Absorption intensities
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص434-435
2025-12-03
19
Absorption intensities
Einstein identified three contributions to the transitions between states. Stimulated absorption is the transition from a low energy state to one of higher energy that is driven by the electromagnetic field oscillating at the transition frequency. We saw in Section 9.10 that the transition rate, w, is the rate of change of probability of the molecule being found in the upper state. We also saw that the more intense the electromagnetic field (the more intense the incident radiation), the greater the rate at which transitions are induced and hence the stronger the absorption by the sample (Fig. 13.5). Einstein wrote the transition rate as
w=Bρ
The constant B is the Einstein coefficient of stimulated absorption and ρdν is the energy density of radiation in the frequency range ν to ν + dν, where ν is the frequency of the transition. When the molecule is exposed to black-body radiation from a source of temperature T, ρ is given by the Planck distribution (eqn 8.5):
For the time being, we can treat B as an empirical parameter that characterizes the transition: if B is large, then a given intensity of incident radiation will induce transitions strongly and the sample will be strongly absorbing. The total rate of absorption, W, the number of molecules excited during an interval divided by the duration of the interval, is the transition rate of a single molecule multiplied by the number of molecules N in the lower state: W = Nw. Einstein considered that the radiation was also able to induce the molecule in the upper state to undergo a transition to the lower state, and hence to generate a photon of frequency ν. Thus, he wrote the rate of this stimulated emission as , w′=B′ρ , where B′ is the Einstein coefficient of stimulated emission. Note that only radiation of the same frequency as the transition can stimulate an excited state to fall to a lower state. However, he realized that stimulated emission was not the only means by which the excited state could generate radiation and return to the lower state, and suggested that an excited state could undergo spontaneous emission at a rate that was independent of the intensity of the radiation (of any frequency) that is already present. Einstein therefore wrote the total rate of transition from the upper to the lower state as , w′=A+B′ ρ , The constant A is the Einstein coefficient of spontaneous emission. The overall rate of emission is , W′=N′ (A+B′ ρ) , where N′ is the population of the upper state. As demonstrated in the Justification below, Einstein was able to show that the two coefficients of stimulated absorption and emission are equal, and that the coefficient of spontaneous emission is related to them by
Justification 13.2 The relation between the Einstein coefficients. At thermal equilibrium, the total rates of emission and absorption are equal, so NB ρ=N′(A+B′ ρ) This expression rearranges into
We have used the Boltzmann expression (Molecular interpretation 3.1) for the ratio of populations of states of energies E and E′ in the last step:
This result has the same form as the Planck distribution (eqn 13.7), which describes the radiation density at thermal equilibrium. Indeed, when we compare the two expressions for ρ, we can conclude that B′=B and that A is related to B by eqn 13.11.
The growth of the importance of spontaneous emission with increasing frequency is a very important conclusion, as we shall see when we consider the operation of lasers (Section 14.5). The equality of the coefficients of stimulated emission and absorption implies that, if two states happen to have equal populations, then the rate of stimulated emission is equal to the rate of stimulated absorption, and there is then no net absorption. Spontaneous emission can be largely ignored at the relatively low frequencies of rotational and vibrational transitions, and the intensities of these transitions can be discussed in terms of stimulated emission and absorption. Then the net rate of absorption is given by , W net = NBρ− N′B′ ρ=(N−N′) Bρ , and is proportional to the population difference of the two states involved in the transition.
Fig. 13.5 The processes that account for absorption and emission of radiation and the attainment of thermal equilibrium. The excited state can return to the lower state spontaneously as well as by a process stimulated by radiation already present at the transition frequency.
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