By Chad Orzel, Union College
The key idea to get the blackbody spectrum comes from thermodynamics, and is called equipartition. Equipartition simply refers to an equal division of energy. We take the energy contained in a hot object and divide it equally among all the possible wavelengths of light that the object could emit. When we add them all up, we should get the blackbody spectrum.
Blackbody Spectrum
No matter what the temperature of an object is, the shape of the spectrum is always the same. However, there’s no explanation of why it should be that particular shape. Through the late 1800s, a lot of really smart people tried to explain the light we see from hot objects, but all of them failed.
Model by Lord Rayleigh and James Jeans, two British physicists, gave a clear illustration of what the problem was. To understand what Rayleigh and Jeans were doing, we need a model of what, exactly, a blackbody is that gives us a way to count possible wavelengths.
You take a box with a small hole in one end, and look at the light that comes out of the hole. As long as the hole is small compared to the size of the box, this is an excellent approximation of a blackbody: A ray of light that enters the hole would need to bounce around a bunch of times, with a little light absorbed each time, before it could find its way back out. If the hole is small enough, the light will be completely absorbed long before it could reach the exit again. (That’s why it’s called a blackbody, because it absorbs light of every possible frequency.)
If we see light coming out from the hole, then, it has nothing to do with the light that comes in from the outside, all of which gets absorbed. The light that comes out is just the light that’s allowed to exist inside the box. We can figure out which wavelengths are allowed in the box, give them each an equal share of the heat energy available, and hopefully their sum will amount to a blackbody spectrum.
Standing Wave
The wavelengths that fit in a box are constrained by the walls of the box. A light wave in a metal box has to go to zero right at the wall of the box, so only certain wave patterns can fit: the ones where the wave starts at zero at one end and returns to zero at the other end. This happens for particular values of the wavelength that are simply related to the size of the box.
The simplest wave pattern that will fit inside a box starts at zero, goes up to some maximum, then comes back down to zero at the other end. This type of pattern is called a standing wave.
This is a transcript from the video series Einstein’s Legacy: Modern Physics All around You. Watch it now, on Wondrium.
Different Standing Wave Patterns
The simplest standing wave pattern looks like half of a sine wave, so the wavelength associated with it is twice the length of the box. There’s also a characteristic frequency associated with it, given by the speed of the wave divided by the wavelength.
But of course, there are a lot of other standing wave patterns that also satisfy the condition of being zero at the ends of the box at the fundamental frequency. The next of these fits one full sine wave into the length of the box and has a frequency twice that of the fundamental. (This adds another fixed node right in the center, where the wave is always zero.)
Then there’s a harmonic that looks like one-and-a-half sine waves, at three times the fundamental frequency, then one that looks like two full sine waves at four times the fundamental frequency, and so on.
We call these wavelengths allowed modes, and we can assign each of them a number: The fundamental is 1, the first harmonic 2, and so on.
The Ultraviolet Catastrophe
There are, in fact, an infinite number of standing wave modes that go to zero at both ends of the box, but it’s the kind of infinity that physicists invented calculus to deal with. This counting of modes is a well-known type of problem, and the most obvious attack on the blackbody spectrum.
However, when we try to combine these modes with the idea of equipartition, as done by Rayleigh and Jeans, it fails spectacularly. The problem is that as we go up the scale of harmonics, the modes get closer and closer together. There’s a big wavelength difference between the fundamental and the first harmonic, but the difference between the first and second harmonics is smaller, and the difference between second and third is smaller still.
If we look at the total amount of light in a narrow band of wavelengths, the way we do in an actual spectrometer, the number of allowed modes that fall within that band just keeps going up and up as we move the center of the band to shorter wavelengths.
This model doesn’t look too bad at long wavelengths, which is why Rayleigh and Jeans were working on it, but it doesn’t produce a peak as we see in the real blackbody spectrum. Instead, the amount of light in a given range of wavelengths just keeps going up and up forever.
The model predicts that any hot object should spew out an infinite amount of light in the ultraviolet and x-ray part of the spectrum. This is not exactly a feature you want in a toaster. This failure is so spectacular that it picked up a colorful name: the ultraviolet catastrophe.
Common Questions about Rayleigh-Jeans Model of the Blackbody and the Ultraviolet Catastrophe
The simplest wave pattern that can fit inside a box starts at zero, goes up to some maximum, then comes back down to zero at the other end. This type of pattern is called a standing wave.
The simplest standing wave pattern looks like half of a sine wave. Then there is one full sine wave. There is also harmonic that looks like one-and-a-half sine waves.
There are infinite number of standing wave modes that go to zero at both ends of the box, but it’s the kind of infinity that physicists invented calculus to deal with. This counting of modes is a well-known type of problem, and the most obvious attack on the blackbody spectrum.
