A1: Deriving Planck's Distribution Law

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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 formula 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, (\(du/d \nu\)). Likewise the probability of an induced transition from level 2 down to level 1 is assumed also to be proportional to (\(du/d\nu\)). These two probabilities are taken to be \(B_{12}(du/d\nu)\) and \(B_{21}(du/d\nu)\), 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.,

\[\underbrace{N_1\left[B_{12}\left(\dfrac{du}{d\nu}\right)\right]}_{\text{flow up}} = \underbrace{N_2\left[B_{21}\left(\dfrac{du}{d\nu} \right)+ A_{21} \right]}_{\text{flow down}} \nonumber \]

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

\[\dfrac{N_2}{N_1} = \dfrac{B_{12}\left(\dfrac{du}{d\nu}\right)}{ B_{21}\left(\dfrac{du}{d\nu}\right) + A_{21}} \label{einstein2} \]

But the occupancies are given by the Boltzmann distribution as

\[N_1 = N_0 \exp \left(− \dfrac{E_1}{kT} \right) \nonumber \]

and

\[N_2 = N_0 \exp \left(−\dfrac{E_2}{kT} \right) \nonumber \]

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

\[\dfrac{N_2}{N_1} = \exp \left(−\dfrac{E_2−E_1}{kT} \right) \label{boltz2} \]

Therefore for radiative equilibrium, Equations \ref{boltz2} and \ref{einstein2} can be set to each other and

\[\exp\left(−\dfrac{E_2−E_1}{kT} \right) = \dfrac{B_{12}\left(\dfrac{du}{d\nu}\right)}{ B_{21}\left(\dfrac{du}{d\nu}\right) + A_{21}} \nonumber \]

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

\[\dfrac{du}{d\nu} = \dfrac{A_{21}}{B_{12}\exp \left( \dfrac{E_2−E_1}{kT} \right)−B_{21}} \nonumber \]

Consider what happens to the above expression for as \(T \rightarrow \infty\). It goes to

\[\lim _ {T \rightarrow \infty} \dfrac{du}{d\nu} = \dfrac{A_{21}}{B_{12}−B_{21}} \nonumber \]

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

\[\dfrac{du}{d\nu} = \dfrac{A_{21}/B_{21}}{\exp \left(\dfrac{E_2−E_1}{kT}\right)−1} \label{eq10} \]

Now Planck's assumption is introduced:

\[E_2−E_1 = hν \nonumber \]

Thus Equation \ref{eq10} becomes

\[\dfrac{du}{d\nu} = \dfrac{A_{21}/B_{21}}{\exp \left(\dfrac{hv}{kT}\right)−1} \label{eq11} \]

The Rayleigh-Jeans Radiation Law says

\[\dfrac{du}{d\nu}= \dfrac{8πkTν^2}{c^3} \label{RJ} \]

The Planck formula must coincide with the Rayleigh-Jeans Law for sufficiently small \(ν\). Note that the exponent in the denominator of Equation \ref{eq11} can be expanded (via a Taylor expansion):

\[\exp\left(\dfrac{hν}{kT}\right) \approx 1 + \dfrac{hν}{kT} \nonumber \]

for sufficiently small \(ν\).

This means that Equation \ref{eq11} simplifies to

\[\dfrac{du}{d\nu} = \dfrac{A_{21}/B_{21}}{1 + (hν/kT) −1} = \dfrac{A_{21}/B_{21}}{hν/kT} \nonumber \]

and hence

\[\dfrac{du}{d\nu}= \left(\dfrac{A_{21}}{B_{21}} \right) \left( \dfrac{kT}{hν}\right) \label{eq20} \]

Equating Equations \ref{RJ} and \ref{eq20} for \((du/dν)\) gives

\[ \left(\dfrac{A_{21}}{B_{21}}\right) \left(\dfrac{kT}{hν}\right) = \dfrac{8πkTν^2}{c^3} \nonumber \]

which reduces to

\[\dfrac{A_{21}}{B_{21}} = \dfrac{8πhν^3}{c^3} \nonumber \]

Thus

\[\dfrac{du}{d\nu} = \dfrac{8πhν^³}{c^³} \dfrac{1}{\exp(hν/kT)−1} \nonumber \]

This is Planck's formula in terms of frequency.

Reference

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