(37)

Suppose that the gas is in equilibrium with a radiation field B(?, T). In a steady state, the number atoms leaving the lower state must equal the number leaving the upper state. Combining the last two equations,

The probability that an atom will be in a state with energy Eu is exp{- Eu/kT) . This is

proved in any book on statistical mechanics, but it is beyond the scope of this chapter. The relative populations of the two states is

Using equation (33),

Therefore, combining equations (37) and (38),

Since the constants and depend only on the structure of the atom, and not on temperature or other aspects of the environment,25

Then

So, using the Plank black-body law [equation (2)],

These coefficients are called the Einstein coefficients; they depend only on the nature of the atom and not on the temperature, the ambient radiant intensity, or any other aspect of the atom’s environment. Therefore, equations (42) and (44) are true in general. In particular, note that this argument shows that stimulated emission must

exist; i.e., . This proved the existence of stimulated emission long before it had been observed in the laboratory; today, of course, it is the basis of the laser (Light Amplification by Stimulated Emission of Radiation) industry. Stimulated emission is necessary, as I have already mentioned, to keep the Second Law of thermodynamics from being violated, because without it, atomic absorption and emission processes could not be in balance with emission from a black body. If there is an enclosure with perfectly black walls with temperature T, then the radiation field inside the enclosure must be equal to the Planck function B(?, T). But now introduce a gas into that enclosure. Without stimulated emission, there would be a continuous transfer of energy either to or from the gas. This, as we know, is impossible.