Universität Heidelberg

Seminar Elektrodynamik

Cherenkov radiation

Julius Eckhard

May 5, 2014

1 Introduction

Fast charged particles may create electromagnetic waves while flying through a medium, which is called Cherenkov radiation. This only happens if the particle is faster than the speed oflight in the particular medium. One can compare this to the supersonic flight and in analogy one can observe a Mach cone. Nowadays there are many areas of application for Cherenkov radiation including the indirect detection of cosmic rays and the measuring of radiation in spent fuel pools.

2 Historic events

Cherenkov radiation was first experimentally discovered and described in 1934 by sowjet physi- cist Pavel Cherenkov and his supervisor Sergei Wawilow. To honour both discoverers the effect is still called Wawilow-Cherenkov radiation in russia. Only three years later sowjet physicists Ilja Frank and Igor Tamm performed the theoretical decription and developped the so called Frank-Tamm formula which describes the whole spectrum completely. In 1958 Cherenkov, Tamm and Frank were eventually awarded the Nobel Prize in Physics "for the discovery and the interpretation of the Cherenkov effect".

1 3 Creation

If charged particles move through an isolator of (initially constant) index of refraction , the latter is polarized by the electromagnetic field of the flying particle. Usually these푛 polarisation superpose destructively and the perturbation can relax while the particle moves on. But if the speed of the particle is higher than the phase velocity of light in the medium −1 the perturbation can no longer훽 decay since the reaction time of the medium is not푐푛 high= 푛 enough and the perturbation remains behind the particle. Due to the Huygens principle there occurs a wave front at but only single spherical waves at . 훽 > 푐푛 훽 < 푐푛

Left: , no wave front Right:훽 < 푐푛 , wave front 훽 > 푐푛

It is now important in which direction the radiation spreads. One can illustrate this via a simple drawing:

The particle moves along the projection of the red arrow and runs during a tim through a distance . In between the푡 Cherenkov 푥푇 = 훽 ⋅ 푡 radiation travels radially by a distance 푡 . Since the wave front represents the tangent푥푆 = on푛 the radiated elementary waves there is a right angle as shown. Using simple trigonometric relations one can

determine the Cherenkov angle 푥푆 . cos(휃푐) = 푥푇

This gives us for the Cherenkov angle:

1 (1) cos(휃푐) = 훽푛 Due to radial symmetry there is a Mach cone around the moving particle.

2 4 Spectrum

The amount of photons d produced in a distance interval d is generally given by the Frank- Tamm formula: 푁 푥

d2 푁 2 휇 2 (2) d d = 푞 sin (휃푐) 휔 푥 4휋 with particle charge and permeability . Frank and Tamm derived푞 it in their original휇 work „Coherent Visible Radiation of fast Electrons Passing Through Matter“ as follows:

1. Go to Fourier space, i.e. define quantities , , , by Fourier transformation. 퐴휔⃗ 휙휔⃗ 퐸⃗휔 퐻⃗휔 2. Derive the polarisation 2 . 푃휔⃗ = (푛 − 1)퐸⃗휔 3. Rewrite Maxwell equations by using : 휕푡 → 푖휔

Lorenz gauge: 2 (3) ∇⃗ 퐴휔⃗ + 푖휔푛 휙휔 = 0 Field definitions: (4) 퐻⃗휔 = ∇⃗ × 퐴휔⃗ 푖 (5) 퐸⃗휔 = − ∇(⃗ ∇⃗ 퐴휔⃗ ) − 푖휔퐴휔⃗ 휔푛2 Wave equation: 2 2 (6) (Δ + 휔 푛 )퐴휔⃗ = −4휋푗휔⃗ 4. Current produced by the particle:

(7) 푗푧(푥, 푦, 푧, 푡) = 푞 훽 훿(푥) 훿(푦) 훿(푧 − 훽푡) 푖휔푧 (8) 푞 − 훽 푗푧(휌, 휙, 푧, 휔) = 푒 훿(휌) 4휋2휌

푖휔푧 5. Solve − 훽 using Bessel functions: 퐴푧(휔) = 푢(휌) 푒

2 휕 1 휕 2 푞 (9) ( + + 푠 ) 푢 = − 훿(휌) 휕휌2 휌 휕휌 휋휌 d 6. Calculate d퐸 by using the Poynting vector: 푥 d ∞ 퐸 1 d (10) d = 2휋휌 ∫ [퐸⃗ × 퐻⃗] 푡 푥 −∞4휋 휌 2 1 d (11) = 푞 ∫휔 (1 − ) 휔 훽2푛2

7. Include 휇 for non-trivial permeability. 4휋

3 This eventually gives the following continuous spectrum:

2 d 푞 1 d d (12) 퐸 = 휇휔 (1 − ) 푥 휔 4휋 훽2푛2 and a total Cherenkov energy of:

d 2 ∞ d (13) d퐸 푞 1 −1 ( )Cherenkov = ∫ 휇(휔)휔 (1 − 2 2 ) 휔Θ (푛(휔) − 훽 ) 푥 4휋 0 훽 푛 (휔) where the Heaviside function −1 appears, because there is no Cherenkov radia- tion for . Θ (푛(휔) − 훽 ) 훽푛 < 1

To further analyse this formula we first have to deal with the properties of the index of re- fraction. It is an important material constant describing the phase velocity of light in matter which can be calculated via √ using permittivity and permeability. In general these values are dependent on the푛 particular = 휖푟휇푟 frequency which is called dispersion. This means also the Cherenkov angle is frequency dependent and there is a different Mach cone for every wave- length.

This becomes very important looking at the Cherenkov energy. If we assume the indexof refraction is universal ( ) one gets: 푛(휔) ≡ 푛 d 2 ∞ d (14) d퐸 푞 휇 1 ( ) =const. = (1 − 2 2 ) ∫ 휔 휔 푥 푛 4휋 훽 푛 0 which obviously does not converge.

To solve this problem dispersion alone is not enough, since 13 does not converge, if ( 푛(휔) − −1 ) for almost all frequencies. We therefor have to demand the existance of a frequency 훽 > 0 (훽)max for every given . 휔 훽

(훽)max −1 (15) 푛(휔 > 휔 ) < 훽 Since this has to hold for every we can also write: 훽

(1)max (16) 푛(휔 > 휔 ) ≤ 1 At first sight this seems unlikely, because it means that the phase velocity of light would ex- ceed the vacuum speed of light. However we can find such maximum frequencies with in the x-ray realm. For even higher frequencies the index of refrection is usually unity. 푛 < 1

Comment: This does not violate SRT, since it only restricts the group velocity whichisre- sponsible for the propagation of information. This group velocity may be (and indeed is) still smaller than unity.

4 5 Application

There are quite a lot of areas of application for the Cherenkov effect. The probably most important one is the detection of cosmic radiation using so called Cherenkov flashs. If high energy photons e.g. from a supernova enter earth's atmosphere they decay into a cascade of (partially charged) secundary particles. Due to the high energy they move with nearly speed of light and Cherenkov radiation occurs for a very short time until the particle energy (and velocity) has decreased sufficiently. Since these flashs have the same direction as the path of the particle one can draw conclusions from spectrum and intensity distribution about the charged particles and finally the -rays. The first project of this kind was designed훾 in 1989 which allowed the observation ofgamma sources for the first time. Nowadays the most important project is the H.E.S.S. in Namibia, which is amongs others supported by the "MPI für Kernphysik" in Heidelberg.

This method is also used in the neutrino research. For example in the IceCube at thesouth pole a block of ice of km3 is used. Neutrinos react with the ice and create muons which then radiate Cherenkov rays1 due to the high index of refractation of ice.

The possibly best-known application is the measuring of remaining radiation in spent fuel pools. The intensity of the character- istic blue Cherenkov light is proportional to the amount of nuclear disintegrations. This is presumably the only possibility to actually see the Cherenkov radiation, since the intensity of a single ray is too weak.

5 6 Sources

• Bolotovskii, Frenkel: „I. Tamm Selected Papers“ Tamm, Frank: „Coherent Visible Radiation of fast Electrons Passing Through Matter“

• Landau, Lifschitz: Elektrodynamik der Kontinua

• Jackson: Klassische Elektrodynamik

• Wikipedia.org

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