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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) - PowerPoint PPT Presentation
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Electromagnetic radiation in the Tamm problem
C.W. James, 5th ARENA Workshop Erlangen, 22nd June 2012
Introduction: the Tamm Problem
(the part without maths)
2 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• 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”
BremssShock 2
C. W. James, ARENA, Erlangen, 22nd June 2012
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* *
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, 22nd June 2012
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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, 22nd June 2012
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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, 22nd June 2012
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Radiation in the Tamm problem: qualitative• Can we distinguish the contributions?
λ
λ
Low frequencies
High frequencies
Nearfield Farfield
* *
Yes! Yes!Yes!
* *
No NoNo
* *
Yes!
No
Yes!
* *
No
No
No
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
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Derivations: ZHS* + endpoints^
• Begin with fields from the Leinard-Weichert potentials:
• Take the radiation term (not interested in the static field)
Radiation termEnergy/area as R-2
Energy transport to infinity
Nearfield TermEnergy/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, 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?
• 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, 22nd 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, 22nd 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, 22nd 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 m1000 m
C. W. James, ARENA, Erlangen, 22nd 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
1 m
C. W. James, ARENA, Erlangen, 22nd 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, 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
• Completely different spectra.• What about the time domain?
L = 10 m
R = 1 m
C. W. James, ARENA, Erlangen, 22nd 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, 22nd June 2012
Starting again: a complete treatment
(the part with so much math, I left it out)
24 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
• 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?
Medium lives here
C. W. James, ARENA, Erlangen, 22nd 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, 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?
ZHS correct
Endpoints needmodification
Far-field Near-field
Nea
r θC
Far f
rom
θC
EP & ZHS agree(correct)
ZHS: works in high-freq, small-track limit
Endpoints: ‘more correct’
Endpoints will not work
ZHS works inhigh-frequency limit
C. W. James, ARENA, Erlangen, 22nd June 2012
(dense media lives here)
(EA
S co
vers
bot
h do
mai
ns)
(do
beam
-test
s ex
plor
e th
is?)
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, 22nd June 2012