Ultraviolet catastrophe

[PG1995]

Hi

I believe the anomalies found in mathematical models of photoelectric emission and black-body radiation were behind the birth of quantum mechanics. I understand the photoelectric emission but I'm having trouble with Rayleigh's law about black-body radiation. What was exact wrong with it? Was the wrong that according to the following formula as the wavelength becomes shorter, the intensity increases but experimentally this wasn't the case?

[latex]I(\lambda )=\frac{2\pi ckT}{\lambda ^{4}}[/latex]

Please help me. Thank you.

Regards
PG

Helpful links:
1: https://en.wikipedia.org/wiki/Ultraviolet_catastrophe
2: https://en.wikipedia.org/wiki/Rayleigh–Jeans_law
3: https://en.wikipedia.org/wiki/Spectral_radiance

 

[steveB]

I'm having trouble with Rayleigh's law about black-body radiation. What was exact wrong with it?

One underlying idea of black body radiation is that it can be well modeled as a large enclosure with a pinhole for the output energy. Such a large enclosure will effectively have an infinite number of allowed cavity modes, effectively because it is very very large. There are more modes with higher frequency. With a small aperture (pin-hole), the radiation will bounce around, get absorbed by the walls and then re-emitted off the interior walls until there is a thermal equilibrium. Then, the small amount of radiation leaking out of the pin-hole from the inside will not drain the internal energy but will emit a spectrum that is representative of the black body emission.

So, there are two issues with Rayleigh's law. First, it assumes that all modes are filled and will have equal thermal energy, which is just an assumption. That's not even a reasonable assumption because the integrated energy then becomes infinite (hence the ultraviolet catastrophe). Also it assumes energy is a continuum, which is a very reasonable assumption given the knowledge at that time.

I'm sure Planck looked at the equal energy assumption as incorrect, but he no doubt was not initially questioning the idea that energy is continuous in such a system (this type of system is basically modeled as a collection of harmonic oscillators, which seems to be a recurring theme in physics, including quantum field theory).

Planck eventually realized that energy must be quantized as E=nhf, where, n is an integer, h is Planck's constant and f is frequency. Combining this with the idea that the distribution of modes with energy should not be equal, he could think of the thermal energy of the cavity needing to fill up the lower energy states allowed by E=nhf. Plank was able to work out the actual energy distribution based on the assumption that energy is quantized, and it happened to match experimental results.

So, you can see that physicists often just make assumptions, and make a prediction from that. Then they see how well it works. In Releigh's case, everyone of that time believed energy is continuous, and then he assumed that modes were equally distributed because that seemed to be true in his work on thermal capacity. But, it didn't work. Planck, assumed energy is quantized, and that was his creativity to consider that possibility when no one else would accept such a silly idea. Even he did not know what to make of his discovery, but then Einstein came along and made the next step. Then, others continued and developed the quantum theory that has been so successful.

 

[Mike odom]

yes, and as your first link points out, Max Planck found the answer by first assuming that electromagnetic radiation didn't follow the classical description, but "could only be emitted in discrete packets of energy proportional to the frequency, as given by Planck's law".

 

[PG1995]

Thank you, Steve, Mike.

Steve: I wasn't able to fully understand your reply because I haven't studied this topic in full detail. When I get free time, I will go through the material in detail so that I can fully understand what you said. Thanks.

I take it that my guess was correct that the wrong with the Rayleigh formula was that according to it as the wavelength becomes shorter, the intensity increases but experimentally this wasn't the case. This makes me wonder that why would Rayleigh propose the formula in the first place when he knew that it became almost completely invalid as frequencies get higher? I don't think in those days generating ultraviolet radiation was that much difficult that he wasn't able to test his half-baked formula before proposing it.

[LATEX]I(\lambda )=\frac{2\pi ckT}{\lambda ^{4}}[/LATEX]
Regards
PG

 

[steveB]

I wasn't able to fully understand your reply because I haven't studied this topic in full detail. When I get free time, I will go through the material in detail so that I can fully understand what you said. Thanks.

That's understandable. This is not an easy topic and the details require careful study.

I take it that my guess was correct that the wrong with the Rayleigh formula was that according to it as the wavelength becomes shorter, the intensity increases but experimentally this wasn't the case. This makes me wonder that why would Rayleigh propose the formula in the first place when he knew that it became almost completely invalid as frequencies get higher? I don't think in those days generating ultraviolet radiation was that much difficult that he wasn't able to test his half-baked formula before proposing it.

I think the way to say it is that the Rayleigh formula was eventually proved to be wrong. What was wrong with it is that the initial assumptions that led to the formula were not correct. The idea that energy could be discrete rather than continuous was never even considered in his time. And, his equal partitioning assumption was just a guess made in the absence of evidence to the contrary. I mentioned that he made that latter assumption based on his earlier work on heat capacity, which did work out. By the way, we still use the Rayleigh Scattering law with it's lambda to the -4 dependence. That's why the sky is blue. It just doesn't work for black body radiation because (as Planck showed) the two assumptions were wrong.

It is hard for us to appreciate the limited knowledge people had in Rayleigh's time. The guy was a genius of a very high level and he did very much work to progress physics and engineering. But, we are all limited by the times we live in. So, "half baked" is a little strong because he would proposed the idea, not realizing the eventual issues that would be shown. A great physicist will have 100 half-baked ideas for every good one, and a good physicist, will have 1000 for every good one. That's the nature of the game, and the human intellectual endeavors should be likened to a relay race where the baton is passed from generation to generation with a finish line 10,000 years in the future.

 

[PG1995]

Thank you, Steve.

It is hard for us to appreciate the limited knowledge people had in Rayleigh's time. The guy was a genius of a very high level and he did very much work to progress physics and engineering. But, we are all limited by the times we live in. So, "half baked" is a little strong because he would proposed the idea, not realizing the eventual issues that would be shown. A great physicist will have 100 half-baked ideas for every good one, and a good physicist, will have 1000 for every good one.

I'm sorry about this. I do agree that "half-baked" was not an apt word to use there. But I would still say that my intention was not to belittle the genius of Rayleigh. My personal favorites have been Archimedes, Michael Faraday, Einstein, and Tesla.

That's the nature of the game, and the human intellectual endeavors should be likened to a relay race where the baton is passed from generation to generation with a finish line 10,000 years in the future.

I really liked what you said above.

Best regards
PG