The Planck's Radiation Formula Derivation Using the Einstein's Coefficients

A number of photons is assigned to every light wave of the frequency ν. Further one assumes that there are discrete atomic (or molecular) energy levels.

| hν = E2 - E1

- Absorption of one photon hν brings an atom or molecule from the ground state E1 to the excited state E2. The number of absorption processes per time unit is proportional to the population number (?) N1 (number of the atoms / molecules in the lower state 1) and the radiation density u(ν). The probability of such process is given by the letter B12:

dN2/dt = B12 · u(ν) · N1

- Emission of one photon hν = E2 − E1 corresponds to the transition from energetically higher state to the lower state. As with the absorption such a transition can be forced by irradiation to photons. With this induced emission the number of photons increases and the population number (?) N2 decreases per time:

dN2/dt = - B12 · u(ν) · N2

- S pontaneous emission occurs from the state E2, so that the population number N2 (?) decreases:

dN2/dt = − A21 N2

There are a lot of transitions occuring equally in both directions (absorption and emission) in the equilibrium position; therefore the population number (?) may not change on average:

dN2/dt = 0 (sum of the three processes)

dN2/dt = B12 u(ν)N1 − B21 u(ν)N2 − A21N2 = 0

® B12 u(ν)N1 = (B21u(ν)+A21)N2

(B12u(ν))/(A21+B21u(ν)) | = | N2/N1 | = | e−E2/kT/e−E1/kT | = | e−hν/kT
|  |  | → | thermal equilibrium, thus Boltzmann distribution |  |

u(ν) = A21/(B12 ehν/kT − B21)

The coefficients A12, B12 and B21 are determined from experiments:

1.) For T ® ¥ should be u(ν) ® ¥.

Þ | B12 = B21 | → | u(ν) = (A21/B12)/(ehν/kT− 1)

2.) For small frequencies, hν << kT, should be hold true experimentally proven Rayleigh-Jeans-Law {u(ν) = 8pn²/c³· kT}:
Series expansion: ehν/kT » 1 + hν/kT + .... → u(ν) ≈ A21/B12 .kT/hν

Comparing with the Rayleigh-Jeans-Law gives:

A21/B12 = 8π h n ³/c ³

The ν3 relation between spontaneous emission and absorption is very important here !
If we summarize all our knowledge about the coefficients B12, B21 and A21 ,we will obtain the Planck Radiation formula:

u(ν) = 8πhν³/c³.1/ehν/kT− 1

So, by the way we have also approached an understanding of operation principle of Lasers; since we must only interpret the above formula {dN2/dt = B12 u(ν)N1 − B21 u(ν)N2 − A21N2} in another way:
Therefore we have lasing whether the photon production is alltogether more than 0. So if u(ν) increases, N2 should decrease, i.e. dN2/dt < 0. This is important for the laser operation:

B12 u(ν)N1 − B21 u(ν)N2 − A21N2 < 0

or, if we neglect the spontaneous emission there should be always the following

N2 > N1,

i.e. the population number (?) of the upper state should be more than the population number of the lower state. However it's not possible in the thermal equilibrium position as one can see from the Boltzmann energy distribution. Thus lasing can be reached only in the thermodynamically non-equilibrium position.