Nonradiative rate

Fig. 1 Jablonski diagram of energy level for describing processes absorption, fluorescence and phosphorescence in complex molecules where kf and /c arc the radiative and nonradiative rates of fluorescence, respectively, kj and kTnr are the radiative and nonradiative rates of phosphorescence, respectively, k-lsc is the interconversion rate, and kmt is the rate of intermolecular processes Av denotes the Stokes shift of fluorescence...

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. 

The long lifetime has important consequences on the decay rates. First, we consider what affects the nonradiative rates (knr) which change the yields of fluorescence and phosphorescence. The nonradiative decay rate is often enhanced in molecules which have flexible constituents (the so-called loose-bolt effect). Therefore, both fluorescence and phosphorescence yields are generally larger for rigid molecules than flexible molecules. For the same reason, a rigid environment will increase the emission yields hence both fluorescence and phosphorescence yields often increase with increasing viscosity. 

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 ... 

EXAMPLE 1.7 The fluorescence lifetime measured from the metastable state Ej/2 ofNd + ions in the laser crystal yttrium aluminum borate (YAl3(B03)4) is 56 lus. If the quantum efficiency from this state is 0.26, determine the radiative lifetime and the radiative and nonradiative rates. 

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

The nonradiative rate. Am, from a (RE) + ion level is also strongly related to the corresponding energy gap. Systematic studies performed over different (RE) + ions in different host crystals have experimentally shown that the rate of phonon emission, or multiphonon emission rate, from a given energy level decreases exponentially with the corresponding energy gap. This behavior can be expressed as follows ... 

Figure 6.5 The measured values of the nonradiative rate, as a function of the energy gap for different trivalent rare earth ions in three host crystals. The straight lines correspond to the best fits to Equation (6.1) (note the log scale on the A axis) (reproduced with permission from Riseberg and Weber, 1975).

EXAMPLE 6.2 Determine the nonradiative rates from the following energy levels of different ions in lanthanum chloride 5/2 (Er )fPo (Pr +) and 2 5/2 (Yif+). 

From this expression, we can estimate the multiphonon emission nonradiative rate, from any particular energy level by simply knowing the energy distance to the next lower energy level (the energy gap), AE. 

Figure 6.6 shows Am versus the number of effective phonons, p, for the same three materials of Figure 6.5. The energy of the effective phonons for each host crystal is indicated in the figure caption. An exponential decrease in the nonradiative rate with...

F ure 6.6 The multiphonon nonradiative rate of (RE) ions as a function of the number of emitted effective phonons for LaCfi (260 cm ), LaEs (350 cm ), and Y2O3 (430-550 cm ). The numbers in brackets indicate the energies of the effective phonons. The shaded area indicates the range of typical radiative rates. 

Finally, it is important to recall that the simple nonradiative rate law described by Equations (6.1) and (6.2) is only vaUd for (RE) + ions. This is a consequence of the weak ion-lattice interactions for these ions, that leads to a Huang-Rhys parameter of... 

The fluorescence lifetime of the /2 metastable state of Nd + ions in LaBGeOs (a solid state laser) is 280 /u.s and its quantum efficiency is 0.9. (a) Calculate the radiative and nonradiative rates from this excited state, (b) If the effective phonons responsible for the nonradiative rate have an energy of 1100 cm, use the Dieke diagram to determine the number of emitted effective phonons from the F3/2 excited state, (c) From which three excited states of the Nd + ions in LaBGeOs do you expect the most intense luminescence emissions to be generated ... 

A Gd + doped crystal is illuminated with a pulsed light source, so that the l7/2 excited state of this ion is populated by absorbing 1 mJ of energy per incident pulse. Determine the heat delivered to the crystal per excitation pulse if the nonradiative rate from this state is 10 s The fluorescence lifetime of the l7/2 state is 30 /xs. 

In Table E7.5, the fluorescence lifetimes and quantum efficiencies measured from different excited states of the Pr + ( Po and D2) and Nd + (" Fs ji) ions in a LiNbOs crystal are listed, (a) Determine the multiphonon nonradiative rate from the 19/2 and In/2 states of the Er + ion in LiNbOs. (b) If a fluorescence lifetime of 535 /us is measured from the excited state Fs/2 of the Yb + ion in this crystal, estimate the radiative lifetime from this state. 

Quenching of narrow-line emissions (as observed for many Ln3+ ions) has been explained by phonon emission to the lattice modes. Moos and co-workers (60) and others (67) have given many examples. Usually the nonradiative rate is described by Kiel s formula (62) for a single-frequency p-phonon process,... 

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. 

The comparison between theory and experiment, particularly for radiative and nonradiative rates, has been of much interest in recent years. The experimental values of tr and x g obtained from the SVL values of p and xp obtained by Miller and Lee (164) are listed in Table 8. 

Franck-Condon factors, rather than energy-level density, are primarily responsible for the variation in nonradiative rates. 

Returning now to the problem of the calculation of nonradiative transition rates, it becomes obvious that much work remains to be done before these calculations can be performed in a routine manner to generate the rates of unmeasured processes. The accurate prediction of a nonradiative rate requires an accurate knowledge of the molecular wavefunctions (both electronic and vibrational), the density of states, and most important, the nuclear-coordinate dependence of the electronic wavefunctions. 

Thayer et al. (240-242) on propynal are the only reasonably complete nonradiative rate calculations done on a carbonyl. Their values for the intersystem crossing and internal conversion rates are low by factors of 10 and 80, respectively, for the vibrationless excited state when compared to the experimental values. They correctly predict the energy dependence of the decay channels, although they fail to predict the large enhancement of the intersystem crossing rate for three vibronic levels. Also, the energy dependence of the collision-induced nonradiative transitions seems to be well reproduced. 

A conclusion which can be drawn from these observations is that a nonhalogen substituent will affect the nonradiative decay rate only when the substitution is directly on the carbonyl chromophore. Deuterium substitution has been shown to have a drastic effect on the nonradiative rates of formaldehyde and glyoxal, with the decrease in kjjR due mainly to a large decrease in Franck-Condon factors. Perdeuteration of acetone, on the other hand, has only a slight effect on kup, the value of kjgc... 

Fluorination and chlorination cause a large decrease in the nonradiative decay rates. The intersystem crossing rate constant drops from 4 x 10 s for acetone to 1.5 x 10 s- - for hexafluoroacetone. For the cyclobutanone/perfluorocyclobutanone pair, the rates are 4.2 x 10 s- - and 1 x 10 s l, respectively. A similar decrease is likely for the glyoxal/oxalyl fluoride pair, where zero-pressure tp values of 2.4 ps and 24 ps, respectively, have been measured (19,25,26). The fluorescence yields of these compounds must be measured before the true extent of this effect is known. Mixed chlorofluoroacetones exhibit nonradiative rates of intermediate value, with k jR increasing at low energy excitation with increasing chlorination from 2.7 x 10 s l for chloropentafluoroacetone to 3.9 x 10 s l and 8.0 x 10 s l for dichlorotetrafluoroacetone and trichloro-trifluoroacetone, respectively (94). 

Since the emission quantum yield accounts for no more than 1.8% of the total depopulation pathways of Sp, the value of tF is mainly dictated by the nonradiative rate constant. By using the f values of Gandini and Kutschke (88), obtained at 265 and... 

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