lower energy
state can occur with the concomitant loss (emission) of a photon
(of frequency equivalent to the energy difference between the two
states).
If the lower state is the ground state of the system then no
further photon emission is possible because there is no state of
lower energy with which to mix and thus to generate the required
oscillatory perturbation.
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However, when the ground state finds itself in the presence of a
photon of the appropriate frequency, the perturbing field can
induce the necessary oscillations, causing the mix to occur.
This leads to the promotion of the system to the upper energy
state and the annihilation of the photon.
This process is stimulated absorption (usually simply absorption).
Einstein pointed out that the Fermi Golden Rule correctly
describes the absorption process, but it was inadequate in
accounting for all contributions to the emission of radiation from
upper electronic states.
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Consider an absorber situated in a radiation field that can be
characterized as an oscillatory electric (and magnetic) field.
The frequency of the radiation is w  2 and the electric vector of
the EMR is aligned in the z-direction.
From our earlier work in perturbation theory
Hˆ  Hˆ (0)  Hˆ (1) (t )
Hˆ (1) (t )  2 Hˆ (1) cos w t
where w represents the angular frequency of the oscillating
perturbation.
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If the perturbation is light with an electric vector in the z-direction
Hˆ (1) (t )   m ze (t )
where mz is the z-component of the dipole moment and e is the
(time varying) field strength, given by
e (t )  2e 0 cos wt
In deriving the Fermi Golden Rule we used the identity
Hˆ (1)
fi  V fi
2
k f i  2 V fi r ( E fi )
k f i 
2
2
m z , fi e 2 r ( E fi )
MODULE 14 (701)
k f i 
2
2
m z , fi e 2 r ( E fi )
r(Efi) is the density of the continuum states with energy E fi  w fi
wfi is the transition frequency
mz,fi is the z-component of the matrix element that generates the
transition dipole moment.
Thus the perturbation matrix element in the Golden Rule is related
to the transition dipole moment.
Recall that FGR was derived for the transition from an initial
discrete state to a continuum of final states, induced by
monochromatic radiation at the transition frequency.
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Our practical interest is a little different.
we are more interested in the transition rate from a discrete initial
state to a discrete final state under the influence of nonmonochromatic radiation.
Borrowing from classical electromagnetic theory:
 m 2 
fi
 r rad ( E fi )
k f i  
2
 6e 0 


 B f i r rad ( E fi )
mfi is an average value over the three Cartesian coordinates
rrad is the density of radiation states at the transition energy
Bf<-i is the Einstein coefficient for stimulated absorption.
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We see that a photon of frequency
  ( E f  Ei ) / h
can provide the perturbation to induce an initial state to generate
a final state, without regard for which of the states is higher in
energy.
Thus when the initial state is higher in energy than the final state
the perturbing influence of a photon of the appropriate frequency
will mix the upper and lower states to create another photon and
leave the system in the lower energy state.
i  h  f
f  h  i  2h
This process is called stimulated emission and is the inverse of
(stimulated) absorption--Einstein coefficient Bf->i
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These two Einstein coefficients, through an argument based on
the hermiticity of the perturbation hamiltonian, are numerically
the same. Thus
B f i  Vif Vif *  V fi *V fi  B f i
Consider an ensemble of molecules in a radiation field.
At some time there are Ni in state i and Nf in state f
At equilibrium there will be the same rate of conversion of upper
to lower as there is from lower to upper, i.e.
N i k f i  N f k f i
Since the Einstein coefficients for the upward and downward
processes are identical, the k values are identical which implies
that Ni and Nf are also identical.
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But at thermal equilibrium the Boltzmann condition requires that
Nf
Ni
e
 E fi / k BT
Einstein proposed that an upper state could also spontaneously
deactivate to create a photon without photon stimulation.
k f i  Af i  B f i rrad ( E fi )
where Af->i is the Einstein coefficient of spontaneous emission.
The equilibrium condition is now
Ni B f i r rad ( E fi )  N f  Af i  B f i rrad ( E fi )
which is now in accord with Boltzmann.
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Inserting the Boltzmann equation into the equilibrium condition
rrad ( E fi ) 
Af i / B f i
( B f i / B f i )e
E fi / kBT
1
In Planck’s description of black body radiation he showed that the
density of states of an electromagnetic field was given by
rrad ( E fi ) 
8 h / c
e
3
fi
E fi / kBT
3
1
Comparing the two equations confirms that Bf->i = Bf<-i and that
A f i 
8 h 3fi
c
3
B f i
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Thus the rates of spontaneous emission and of stimulated
emission for a particular system are related.
A f i 
8 h 3fi
c
3
B f i
The value of the multiplier of Bfi on the RHS for 500 nm photons
is ca 1.3 x 10-13, showing that spontaneous emission is not a very
efficient process
Moreover, because of the equality Bf->i = Bf<-i the rate constant of
spontaneous emission from an upper state is proportional to the
rate constant for (stimulated) absorption of photons by the lower
state (more later).
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One important fact to note from the last equation is that the
efficiency of the spontaneous emission process increases as the
third power of the frequency.
Thus at very high frequencies, e.g. X-rays, the excited states are
so short lived that significant populations of excited states are
hard to maintain. Thus stimulated emission events are disfavored
and lasing in the X-ray region is difficult to achieve.
Quantum electrodynamics theory tells us that the spontaneous
emission process is in reality induced by the presence of zeropoint fluctuations in the radiation field, even in the absence of any
“photon gas”, these fluctuations exist
MODULE 14 (701)
We have seen that the rate of photon absorption to promote a
transition between electronic states is governed by the transition
dipole moment, or the matrix element of the electric dipole
moment taken between the initial and final states
m fi  f mˆ fi i
Our goal of finding some quantity to substitute for Vfi in the FGR
and thence to calculate the transition rate is in sight.
We need to establish the form of the dipole moment operator
(using the Einstein coefficients), and of the wave functions of the
initial and final states.
Calculations such as these can be done for simple situations.
There is an experimental approach also that we shall examine.
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It is important to recognize that even though there is a
correspondence between a photon energy and the energies of a
pair of molecular states, an electronic transition will not
necessarily occur (the energy condition is necessary but not
sufficient).
This is because the magnitude of the integral in the matrix
element can be zero, or near zero.
This leads to the idea of allowed and forbidden transitions and to
the concept of selection rules.
Selection rules will be examined later.