Details

Snapshot 1: quantum mechanics: Planck's law is applied to the energy quantization; the resulting curve follows the blackbody radiation prediction

Snapshot 2: quantum mechanics: increasing the temperature leads to a shift of the maximum toward the visible spectrum frequencies, according to Wien's law

Snapshot 3: intermediate case: by applying an factor to the quantization, we find an intermediate solution between the Planck and the Rayleigh–Jeans distributions

A blackbody is an idealized object absorbing all incident electromagnetic radiation, regardless of frequency or angle of incidence. Consequently, it absorbs all colors of light as well and thus appears completely black.

A blackbody in thermal equilibrium (at a constant temperature) emits electromagnetic blackbody radiation. The first theoretical analysis resulted in the Rayleigh–Jeans formula, based on the classical hypothesis that the energies of the photons inside the blackbody could have any values from an energy continuum.

We begin with the Boltzmann distribution for thermal equilibrium with noninteracting particles distributed over energy states:

.

The average photon energy at an equilibrium temperature is then given by:

and thus, for the blackbody spectral radiance:

,

which is, as noted, the Rayleigh–Jeans formula.

This equation predicts a spectral radiance that diverges for high frequencies, leading to the so-called "ultraviolet catastrophe."

To solve the problem, Planck discarded the continuum hypothesis for the photon energy and assumed instead that photons at frequency could carry energies from a discrete spectrum:

,

.

In this way, as shown, we get a new average photon energy value:

so that

.

Now we parametrize a smooth passage from Planck's to Rayleigh–Jeans's distribution. Introducing a real parameter , we evaluate the average photon energy assuming a minimum energy of:

,

,

.

The average energy value becomes

,

where has been inserted as a normalization factor.

Then for the spectral radiance, we arrive at an equation interpolating between Planck's law and the Rayleigh–Jeans's formula, using the parameter:

.

References

[1] R. A. Serway, C. A. Moses and C. J. Moyer, Modern Physics, 2nd ed., Philadelphia: Saunders College Publishing, 1997.

[2] SpectralCalc.com. "The Planck Blackbody Formula in Units of Frequency." (Jul 15, 2021) www.spectralcalc.com/blackbody/planck_blackbody.html.