Decay rate electronic example

This behavior of the ID is directly related to experimental data such as optical spectra of polyatomic molecules or nonradiative decay rates. For example, the spectral line shape of a spectrum may be deduced by convoluting the ID with Lorentzian functions centered around Q q), WiCOj—where Q corresponds to the energy gap between the two molecular (electronic) states and coj, are the vibrational frequencies of the final state. This point will be addressed in greater detail in Chapter 7. Here, we merely note that the effect shown in Figure 4.10 leads to a certain mode selection. If the angle

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Morishima et al. [75, 76] have shown a remarkable effect of the polyelectrolyte surface potential on photoinduced ET in the laser photolysis of APh-x (8) and QPh-x (12) with viologens as electron acceptors. Decay profiles for the SPV (14) radical anion (SPV- ) generated by the photoinduced ET following a 347.1-nm laser excitation were monitored at 602 nm (Fig. 13) [75], For APh-9, the SPV- transient absorption persisted for several hundred microseconds after the laser pulse. The second-order rate constant (kb) for the back ET from SPV- to the oxidized Phen residue (Phen+) was estimated to be 8.7 x 107 M 1 s-1 for the APh-9-SPV system. For the monomer model system (AM(15)-SPV), on the other hand, kb was 2.8 x 109 M-1 s-1. This marked retardation of the back ET in the APh-9-SPV system is attributed to the electrostatic repulsion of SPV- by the electric field on the molecular surface of APh-9. The addition of NaCl decreases the electrostatic interaction. In fact, it increased the back ET rate. For example, at NaCl concentrations of 0.025 and 0.2 M, the value of kb increased to 2.5 x 108 and... 

In contrast, the nonradiative decay rate k r may be viewed to be determined by the localized environment of the luminescent molecule. The localized environment perturbs the natural electronic configuration of the sensor molecule increasing the probability of its decay. The functional form of knr is determined by the nature of the interaction between the excited sensor and its surrounding perturbation. For example, the knr may be proportional to the concentration, partial pressure, or value of a [Parameter] of interest ... 

Fortunately, the partial decoupling of the ET and conformational processes afforded by the absence of synchronous events in principle and in practice allows for the identification of an observed decay rate constant. For example, if one constructs a series of systems in which the ET energetics (or electronic coupling) is modified without change in the conformational equilibrium, thus leaving the conformational rates unchanged, then the observed rate constants will be unchanged if the reaction is controlled by a conformational rate, but will vary if this is not so. 

For use in geochronology, the decay constant of a radioactive nuclide must be constant and must be accurately known. For a-decay and most (3-decays, the decay constant does not depend on the chemical environment, temperature, or pressure. However, for one mode of 3-decay, the electron capture (capture of K-shell electrons), the decay "constant" may vary slightly from compound to compound, or with temperature and pressure. This is because the K-shell (the innermost shell) electrons may be affected by the local chemical environment, leading to variation in the rate of electron capture into the nucleus. The effect is typically small. For example, for Be, which has a small number of electrons and hence the K-shell is easily affected by chemical environments, Huh (1999) showed that the decay constant may vary by about 1.5% relative (Figure l-4b). Among decay systems with geochronological applications, the branch decay constant of °K to °Ar may vary very slightly (<1% relative). 

For example, supported TiCl4/MgCl2 catalysts show a short period of acceleration, followed by a prolonged steady period 92,93). However, in the presence of electron donors, they may show the typical decay rate kinetics observed during propylene polymerization 93). Bulk catalysts prepared by interaction of TiCU with Mg(OR)2 show either a stationary rate, or a non-stationary rate, according to the titanium content 88,94). Bulk catalysts prepared by reduction of TiCl4 with organomagnesium compounds show a decay type rate 92-95>. 

An example of a nearly pure first-order irreversible reaction is the decay of radioactive isotopes. These reactions depend on instabilities within the nucleus rather than the electronic configuration of the atom and are thus totally uninfluenced by the surroundings. The reaction rate is dependent on the concentration of the isotope and the probability of spontaneous decay. An important example in aquatic systems is the decay of carbon-14 (Fig. 9.5), which is formed in the upper atmosphere by interaction between cosmic ray particles and nitrogen atoms. is then mixed into the lower atmosphere and... 

ESR experiments were used to measure the kinetics of several types of reactions, those that can be monitored only by ESR, such as proton exchange or electron exchange reactions of radicals, and some that can be measured by other techniques as well, e.g. decay kinetics. Although most decay kinetics of phenoxyl radicals were followed by pulse radiolysis or flash photolysis by monitoring optical absorption, kinetics for some long-lived radicals were frequently monitored by ESR. For example, the second-order decay rates of 4-alkyl-2,6-di-f-butylphenoxyl radicals were measured to be 2200, 500 and 2 s ... 

Electronic Feshbach resonances are often very long lived and hence have narrow (often < 0.01 eV) widths. Their lifetimes are determined by the coupling between the quasibound and asymptotic components of their electronic wavefunctions. Because the Feshbach decay process Involves ejection of one electron and deexcltatlon of a second, It proceeds via the two-electron terms e /r.. In the Hamiltonian. For example, the rate of electron loss lil H (2s2p, P°) Is proportional to the square of the two-electron Integral <2s2p(e /r.21 Is kp>, where kp represents the continuum p-wave orbital. This Integral, and hence the decay rate. Is often quite small because of the size difference between the 2s or 2p and Is orbitals and because of the oscillatory nature of the kp orbital. 

Another complication in plutonium solution is the gradual, spontaneous reduction of Pu(VI) to Pu(IV), and Pu(IV) to Pu(III), caused by ionization products of alpha particles emitted in radioactive decay [SI]. The rate of alpha reduction is slow, however. For example, the observed rate of reduction of Pu(VI) in 0.5 M HCl at 25°C is 0.0035 g-equiv/day per mole of plutonium, which corresponds to a half-life of 199 days for reduction of Pu(VI) to Pu(TV). From these rates and the known alpha-decay rate and decay energies of plutoniiun, it is estimated that approximately 80 eV of dissipated alpha energy in this solution brings about the addition of one electron in reducing plutonium ions. After several hundred days the plutonium reaches an average oxidation state intermediate between Pu(III) and Pu(IV). 

These limits are rather simply understood by the consideration of even a simple diatomic molecule example. Suppose first that and , represent two bound electronic states of the diatomic molecule. Then under isolated molecule conditions F, corresponds to the radiative decay rate of the individual level. F is typically on the order of 10" -10" sec". However, in the diatomic molecule the average spacing between vibronic levels, /, is on the order of hundreds or thousands of cm". Thus the simple diatomic molecule example is definitely within the small molecule limit of (2.2). 

Addition of various concentrations of [60]fullerene, for example, to a ZnTPP solution, resulted in an accelerated decay of the 7t-radical anion (ZnTPP "). The observed rate was linearly dependent on the [60]fullerene concentration, which, in turn, has led to the assumption that the ZnTPP tt-radical anion reacts with [60]fullerene. To confirm a probable electron transfer, the formation of the characteristic C60 absorption in the NIR ( ax = 1080 run) was also monitored. The grow-in rate of the C o " absorption at various wavelengths in the 980-1060 nm range was nearly identical to the decay rate of the MP absorption at 650-750 nm. For example, in the case of ZnTPP 7t-radical anion (ZnTPP "), a bimolecular rate constant of (2.5+1.0) x 10 M s was derived from the ZnTPP " decay (720 nm) and (1.4 1.0) x 10 M s from the Ceo formation (970 nm). These two values are in reasonable agreement and confirm unmistakably the electron transfer from the one-electron reduced metalloporphyrin (ZnTPP) to the singlet ground state of the fullerene ... 

Radiative decay rates are also affected when an emitting state is comprised of a quantum mechanical mixture of two or more pure electronic states that have different radiative decay rates and different shift behavior with pressure. Many emitting states, for example, consist of two or more electronic states coupled... 

The mean fluorescence lifetime may also be determined by continuous intensity measurements, if the exciting light intensity is modulated at a high frequency. Fluorescence is excited by light modulated sinusoidally at a known frequency (ajln Hz). The emission is a forced response to the excitation, and is therefore modulated at the same frequency, but with a phase shift, due to the time-lag between absorption and emission. The intensities of the two beams are monitored by photomultipliers. The difference in phase (0) between the two intensities is determined electronically. The lifetime r is given by cox = tan<. The modulation frequency must be made comparable to the decay rate, e.g., around 30 MHz for a mean lifetime of 30 ns. Such frequencies can be achieved by using a hydrogen lamp actuated by a suitably modulated current source. Commercial equipment is available. The method has been applied to quinine sulphate, fluorescein, and acridine, for example, with a precision of 1-2%. It is especially useful for very short (sub-nanosecond) lifetimes. 

As an example of scattering by adsorbates of an electron in image-potential states. Figure 6.9 shows the adsorbate-induced decay rate of the n = 1 and n = 2 image-potential states on a Cu(lOO) surface with Cs adsorbates [15]. The decay rate is given for a Cs coverage of the surface equal to 1 Cs adsorbate per 1000 Cu surface atoms. 

The methods we have described for studying electronic relaxation and decay of electronic states may be applied to a variety of problems. These include, for example, the study of vibrational relaxations (VR) which will be illustrated with a simple application to the decay of an initially prepared harmonic molecular oscillator state t >= l,ria > of a macroscopic system to the final state f >= 0,n a >The system is supposed to contain a guest, represented by a harmonic molecular oscillator of frequency co, coupled to a very large number of medium (phonon) states n > of considerable lower frequencies < Then, for the VR rate w , we have [130]... 

Therefore the fluorescence lifetime, Tp, is a measure of the fluorescence quantum yield. Op. This indicates that the rate constant for the radiative decay processes kg, is constant for a particular fluorophore as it is an intrinsic electronic property of the molecule. Consequently the fluorescence lifetime is influenced by changes in the nonradiative decay pathways. For example, a subsequent increase in nonradiative decay rates will reduce the fluorescence lifetime. As such, fluorescence lifetimes are extremely sensitive to the molecular environment surrounding the fluorophore. Therefore, this makes fluorescence lifetime measurement of individual fluorophores present in complex aquatic samples difficult to interpret. 

For some experiments, the solar neutrino flux and the rate of decay of the proton being extreme examples, tire count rate is so small that observation times of months or even years are required to yield rates of sufficiently small relative uncertainty to be significant. For high count rate experiments, the limitation is the speed with which the electronics can process and record the incoming infomiation. 

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