Miscellaneous Topics
Curvature Radiation
Cernkov Radiation
COHERENT CURVATURE RADIATION
When we discussed synchrotron and cyclotron radiation, we said that the component
of the electron velocity parallel to the B-field didn’t change, and that the particle’s
motion could be thought of as (1) constant velocity motion parallel to the B-field, and
(2) Circular motion around the B-field lines.
If, however, the field lines CURVE significantly, then the electron “follows” the lines,
It will move in a curve, even if the v(perp)~0.
The electrons thus experience an acceleration, and therefore radiate.
This radiation is called “CURVATURE RADIATION”.
In general the curvature radiation is weak compared to the synchrotron radiation we
Discussed earlier.
However, if the electrons have very high energy, and move in bunches that are
Smaller than the wavelength of the radiation – each bunch
radiates like a single large charge, the power radiated can be large. This case is
called COHERENT CURVATURE RADIATION.
A place where this process is thought to be important is at the poles of
rotating neutron stars (pulsars):
Some simple estimates:
Recall that for cyclotron radiation, we have Larmor’s formula for the power radiated by
an accelerating electron, summed over solid angle and frequency:
P q a
2 2
If there are N charges, each with charge q, and they radiate randomly with respect
to each other, then the total power will be N times the power radiated by a single
electron, or

P  Nq 2 a2
On the other hand, if the N electrons are traveling together in a bunch which is
Small compared to the wavelength of the radiation they are emitting, they can
Radiate in phase with each other, and therefore radiate as if they are a single charge
=Nq
 P (Nq)2 a2  N 2q2a2
Note the N2
So 100 electrons moving together will radiate 10,000x more power than a single electron
A similar result holds for synchrotron radiation.
CHERENKOV RADIATION
Remember the definitions of “Phase Velocity” and “Group Velocity” for EM waves:
The speed at which the sine moves is the phase velocity

The group velocity is
vphase
 c
k

vg 
k
This is usually discussed when you have several waves superimposed,
which make a modulated wave:
the modulation envelope travels with the group velocity
In a dispersive medium ω=ω(k) so
However, in a vacuum, vgroup= c

v necessa
y
g
k
Group and Phase Velocities
See Rybicki & Lightman, Chapter 8
Plasma Effects
Recall that when we found the wave solutions to the wave equations, we set
the dielectric constant = 1
But in a plasma, you can have dielectric constant, and no “sources” – solve
Wave equations and get E&M waves
2
4

ne
p2 
m
Define plasma frequency
Then the wave solutions for E & B have
1
k c
 p
2
2
 2   p 2  k 2c 2
Instead of
  ck
Phase velocity:

c
v ph  
k nr
Where nr = index of refraction

Group velocity:

nr  1
p
c
2
2
p

vgr 
 c 1 2
k

2
Is always < c

Cherenkov Radiation:
If a charged particle moves faster than the speed of light in the plasma, then
the E-field can bunch up – the wave fronts catch up with each other.
Then you can get radiation produced even if the particle isn’t accelerating—into a
characteristic cone of radiation
Seen in nuclear reactors as a faint blue glow:
Super-Kamiokande Neutrino Detector
In a zinc mine outside Tokyo, there is a tank of 50,000 tons of water, in a tank lined with
11,000 photomultiplier tubes to detect the Cherenkov radiation from neutrinos, including
solar neutrinos interacting with the water.
VERITAS: Very energetic Radiation Imaging Telescope Array
Energetic gamma rays (GeV-TeV) ionize particles in the upper atmosphere which
produces a shower of particles which then emit Cherenkov radiation.
Detected with 12-m dishes of circular segments.
Cosmic Ray