The Probability of Spontaneous Emission

By Equation (5.17), we see that the probability of spontaneous emission is proportional to I/LXIso we can use the same selection rules as previously established for absorption and stimulated emission to predict the allowance of spontaneous emission. Moreover, it can be noted that A is proportional to ci l and so, for a small energy separation between the two levels of our system, the radiative emission rate A should also be small. In this case, noimadiative processes, described by A r (see Equation (1.17)), can be dominant so that no emitted light is observed. 

EXAMPLE 5.3 Radiative lifetimes for electric dipole and magnetic dipole transitions. 

The order of magnitude of the probability of spontaneous emission in the visible range can be estimated from Equation (5.17). We consider a typical dielectric medium with = 1.5 at a wavelength in the middle of the visible range, Xq = 500 nm (coq = 3.8 x 10 s ). If we make the approximation l/ l = ea, where e = 1.6 x 10 C is the electroific charge and a = 1 A is the atomic radius, then Equation (5.17) gives an electric dipole probability of spontaneous emission of 

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, 

Recall that the radiative lifetime, tq = 1 /A, can be determined from Equation (1.20) by measuring the fluorescence lifetime t from a luminescence decaytime experiment, and provided that the nom-adiative rate Am is known. Eor pro-ces ses where the nom-adiative rate is negligible (Am 0), t = tq and so we will measure lifetimes in the range of nanoseconds for electric dipole transitions and lifetimes in the range of microseconds for magnetic dipole transitions. 

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Appendix A3 The Calculation of the Probability of Spontaneous Emission by Means of Einstein s Thermodynamic Treatment... 

THE PROBABILITY OF SPONTANEOUS EMISSION 273 After integrating this equation, we obtain... 

Finally, using the relationship between the Einstein A and B coefficients (A3.8) together with the previous expression, we obtain the following expression for the probability of spontaneous emission ... 

The possibility of deactivation of vibrationally excited molecules by spontaneous radiation is always present for infrared-active vibrational modes, but this is usually much slower than collisional deactivation and plays no significant role (this is obviously not the case for infrared gas lasers). CO is a particular exception in possessing an infrared-active vibration of high frequency (2144 cm-1). The probability of spontaneous emission depends on the cube of the frequency, so that the radiative life decreases as the third power of the frequency, and is, of course, independent of both pressure and temperature the collisional life, in contrast, increases exponentially with the frequency. Reference to the vibrational relaxation times given in Table 2, where CO has the highest vibrational frequency and shortest radiative lifetime of the polar molecules listed, shows that most vibrational relaxation times are much shorter than the 3 x 104 /isec radiative lifetime of CO. For CO itself radiative deactivation only becomes important at lower temperatures, where collisional deactivation is very slow indeed, and the specific heat contribution of vibrational energy is infinitesimal. Radiative processes do play an important role in reactions in the upper atmosphere, where collision rates are extremely slow. 

The lifetime of the upper state as determined by the probability of spontaneous emission between the two spin levels can be calculated from the Einstein coefficient... 

Einstein transition probability - A constant in the Einstein relation A.. + B.p for the probability of a transition between two energy levels i and j in a radiation field of energy density p. The A., coefficient describes the probability of spontaneous emission, while and B.. govern the probability of stimulated... 

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