The Rates of Absorption and Stimulated Emission

Suppose that before we turn on the light our electron is in a state described by wavefunction f. In the presence of the oscillating radiation field, this and the other solutions to the Schrodinger equation for the unperturbed system become unsatisfactory they no longer represent stationary states. However, we can represent the 

The probability that the electron has made a transiticHi from Pa to Fi, by time r is obtained by integrating Eq. (4.6b) from time t = 0 to t (Eq. 2.55) and then evaluating IC (t)P. Integrating Eq. (4.6b) is straightforward, and gives the following result  

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The coefficients Asi, B i, and Bu. came to be known as the Einstein coefficients for spontaneous emission and for stimulated emission and absorption of radiation, respectively. The rates of spontaneous and stimulated emission and Wei are connected by the expression... 

In addition to the spontaneous emission of excited molecules, fluorescence and phosphorescence (Section 2.1.1), the interaction of electromagnetic radiation with excited molecules gives rise to stimulated emission, the microscopic counterpart of (stimulated) absorption. Albert Einstein derived the existence of a close relationship between the rates of absorption and emission in 1917, before the advent of quantum mechanics (see Special Topic 2.1). 

Here, Ri f and Rf i are the rates (per moleeule) of transitions for the i ==> f and f ==> i transitions respeetively. As noted above, these rates are proportional to the intensity of the light souree (i.e., the photon intensity) at the resonant frequeney and to the square of a matrix element eonneeting the respeetive states. This matrix element square is oti fp in the former ease and otf ip in the latter. Beeause the perturbation operator whose matrix elements are ai f and af i is Hermitian (this is true through all orders of perturbation theory and for all terms in the long-wavelength expansion), these two quantities are eomplex eonjugates of one another, and, henee ai fp = af ip, from whieh it follows that Ri f = Rf i. This means that the state-to-state absorption and stimulated emission rate eoeffieients (i.e., the rate per moleeule undergoing the transition) are identieal. This result is referred to as the prineiple of microscopic reversibility. 

By Equation (5.17), we see that the probability of spontaneous emission is proportional to I/LXIso we can use the same selection rules as previously established for absorption and stimulated emission to predict the allowance of spontaneous emission. Moreover, it can be noted that A is proportional to ci>l and so, for a small energy separation between the two levels of our system, the radiative emission rate A should also be small. In this case, noimadiative processes, described by A r (see Equation (1.17)), can be dominant so that no emitted light is observed. 

Since our system is in equilibrium, the number of absorption transitions i f per unit time must be equal to the number of emission transitions / / per unit time. Considering that the light-matter interaction processes described in Chapter 2 (Figure 2.5) are taking place, in equilibrium the rate of absorption must be equal to the rate of (stimulated and spontaneous) emission. That is ... 

These equations are similar to those of first- and second-order chemical reactions, I being a photon concentration. This applies only to isotropic radiation. The coefficients A and B are known as the Einstein coefficients for spontaneous emission and for absorption and stimulated emission, respectively. These coefficients play the roles of rate constants in the similar equations of chemical kinetics and they give the transition probabilities. 

In addition to absorption and stimulated emission, a third process, spontaneous emission, is required in the theory of radiation. In this process, an excited species may lose energy in the absence of a radiation field to reach a lower energy state. Spontaneous emission is a random process, and the rate of loss of excited species by spontaneous emission (from a statistically large number of excited species) is kinetically first-order. A first-order rate constant may therefore be used to describe the intensity of spontaneous emission this constant is the Einstein A factor, Ami, which corresponds for the spontaneous process to the second-order B constant of the induced processes. The rate of spontaneous emission is equal to Aminm, and intensities of spontaneous emission can be used to calculate nm if Am is known. Most of the emission phenomena with which we are concerned in photochemistry—fluorescence, phosphorescence, and chemiluminescence—are spontaneous, and the descriptive adjective will be dropped henceforth. Where emission is stimulated, the fact will be stated. 

As we saw in Section 5.5, the rate of absorption between two molecular energy levels E] and E2 is exactly equal to the rate of stimulated emission, for a given density of resonant photons, with co = E2 — E )/h. Whether there will be net absorption or emission depends on the relative populations, N and N2, of the two levels. At thermal equilibrium, when the populations follow a Boltzmann distribution, with the lower level E more populated than the upper level E2, a net absorption of radiation of frequency w can occur. If the two populations are equal, there will be neither net absortion or emission since the rates of upward and downward transitions will exactly balance. Only if we can somehow contrive to achieve a population inversion, with N2> N, can we achieve net emission, which amounts to amplification of the radiation (the A in LASER). Thus, to construct a laser, the first requirement is to produce a population inversion. Laser action can then be triggered by a few molecules undergoing spontaneous emission. 

A simple relation will be derived between the probabilities for spontaneous and stimulated emission and absorption of radiation using well-known statistical distribution laws. Consider a system such as that illustrated in Fig.4.3 with two energy levels, E and E2, populated by and N2 atoms, respectively. Three radiative processes can occur between the levels, as discussed above. In the figure the processes are expressed using the so-called Einstein coefficients 6 2, B21 and A21, which are defined such that the rate of change in the population numbers is... 

Spin-orbit state selection by optical pumping is exemplified in Fig. 1 by the barium atom. The distribution in the spin-orbit levels of the metastable 6s5d term can be altered by irradiation on 5 of the lines of the 5d6p po <- 6s5d multiplet near 600 nm. If a cw laser is tuned to one of these atomic lines, then absorption and stimulated emission will occur at a rate proportional to the laser power. The upper level can also decay spontaneously to other lower spin-orbit levels, hence transferring jx)pulation permanently out of the initially pumped state. 

Fig. 5.1 The rate of absorption is proportional to p(y)Na, the rate of stimulated emission is proportional to p(yWb- Fot the rates of upward and downward transitions to balance, downward transitions must occur by an additional mechanism that is independent of the light intensity. This is fluorescence...

From the general considerations of transition intensities in Section 12.2, we know that the rate of absorption of electromagnetic radiation is proportional to the population of the lower energy state (N in the case of a proton NMR transition) and the rate of stimulated emission is proportional to the population of the upper state (Np). At the low frequencies typical of magnetic resonance, we can neglect spontaneous emission as it is very slow. Therefore, the net rate of absorption is proportional to the difference in populations, and we can write... 

Again, note the involvement of the optical field in the absorption and stimulated emission by the term p( 2i) but not in the spontaneous emission process. Under thermal equilibrium, as in the case of Equation 3.12 but modified for a semiconductor, the downward transition rate must be equal to the upward transition rate... 

We now consider the transition rates for absorption of and stimulated emission induced by the black body radiation and compare these rates to the spontaneous... 

The above treatment is valid quite generally, even if f2 — fi > huj. when the denominator in (4.17) is negative. Under the same conditions, the absorption coefficient in (4.15) is also negative, and the spontaneous emission rate in (4.17) remains positive. When the absorption coefficient is negative, stimulated emission overcompensates the rate of upward transitions and the 2-level system amplifies the incident light exactly as in a laser. The condition f2 — fi > tko is also known as the lasing condition and is called inversion. 

In the previous sections, it has been shown how powerful the time-resolved fluorescence techniques are in real time probing of photoinduced processes and in allowing the determination of reaction rates from fluorescence lifetimes. The present section is devoted to the method of UV/vis transient absorption spectroscopy, which is a key method in probing non emissive species and is thus crucial to detect photoreaction products or intermediates following optical excitation of molecules in their electronic excited states. When carried out on short time scales, i.e. with femtosecond to subnanosecond excitation sources, fluorescent species can also be detected by their stimulated emission. Combining time-resolved fluorometry and transient absorption spectroscopy is ideal for the study of photochemical and photophysical molecular processes. 

The probability of stimulated emission by an excited molecule, Bpv, is the same as that of the reverse process, absorption by the ground-state molecule. This is a consequence of the law of microscopic reversibility if the number of excited molecules Nm is equal to the number of ground-state molecules N , then the rates of stimulated absorption and emission must be equal. If by some means a population inversion can be produced, Nm > Nn, the net effect of interaction with electromagnetic radiation of frequency v will be stimulated emission (Figure 2.4). This is the operating principle of the loser. LASER is an acronym for light amplification by stimulated emission of radiation (Section 3.1). 

Eq. (1.125) shows that the transition rate of absorption, fVab, is proportional to the number of photons, nkv, while Eq. (1.126) suggests that, for the transition rate of emission, IVem, there exist two terms, one which is proportional to the number of photons and one independent of the number of photons emitted. Thus, absorption and emission of light are not symmetric phenomena, as suggested already by Einstein in his kinetic formulation for absorption and emission of light. The term proportional to the number of photons in the emission process is called stimulated emission. The term independent of nh. is called spontaneous emission. The stimulated emission is the basis of the laser oscillation, as will be described in Section 1.4.5. 

The light emitted in the spontaneous recombination process can leave the semiconductor, be absorbed or cause additional transitions by stimulating electrons in the CB to make a transition to the VB. In this stimulated recombination process another photon is emitted. The rate of stimulated emission is governed by a detailed balance between absorption, and spontaneous and stimulated emission rates. Stimulated emission oecurs when the probability of a photon causing a transition of an electron from the CB to VB with the emission of another photon is greater than that for the upward transition of an eleetron from the VB to the CB upon absorption of the photon. These rates are eommonly described in terms of Einstein s A and B eoeffieients [8, 43]. For semiconductors, there is a simple eondition describing the carrier density neeessary for stimulated emission, or lasing. This earner density is known as... 

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