ZHS and EP theory
C. W. James, Columbus, Ohio, Feb 23rd,
2012
Step 1: Liénard-Weichert Potentials
-
Begin with Maxwell’s equations
Add a single (monopolar) particle as a source
Allow for finite light propagation speed
Use Lorentz gauge
æ
ö
1 ç
q
÷
j R, t =
4pe ç 1- n r̂ × b R ÷
è
øt¢
æ
ö
m ç
qv
÷
A R, t =
4p ç 1- n r̂ × b R ÷
è
øt¢
t¢ = t - n R(t¢) / c
( )
( )
(
(
C. W. James, Columbus, Ohio, Feb 23rd 2012
)
)
q
R
r̂
particle charge
observer distance
unit vector to observer
b
particle velocity /c
medium refractive index
observer time
retarded time
n
t
t¢
2
Step 2:
¶A
• Apply
to get:
E r, t = -Ñf ¶t
æ
r̂ ´ r̂ - nb ´ b
r̂ - b
q ç
1
E R, t =
+
ç 2
3
4pe ç g 1- n r̂ × b R 2
c 1- n r̂ × b 3 R
è
( )
( )
(
(
)
)
(
Nearfield Term
Energy/area as R-4
Energy decreases with distance
{(
) }
)
Radiation term
Energy/area as R-2
Energy transport to infinity
“Accelerating charged particles radiate”
Get rid of this
(no Frank-Tamm VC)
C. W. James, Columbus, Ohio, Feb 23rd 2012
ö
÷
÷
÷
øt¢
b º 0 (no acceleration)
ÞEº0
3
Do some maths…
x1, t1, b1
*
• Endpoints
E ( x̂, n )
ö
q æ e öæ
e
= + ç
֍
÷ b1.5 sinq1
c è R1 øè 1- n1b1.5 cosq1 ø
ö
q æ eikR2 öæ
e2 p in t2
- ç
÷ b1.5 sinq 2
֍
c è R2 øè 1- n2 b1.5 cosq 2 ø
ikR1
x1.5, t1.5
*
b1.5
*
x2 , t2 , b 2
2 p in t1
b1.5 = t 1-t ( x2 - x1 )
2
1
• ZHS formula
q æ eikR1.5 öæ e2 p in (1-n1.5b1.5 cosq1.5 )t2 - e2 p in (1-n1.5b1.5 cosq1.5 )t1 ö
E ( x, n ) = ç
֍
÷ b1.5 sinq1.5
c è R1.5 øè
1- n1.5b1.5 cosq1.5
ø
R1.5 = R ( x1.5 )
• Endpoints -> ZHS:
R1 = R2 = R1.5
(the far-field approximation) q1 = q 2 = q1.5
q1.5 = q ( x1.5 )
x1.5 = 1 2 ( x2 - x1 )
n1 = n2 = n1.5
C. W. James, Columbus, Ohio, Feb 23rd 2012
4
Toy experiments
• Take a straight particle track:
x̂
b
z1 = -z, t1
R
q
ẑ
z2 = +z, t2
ŷ
*
*
• Place an observer in x-z plane
• Calculate emission via…
- Endpoints
- ZHS (single track)
- ZHS (very many sub-tracks)
C. W. James, Columbus, Ohio, Feb 23rd 2012
5
What do we expect to see?
•
Afanasiev, Kartavenko, Stepanovsky J Phys D, 32 (1999)
Bremsstrahlung from
endpoints dominates
Bremsstrahlung from
endpoints dominates
C. W. James, Columbus, Ohio, Feb 23rd 2012
6
b = 0.999
Far-field, far from theta_C
spectral strength (V/Hz)
1e-06
1e-07
n=2
ZHS single track
ZHS inf tracks
endpoints
o
q=110
R=1000 m
q =110
1e-08
1e-09
1e-10
1e-11
1e-12
1e+06
1e+07
1e+08
1e+09
1e+10
frequency (Hz)
• Endpoints, ZHS agree perfectly.
• No ZHS track sub-division needed (1m source at 1 km
unresolved)
C. W. James, Columbus, Ohio, Feb 23rd 2012
1m
7
Low-frequency-limit
• Endpoints reduce to:
æ
ö
q
sinq1
sinq 2
÷÷
b1.5 çç
c
è R1 (1- n1b1.5 cosq1 ) R2 (1- n2 b1.5 cosq 2 ) ø
Tends towards a constant term at low frequencies
• ZHS low-phase limit:
é q
ù
æ eikR1.5 ö
® ê b1.5 sin q1.5 ç
÷ 2p i ( t2 - t1 )ún
è R1.5 ø
ë c
û
lim n ® 0 : 0
Tends towards zero at low frequencies
C. W. James, Columbus, Ohio, Feb 23rd 2012
8
1m
Difference in the near-field
1e-06
spectral strength (V/Hz)
1e-07
ZHS single track
ZHS inf tracks
endpoints
q =110
o
q=110
R=1 m
b = 0.999
n=2
1e-08
1e-09
1e-10
1e-11
1e-12
1e+06
1e+07
1e+08
1e+09
1e+10
frequency (Hz)
• Observer much closer to track start than track end
• Endpoints accounts for this, ZHS can not
C. W. James, Columbus, Ohio, Feb 23rd 2012
9
Why?
• ZHS formula:
- Accounts for distance difference in phase, but not magnitude
- true no matter how tracks are subdivided
• Endpoints:
- Distance affects both magnitude and phase
• Clearly, an observer in the nearfield should see a
monopolar component to the pulse
- [total net change in potential]
• Important for:
- Lunar Cherenkov? No! (very far field)
- Important for air-showers? Perhaps (REAS3 vs ZHAires).
- Important for dense media?...
C. W. James, Columbus, Ohio, Feb 23rd 2012
10
What about near the Cherenkov angle?
• Endpoint formulation: 1- n1b1.5 cosq1 ® 0
eikR1
e2 p in t1
® ±¥,
R1 1- n1b1.5 cosq1
eikR2
e2 p in t2
® [something]
R2 1- n2 b1.5 cosq 2
Result can be arbitrarily large (it blows up)
• In ZHS: 1- n1.5b1.5 cosq1.5 ® 0
qb1.5 sin q1.5 æ eikR1.5 öæ e2 p in (1-n1.5b1.5 cosq1.5 )t2 - e2 p in (1-n1.5b1.5 cosq1.5 )t1 ö
ç
֍
÷
c
1- n1.5b1.5 cosq1.5
è R1.5 øè
ø
®
qb1.5 sinq1.5 æ eikR1.5 ö
ç
÷ 2p in ( t2 - t1 )
c
è R1.5 ø
Result is always finite (more sensible)
C. W. James, Columbus, Ohio, Feb 23rd 2012
11
1m
Behaviour near the Cherenkov angle
spectral strength (V/Hz)
1e-06
1e-07
q = 60
o
q= qC (60 )
R=1000m
1e-08
1e-09
b = 0.999
n=2
1e-10
1e-11
1e-12
1e+06
ZHS single track
ZHS inf tracks
endpoints
1e+07
1e+08
1e+09
1e+10
frequency (Hz)
• Endpoints produce a larger contribution (can be arbitrarily
large)
C. W. James, Columbus, Ohio, Feb 23rd 2012
12
Why do endpoints blow up?
• Endpoints allow:
-
Infinitely small acceleration zone
Infinitely small source particle
Infinitely small detector
[time-domain only] constant refractive index
• Result: potentially infinite field
• This should not be unexpected!
- Very common to see infinities in the literature
- This is why textbooks always derive the total radiated
power and not the field strengths.
• This is small consolation.
C. W. James, Columbus, Ohio, Feb 23rd 2012
13
What happens in the near-field
in the Cherenkov regime?
OR:
When good techniques go bad
C. W. James, Columbus, Ohio, Feb 23rd 2012
14
Toy experimental set-up
• Place the observer firmly in the Cherenkov
regime
10 m
b = 0.999
q = 60
n = 1 or 2
C. W. James, Columbus, Ohio, Feb 23rd 2012
15
Spectrum: n=2
• Now we see differences…
refN=2., R=1m, theta=60
Signal strength (V/m/Hz)
1e-16
1e-17
1e-18
1e-19
1e-20
1e-21
ep
ep
ZHS 5k
ZHS 5k
ZHS 20k
ZHS 20k
1e-22
1e+06
z
x
z
x
z
x
1e+07
1e+08
1e+09
1e+10
Frequency (Hz)
C. W. James, Columbus, Ohio, Feb 23rd 2012
16
Time-domain (no band limit)
• Time-domain output (ZHS vs EP) (n=2):
(note different y-axis scales)
refN=2., R=1m, theta=60
refN=2., R=1m, theta=60
8e-15
3e-12
6e-15
2.5e-12
4e-15
2e-12
V/m
V/m
z
x
z
x
1.5e-12
2e-15
0
-2e-15
1e-12
5e-13
0
-4e-15
-5e-13
-6e-15
-8e-15
ZHS 5k
ZHS 5k
ZHS 20k
ZHS 20k
-1e-12
ep z
ep x
0
20
40
60
80
time (ns)
100
-1.5e-12
0
20
40
60
time (ns)
• Large contribution from ZHS NOT in endpoints!
• Could this be a ‘true’ Vavilov-Cherenkov
emission? (or a numerical artefact?)
C. W. James, Columbus, Ohio, Feb 23rd 2012
17
80
100
Quick check: in vacuum
• We do not expect and Cherenkov shock
• But we do expect two bremsstrahlung shocks…
refN=2., R=1m, theta=60
4e-13
ep z
ep x
3e-13
2e-13
2e-13
1e-13
1e-13
V/m
V/m
3e-13
refN=2., R=1m, theta=60
4e-13
0
0
-1e-13
-1e-13
-2e-13
-2e-13
-3e-13
-4e-13
ZHS 5k
ZHS 5k
ZHS 20k
ZHS 20k
-3e-13
450
460
470
480
490
500
-4e-13
450
460
470
time (ns)
480
time (ns)
• I do not understand this ZHS behaviour
C. W. James, Columbus, Ohio, Feb 23rd 2012
18
z
x
z
x
490
500
1cm from the vacuum track
• Large ZHS pulse… in a vacuum.
refN=1., R=1cm, theta=60
8e-15
2.5e-11
ep z
ep x
6e-15
2e-11
4e-15
1.5e-11
2e-15
1e-11
0
5e-12
V/m
V/m
refN=1., R=1cm, theta=60
-2e-15
0
-4e-15
-5e-12
-6e-15
-1e-11
-8e-15
-1.5e-11
-1e-14
15
20
25
30
35
40
45
50
time (ns)
55
60
-2e-11
ZHS 5k
ZHS 5k
ZHS 20k
ZHS 20k
15
20
25
30
35
time (ns)
• This is not V-C radiation!
• It is a numerical artefact OR a static term.
C. W. James, Columbus, Ohio, Feb 23rd 2012
40
19
45
z
x
z
x
50
55
60
Summary from toy experiments
• Theoretical expectation:
- EP theory models only bremsstrahlung
• Handles near-field
• Breaks down near theta_C
- ZHS models only bremsstrahlung + far-field approx
• Breaks down in near-field
• Handles theta_C
• What we see:
- EP theory matches expectation
- ZHS: some strange results…
• Produces phantom Vavilov-Cherenkov-like pulse
• Somehow misses bremsstrahlung
C. W. James, Columbus, Ohio, Feb 23rd 2012
20
Main conclusion
• Neither endpoints nor ZHS get it completely right
Near θC
Far from θC
Far-field
Near-field
EP & ZHS agree
(probably correct)
EP theory is better
(probably correct)
ZHS is better
(probably not correct)
ZHS crazy
EP misses VC (main)
(probably both crap)
C. W. James, Columbus, Ohio, Feb 23rd 2012
21
Philosophical aside
• What about smooth particle motion?
• Radiation is emitted constantly
• Limit (description -> perfection) [inf points]:
- Endpoints have contributions equal-and-opposite sides
of the Cherenkov angle
- Divergences are expected to cancel
- Hence tendency towards ZHS treatment in REAS3
C. W. James, Columbus, Ohio, Feb 23rd 2012
22
What does the ZHS formula produce
• ZHS formula approximates:
1
1
1
1
»
R1 1- n1b1.5 cosq1 R2 1- n2 b1.5 cosq 2
• This approximation can not be made near the
Cherenkov angle
- Same approximation as Tamm (1939)
- Shown to exclude Frank-Tamm Cherenkov
• And yet…
- ZHS formula produces something sensible.
- Endpoints do not.
• We do not know what ZHS produces at the
Cherenkov angle
C. W. James, Columbus, Ohio, Feb 23rd 2012
23
Is the divergence physical?
• If:
- n is constant
- The acceleration event is truly instantaneous
- The particle and detector are both infinitely small
• Then yes!
• Divergence/magnification at the Cherenkov angle does
NOT necessarily mean Vavilov-Cherenkov radiation!
• Q: Why do we often see total radiated power calculated,
but not the fields?
• A: Because this can hide nasty divergences (integrate
away this divergence over finite spatial angles)
C. W. James, Columbus, Ohio, Feb 23rd 2012
24