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) 2 C. W. James, ARENA, Erlangen, 22 nd 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, 22 nd 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, 22 nd June 2012

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

6 * * 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. Radiation in the Tamm problem: qualitative C. W. James, ARENA, Erlangen, 22 nd June 2012

7 Radiation in the Tamm problem: qualitative ‘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 * * ‘information’ shock wave: - “all this stuff happened” C. W. James, ARENA, Erlangen, 22 nd June 2012

8 Radiation in the Tamm problem: qualitative  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 * * B1, charge motion, B2 VC, B1, B2 VC, B2, B1 B2, charge motion, B1 C. W. James, ARENA, Erlangen, 22 nd June 2012

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

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

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

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

14 Are these expectations correct? 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’) * * R C. W. James, ARENA, Erlangen, 22 nd June 2012

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

16 Low-frequency-limit 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? Tends towards a constant term at low frequencies Tends towards zero at low frequencies C. W. James, ARENA, Erlangen, 22 nd June 2012

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

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

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

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

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

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

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

27 Complete fields in the Tamm problem Frequency-domain expression:  K 0,1 (rq) modified Bessel functions of the second kind  e is the charge: ~ -1.6x for an electron  Can not in general reduce the integral over k z : numerics! Assumptions:  Instantaneous acceleration at start/end  Infinitely small source (breaks down after UV) Can we reduce the integrals for specific cases? Medium lives here C. W. James, ARENA, Erlangen, 22 nd June 2012

28 Analytic approximations… IF: And: And: [a few more criteria are satisfied] Then we find: Or in spherical coordinates… 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! C. W. James, ARENA, Erlangen, 22 nd 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, 22 nd 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, 22 nd June 2012

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

32 Summary 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 (experimentalists can stop listening now) C. W. James, ARENA, Erlangen, 22 nd June 2012