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.

MODULE 14 (701) The Fermi Golden Rule (FGR) governs the rate of a transition between a pair of states situated in a radiation field.  (E fi ) 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]  (E fi ) is relatively straightforward – a count of the total number of states at energy E fi that can be accessed by the perturbation. The V fi quantity requires some further development.

MODULE 14 (701) In the FGR derivation the perturbation was H fi which naturally (through the hamiltonian) leads to an energy term, V fi. 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.

MODULE 14 (701) 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.

MODULE 14 (701) 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.

MODULE 14 (701) 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 E 1. The state can be described as where  1 is the time-independent wave function.

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

MODULE 14 (701) 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 E 1 or E 2, and the wavefunction describing the system would be a linear combination of the eigenstate functions.

MODULE 14 (701) 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.

MODULE 14 (701) The Euler relationships 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 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

MODULE 14 (701) 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 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.

MODULE 14 (701) 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 (  = (E i –E f )/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 This matrix element is named the transition dipole moment.

MODULE 14 (701) 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.

MODULE 14 (701) 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.

MODULE 14 (701) 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 where  represents the angular frequency of the oscillating perturbation.

MODULE 14 (701) 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

MODULE 14 (701)  (E fi ) 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.

MODULE 14 (701) 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:  fi is an average value over the three Cartesian coordinates  rad is the density of radiation states at the transition energy B f<-i is the Einstein coefficient for stimulated absorption.

MODULE 14 (701) 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. 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 B f->i

MODULE 14 (701) 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 N i in state and N f 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. Since the Einstein coefficients for the upward and downward processes are identical, the k values are identical which implies that N i and N f are also identical.

MODULE 14 (701) 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. where A f->i is the Einstein coefficient of spontaneous emission. The equilibrium condition is now which is now in accord with Boltzmann.

MODULE 14 (701) 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 Comparing the two equations confirms that B f->i = B f<-i and that

MODULE 14 (701) Thus the rates of spontaneous emission and of stimulated emission for a particular system are related. The value of the multiplier of B f  i on the RHS for 500 nm photons is ca 1.3 x , showing that spontaneous emission is not a very efficient process Moreover, because of the equality B f->i = B f<-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).

MODULE 14 (701) 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

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 Our goal of finding some quantity to substitute for V fi 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.

MODULE 14 (701) 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.