Einstein Coefficient
The absorption and emission of radiation are related through the Einstein coefficients.
A molecule of a liquid, like the 2butanone molecules that produce the absorption band is involved in nearly continuous energy exchanging processes with neighboring molecules. The energy that a molecule gains from the incident radiation is quickly degraded into the thermal energy of the solution.
The consequences of the absorption into the thermal energy can be studied if we imagine the absorbing molecules to be noninteracting gas phase species in a radiation environment. To investigate this situation, we first describe the radiation environment in a nearly enclosed oven, or cavity.
A cavity acts as a blackbody, a term that implies that no frequency regions have particular tendencies to absorb or emit radiation, and tendencies that would lead to a particular colour. The nature of the radiation emitted from such a cavity was deduced to a particular colour. The nature of the radiation was deduced by Max Planck in the early stages of the development of quantum mechanics. The idea that the cavity radiation is in equilibrium with the particles led Planck to express the density of radiation in the cavity as
pv = 8∏hv^{3}/c^{3}/e^{hv/(kT)} – 1
This equation leads to an expression for the distribution of energy in the radiation that emerges from a cavity that is in complete agreement with experimentally obtained blackbody radiation is completely specified by the temperature. An example of the radiation emitted from a cavity at 6000 K, the approximate temperature of the surface of the sun, is illustrated.
Einstein coefficients: now we investigate, as Einstein did, the affairs of a molecule that is in, and is affected only by, the radiation of the cavity treated above. The particles with which we deal are in equilibrium with the radiation and, like the wall particles, are distributed throughout their allowed states according to the Boltzmann distribution. The population of the l and m states is given by:
Nm/ Nt = e ^{– hv/m/kT} Where hvtm is the energy separation between these states.
The population of the higher energy m state will be less than that of the lower energy l state.
Assume that the molecules within the cavity can absorb radiation and are thereby brought from the l quantum state. Einstein recognized that the equilibrium reached by the molecules in the cavity can be described on the basis of the three processes. The rtes of the processes are expressed in terms of Einstein coefficients. The Einstein coefficients for induced absorption and for included emission are represented by B, that for spontaneous emission by A. the coefficients for the induced transitions depend, as the two quantum states and on a transition operator that is not dependent on the direction of the radiation.
The Einstein coefficients determine the rate of the various transition processes by the expressions. If the three processes are the only effective processes, at equilibrium the rate of l = m transitions must equal the rate of m ltransitions. We can write with B_{ml} = B_{lm},
B_{lm} N_{i} p (v_{lm}) = B_{lm }N_{m} p (v_{lm}) + A_{ml} N_{m}
Rearrangements lead to:
N_{m}/N_{i} = B_{lm} p(v_{lm})/B_{lm} p(v_{lm}) + A_{ml}
Relative rates of spontaneous and induced transitions for molecules in a cavity at 6000 K:

Rotational Transitions
(vlm = 10^{10} Hz) 
Electronic transitions
(vlm = 10^{15} Hz) 
A_{ml}//B_{lm} 
6.2 × 10^{28} 
6.2 × 10^{13} 
P(v_{lm}) 
7.7 × 10^{24} 
2.1 × 10^{16} 
A_{ml}/[p(v_{lm})B_{lm}] 
8.0 × 10^{5} 
3.0 × 10^{3} 
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