Rate of Spontaneous Emission

The calculation of the rate of spontaneous anission from quantum electrodynamical theory is a rather complicated problem. We wiU use instead a treatment based on statistical equilibrium, originally derived by Einslein. We assume that the number of molecules in the ground state is Nq and in the excited state N,. For simplicity, we assume that both states are nondegenerate. The Boltzmann distribution ratio at equilibrium is equal to 

We assume that the radiation field is so weak that its presence does not require any modification of Equation 12.32. The rate of the induced absorption may be written as 

The rate of induced emission is derived in the same way as induced absorption and can be rewritten with the help of an expression equivalent to Equation 12.31  

Obviously, if there were no spontaneous emission, we would have Nj = Nq at equilibrium, which would contradict Equation 12.32. The spontaneous emission rate must be proportional to Nj and thus Equation 12.35 has to be replaced by 

At equilibrium, the sum of the absorption rate and the emission rate must be the same therefore, using Equations 12.32 and 12.36, we obtain 

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When the light souree s intensity is so large as to render gBf i Af i (i.e., when the rate of spontaneous emission is small eompared to the stimulated rate), this population ratio reaehes (Bi f/Bf i), whieh was shown earlier to equal (gf/gi). In this ease, one says that the populations have been saturated by the intense light souree. Any further inerease in light intensity will result in zero inerease in the rate at whieh photons are being absorbed. Transitions that have had their populations saturated by the applieation of intense light sourees are said to display optieal transparency because they are unable to absorb (or emit) any further photons because of their state of saturation. 

Lasers are devices for producing coherent light by way of stimulated emission. (Laser is an acronym for light amplification by stimulated emission of radiation.) In order to impose stimulated emission upon the system, it is necessary to bypass the equilibrium state, characterized by the Boltzmann law (Section 9.6.2), and arrange for more atoms to be in the excited-state E than there are in the ground-state E0. This state of affairs is called a population inversion and it is a necessary precursor to laser action. In addition, it must be possible to overcome the limitation upon the relative rate of spontaneous emission to stimulated emission, given above. Ways in which this can be achieved are described below, using the ruby laser and the neodymium laser as examples. 

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. 

Consider now the case of bound molecular dynamics wherein classical mechanics is often tacitly assumed valid. If the rate of spontaneous emission... 

In these equations and hereafter we use simplified symbols for quantities referring to the P-branch line v, 7—l- c—I, /. Namely, Xvy instead of Xu,y-i etc. The second term in (2) represents the rate of spontaneous emission into the oscillating cavity modes. 1 is approximately the effective solid angle subtended by the mirrors after several reflections. (Alternatively, e is the fraction of stable transverse modes.) After threshold eS j is negligible, eS j- Xvj- i i i " i important only before threshold —as a source of noise photons to trigger-on the lasing process. The spontaneous emission terms in (1) are given by =A j+iN y, A j is the Einstein coefficient. In infrared lasers where typically A ) 10s. S, ... 

The radiation field induces also a transition from the upper to the lower state (stimulated emi.ssion). The rate is w = B p, where B is the Einstein coefficient of stimulated emission. The rate of spontaneous emission is w" = A, with A the coefficient of spontaneous emission (note the absence of p in this expression). This is... 

A second problem compounds the difficulty. The ratio R of the rate of spontaneous emission to stimulated emission under conditions of thermal equilibrium is given by ... 

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... 

Fig. 12. Ratio R3 of coherent/incoherent emission measured for AX = 0.7 A and a solid angle of 0.2 mrad. Experimental points are compared with theoretical curve a. Data in b and c are of f3, the modulation rate of spontaneous emission measured respectively with and without the laser. Abscissa is stored current. E = 166 MeV X] = 1.06 nm, X3 = 0.355 nm.

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. 

Determination of the intrinsic quantum yield with (16) requires evaluation of the radiative lifetime which is related to Einstein s rates of spontaneous emission A from an initial state R/), characterized by a quantum number /, to a final state... 

The rate of spontaneous emission of radiation for a transition from a state n to n is given by the Einstein A coefficient ... 

The enhanced spectral broadening observed in SL s is contained in the final three terms in Eq. (1). The first, ngp, is the spontaneous emission factor and gives the ratio of the rate of spontaneous emission into the laser mode to that of stimulated emission per photon in the mode. In many laser systems ngp is close to unity. However, this is not true of SL s due to the finite population of the lower level of the laser transition. In SL s, ngp approaches unity only at very low temperatures where the carriers are distributed according to Fermi-Dirac statistics. At room temperature ngp 2.5 for (GaAl)As lasers. 

The Einstein rate equation picture of the steady state atomic inversion, Eq. (5), relies on the fact that the vacuum density of electromagnetic modes N(o)) is relatively smooth on the scale of A. That is to say, the Einstein picture assumes that the rate of spontaneous emission in the Mollow sidebands at [Pg.328]

Due to the low rate of spontaneous emission, a number of processes appear before emission. The processes depend strongly on the molecule therefore, the content of the next section is referred to as molecular photophysics. 

The transition probability for spontaneous emission shows that the rate of spontaneous emission depends linearly on the oscillator strength for the absorption... 

In addition, Einstein recognized that a system can go from an excited state to a lower-energy state spontaneously if there is a spontaneous emission of a photon of the right frequency. This spontaneous emission is dependent on the concentration of species in the excited state but not dependent on the photon density p v). The rate of spontaneous emission is characterized by A, the Einstein coefficient of spontaneous emission ... 

The quantity Trad expresses the rate of spontaneous emission per molecule per unit of angular frequency between E/h and E + d ) jh [2]. For a single fluorescence band, in the absence of non-radiative processes, the radiative decay rate (iCrad) the maximum lifetime (tq) are given by... 

Two processes ue principcJly responsible for the finite lifetimes of excited states md hence for the widths of trcuisitions to or from them. One is coUisional deactivation, which cuises from coUisions between molecules. The second is spontcmeous emission. Because the rate of spontaneous emission cannot be chcmged (without changing the molecule), it is a natural limit to the lifetime of cm excited state. The resulting lifetime broadening is the natural linewidth of... 

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