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The Derivation of the Planck
Formula for Thermal Radiation

Albert Einstein developed a simple but effective analysis of induced emission and absorption of radiation along with spontaneous emission that can be used to derive the Planck formular for thermal radiation.

Consider two energy levels for the molecules in a material. The lower of the two is denoted as E1 and the higher as E2. The probability of a transition from level 1 up to level 2 through induced absorption is assumed to be proportional to the energy density per unit frequency interval, (du/dν). Likewise the probability of an induced transition from level 2 down to level 1 is assumed also to be proportional to (du/dν). These two probabilities are taken to be B12(du/dν) and B21(du/dν), respectively, where B12 and B21 are constants. The probability of a spontaneous emission is assumed to be a constant A21.

Let N1 and N2 be the number of molecules in energy states 1 and 2, respectively. For equilibrium the number of transitions from 1 to 2 has to be equal to the number from 2 to 1; i.e.,

N1(B12(du/dν)) = N2(B21(du/dν) + A21)

This means that the ratio of the occupancies of the energy levels must be

N2/N1 = B12(du/dν)/[B21(du/dν) + A21]

But the occupancies are given by the Boltzmann distribution as

N1 = N0exp(−E1/kT)
and
N2 = N0exp(−E2/kT)

where k is Boltzmann's constant and T is absolute temperature. N0 is just a constant that is irrelevant for the rest of the analysis.

Thus according to the Boltzmann distribution

N2/N1 = exp(−(E2−E1)/kT)

Therefore for radiative equilibrium it must be that

exp(−(E2−E1)/kT) = B12(du/dν)/[B21(du/dν) + A21]

This condition can be solved for (du/dν); i.e.,

(du/dν) = A21/[B12exp((E2−E1)/kT)−B21]

Consider what happens to the above expression for (du/dν) as T→∞. It goes to

(du/dν) = A21/[B12−B21]

Einstein maintained that (du/dν) must go to infinity as T goes to infinity. This requires that B12 be equal to B21.

Thus

(du/dν) = (A21/B21)/[exp((E2−E1)/kT)−1]

Now Planck's assumption that (E2−E2) is equal to hν is introduced. Thus

(du/dν) = A21/B21/[exp((hν/kT)−1]

The Rayleigh-Jeans Radiation Law says

(du/dν) = 8πkTν²/c³

The Planck formula must coincide with the Rayleigh-Jeans Law for sufficiently small ν. Note that

exp(hν/kT) ≅ (1 + hν/kT)

for sufficiently small ν.

This means that

(du/dν) = (A21/B21)/[1 + (hν/kT) −1] = (A21/B21)/[(hν/kT)]
and hence
(du/dν) = (A21/B21)(kT/hν)

Equating the two expressions for (du/dν) gives

(A21/B21)(kT/hν) = 8πkTν²/c³
and this reduces to
A21/B21 = 8πhν³/c³

Thus

(du/dν) = (8πhν³/c³)/[exp(hν/kT)−1]

This is Planck's formula in terms of frequency.


Reference:

K.D. Möller, Optics, University Science Books, Mill Valley, California, 1988.

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