Čerenkov Radiation Application

Last week I gave a quick run-down of the justification of Cherenkov radiation from classical electrodynamics.  In practice, this effect is the centerpiece of many experiments, and I'll run through how and what we need to know in this post.

We recall the velocity of the charge particle must exceed the phase velocity of the E&M field to emit Cherenkov radiation, or with index of refraction n:



Cherenkov radiation comes in the form of an electromagnetic shock wave, a conical wavefront formed following the particle, emitted at angle:



Since we'd like to pick up this radiation in photomultiplier tubes to observe the effect and therewith the incident charge particle, it's good to know how many photons to expect.  Following the math of the last post on Cherenkov radiation (or just following along in Jackson), we take the energy differential over frequency and divide by de Broglie's h bar omega and then by L we get the number of photons per unit length:



Cherenkov radiation contributes to the mechanism of several types of particle detectors, including electromagnetic calorimeters and non-scintillating hodoscopes; there are two types of detectors specifically designed to directly take advantage of the Cherenkov radition emitted by high energy charge particles: threshold Cherenkov detectors and differential Cherenkov detectors.

In the threshold Cherenkov detector, the medium in the tank is chosen carefully for a refractive index that indicates the passage of charge particles that exceed a given velocity threshold.  Usually this is done with a gas such as hydrogen, nitrogen or carbon dioxide, and further control of the refractive index comes through the pressure of the gas in the tank.

The differential Cherenkov detector allows the measurement of a particle's velocity while rejecting particles outside a given mass range.  This is accomplished by accepting a small annulus around the track of incident particles at some angle theta.  This corresponds, at a given refractive index, to a velocity resolution:



The minimum velocity resolution is often constrained by the minimum angular resolution, which tends to be limited by dispersion in the slit.

Sources:  Leo's Techniques and Fernow's Intro to Experimental Particle Physics

Čerenkov Radiation

This phenomena, aside from providing the eerie lighting in this picture from Idaho National Lab, is a central tool for many particle physics experiments.  Actually, it's so important, it won Chernkov the Noble prize in 1958.  We recall that it's analogous to a sonic boom, but for light, and we can explain it in purely in terms of classical electrodynamics.

The derivation is in full in Jackson, but I'll sketch it here.  We consider a charged particle moving through a medium, where all the distant interactions of the particle with the atoms in the material are represented by a macroscopic dielectric constant .   Taking the Fourier transforms of the E&M wave equations, we get the E and B fields as follows:

Then, after integrating in k and recognizing the modified Bessel function, we get three field components in terms of  , in each direction, where here 1 is parallel to the particle's velocity.  I'll skip straight to the limit where    (a and b are impact parameters, and lambda is  ), which gives:

We integrate over frequency to get energy over distance , which has Jackson explains is done rather elegantly by finding the energy flow through a cylinder radius a.  After this integration and applying our limit, we get an expression which is multiplied by:

If lambda has a positive real part, this vanishes at large distance as the energy is deposited near the path.  If lambda is purely imaginary, the exponential is unity and the expression is independent of scattering distance a, so some of the energy escapes to infinity as radiation!  When is lambda imaginary?  When epsilon is real (there is little absorption) and  

 .  

Or, the speed of the particle is greater than the phase velocity of the E&M fields in the material.  That's it!  

We've now seen that with the right circumstances, namely the speed of the particle is greater than the phase velocity in the material, some of the energy of the particle escapes as "Cherenkov" radiation.  You can go on to calculate the angle of emission and the photon yields, but I'll leave that to another post in which I'll describe the practical applications of this effect.

Source: Jackson's Classical Electrodynamics