Rate constant spontaneous emission

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

The band gap of the semiconductor HgTe is 0.06 eV. (a) What is the ratio of spontaneous to stimulated emission from this semiconductor at 300 K, for transitions from the valence band to the conduction band (b) If the band gap is constant, at what temperature does the rate of spontaneous emission equal the rate of stimulated emission ... 

Similarly, the rate of stimulated emission from state m to n equals mu(vmn) where is another constant. The spontaneous emission probability is independent of the presence or absence of radiation. Hence the rate of spontaneous emission from m to n is Am nNm, where Am n is still another constant. 

In addition to absorption and stimulated emission, a third process, spontaneous emission, is required in the theory of radiation. In this process, an excited species may lose energy in the absence of a radiation field to reach a lower energy state. Spontaneous emission is a random process, and the rate of loss of excited species by spontaneous emission (from a statistically large number of excited species) is kinetically first-order. A first-order rate constant may therefore be used to describe the intensity of spontaneous emission this constant is the Einstein A factor, Ami, which corresponds for the spontaneous process to the second-order B constant of the induced processes. The rate of spontaneous emission is equal to Aminm, and intensities of spontaneous emission can be used to calculate nm if Am is known. Most of the emission phenomena with which we are concerned in photochemistry—fluorescence, phosphorescence, and chemiluminescence—are spontaneous, and the descriptive adjective will be dropped henceforth. Where emission is stimulated, the fact will be stated. 

Ln-L distance, energy transfer occurs as long as the higher vibrational levels of the triplet state are populated, that is the transfer stops when the lowest vibrational level is reached and triplet state phosphorescence takes over. On the other hand, if the Ln-L expansion is small, transfer is feasible as long as the triplet state is populated. If the rate constant of the transfer is large with respect to both radiative and nonradiative deactivation of T, the transfer then becomes very efficient ( jsens 1, eqs. (11)). In order to compare the efficiency of chromophores to sensitize Ln - luminescence, both the overall and intrinsic quantum yields have to be determined experimentally. If general procedures are well known for both solutions (Chauvin et al., 2004) and solid state samples (de Mello et al., 1997), measurement of Q is not always easy in view of the very small absorption coefficients of the f-f transitions. This quantity can in principle be estimated differently, from eq. (7), if the radiative lifetime is known. The latter is related to Einstein s expression for the rate of spontaneous emission A from an initial state I J) characterized by a / quantum number to a final state J ) ... 

R = Bp, where p is the density of electromagnetic radiation and Bis the Einstein B coefficient associated with absorption. The rate of induced emission is also given by Bp, with the coefficient B of induced emission being equal to the coefficient of absorption. The rate of spontaneous emission Is given by A, where A is the Einstein A coefficient of spontaneous emission. The A and B coefficients are related byA = 8nhv B/( , where h is the Planck constant, v is the frequency of electromagnetic radiation, and c is the speed of light. The coefficients were put forward by... 

The maximum fluorescence quantum yield is 1.0 (100 %) every photon absorbed results in a photon emitted. Compounds with quantum yields of 0.10 are still considered quite fluorescent. The fluorescence lifetime is an instance of exponential decay. Thus, it is similar to a first-order chemical reaction in which the first-order rate constant is the sum of all of the rates (a parallel kinetic model). Thus, the lifetime is related to the facility of the relaxation pathway. If the rate of spontaneous emission or any of the other rates are fast, the lifetime is short (for commonly used fluorescent compounds, typical excited state decay times for fluorescent compounds that emit photons with energies from the UV to near infrared are within the range of 0.5-20 ns). The fluorescence lifetime is an important parameter for practical applications of fluorescence such as fluorescence resonance energy transfer. There are several rules that deal with fluorescence. 

We now make two coimections with topics discussed earlier. First, at the begiiming of this section we defined 1/Jj as the rate constant for population decay and 1/J2 as the rate constant for coherence decay. Equation (A1.6.63) shows that for spontaneous emission MT = y, while 1/J2 = y/2 comparing with equation (A1.6.60) we see that for spontaneous emission, 1/J2 = 0- Second, note that y is the rate constant for population transfer due to spontaneous emission it is identical to the Einstein A coefficient which we defined in equation (Al.6.3). 

The interpretation of emission spectra is somewhat different but similar to that of absorption spectra. The intensity observed m a typical emission spectrum is a complicated fiinction of the excitation conditions which detennine the number of excited states produced, quenching processes which compete with emission, and the efficiency of the detection system. The quantities of theoretical interest which replace the integrated intensity of absorption spectroscopy are the rate constant for spontaneous emission and the related excited-state lifetime. 

Figure B2.3.13. Model 2-level system describing molecular optical excitation, with first-order excitation rate constant W 2 proportional to the laser power, and spontaneous (first-order rate constant 21) stimulated (first-order rate constant 1 2 proportional to the laser power) emission pathways.

For a typical sodium atom, the initial velocity in the atomic beam is about 1000 m s1 and the velocity change per photon absorbed is 3 crn-s. This means that the sodium atom must absorb and spontaneously emit over 3 x 104 photons to be stopped. It can be shown that the maximum rate of velocity change for an atom of mass m with a photon of frequency u is equal to hu/lmcr where h and c are Planck s constant and the speed of light, and r is the lifetime for spontaneous emission from the excited state. For sodium, this corresponds to a deceleration of about 106 m s"2. This should be sufficient to stop the motion of 1000 m-s 1 sodium atoms in a time of approximately 1 ms over a distance of 0.5 m, a condition that can be realized in the laboratory. 

The luminescence of an excited state generally decays spontaneously along one or more separate pathways light emission (fluorescence or phosphorescence) and non-radiative decay. The collective rate constant is designated k° (lifetime r°). The excited state may also react with another entity in the solution. Such a species is called a quencher, Q. Each quencher has a characteristic bimolecular rate constant kq. The scheme and rate law are... 

The radiative mechanism can be summarized by the reaction scheme in Equation (23) where and represent the rate constants for absorption and spontaneous emission of the infrared radiation. 

This effective dye relaxation time rp is the spontaneous fluorescence decay time shortened by stimulated emission which is more severe the higher the excitation and therefore the higher the population density w j. The dependence of fluorescence decay time on excitation intensity was shown in 34 35>. Thus, fluorescence decay times measured with high intensity laser excitation 3e>37> are often not the true molecular constants of the spontaneous emission rate which can only be measured under low excitation conditions. At the short time scale of modelocking the reorientation of the solvent cage after absorption has occurred plays a certain role 8 > as well as the rotational reorientation of the dye molecules 3M°)... 

Emission bands from the 42Z, B2n, C2n, and D2X states have been observed and decay rates of fluorescence have been measured extensively [Callear et al. (167-171, 174, 175)]. Various spontaneous processes of electronically excited NO are given in Table V-5. These states are quenched to a different degree by various gases. Quenching half pressures, p1/2, in torr defined as P112 — where is the quenching rate constant in sec-1 torr-1... 

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. 

Luminescence lifetime depends upon radiative and nomadiative decay rates. In nanoscale systems, there are many factors that may affect the luminescence lifetime. Usually the luminescence lifetime of lanthanide ions in nanociystals is shortened because of the increase in nomadiative relaxation rate due to surface defects or quenching centers. On the other hand, a longer radiative lifetime of lanthanide states (such as 5Do of Eu3+) in nanocrystals can be observed due to (1) the non-solid medium surrounding the nanoparticles that changes the effective index of refraction thus modifies the radiative lifetime (Meltzer et al., 1999 Schniepp and Sandoghdar, 2002) (2) size-dependent spontaneous emission rate increases up to 3 folds (Schniepp and Sandoghdar, 2002) (3) an increased lattice constant which reduces the odd crystal field component (Schmechel et al., 2001). 

Figure 1. Four-level molecular model. QiS is the collisional-transfer rate constant from level i to level j, TV is the sum of the electronic quenching and spontaneous emission rate constants, W,t is the absorption rate constant, and Wlt is the stimulated emission rate constant. WIt and WtI are proportional to the laser power PL. The dashed vertical line separates levels le and 2e, which are treated as an isolated system, from those levels not affected directly by the laser radiation.

It is to be expected that the radiative rate constants for spontaneous emission of exciplexes and electroplexes will resemble those for excimer and electromers [see discussion of Eqs. (20) and (21) in Sec. 2.3.1)]. This implies that, dependent on the intermolecular configuration, the fluorescence lifetimes of exciplexes range between 10 and 200 ns, and that the emission lifetime of electroplexes is limited by the intermolecular electron hopping time. Though there are no at present direct lifetime measurements on electroplexes, the measurements on a series of donor-acceptor species in solid-state solutions of PC prove these predictions for exciplexes (Table 1). 

Each process competing with spontaneous emission reduces the observed lifetime x relative to the natural lifetime x . In the case where only unimo-lecular processes i with rate constant A , compete with emission, one has... 

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