Radiative rates

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

Once the excited molecule reaches the S state it can decay by emitting fluorescence or it can undergo a fiirtlier radiationless transition to a triplet state. A radiationless transition between states of different multiplicity is called intersystem crossing. This is a spin-forbidden process. It is not as fast as internal conversion and often has a rate comparable to the radiative rate, so some S molecules fluoresce and otliers produce triplet states. There may also be fiirther internal conversion from to the ground state, though it is not easy to detemiine the extent to which that occurs. Photochemical reactions or energy transfer may also occur from S. ... 

Table 1. Activation energies and ratios of the preexponential factors to the radiative rate constants (A/kp) for the photoisomcrization of BMPC in several solvents. Solvent dielectric constants at room temperature, e [65], and viscous flow activation energies, Eri [66], are shown too.

This value of P explains also our Ce(III)-Eu(III) results. The radiative rate of Ce(III) is about 10 s E Since the emission is practically quenched by a Eu(III) neighbor, the MMCT rate will be about 10 s for r = 4 A, which is a realistic nearest neighbor distance. For P = 1.2 A and r = 12 A this rate has dropped to 10 s i.e. very small compared to the radiative rate. 

We now focus our attention on the presence of the unperturbed donor quantum yield, Qd, in the definition of R60 [Eq. (12.1)]. We have pointed out previously [1, 2] that xd appears both in the numerator and denominator of kt and, therefore, cancels out. In fact, xo is absent from the more fundamental expression representing the essence of the Forster relationship, namely the ratio of the rate of energy transfer, kt, to the radiative rate constant, kf [Eq. (12.3)]. Thus, this quantity can be expressed in the form of a simplified Forster constant we denote as rc. We propose that ro is better suited to FRET measurements based on acceptor ( donor) properties in that it avoids the arbitrary introduction into the definition of Ra of a quantity (i />) that can vary from one position to another in an unknown and indeterminate manner (for example due to changes in refractive index, [3]), and thereby bypasses the requirement for an estimation of E [Eq. (12.1)]. 

Fluorescence Lifetimes. The fluorescence decay times of TIN in a number of solvents (11.14.16.18.19), low-temperature glasses (12.) and in the crystalline form (15.) have been measured previously. Values of the fluorescence lifetime, Tf, of the initially excited form of TIN and TINS in the various solvents investigated in this work are listed in Table III. Values of the radiative and non-radiative rate constants, kf and knr respectively, are also given in this table. A single exponential decay was observed for the room-temperature fluorescence emission of each of the derivatives examined. This indicates that only one excited-state species is responsible for the fluorescence in these systems. 

In solution at room temperature, non-radiative de-excitation from the triplet state Ti, is predominant over radiative de-excitation called phosphorescence. In fact, the transition Ti —> S0 is forbidden (but it can be observed because of spin-orbit coupling), and the radiative rate constant is thus very low. During such a slow process, the numerous collisions with solvent molecules favor intersystem crossing and vibrational relaxation in So-... 

Regarding the two latter non-radiative pathways of de-excitation from Si, it is convenient to introduce the overall non-radiative rate constant kfu such that... 

It is interesting to note that when the fluorescence quantum yield and the excited-state lifetime of a fluorophore are measured under the same conditions, the non-radiative and radiative rate constants can be easily... 

In solvents of medium and high viscosity, an empirical relation has been proposed (Loutfy and Arnold, 1982) to link the non-radiative rate constant for deexcitation to the ratio of the van der Waals volume to the free volume according to... 

Two principal ways exist to use a dye as a sensor of local polarity (or of microscopic electric fields) (1) monitoring the polarity-induced shift of the energy levels, e.g., the red shift of the fluorescence and (2) monitoring changes in fluorescence intensity induced by the polarity- or field-induced modulation of nonradiative rates. As these compete with the fluorescence emission, the fluorescence intensity (and lifetime) is correspondingly modulated. (3) In some cases, the radiative rates are also solvent sensitive this is usually connected with the formation of luminescent products. 

Absorption and extinction coefficients are generally less pH dependent than fluorescence spectra and quantum yields because the radiative rates often compete with intra- and intermolecular relaxation precesses. 

Dispersion of the radiative rate constant by local variations of the refractive index at the solid/gas interface. This could explain the tailing of the decay curves even at very low loadings, with lifetime components that are two to three times as long as the intrinsic radiative lifetimes in solution/85 This could also explain the disappearance... 

To test the above ideas, Weitz etal.(i2) performed experiments on the fluorescence decay from a thin layer of europium(III) thenoyltrifluoracetonate (ETA) deposited on a glass slide covered with Ag particles approximately 200 A in diameter. The fluorescence decay rate was found to increase by three orders of magnitude in comparison with that of ETA in solid form. In addition to the large increase in decay rate, there was also evidence for an increase in overall fluorescence quantum efficiency. It is not possible from Eq. (8.11) to say anything about the manner in which is partitioned between radiative and nonradiative processes, y should be written in terms of a radiative part yr and a nonradiative part ynr y = yr + y r. Since the radiative rate for dipole emission is given by... 

A being the radiative rate (labeled in such a way because it coincides with the Einstein coefficient of spontaneous emission) and Anr being the nonradiative rate, that is, the rate for nonradiative processes. The solution of the differential equation (1.16) gives the density of excited centers at any time r ... 

The previous formula indicates that the radiative lifetime tq (and hence the radiative rate A) can be determined from luminescence decay-time measurements if the quantum efficiency rj is measured by an independent experiment. Methods devoted to the measurement of quantum efficiencies are given in Section 5.7. 

The nonradiative rate is much larger than the radiative rate A as aresult,... 

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