Einstein coefficients of emission and absorption

Special Topic 2.1 Einstein coefficients of absorption and emission... 

Polyatomic molecules have broad absorption and emission bands due to the contributions of vibronic transitions. Einstein s equation (Equation 2.10) relates the coefficients of absorption and emission in a two-level system. Several extensions of... 

Lissamine molecules in the vicinity of gold nanoparticles because the [H oportionality of Einstein coefficients of absorption and emissiai must hold as long as absorption and emission occur without a Stokes shift. We find that the absorption cross section is changed by 30% for nanoparticles of 30 nm radius. This effect is ready included in the fluorescence quenching efficiency above and will be included in a similar way in the analysis of the following time resolved measurements. 

In the above rather simplified analysis of the interaction of light and matter, it was assumed that the process involved was the absorption of light due to a transition m - n. However, the same result is obtained for the case of light emission stimulated by the electromagnetic radiation, which is the result of a transition m -> n. Then the Einstein coefficients for absorption and stimulated emission are identical, viz. fiOT< n = m rt. 

P+ and P are the probabilities for absorption and emission, respectively B+ and B are the coefficients of absorption and of induced emission, respectively A- is the coefficient of spontaneous emission and p v) is the density of radiation at the frequency that induces the transition. Einstein showed that B+ = B, while A frequency dependence, spontaneous emission (fluorescence), which usually dominates in the visible region of the spectrum, is an extremely improbable process in the rf region and may be disregarded. Thus the net probability of absorption of rf energy, which is proportional to the strength of the NMR signal, is... 

The UV-VIS spectrum, usually an A or log A vs. plot, in a first approximation reflects the discrete electronic states as absorption maxima at different v ,ax positions which are correlated with the molecular structure and geometry. The extinction coefficient e or more likely the integral absorption / e(v) dv, which is approximately the product of times the halfwidth AVi/2> gives information on the transition dipole moment or the Einstein coefficient of absorption or induced emission B n, which are interrelated by... 

Hence, the Einstein coefficients for absorption, spontaneous emission, and stimulated emission are all simply related. The factor that enters in the spontaneous emission coefficient (Eq. 8.35) has had historical importance in the development of lasers, since it implies that spontaneous emission competes more effectively with stimulated emission at higher frequencies. High-frequency lasers have therefore been more difficult to construct. This is one of the reasons why X-ray lasers have only recently been built, and why the first laser was an ammonia maser operating on a microwave umbrella-inversion vibration rather than a visible laser. 

Einstein coefficients for absorption and stimulated emission, denoted by and respectively. The expressions for B j, and Bj are then confirmed by means of quantum mechanics using time-dependent perturbation theory. This enables the probability of stimulated emission and absorption of radiation to be given in terms of the oscillator strengths of spectral lines. Finally we show that there is close agreement between the classical and quantum-mechanical expressions for the total absorption cross-section and explain how the atomic frequency response may be introduced into the quantum-mechanical results. 

It is interesting to note that for a resonant transition (i.e. coinciding absorption and emission frequencies), the reciprocal of the radiative lifetime is equal to the Einstein coefficient Ai for spontaneous emission (see Box 3.2). 

Equation (A3.7) shows the equality between the probabilities of absorption and stimulated emission that we have already established for monochromatic radiation in Equation (5.15). Equation (A3.8) gives the ratio of tlie spontaneous to the induced transition probability. It allows us to calculate the probability A of spontaneous emission once the Einstein B coefficient is known. 

The molecule-intrinsic factor in the intensities of emission spectra can be obtained from the well-known Einstein coefficients (see, for example. Refs. [20, 21]). For the two states i and f considered above, whose energies are Ei and Ef, respectively, with Ei < Ef, we define as the Einstein coefficient for absorption, Bfl as the Einstein coefficient for stimulated emission, and Afl as the Einstein coefficient for spontaneous emission. We denote by Ni and Nf the number of molecules with energies Ei and Ef, respectively, and the Einstein coefficients are defined such that, for example, the change in Nf caused by electric dipole transitions to and from i is given by... 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

Whereas absorption spectra can be obtained at a given temperature via Monte-Carlo t q)e simulations, the reach of equilibrium in an excited state of an isolated cluster is less obvious, and even less is the definition of a relevant temperature. In any case, the final state may be strongly dependent on the excitation process. Here we will ignore the vibrations of the Na(3p)Arn cluster. We assume a Franck-Condon type approximation and that emission takes place from relaxed equilibrium geometry structures on the Na(3p)Arn excited PES. The Einstein coefficients of the lines of emission towards the ground state at energy AE are given by... 

The measurable quantities of interest for absorption and emission are oscillator strengths and Einstein A coefficients, respectively (e.g., see Henderson and Imbusch, 1989 Reid, 2005). The starting point for the calculations is the electric dipole strength for a particular polarization, q,... 

Inasmuch as a thoroughly satisfactory quantum-mechanical theory of systems containing radiation as well as matter has not yet been developed, we must base our discussion of the emission and absorption of radiation by atoms and molecules on an approximate method of treatment, drawing upon classical electromagnetic theory for aid. The most satisfactory treatment of this type is that of Dirac,1 which leads directly to the formulas for spontaneous emission as well as absorption and induced emission of radiation. Because of the complexity of this theory, however, we shall give a simpler one, in which only absorption and induced emission are treated, prefacing this by a general discussion of the Einstein coefficients of emission and absorption of radiation in order to show the relation that spontaneous emission bears to the other two phenomena. 

B m is called Einstein s coefficient of absorption. The probability of absorption of radiation is thus assumed to be proportional to the density of radiation. On the other hand, it is necessary in order to carry through the following argument to postulate2 that the probability of emission is the sum of two parts, one of which is independent of the radiation density and the other proportional to it. We therefore assume that the probability that the system in the upper state m will undergo transition to the lower state with the emission of radiant energy is... 

Here, ijJ v)) is the mean and angle averaged value of the local radiation field, weighted with the profile function of the local absorption coefficient. The Aij and Bij are the Einstein coefficients for spontaneous and induced transitions, while denotes the probability for a collisional transition from state j —> i. Accordingly, the first row in eq. (10.20) accounts for spontaneous emission and collision of the molecule considered with H2, whereas in the second row induced emission processes are described. This system of rate equations has to be solved simultaneously with the generalized radiative transfer equation for every point in physical and velocity space. 

It should be realized that absorption and emission as discussed in Chapters 2 tuid 3 are different processes, since the former needs a radiation field and the latter not. Einstein considered the problem of transition rates in the presence of a radiation field [31]. For the transition rate from lower to upper level he wrote w = Bp, where B is the Bin.stein coefficient of (.stimulated) absorption and p is the radiation density. 

In similar fashion, two other rates can be defined for the cases of stimulated emission and for absorption. These can be expressed in terms of two rate constants (or Einstein coefficients), one for stimulated emission and one for absorption. The rates are proportional to the numbers of atoms in the relevant state and the number of photons present. Thus the rate at which atoms in state Eq are excited to state El is then given by ... 

Using highly correlated MCSCF-Cl wave functions for the A rij and X states, the transition moment function for the A - X transition has been calculated which in turn allowed the evaluation of Einstein coefficients of spontaneous emission A, (v = 0,1 v" = 0,1,2), absorption oscillator strengths f v (v = 0,1 v" = 0,1), and radiative lifetimes for A Ili, v = 0,1 of PH and PD. The v = 0 lifetime Xrad = 399 ns for PH (390 ns for PD) is shorter than the experimental value, probably because the large correlation energy contributions to the transition moment have not been sufficiently accounted for in the calculation [32]. 

R = Bp, where p is the density of electromagnetic radiation and Bis the Einstein B coefficient associated with absorption. The rate of induced emission is also given by Bp, with the coefficient B of induced emission being equal to the coefficient of absorption. The rate of spontaneous emission Is given by A, where A is the Einstein A coefficient of spontaneous emission. The A and B coefficients are related byA = 8nhv B/( , where h is the Planck constant, v is the frequency of electromagnetic radiation, and c is the speed of light. The coefficients were put forward by... 

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