### Reference Material

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\section*{Introduction} While Einstein's Theory of Special Relativity prevents particles from moving faster than $c$, the speed of light in a vacuum, it does allow particles to move faster than light in media with refractive index $n >1$. That's because light itself propagates at speed $c/n < c$. \newline

This has some interesting physical consequences. A charged particle moving through a medium will generally interact with its environment and this process will emit radiation. Now suppose that this particle happens to move through the medium at a speed faster than $c/n$. What happens is the electromagnetic equivalent of the sonic boom and is known as Cherenkov Radiation. \newline

Because the circular wavefronts propagate slower than the particle, radiation is confined to a cone behind the moving particle.

Dealing with relativistic charges in the proximity of molecules and atoms is a complex problem. But we can still make some powerful predictions about the geometry of emitted radiation with classical physics. \newline The picture we should have in mind is shown on the right. The diagram shows wavefronts produced by a superluminal particle. Because these wavefronts propagate slower than the particle itself, the radiation will always be confined to a cone behind the particle, as is shown in the diagram. Note that inside the cone wavefronts cross, leading to constructive interference. \newline

The radiation will move outward perpendicular to the walls of the cone, at an angle $\theta$ relative to the particle. This angle is fixed by the cone's geometry and can be found by comparing the distance traveled by the wavefronts and particle. From the right triangle shown in the diagram we see that $$\cos(\theta) = \frac{AB}{AC} = \frac{\frac{c}{n} \cdot \Delta t }{v \cdot \Delta t}= \frac{1}{\beta n} \ ,$$ $$\Rightarrow \theta = \cos^{-1} (\frac{1}{\beta n}) \ ,$$ where $\beta$ is the relativistic parameter defined as $\beta = \frac{v}{c}$. We would like $\frac{1}{\beta n} <1$. As a result, for Cherenkov Radiation the particle's speed must be in the range $$\frac{c}{n} < v < c \ .$$

The direction of propagation $\theta$ increases with increasing refractive index and speed. \newline For ultrarelativistic particles with $\beta \approx 1$, $\theta \approx \cos^{-1}(\frac{1}{n})$. For water with $n =1.33$ this yields $\theta_{water} \approx 41^{\circ}$. For air with n = 1.0003 we get $\theta_{air} \approx 1^{\circ}$, so Cherenkov Radiation propagates almost parallel to the charged particle.

\section*{Cherenkov Spectrum} The amount of energy emitted in a frequency range $d\omega$ per unit distance traveled by a particle of charge $Ze$ is given by the Frank-Tamm formula: $$\frac{dE}{dx}= \frac{Z^2e^2}{c^2} (1-\frac{1}{\beta^2 n(\omega)^2}) )\ \Theta(v-\frac{c}{n(\omega)}) \ \omega \ d\omega \ ,$$ where $\Theta(x)$ is the Heaviside step function:

$$\Theta(x) = \begin{cases} 0 & x<0 \\ 1 & x > 0 \end{cases}$$


Note that we have included a frequency dependence in the refractive index. By using the chain rule $\frac{dE}{dx} = \frac{dE}{dt}\frac{dt}{dx}$, this can be converted into a formula for power: $$\frac{dP}{d\omega} = \frac{Z^2e^2 v}{c^2} (1-\frac{1}{\beta^2 n(\omega)^2}) \ \Theta(v-\frac{c}{n(\omega)}) \ \omega \ .$$ If we ignore the dependence on $\omega$ in $n(\omega)$, we see that $\frac{dP}{d\omega} \propto \omega$ . So the Frank-Tamm formula tells us that more energy is emitted at higher frequencies. This is why Cherenkov radiation typically has a blueish color. The frequency dependence in $n(\omega)$ is especially important at high energies and prevents the total power from diverging. \section*{Astrophysical Applications} Cherenkov Radiation detectors can be used for measuring energy spectra of cosmic rays and gamma rays. As they hit Earth's upper atmosphere, cosmic rays produce showers of charged particles. The relativistically moving particles created in this cascade emit Cherenkov Radiation at a small angle ($\approx 1^{\circ}$) to their direction of motion. Measurements of Cherenkov flashes reaching the ground can be used to gain information about the energy and direction of propagation of the original cosmic/gamma ray. \newline

Cherenkov detectors are also a very important class of neutrino detectors. Water based Cherenkov detectors recorded flashes of radiation due to neutrinos coming from SN 1987A hours before the first photons from SN 1987A arrived.

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