Electromagnetic radiation in the Tamm problem

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

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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

**

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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.

*

**

*


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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


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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


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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”


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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


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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


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Current treatments of the Tamm problem:

ZHS & Endpoints

(the part with a bit of math)


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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 ( )


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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)

* **


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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):


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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


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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

✓


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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


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Farfield, near the Cherenkov angle

• Endpoints ‘low-frequency’ effect becomes more significant!

• But ‘near-to-start’ effect is no more important…

1 m1000 m


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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


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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?)


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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


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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


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Cherenkov zone

• Completely different spectra.• What about the time domain?

L = 10 m

R = 1 m


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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


Starting again: a complete treatment

(the part with so much math, I left it out)

24 C. W. James, ARENA, Erlangen, 22nd June 2012

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Ingredients:

• Get the source function

• Work in the Fourier domain:

• Use the potentials


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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

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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


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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!


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Example of full integration• Extreme near field

- end-points obviously will not work- ZHS correct at high frequencies- Care needed with correct integration


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Attempts at improved analytic treatments• Really *@&&#*! difficult

- Also face low/high frequency limits- Also require small tracks- Minor improvements on EP & ZHS


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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


(dense media lives here)

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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)


Electromagnetic radiation in the Tamm problem

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