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.

(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]

(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.

22 ( )f i fi fik V E

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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.

MODULE 14 (701)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

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

If it is an eigenstate a measurement of energy provides only E1.

The state can be described as

1 2h

1 /1 1( ) iE tx e

where 1 is the time-independent wave function.

1

MODULE 14 (701)Then

and the probability density is time-independent.

Thus eigenstate 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

1 1/ /1 1 1 1 1 1* *( ) ( ) * ( )iE t iE tx e x e x x

1

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. 1 2/ /1 1 2 2

iE t iE tc e c e

2 1

2 1

* * * *1 1 1 1 2 2 2 2

( ) /* *2 1 2 1

( ) /* *1 2 1 2

*i E E t

i E E t

c c c c

c c e

c c e

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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.

2 1

2 1

* * * *1 1 1 1 2 2 2 2

( ) /* *2 1 2 1

( ) /* *1 2 1 2

*i E E t

i E E t

c c c c

c c e

c c e

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The Euler relationships

2 1 2 1

2

E E E E

h

The time-dependent changes of the electronic charge distribution that are set up during a transition are

oscillations of the electric dipole moment (),

This is given by

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 operatorˆ 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.

( ) / ( ) /

ˆ ˆ ˆ

ˆ ˆi f i f

f f i i

i E E t i E E ti f f ie e

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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 ˆfi f i fid f i This matrix element is named the transition dipole

moment.

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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 and

Thus when our absorber undergoes a transition induced by a 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.

i

i

i f

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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.

MODULE 14 (701)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 and the electric vector of the EMR is aligned in the z-direction.

From our earlier work in perturbation theory(0) (1)ˆ ˆ ˆ ( )H H H t

(1) (1)ˆ ˆ( ) 2 cosH t H t

where 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

where z is the z-component of the dipole moment and is the (time varying) field strength, given by

In deriving the Fermi Golden Rule we used the identity

(1)ˆ ( ) ( )zH t t

0( ) 2 cost t

(1)ˆfi fiH V

22 ( )f i fi fik V E

22

,

2( )f i z fifi

k E

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(Efi) is the density of the continuum states with energy

fi is the transition frequency

z,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.

fi fiE

22

,

2( )f i z fifi

k E

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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 non-monochromatic radiation.

Borrowing from classical electromagnetic theory:2

20

( )6

( )f i

fi

f i rad fi

rad fi

k E

B E

fi is an average value over the three Cartesian coordinates

rad 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

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.

( ) /f iE E h

2

i h f

f h i h

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.

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

Consider an ensemble of molecules in a radiation field.

At some time there are Ni in state and Nf in state

At equilibrium there will be the same rate of conversion of upper to lower as there is from lower to upper, i.e.

* *f i if if fi fi f iB V V V V B

i f i f f iN k N k

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.

i f

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But at thermal equilibrium the Boltzmann condition requires that

Einstein proposed that an upper state could also spontaneously deactivate to create a photon without

photon stimulation.

/fi BE k Tf

i

Ne

N

( )f i f i f i rad fik A B E

where Af->i is the Einstein coefficient of spontaneous emission.

The equilibrium condition is now ( ) ( )i f i rad fi f f i f i rad fiN B E N A B E

which is now in accord with Boltzmann.

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Inserting the Boltzmann equation into the equilibrium condition

In Planck’s description of black body radiation he showed that the density of states of an electromagnetic field was

given by

/

/( )

( / ) 1fi B

f i f irad fi E k T

f i f i

A BE

B B e

3 3

/

8 /( )

1fi B

firad fi E k T

h cE

e

Comparing the two equations confirms that Bf->i = Bf<-i and that

3

3

8 fif i f i

hA B

c

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Thus the rates of spontaneous emission and of stimulated emission for a particular system are related.

The value of the multiplier of Bfi 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).

3

3

8 fif i f i

hA B

c

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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 zero-point fluctuations in the radiation field,

even in the absence of any “photon gas”, these fluctuations exist

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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 ˆfi fif 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.