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Electromagnetic radiation in the Tamm problem

Electromagnetic radiation in the Tamm problem. C.W . James, 5 th ARENA Workshop Erlangen, 22 nd June 2012. Introduction: the Tamm Problem. (the part without maths ). The ‘Tamm problem’:. *. *. First treated by Tamm (1939)

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Electromagnetic radiation in the Tamm problem

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  1. Electromagnetic radiation in the Tamm problem C.W. James, 5th ARENA Workshop Erlangen, 22nd June 2012

  2. Introduction: the Tamm Problem (the part without maths) C. W. James, ARENA, Erlangen, 22nd June 2012

  3. The ‘Tamm problem’: * * • First treated by Tamm (1939) • Frank and Tamm (1937): Vavilov-Cherenkov radiation from an infinite particle track • What about a finite track? • Calculate the radiation from the following: • Single charged particle (~electron) • Uniform velocity • Finite track length • In a medium (refractive index n != 1) C. W. James, ARENA, Erlangen, 22nd June 2012

  4. Why do we care about it? Cascade simulations * * * * • Many codes use (Monte-Carlo) cascade simulations to generate particle motion • Simulation output: • ‘Join the dots’: we get finite tracks • The Tamm problem! • All current MC programs (ZHS, ZHAireS, REAS3, COREAS, SELFAS2) implement an algorithm which gives the radiation from the Tamm problem. C. W. James, ARENA, Erlangen, 22nd June 2012

  5. Radiation in the Tamm problem: qualitative Bremss shock 1: - “a charge accelerated” - “a charge just moved here” Cherenkov shock wave: - “a charge is now here” Bremss Shock 2 * * • Three contributions: • B1: bremsstrahlung shock from initial acceleration • B2: bremsstrahlung shock from final deceleration • VC: Vavilov-Cherenkov contribution from superluminal motion C. W. James, ARENA, Erlangen, 22nd June 2012

  6. Radiation in the Tamm problem: qualitative Field of a static charge: “a charge sits here” * * In the idealised Tamm problem, the charge was at rest for infinite time, then remains at rest for infinite time. In a Monte-Carlo, connecting tracks keeps the charge moving. C. W. James, ARENA, Erlangen, 22nd June 2012

  7. Radiation in the Tamm problem: qualitative ‘information’ shock wave: - “all this stuff happened” * * • ‘Pure’ VC radiation: no magic! • Simply the sudden arrival of information from a non-accelerating particle in a uniform medium • “Cherenkov effects”: coincident arrival of any sort of information due to >1 refractive index C. W. James, ARENA, Erlangen, 22nd June 2012

  8. Radiation in the Tamm problem: qualitative B1, charge motion, B2 VC, B1, B2 VC, B2, B1 B2, charge motion, B1 * * • Viewing order depends on observer location • So does relative strength • Only a true VC shock in a particular spatial region • Charge-motion significantly doppler-enhanced near VC shock C. W. James, ARENA, Erlangen, 22nd June 2012

  9. Radiation in the Tamm problem: qualitative Farfield Nearfield Low frequencies λ High frequencies No No Yes! No No Yes! No Yes! λ No Yes! Yes! No * * * * * * * * Can we distinguish the contributions? C. W. James, ARENA, Erlangen, 22nd June 2012

  10. Current treatments of the Tamm problem:ZHS & Endpoints (the part with a bit of math) C. W. James, ARENA, Erlangen, 22nd June 2012

  11. Derivations: ZHS* + endpoints^ Nearfield Term Energy/area as R-4 Energy decreases with distance Radiation term Energy/area as R-2 Energy transport to infinity * * Instantaneous charge acceleration ( ) *implied/described from Zas, Halzen, Stanev, PRD 45, (1992) ^James, Falcke, Heuge, Ludwig, PRE 84 (2011) Begin with fields from the Leinard-Weichert potentials: Take the radiation term (not interested in the static field) C. W. James, ARENA, Erlangen, 22nd June 2012

  12. Do some maths to get… * * * • Endpoints • Independent contributions from each acceleration point • ZHS formula • Assumes a farfield observer (constant angle, 1st-order phase differences) • In the far-field, endpoints and ZHS are identical(the far-field approximation) C. W. James, ARENA, Erlangen, 22nd June 2012

  13. Philosophically… • Endpoints: • radiation comes from the ‘end points’ of the track • No ‘near-field’: sources infinitely small • ZHS: • radiation is emitted by the track • Nearfield: when ‘equal-angle’ approximation breaks down • Expectations from derivations • Both should model radiation from accelerated particles • Neither should model the Vavilov-Cherenkov radiation from an infinite track (Frank-Tamm case): C. W. James, ARENA, Erlangen, 22nd June 2012

  14. Are these expectations correct? * * R • Take a straight particle track: • Place an observer in x-z plane • Calculate emission via… • Endpoints • ZHS (single track) • ZHS (very many sub-tracks: ‘divide and conquer’) C. W. James, ARENA, Erlangen, 22nd June 2012

  15. 1 m Far-field, far from theta_C 1000 m ✓ Good agreement, except at very low frequencies. No ZHS track sub-division needed. Endpoints account for being closer to start than end C. W. James, ARENA, Erlangen, 22nd June 2012

  16. Low-frequency-limit • Tends towards a constant term at low frequencies • Tends towards zero at low frequencies • Endpoints reduce to: • ZHS low-phase limit: • Observer closer to track start than track end • Endpoints accounts for this, ZHS can not • But does this mean that endpoints are correct? C. W. James, ARENA, Erlangen, 22nd June 2012

  17. 1 m Farfield, near the Cherenkov angle 1000 m Endpoints ‘low-frequency’ effect becomes more significant! But ‘near-to-start’ effect is no more important… C. W. James, ARENA, Erlangen, 22nd June 2012

  18. 1 m Difference in the near-field 1 m Observer at the Cherenkov angle for the initial point Causes divergence for endpoints: NOT for ZHS C. W. James, ARENA, Erlangen, 22nd June 2012

  19. What happens with endpoints? • Result can be arbitrarily large (can not be correct) • Result is always finite • (and correct?) • Endpoint formulation: • In ZHS: • Here, the near-field term is important – does ZHS somehow include it? (how???) C. W. James, ARENA, Erlangen, 22nd June 2012

  20. Endpoints in practice (REAS3 & COREAS) • IF: • Boosting of the static field important (close to the Cherenkov angle) • THEN: • revert to small-phase limit of ZHS • ELSE: • use endpoints unmodified C. W. James, ARENA, Erlangen, 22nd June 2012

  21. ZHS in practice (ZHAireS) • IF: • angle to observer changes significantly over the track • THEN: • sub-divide the track accordingly • ELSE: • use a single ZHS track C. W. James, ARENA, Erlangen, 22nd June 2012

  22. Cherenkov zone L = 10 m R = 1 m Completely different spectra. What about the time domain? C. W. James, ARENA, Erlangen, 22nd June 2012

  23. Time-domain (1 MHz – 1 GHz band limit) 0-10 GHz • Large contribution from ZHS NOT in endpoints! • Consistent with VC polarisation and arrival time • Bremsstrahlung shocks agree: (ZHS bremss shocks suppressed by ringing) C. W. James, ARENA, Erlangen, 22nd June 2012

  24. Starting again: a complete treatment (the part with so much math, I left it out) C. W. James, ARENA, Erlangen, 22nd June 2012

  25. Ingredients: Get the source function Work in the Fourier domain: Use the potentials C. W. James, ARENA, Erlangen, 22nd June 2012

  26. Do some math: Work in spherical coordinates: Get the potentials in frequency domain Apply (in the x-t domain): And we get… C. W. James, ARENA, Erlangen, 22nd June 2012

  27. Complete fields in the Tamm problem Medium lives here • Frequency-domain expression: • K0,1(rq) modified Bessel functions of the second kind • e is the charge: ~ -1.6x10-19 for an electron • Can not in general reduce the integral over kz: numerics! • Assumptions: • Instantaneous acceleration at start/end • Infinitely small source (breaks down after UV) • Can we reduce the integrals for specific cases? C. W. James, ARENA, Erlangen, 22nd June 2012

  28. Analytic approximations… Duration of event as seen by observer: of course they see a static field during that time! Agrees with small-phase limit of the ZHS formula! IF: And: And: [a few more criteria are satisfied] Then we find: Or in spherical coordinates… C. W. James, ARENA, Erlangen, 22nd June 2012

  29. Example of full integration • Extreme near field • end-points obviously will not work • ZHS correct at high frequencies • Care needed with correct integration C. W. James, ARENA, Erlangen, 22nd June 2012

  30. Attempts at improved analytic treatments • Really *@&&#*! difficult • Also face low/high frequency limits • Also require small tracks • Minor improvements on EP & ZHS C. W. James, ARENA, Erlangen, 22nd June 2012

  31. What works where? Far-field Near-field (dense media lives here) ZHS: works in high-freq, small-track limit Endpoints: ‘more correct’ EP & ZHS agree (correct) Far from θC (do beam-tests explore this?) (EAS covers both domains) Endpoints will not work ZHS works in high-frequency limit ZHS correct Endpoints need modification Near θC C. W. James, ARENA, Erlangen, 22nd June 2012

  32. Summary (experimentalists can stop listening now) • For all current experimental geometries, endpoints and ZHS (as currently used in practice) are equivalent • REAS3, ZHAireS, COREAS: do it correctly! • Of interest to theory: • There is a low-frequency limit at which ZHS will always break down. • In this limit, endpoints are qualitatively correct, but the quantitative results generally are not. • Producing analytic solutions to the equations is tough • Konstantin, being able to read Russian and having been taught math in school, will now probably tell you what they are C. W. James, ARENA, Erlangen, 22nd June 2012

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