### Relation between Einstein's Coefficients

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

 (2.148)

In the thermal equilibrium, the population of atoms is given by the Boltzmann distribution as
 (2.149)

where represents the statistical weight for the state n.

Equation (2.148) gives

 (2.150)

In the case of thermal equilibrium, the average specific intensity agrees with the Planck function as
 (2.151)

Using equations (2.149) and (2.151), equation (2.148) reduces to
 (2.152)

Lefthand-side of the equation is not dependent on the temperature. For this equation to be valid for various temperature,
 (2.153)

and
 (2.154)

Kohji Tomisaka 2012-10-03