Analysis using the Einstein's coefficients is applicable to
the state in which the thermal equilibrium is not achieved.
However, if this is applied to the thermal equilibrium,
we can obtain relations between these coefficients.
Assume the detailed balance is achieved between
the spontaneous emission, the induced emission and the absorption as
In the thermal equilibrium,
the population of atoms is given by the Boltzmann distribution as
where represents the statistical weight for the state n.
Equation (2.148) gives
In the case of thermal equilibrium, the average specific intensity
agrees with the Planck function as
Using equations (2.149) and (2.151),
equation (2.148) reduces to
Lefthand-side of the equation is not dependent on the temperature.
For this equation to be valid for various temperature,