Rate constant radiative

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. 

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.

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... 

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... 

Again the radiative association kinetics described above allow a direct comparison for some realistic values of k and k. For most chemically activated systems at the threshold for unimolecular dissociation, the observed radiative rate constants are of the order of 10-100 s and hence are much below the values expected for k of about 10 s . Therefore, the first limit is most likely to be valid, with the interesting conclusion that the observed unimolecular dissociation rate constant will depend only on the photon density and the absorption cross section (rate constant) at a given wavelength. 

Fig. 34 Photosensitized singlet oxygen production 1/ r is the general (radiative and non-radiative) rate constant of the transition Si So fcsi is the rate constant of singlet-triplet conversion tt is the lifetime of the triplet, T1, electronic state of PS kj is the second-order rate constant of singlet oxygen quenching of the Ti state of PS tl and nr are the radiative lifetime and rate constant of all intramolecular nonradiative energy relaxation processes of O2 ( Ag)...

Values of the radiative rate constant fcr can be estimated from the transition probability. A suggested relationship14 57 is given in equation (25), where nt is the index of refraction of the medium, emission frequency, and gi/ga is the ratio of the degeneracies in the lower and upper states. It is assumed that the absorption and emission spectra are mirror-image-like and that excited state distortion is small. The basic theory is based on a field wave mechanical model whereby emission is stimulated by the dipole field of the molecule itself. Theory, however, has not so far been of much predictive or diagnostic value. 

They should have large radiative rate constants (k ) so that emission can be observed on the femtosecond time scale. 

For ideal probes the shape of the fluorescence spectrum should be structureless and smooth without vibronic features in a range of solvents of different polarity. Another important observable is the radiative rate constant (Eq. (22))... 

Rate Constants of Radiative Transitions. The natural radiative rate constant kr of an electronic transition from a state to a state Sf is related to the transition moment M and thereby to the oscillator strength f. It is convenient to factorize f to highlight the various factors which determine to what extent a transition is allowed if near 1) or forbidden (f near 0). The transition moment includes the displacement of all particles of the molecule during the transition, nuclei as well as electrons. The heavy nuclei move much more slowly than the light electrons so that their motions can be considered to be independent. Within this approximation the transition moment is given as... 

This is similar to a first-order reaction in chemical kinetics and follows the same law as radioactive decay. The rate constant kv defined in this manner is the natural radiative rate constant which also defines the natural radiative lifetime... 

In the case of 27-29, with the exception of the borderline 28, the radiative rate constant is one order of magnitude lower than the value found for 24-26. This observation, together with the longer values obtained for the natural radiative lifetime (see t°f = tf/< )f)/ is clearly compatible with the forbidden nature of the lowest lying transition/state. The nature of this transition seems to be a 71,71 symmetry forbidden transition/state. 

The spin-orbit coupling matrix element <1MLCT Hso 3LC) can be calculated from the radiative rate constants and the energies of the states involved [64] ... 

We have seen from the above work that the nonradiative rate constants dominate the luminescence behavior of ruthenium(II) complexes. If one can increase the value of the radiative rate constant, kr, without substantial increases in knr, then emission efficiency can be improved. The radiative rate constant is, in theory at least, related to the molar absorption coefficient, epsilon187. Demas and Crosby188, made a number of assumptions and calculated radiative lifetimes based on observed epsilon values, which were in good agreement with the experimental kr values. Watts and Crosby1895 went on to comment on the possible implications of the epsilon value. 

Figure 3 Excited state diagram for (a) intramolecular and (b) intermolecular decay processes of a molecule in an excited state M. kr and km are the radiative and non-radiative rate constants, respectively. Superscript M refers to processes that return the excited state to ground state without chemical change. Superscript P refers to processes that result in the formation of a product, chemically distinct from M. Quenching rate constants depend on the concentration of quencher Q or P.

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