Einstein coefficients for

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. 

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... 

On the basis of these formulae one can convert measurements of area, which equals the integral in the latter formula, under spectral lines into values of coefficients in a selected radial function for electric dipolar moment for a polar diatomic molecular species. Just such an exercise resulted in the formula for that radial function [129] of HCl in formula 82, combining in this case other data for expectation values (0,7 p(v) 0,7) from measurements of the Stark effect as mentioned above. For applications involving these vibration-rotational matrix elements in emission spectra, the Einstein coefficients for spontaneous emission conform to this relation. 

By completely analogous treatment in which values of aH (0) = 0 and am (0) = 1, are used, the Einstein coefficient for induced emission Jm is found to be given by the equation ... 

Consider a system in which matter and radiation are in equilibrium in a closed cavity at temperature T. (This equilibrium situation does not generally hold in spectroscopy, but the transition probabilities are fundamental properties of the interaction between radiation and matter and cannot be affected by the presence or absence of equilibrium.) As before, let be greater than (0). The probability of absorption from state n to state m is proportional to the number of photons with frequency near vmn the number of photons is proportional to the radiation density u(vmn). Hence the rate of absorption is given by Bn t,mNnu(i mn)t where Nn is the number of molecules in state n and Bn m is a proportionality constant called the Einstein coefficient for absorption. From the discussion following (3.46) and from (3.47), it follows that... 

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. 

N, Ng...population of excited, ground states in two-level systems Bu, B. ..Einstein coefficient for absorption, stimulated emission A.Einstein coefficient for spontaneous emission... 

We now finally define the Einstein coefficient for induced absorption as ... 

Therefore the Einstein coefficient for spontaneous emission becomes, using Eq. (3.33.11), ... 

In Fig. 7 we recapitulate the spin-averaged Einstein coefficients for the Vegard-Kaplan emission from the lowest vibrational state of the triplet as well as the corresponding values reported by Piper [89]. The relative transition probabilities for different vibronic phosphorescence bands are quite good [26]. The absolute and the relative intensities of the higher vibrations v" are very sensitive to the transition moment curve... 

Einstein coefficients for the Vegard-Kaplan system in a logarithmic scale. Valence + Rydberg CAS (CAS-2, solid line), experimental data (dashed line) [89]. Prom ref. [26]. 

The electric dipole T-S transition moments calculated as functions of the internuclear distances were used for the estimation of the vibronic transition probabilities by a vibrational averaging procedure. The calculated Einstein coefficients for emission from... 

Solution of the steady-state equations for [X ] (i.e. with d[X ]/df = 0) provides an expression for the luminescence emission intensity, Iium, in terms of the intensity of absorbed radiation, iabs, where A is the Einstein coefficient for spontaneous emission ... 

V is the resonator volume, c is the speed of light, rp is the photon lifetime in the cavity, a% j is the cross-section for stimulated emission, Av>v is the Einstein coefficient for spontaneous emission, and Ais the population inversion. It is assumed that emission occurs in only one v/-line in a given vibrational band, thus draining the entire population inversion of this band. 

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. 

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