Cherenkov Radiation Bradley J. Wogsland August 3, 26 Contents 1 Cherenkov Radiation 1 1.1 Cherenkov History Introdution................... 1 1.2 Frank-Tamm Theory......................... 2 1.3 Dispertion............................... 5 1.3.1 Complex Indies of Refration............... 5 1.4 Diffration............................... 5 1.5 Sattering............................... 6 1.6 Edge Effets.............................. 6 1.7 Radiation Below the β Threshold.................. 6 1.8 Quantum Mehanial Modifiations................. 6 1.9 Cherenkov Radiation is not Bremsstrahlung!............ 6 1 Cherenkov Radiation 1.1 Cherenkov History Introdution Everyone is familiar with Einstein s postulate that the speed of light in a vauum is an absolute limit on veloity, however, it is also true that light travels through any medium at a slower rate. Heaviside was the first to realize in the late nineteenth entury that a partile passing through a medium faster than the speed of light in that medium ought to emit radiation. Although, not yet having Einstein s postulate he also alulated what this would be for a partile traveling through the aether. The angle away from the partile s path with whih that light will be emitted is osθ C = 1/βn 1) where θ C is the angle, β is the veloity of the partile relative to the speed of light in a vauum β = v/), and n is the index of refration of the medium through whih the partile is passing. Unaware of Heaviside s theory, in 1937 Cherenkov and Vavilov working in Russia disovered this radiation while studying nulear deay. Pavel Cherenkov won the 1958 Nobel Prize for the disovery of the effet, and surely would have shared it with Vavilov if he was still alive. 1
1.2 Frank-Tamm Theory Frank and Tamm were ollegues of Vavilov and Cherenkov at the Lebedev Physial Institute in Mosow, so they naturally had a head start in formulating the theory. In 1938 they published their theory of Cherenkov radiation [itation?] for whih they would reeive the Nobel Prize along with Cherenkov in 1958. Pedagogial introdutions to their theory an be found in the books by Jelley [1] and Zrelov [2]. Developing along lines whih might have been familiar to Heaviside, the starting point is the relation of the eletri field E and its assoiated polarization vetor P in a medium with refrative index n: P = n 2 1)E 2) Note also that there is a frequeny dependene of this equation arising from the impliit Fourier expansions P = + Pω)e iωt dω, n = + nω)e iωt dω, et. 3) of all the variable quantities. Now to write Maxwell s equations in this medium we onvert to D = n 2 E 4) whih yields D = 4πρ 5) E = 1 H 6) t H = 7) H = 4π j + 1 D t. 8) Then one an redue Maxwell s equations to a more suitable form using the potentials whih are given in the Lorentz gauge, E = ϕ 1 A 9) t H = A 1) A = ε ϕ t. 11) Note that the two potential equations are equivalent to 6) and 7). Taking the gradient of 11) and adding it to ε t operated on 9) yields one 1 D t = A) ε 2 A 2 t 2 12) 2
and plugging 1) into 8) yields A) 2 A = 4π j + 1 D t whih ombined with the previous gives the final result 13) 2 A ε 2 2 A t 2 = 4π j 14) for the vetor potential and urrent. Similarly, taking the partial derivative of 11) with respet to time and adding it to ε operated on 9) yields D = ε 2 ϕ + ε2 Combining this with 5) finally gives 2 ϕ t 2. 15) 2 ϕ ε 2 ϕ 2 t 2 = 4π ρ. 16) ε Now sine we are interested in only the optial properties of our medium, we reall from 4) that ε = n 2 and that Fourier transforming the equations simply means t iω. Thus if we apply this knowledge to 14) and 16) we get 2 A n2 ω 2 2 A = 4π j 17) 2 ϕ n2 ω 2 2 ϕ = 4π n 2 ρ 18) It is at this junture where we part from Heaviside so as to study the simplest possible ase of a single eletron passing through the medium in question. To the resolution of urrent experiments the eletron is a point partile so we use Dira delta funtions, j = evδx)δy)δz vt)ẑ 19) where v is the eletron s veloity and the z-axis orresponds to the beamline. Now to Fourier transform this equation we reall that δax) = 1 a δx) whih means that j = eδx)δy) + δ z v t)eiωt dωẑ 2) = eδx)δy)e iωz v ẑ 21) assuming, as we have, a positive veloity v. Hopefully it is lear that the first e is the eletron s harge whereas the seond is 2.718... If we now plug 21) into 17) to get an equation for the vetor potential in the medium through whih the eletron passes, this yields 2 A n2 ω 2 2 A = 4eπ iωz δx)δy)e v ẑ. 22) 3
Clearly in the x and y diretions there are idential Helmholtz equations, but we are only interested in the trivial solutions sine????. Thus we onentrate on the beam diretion, where 2 A z n2 ω 2 2 A z = 4eπ iωz δx)δy)e v 23) and with presient forsight we guess a solution of the form A z swithing to ylindrial oordinates as well. Simplifying, = uρ)e iωz v, 2 u ρ 2 + 1 u ρ ρ + ω2 v 2 β2 n 2 1)u = 2e δρ) 24) ρ sine δx)δy) = 1 2πρ δρ). This Bessel equation s solution will have different harateristis depending on whether β 2 n 2 = 1, β 2 n 2 < 1 or β 2 n 2 > 1. Letting ω 2 v 2 β2 n 2 1) = k 2 25) to simplify matters, one sees that the solutions to the homogeneous part of 24) are the usual Bessel funtions J kρ) = 1 2π Y kρ) = 2 π 2π 1 e ikρsinθ dθ 26) oskρt) dt. 27) t2 1 Returning to the inhomgeneaity in 24) we multiply through by ρ and integrate, k 2 ρudρ+ ρ 2 u ρ 2 + u u ρ + u lim ρ ρ + k2 ρu)dρ = ρ + k2 ρu)dρ + ρ u ) ρ lim ρ 2e δρ)dρ 28) ) = 2e 29) ρ ρ u ) = 2e 3) ρ ρ u ) = 2e 31) ρ ρ u lim ρ where one an eliminate the first two terms in the penultimate equation beause?????. Atually, it seems to me they ought to diverge sine J, J, Y and Y all go like 1 ρ at large ρ.) Thus far we have followed the usual proedure to solve a differential equation of known form. Herefollows the gimmik that earned Frank and Tamm their Nobel prize. Instead of ontinuing on to find a partiular solution of the inhomogeneous equation and then using initial onditions to find the oeffeients of 26) and 27) in the general solution, they used 31) to get the asymptoti oeffiients of the solution uρ). Sine the length sale of the eletromagneti 4
fore is mirosopi, it is not unreasonable to use an asymptoti approximation of the Bessel funtions for large ρ, uρ) = AJ kρ) + BY kρ) 32) 2 Aoskρ π kπρ 4 ) + Bsinkρ π ) 4 ), 33) as we are interested in marosopi effets. Now it is lear that the β 2 n 2 = 1 is nonphysial sine k = in this ase. For the seond ase it makes sense to rotate the oeffiients to 2 uρ) Ce ikρ π 4 ) + De ikρ π )) 4. 34) kπρ Considering the β 2 n 2 < 1 ase, k is imaginary whih means that D must be zero to avoid divergenes. All that s left is a deaying exponential whih will leave no appreiable vetor potential at large distanes. This means that low veloity partiles don t emit Cherenkov radiation. The final ase involves those partiles with high enough energy for whih β 2 n 2 > 1. Here the Bessel funtions naturally yield a nie outgoing ylindrial wave. skipping ahead... Finally this yields the Frank-Tamm equation, 1.3 Dispertion dw dl = e2 2 βn>1 1 1 β 2 n 2 ) ωdω 35) The refrative index of a given substane is atually a funtion of the wavelength λ of the light passing through it. 1.3.1 Complex Indies of Refration ɛω) = ɛ R ω) + iɛ I ω) 36) Whih modifies the Frank-Tamm equation to ) dw = e2 dl 2 1 ɛ Rω) β 2 ɛω) 2 ωdω 37) and leads to damping. 1.4 Diffration βn>1 Main ause of width of Cherenkov Ring: θ C λ Lsinθ 38) where L is the length of the partile s path in the radiator and λ is the wavelength of light being emitted. 5
1.5 Sattering 1.6 Edge Effets 1.7 Radiation Below the β Threshold 1.8 Quantum Mehanial Modifiations There is a small orretion to 1), namely osθ C = 1 βn + Λ λ n 2 ) 1 2n 2 39) 1.9 Cherenkov Radiation is not Bremsstrahlung! Cherenkov radiation is a marosopi effet of the whole medium on a harged partile, whereas Bremsstrahlung is the interation of a harged partile with a individual sreened nuleus. Referenes [1] J. V. Jelley, Cherenkov Radiation and Its Appliations, 1958. [2] V. P. Zrelov, Cherenkov Radiation in High-Energy Physis, 197. 6