How does the number of photons determine wavelength?

[vhbelvadi]

While reading on Cerenkov radiations I've been coming across statement to this effect:

It has been noted that the number of photons at a particular frequency or wavelength, as it turns out, is proportional to 1/l²

How is that (mathematically) so?

 

[Simon Bridge]

That's the shape of the spectrum.
http://en.wikipedia.org/wiki/Frank–Tamm_formula

It's not that the number of photons depends on the wavelength it is that the wavelength of the photon and the rate that photons are produced both depend on the same thing.

 

[vhbelvadi]

@Simon - So the wavelength of a photon and rate of emission are inter-dependent?

I've been looking for a neat derivation for the Frank-Tamm formula. Any ideas/pages that have it?

And thanks for your reply!

 

[Simon Bridge]

Um - not off hand - deriving the F-T formula is what you need to do though.
You should be able to do it from first principles though. Start with how Cherenkov radiation happens in the first place.

It's easy to understand by analogy: look at other common spectra - black-body radiation for eg. I take it you have no problem with the idea that a hot body radiates more in some parts of the EM spectrum than in others? The rate of emmission and the wavelength of the photons both depend on temperature - so we can eliminate the common variable and plot one against the other.

 

[Naty1]

some information and references here:

http://en.wikipedia.org/wiki/Frank-Tamm_formula