The Planck's Radiation Formula Derivation Using the Einstein Coefficients

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

hν  =  E2 - E1
dN2/dt  =  B12 · u(ν) · N1
dN2/dt  - B12 · u(ν) · N2
dN2/dt  =  − A21 N2

Many transitions occur equally in both directions (absorption and emission) in the equilibrium position; therefore the population number (?) may not have any net change:

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

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

®  B12 u(ν)N(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: 

Þ B12 = B21 u(ν)  =  (A21/B12)/(ehν/kT− 1)
    Comparing with the Rayleigh-Jeans-Law  gives:
  A21/B12  =  8πhn ³/c³ 

The ν3 relation between spontaneous emission and absorption is very important here !
By combining what weve learned about the coefficients B12, B21 and A21 ,we obtain the Planck Radiation formula:
 

u(ν)  =  hν³/.1/ehν/kT− 1


Here we have also approached an important principle in the operation of Lasers; we must only interpret the above formula {dN2/dt = B12 u(ν)N1 − B21 u(ν)N2 − A21N2} in a different way:
Therefore we have laser operation when the photon production is alltogether more than 0. 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 will always be:

N2 > N1,

i.e. the population number (?) of the upper state should be more than the population number of the lower state. However this is not possible at thermal equilibrium position as one can see from examining the Boltzmann energy distribution. Thus lasers can be only operate in the non-equilibrium thermodynamic state.

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