# Black-body radiation

#### Prerequisite

## The thermodynamics of light: Equipartition and the black-body radiation puzzle

Once researchers were convinced that they understood what light was, and how it might be absorbed and emitted by matter, it was natural to consider how light played into thermodynamics and statistical physics. Suppose you took a box made of matter and filled it with a gas and let it come to thermodynamic equilibrium. The box and the gas would exchange energy at random and eventually come to the same temperature. It was understood that this meant that the gas molecules would have a certain average speed and that the distribution of speeds would be governed by the Maxwell-Boltzmann distribution (A Boltzmann factor, weighted by the energy of the molecule: $Ee^{-E/k_BT}$). This proved to work extremely well.

Now suppose instead you took a box made of matter and emptied everything out — "filled the box with vacuum". The matter in the walls contain vibrating charges so they should emit electromagnetic radiation into the box. Eventually, the electromagnetic radiation should exchange energy with the matter of the walls and come to a thermodynamic equilibrium — a common temperature. The question asked was:

*What does it mean for a box of light to have a particular temperature? *

Electromagnetic radiation can be made up of lots of different frequencies. What would be the distribution of electromagnetic energy in the box as a function of frequency?

This may sound like a silly question, but if you heat the box up until it starts to glow (to a few thousand degrees Kelvin), it's clear that it emits light. If you now cut a small hole in the box and let light come out from inside, you could take the light coming out and put it through a prism (or diffraction grating) to get the spectrum. If the light comes into thermal equilibrium with the matter, the spectrum should represent the distribution of the wavelengths of light "when it has a particular temperature" and should be independent of the material the box was made of. This proved to be true. (A hole in a box is referred to as a "black-body" since it is a perfect absorber: any light that falls on it will be absorbed.)

The natural assumption was made that there would be an equipartition of energy — and equal amount of energy (on the average) in each degree of freedom. The result is a simple function of the wavelength and the scale of the distribution is determined by the temperature.

The result of the classical theory (the Rayleigh-Jeans law) is shown for 5000 K in the figure at the right along with the observational results at three different temperatures.

It's clear that the classical result is terrible. It agrees for very long wavelengths, but the experimental curves cut of for short wavelengths. The experimental curves have a peak that moves to shorter wavelengths as the temperature goes up. What's going on?

## Planck's suggestion: Quantize matter

Researchers struggled with this problem. Some hypothesized that ordinary matter could not confine short wavelength radiation in a box and so there were necessarily leaks. Models of the leakage didn't do too badly. But the turn of the century brought a hypothesis that wound up transforming our view of matter entirely (and having powerful implications for the technology of the 20^{th} century).

Planck did some playing with the theoretical thermodynamics of electromagnetism. He began with the relation between the entropy, the internal energy, and the temperature. In one form, the result for classical electromagnetism was linear in the internal energy: $U$. Planck tried adding a quadratic term: $U +aU^2$. Using some other results of classical physics he showed that his "a" had to be proportional to frequency. After some transformations he was able to construct a function for the energy distribution that looked right and, by adjusting his single constant, a, he was able to fit the curves for all observations.

His result was that the density of energy in light at thermodynamic equilibrium depended on frequency ($f$) and temperature ($T$) in a more complex way then just a Boltzmann factor. It looked like this:

$$\rho(f,T) = \frac{8 \pi hf}{k_B c^3} \frac{1}{e^{\frac{hf}{k_BT}} - 1}$$

where $k_B$ is Boltzmann's constant and $h$ is a new constant — ** Planck's constant**. Note that it has dimensionality of energy times time, since $k_BT$ has dimensionality of energy and $f$ has dimensionality of 1/time. It sort of looks like a Boltzmann factor if you dropped the "-1" in the denominator. Then it would reduce to the classical law.

Interestingly enough, Planck did not stop there but tried to figure out what this new structure might mean physically. It's a complex story, but in the end, Planck reasoned that the walls of the box could only absorb or emit light in packets of particular sizes. (For details for the technically inclined, I recommend W. H. Cropper's book, *The Quantum Physicists and an Introduction to Their Physics.*) This was reasonable, given our understanding of resonance. Various bits of connected matter can oscillate at various natural frequencies and they preferentially absorb and emit energies at those frequencies.

## Einstein's better idea: Quantize the light

Einstein made the next big leap in 1905: he suggested that the limitation to absorption and emission didn't reside in the matter but in the light itself. Interestingly, Einstein also began with thermodynamics. Planck's curve yielded a function for the entropy of the electromagnetic energy. Einstein played with that and showed that it could be rearranged to look like the entropy for an ideal gas of particles if you took the number of particles corresponding to a frequency of light, $f$ as being

$$N = (\mathrm{amount\;of\;energy\;in\;frequency\;}f ) / (hf )$$

where $h$ was Planck's new constant. This could be rewritten

$$\mathrm{amount\;of\;energy\;in\;frequency\;}f = N (hf)$$

This is easy to interpret: the light having frequency $f$ comes in packets that have energy proportional to $f$*. *

Einstein took the next step and said, "If the quantization is in the light and not in the matter, this should mean that even low intensity light (few photons) should be able to knock out electrons from metals." The situation and his analysis is described in the page, The photoelectric effect.

Joe Redish 5/12/12 and 7/10/19

#### Follow-on

Last Modified: July 10, 2019