Transition probabilities spontaneous emission

As discussed above, for a single element planar transition tensor the angle-dependent two-photon transition probability is the squared modulus of the single photon cos 6 transition probability. Spontaneous emission from a two-photon excited molecular population occurs via the same —> So transition as in single photon fluorescence ... 

Large scale ab initio wave functions have been used to study the transition probabilities between low-lying states of the BN molecule. The square of the electronic transition dipole moment 2 iRe was computed as a function of the internuclear distance r(B-N) from which the radiative lifetimes for the low vibrational levels are calculated. Calculated electronic transition moments are presented in Fig. 4-9, p. 34, Fig. 4-10, p. 34, and Fig. 4-11, p. 35 for extensive tabulation of transition energies, spontaneous emission probabilities, and radiative lifetimes of the transition, see [7]. 

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

The Einstein coefficients characterize the probability of transition of a molecule between two energy levels Ei and E2 (Scheme B3.2). Bu is the induced absorption coefficient (see Chapter 2), B21 is the induced emission coefficient and A21 is the spontaneous emission coefficient. The emission-induced process E2 —> Ei occurs at exactly the same rate as the absorption-induced process Ei —> E2, so that B12 = B 21. 

As shown in Example 5.2, it is easy to obtain that (A)(./(A) 10, where (A)m is the probability of spontaneous emission for a magnetic dipole transition. Thus, using the previous estimation of (A)e, we obtain that, for a magnetic dipole transition,... 

Equation (A3.7) shows the equality between the probabilities of absorption and stimulated emission that we have already established for monochromatic radiation in Equation (5.15). Equation (A3.8) gives the ratio of tlie spontaneous to the induced transition probability. It allows us to calculate the probability A of spontaneous emission once the Einstein B coefficient is known. 

From the data of Hoogschagen and Gorter (104), the oscillator strength of the 5D4-+7F6 transition was obtained. By means of the Ladenburg formula, the spontaneous coefficient A46 was calculated. Using the relative-emission intensities, the rest of the A4J spontaneous-emission coefficients could be calculated. From these and a measured lifetime of 5.5 x 10 4 sec at 15°C, he calculated a quantum efficiency of 0.8 per cent. Kondrat eva concluded that the probability of radiationless transition for the trivalent terbium ion in aqueous solution is approximately two orders of magnitude greater than for the radiation transition. 

Changing the base glass caused considerable changes in the -state mean life. To ascertain whether this was the result of variations in spontaneous emission matrix elements or radiationless transition probabilities, the peak absorption coefficient of the 4/9/2 —4F3/2 transition was plotted... 

The first term on the right-hand side is a gain term due to transitions between level m and n, the second a loss term Nn is the number of atoms in level n. The important new element introduced by Einstein was the discovery of spontaneous emission. The transition probability is the sum of two contributions ... 

Einstein s treatment of spontaneous emission uses occupation numbers and transition probabilities. On the other hand, quantum mechanics is based on probability amplitudes. The difference between these two points... 

A, A", symmetry species of Cg, 75 Amn, Einstein transition probability of spontaneous emission, 24, 38... 

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 Kinetics, Luminescence Lifetimes. The Einstein coefficient A for spontaneous emission gives the probability of radiative transition. Since this probability is the same for all molecules of the same excited species, it follows that the decrease in the number of excited molecules within a differential time increment is simply proportional to the number of excited... 

For systems in which no electron migration or stabilization can occur after excitation, the excited electron returns to its ground state by either luminescence or radiationless transition. Since most charge transfer transitions occur with very high probability, the excited state persists for only about 10 8 sec. Therefore, the secondary process must be extremely fast to compete with spontaneous emission of the excited state. 

The interesting feature of Eq. (2.11) is that the induced and spontaneous emission combine into a factor as simple as Nv + 1. This is strongly suggestive of the factors N3- + 1, which we met in the probability of transition in the Einstein-Bose statistics, Eq. (4.2), of Chap. VI. As a matter of fact, the Einstein-Bose statistics, in a slightly modified form, applies to photons. Since it does not really contribute further to our understanding of radiation, however, we shall not carry through a discussion of this relation, but merely mention its existence. 

P+ and P are the probabilities for absorption and emission, respectively B+ and B are the coefficients of absorption and of induced emission, respectively A- is the coefficient of spontaneous emission and p v) is the density of radiation at the frequency that induces the transition. Einstein showed that B+ = B, while A frequency dependence, spontaneous emission (fluorescence), which usually dominates in the visible region of the spectrum, is an extremely improbable process in the rf region and may be disregarded. Thus the net probability of absorption of rf energy, which is proportional to the strength of the NMR signal, is... 

Experimentally the possibility of observing the transitions depends on the population difference between the two spin states, and the lifetime of the upper state which is determined by the spontaneous emission probability and by the various relaxation processes available. ESR emissions have been observed (19) in several systems in which the upper spin state has a higher population. 

Because of the small mass of the proton, the decrease of the transition dipole moment as we move to higher overtone bands of H3 is not as drastic as in ordinary molecules. The band origins, transition moments, relative intensities and Einstein s spontaneous emission probabilities theoretically calculated by Dinelli, Miller and Tennyson are listed in Table 1. Note that the value of Aij is larger for the 2v2(2) overtone band than for the Vj fundamental band because the factor in the Einstein formula overrides the reduction of j n. This explains the strong 2 pm overtone emission observed in Jupiter. ... 

The transition probability for the upward transition (absorption) is equal to that for the downward transition (stimulated emission). The contribution of spontaneous emission is neglible at radiofrequencies. Thus, if there were equal populations of nuclei in the a and f spin states, there would be zero net absorption by a macroscopic sample. The possibility of observable NMR absorption depends on the lower state having at least a slight excess in population. At thermal equlibrium, the ratio of populations follows a Boltzmann distribution... 

For the x-ray emission process, the transition probability( is also calculated from the dipole matrix similar to the case of the x-ray absorption, but the molecular state f in eq.(lO) is of occupied in this case. The transition probability corresponds to the spontaneous emission rate, then is given by Einstein formula as... 

B12 and 21 are the transition probabilities for absorption and stimulated emission, respectively, and 1 n2, n, and wT the analyte atom densities for the lowest state, the excited state, the ionized state and the total number densities, gi and g2 are the statistical weights, A2i is the transition probability for spontaneous emission and 21 the coefficient for collisional decay. Accordingly,... 

---