Both absorption and stimulated emission
are induced transitions,
whose rates are proportional to the intensity I (T) of the
surrounding radiation
(a density of energy expressed in pascals (Pa) or joules per cubic meter).
Being exact timereversal of each other, they must occur at the same rate:
B_{12} =
B_{21} = B
The coefficients A and B are properties of the atom and,
thus, do not depend on the temperature T of the surrounding
photon gas. When thermal equilibrium is achieved at a certain
temperature T, the numbers of atoms in both states
(which may depend on T) remain constant.
So, the total transition rates from either energy level to the other are equal:
N_{1} [ B_{12} I (T) ] =
N_{2} [ A + B_{21} I (T) ]
Solving for I, this yields: I (T) =
( A / B ) / [ N_{1} / N_{2}  1 ]
On the other hand, the ratios of the populations of the two energy levels is
an exponential function of the ratio of the energy difference to the
thermal energy (kT)
according to Boltzmann's statistics:
N_{2} / N_{1} =
exp [ ( E_{2} E_{1 }) / kT ]
Einstein's equation:
A / B =
8p hn^{3} dn
/ c^{3}
In this, dn stands for the width of
the atomic transistion spectrum which is very narrow compared to the whole blackbody
spectrum and is thus adequately represented by a delta distribution
(instead of properly convoluting the blackbody spectrum with the socalled
atomic lineshape function ).
Absorption (electromagnetic radiation)

Spontaneous emission

Stimulated emission
Detailed balance

Boltzmann distribution

Planck's law

Einstein coefficients

Atomic spectral line