Spontaneous emission probability

Inadequacy of simple quantum theory. The quantum theory of atoms outlined in Chapter 3 provides a time-dependent description of excited states of atoms. An excited hydrogen-atom, for example, in the level n=2, 7 = 1, m=0 is 

In the absence of external radiation, however, the theory predicts that if the atom is prepared in this state at time t=0 then it will remain in that state indefinitely. This is clearly incorrect, since hydrogen atoms in the 2p state are observed to decay rapidly to the ground state Is with the spontaneous emission of Lyman a radiation, A = 1216 8. 

Thus the simple quantum theory is inadequate and to account for the spontaneous emission of radiation we adopt alternative techniques the first of these takes the classical result of section 4.2 and attempts to convert it into a quantum-mechanical form the second develops a complete quantum theory in which not only the atoms but also the radiation fields are properly quantized. We examine each of these methods in the following sections. 

Conversion of the classical result into quantum-medianical terms. We start from the classical result for the time-averaged rate of radiation of energy by an oscillating electric dipole moment, equation (2.74)  

We now identify the frequency of the classical oscillator with that of the spontaneous electric dipole transition from level k to level i  

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Using Equations (5.17) and (A4.2), the cross section can be written in terms of the spontaneous emission probability ... 

Purcell, E.M., 1946, Spontaneous emission probabilities at radio frequencies, Phys. Rev. 69 681. 

The semiclassical treatment just given has the defect of not predicting spontaneous emission. According to (3.13), if there is no outside perturbation, that is, if // (0 = 0, then dcm/dt = 0 for all m if the atom is in the nth stationary state at / = 0, it will persist in that state forever. However, experimentally we find that unperturbed atoms in excited states spontaneously radiate energy and drop to lower states. Quantum field theory does predict spontaneous emission. Since quantum field theory is beyond us, we shall use an argument given by Einstein in 1917 to find the spontaneous-emission probability. 

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. 

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. ... 

While the overall quantum yield is relatively easy to measure, Q, which is needed to evaluate /sens/ is quite difficult to determine experimentally in view of the weakness of the f-f transitions. One way to estimate it is by means of Eq. (10) where Tobs is the actual lifetime of the emitting excited state and Trad is its natural radiative lifetime which obey Einstein relation for spontaneous emission probability between two states with quantum numbers / and /, A( Pj, Pj ) (Gorller-Walrand and Binnemans, 1998) ... 

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]. 

A,W spontaneous emission probability d electric dipole operator... 

Since we are interested mainly in the roots of spontaneous emission, we shall quantize the electromagnetic field because we know that the semi-classical description where the atom is quantized and the field is classical, does not provide aity spontaneous emission it is introduced phenomenologically by a detailed balance of the population of the two-states atom and comparison with Planck s law. This procedure introduced by Einstein gave the well-known relationship between induced absorption (or emission), and spontaneous emission probabilities, the B12, B21 and A21 coefficients, respectively, but caimot produce the coherent aspect and its link with spontaneous emission. 

With the pump rate P (which equals the number of atoms that are pumped per second and per cm into the upper laser level 2)), the relaxation rates RiNi (which equal the number of atoms that are removed per second and cm from the level /) by collision or spontaneous emission), and the spontaneous emission probability A21 per second, we obtain from (2.21) for equal statist -... 

Radiative relaxation from an excited state /J of a lanthanide ion usually occurs in various lower lying state giving rise to several lines in the emission spectrum. For example, the red luminescence of Eu is a result of transitions from its Dq state to all of the lower lying Fj levels. The spontaneous emission probability, A, of the transition /J is related to its dipole strength according to... 

Purcell E.M. Spontaneous emission probabilities at radio frequencies. Phys. Rev. 1946 69 681 Quimby R.S., Miniscalco W.J., Thompson B. Clustering in erbium-doped sihca glass fibers analyzed using 980 nm excited-state absorption. J. Appl. Phys. 1994 76 4472-4478... 

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