In 1934, Cherenkov discovered that electrons moving with large constant velocities through polarizable media caused a faint bluish glow. 'Why is the radiation blue in colour? How can a charged body moving with constant velocity emit an electromagnetic radiation?'
[Physics] Why is Cherenkov Radiation faint blue in colour
cherenkov-radiationelectromagnetic-radiationvisible-light
Related Solutions
I'm by no means an expert on this, but since I found this question interesting I searched in the infinite vastness of the internet for a hint and came up with three links:
- Number 1. (search for the word "celebrated")
- Number 2. eq. 1.10 on p.6 (warning: it's a.) a pdf and b.) it's in german)
- Number 3. eq. 2.5 on p.7 (same warning)
Have I interpreted this equation correctly?
Basically yes. You should write it as follows:
$$ N = \dfrac{ 2πα Z^2 L\sin^2θ d\lambda}{\lambda^2}$$
(so there's no confusion between the length L and the wavelength lambda)
In the links they write $\frac{dN}{dx}$, so when you write it the way you did you have already integrated over dx. But you still have to integrate over the range of wavelenghts of emitted photons, this is why you can't cancel the $d\lambda$-factor.
Since $\cos(\theta)=\frac{1}{n \beta}$, you can write the $sin(\theta)^2$ as:
$\left(1-\frac{1}{\beta^2 n^2}\right)$ which then yields $$\frac{dN}{dx} = \dfrac{ 2πα Z^2 d\lambda }{\lambda^2}\left(1-\frac{1}{\beta^2 n^2}\right)$$ as given in the second two links.
edit: I just realized that I completely ignored that there's no Z in your formula. So Z is the charge of the particle traveling through the medium. When you want to calculate the number of photons emitted by an electron, then Z would obviously be 1.
General Considerations
The place to start is with the Frank-Tamm formula for the quantity and spectrum of Cerenkov light. $$ \frac{\mathrm{d} E}{\mathrm{d}x\,\mathrm{d}\omega} = \frac{q^2}{4 \pi} \mu(\omega) \omega \left(1 - \frac{c^2}{v^2 n^2(\omega)} \right) \,, $$ where $v$ is the particle's speed, $q$ it's charge, and $\mu(\omega)$ and $n(\omega)$ are the frequency dependent permeability and index of refraction of the material respectively.
This depends on both the speed of the charged particles and the radiating material, but the explicit linear dependence on $\omega$ means that it tends to be brighter at higher frequency, leading to the characteristic blue appearance.
To find a material in which it had a different appearance would require seeking one for which $\mu$ or $n$ were strong functions of frequency in the higher frequency half of the human visual spectrum.
A Couple of Asides
Cerenkov radiation occurs whenever a charged particle passes through a medium at speeds faster than the $c/n$. There is nothing special about electron, water or nuclear reactors in that regard; those are just one of the very few situations in which you can see the light with the unaided eye.
The correct Anglicanism of the name is a matter of occasional debate. I had a Russian professor in grad school who preferred "Cerenkov" and I follow his lead.
Best Answer
From the wiki article
Italics mine.
It moves with constant velocity until it meets and interacts with the field of an atom/molecule.
From the wiki link again: