The connection with experiment

The central idea is that rates of spectroscopic processes determine the intensity of the spectral response. We begin by determining those rates and then show how these are connected to the transition dipole moment, a molecular property.

Einstein coefficients

The classical treatment of the radiation field can account for absorption and stimulated emission, but not spontaneous emission. The rates of absorption and stimulated emission are proportional to the energy per unit volume per unit frequency interval, r(n) = du/dn. We use the blackbody distribution for the form of r(n), but the model is generally valid for any distribution. The rate of transitions out of a state is proportional to the number of molecules in that state. We designate the number of molecules in the lower and upper levels N₁ and N₂, respectively.

The rate of the transition from state 1 to state 2 is called W₁₂, and vice versa. We can also equate the rates with the rate of change of the population in each state.

The rates are illustrated below.

We make the following definitions

B₁₂ is the coefficient for absorption.

B₂₁ is the coefficient for stimulated emission.

A₂₁ is the coefficient for spontaneous emission.

The transition rates are:

In the absence of radiation we have

We can exponentiate both sides to obtain

The decay is a single exponential. This leads to the conclusion the radiative lifetime (corresponding to spontaneous emission) is

We will show that A₂₁ is proportional to the square of the transition dipole moment. We will assume the thermal equilibrium is maintained. Thus,

where E₂ and E₁ are the energies and g₂ and g₁ are the degeneracies of the upper and lower states, respectively. The Einstein condition is E₂ - E₁ = hn. So we have

To maintain equilibrium the rate of upward and downward transitions must be equal. This means W₁₂ = W₂₁. Therefore,

We solve for r(n).

This expression can be compared with the expression for blackbody radiation:

The two are identical provided

The significance of these relations are:

The rate constant for absorption is equal to the rate constant for stimulated emission for two nondegenerate states. If they are degenerate we can still calculate these coefficients readily.
The rate constant for spontaneous emission is a strong function of frequency. As the frequency increases spontaneous emission competes more effectively with stimulated emission. In the optical region of the spectrum spontaneous emission greatly exceeds stimulated emission. However, in the far infrared stimulated emission effectively competes spontaneous emission. This has important consequences for the observation of spectra since states are typically thermally populated in the far infrared region as well.

To achieve laser emission we must create a situation where stimulated emission can occur. To do this we create a population inversion N₂ > N₁. (Note: this is not possible in a two level system considered and we shall see how to do this in three or four level system.) However, even if we create a population inversion we must remember that spontaneous emission is always present and must be separated from the stimulated emission that gives rise to the laser effect.

Relationship between the Fermi Golden Rule and the Einstein rate constants

Absorption rate constant

Starting with the definition of energy density

and the Fermi Golden Rule

we can obtain the transition probability per unit time w₁₂. In the FGR expression m₁₂ is the transition moment of the molecule and E₀ is the electric field vector of the light interacting with the molecule. The delta functions guarantee that the energy the incident light must equal that of the transition n₂₁. The transition probability per unit time as a function of the incident energy density is:

where e (hat) is a unit vector along the direction of the electric field of the light. What happened to the delta function? We have used the property of a delta function that the integral over d(x - x₀) is equal to the function evaluated at x₀.

The energy density is typically polarized along one direction, x, y, or z. Therefore, the transition probability per unit time is 1/3 as large as the above derivation.

We can compare this expression with Einstein rate

Thus,

Emission rate constant

Based on the relationship between A₂₁ and B₁₂ we have

We have used c/n for the speed of light in a medium with index of refraction n. This derivation assumes that all of the radiation along a given direction (e.g. x-polarized) is collected. In reality, we must account for the experimental geometry. The collection optic has a finite size and is at a fixed distance from the sample. To take this properly into account we use a photon density of states as a function of the solid angle.

For emission we can make a substitution similar to the above

This is valid because the energy density is the number of photons N per unit volume V times the energy per photon.

Here we have made use of the non-intuitive property of a delta function

We make the association between the delta function and the photon density of states

We express the emission rate as a differential rate per unit solid angle

Again as above we can assume that radiation is being detected along one polarization and we divide the above expression by 3.

Furthermore, we can substitute c/n for c as above and w = 2pn. Thus,

which is the same as the expression above if we integrate over dW.

Since there are two polarizations of light there is an additional factor of two required. With this factor the expression is identical to the expression for A₂₁ given above.

Note that (as above) the rate of upward transitions is equal to the rate of downward transitions at equilibrium, thus W₂₁ = W₁₂. The number of molecules N represents the number in the particular state and can N₁ or N₂ depending on the context.

To summarize, the point here is that we can determine the emission a function of the solid angle of the collection optics as shown below.

Relationship of the transition moment to experimental observables

Fluorescence Lifetime and Yield

These expressions show that the rate of absorption and spontaneous emission are proportional to the square of the transition dipole moment. Since A₂₁ is proportional to 1/t_rad we see that the radiative lifetime gets shorter the larger the transition moment. Thus, we have an experimental connection between the measured rate of decay of an excited state and the transition moment. However, this is not the best way to estimate the transition moment. First it must be realized that the observed fluorescence lifetime may not correspond to the intrinsic radiative lifetime.

If there are both non-radiative and radiative pathways to the ground state then the observed rate of the fluorescence decay will be

k_fluorescence = k_nonradiative + k_radiative

1/t_fluorescence = 1/t_nonradiative + 1/t_radiative

The intrinsic fluorescence lifetime can be determined from a measurement of both the decay kinetics and the fluorescence quantum yield. The emission yield is

The absolute fluorescence quantum yield is still not easy to measure. The amount of fluorescence collected depends on the solid angle of the collection optics in a fluorometer as shown in the Figure above. In practice one uses a standard fluorescent dye and compares this to the unknown of interest. The relative yield of the known compared to the standard dye can be used to determine the absolute quantum yield.

There are additional complications if other states are present. We have thus far only discussed a two level system. There are several other processes that can compete with radiative decay including:

Intersystem crossing (e.g. conversion from a singlet to triplet state)
Electron transfer or energy transfer to a neighboring molecule.

Integrated Absorption Band

The transition moment can also be estimated from the integrated absorption and emission spectra. To see this we start with Beer's law.

The concentration is C and the extinction coefficient is Î in units L mol^-1 cm^-1. The factor of 2.303 is the conversion from base e to base 10. Usually we see Beer's law as

If there are N molecules per cm³ then we can also express the change in intensity as

The above way of writing the change in intensity considers the photons of energy hn absorbed or emitted by N molecules each with a probability w₁₂. We can write

It is convenient to replace I by cu/n and the molar concentration C by 1000N/N_A, where N is the number of molecules per cm³ and N_A is Avagadro's number.

using the definition

This expression can be related the B₁₂ coefficient.

Finally, this expression can be related to the square of the transition moment.

This expression shows that the magnitude of the square of the transition moment can be related to the integrated band regardless of the band shape. This suggests that there is a shape factor that does not change the overall strength of the absorption.