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The Derivation of the Planck Formula for Thermal Radiation |
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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 E_{1} and the higher as E_{2}. 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 B_{12}(du/dν) and B_{21}(du/dν), respectively, where B_{12} and B_{21} are constants. The probability of a spontaneous emission is assumed to be a constant A_{21}.
Let N_{1} and N_{2} 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.,
This means that the ratio of the occupancies of the energy levels must be
But the occupancies are given by the Boltzmann distribution as
where k is Boltzmann's constant and T is absolute temperature. N_{0} is just a constant that is irrelevant for the rest of the analysis.
Thus according to the Boltzmann distribution
Therefore for radiative equilibrium it must be that
This condition can be solved for (du/dν); i.e.,
Consider what happens to the above expression for (du/dν) as T→∞. It goes to
Einstein maintained that (du/dν) must go to infinity as T goes to infinity. This requires that B_{12} be equal to B_{21}.
Thus
Now Planck's assumption that (E_{2}−E_{2}) is equal to hν is introduced. Thus
The Rayleigh-Jeans Radiation Law says
The Planck formula must coincide with the Rayleigh-Jeans Law for sufficiently small ν. Note that
for sufficiently small ν.
This means that
Equating the two expressions for (du/dν) gives
Thus
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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