Miscellaneous Topics

Curvature RadiationCernkov Radiation

COHERENT CURVATURE RADIATIONWhen we discussed synchrotron and cyclotron radiation, we said that the componentof the electron velocity parallel to the B-field didn’t change, and that the particle’smotion 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 bunchradiates like a single large charge, the power radiated can be large. This case iscalled COHERENT CURVATURE RADIATION.

A place where this process is thought to be important is at the poles ofrotating neutron stars (pulsars):

Some simple estimates:

Recall that for cyclotron radiation, we have Larmor’s formula for the power radiated byan accelerating electron, summed over solid angle and frequency:

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P ∝ q2a2

If there are N charges, each with charge q, and they radiate randomly with respectto each other, then the total power will be N times the power radiated by a single electron, or

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P ∝Nq2a2

On the other hand, if the N electrons are traveling together in a bunch which isSmall compared to the wavelength of the radiation they are emitting, they canRadiate in phase with each other, and therefore radiate as if they are a single charge=Nq

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P ∝ (Nq)2a2 = N 2q2a2 Note the N2

So 100 electrons moving together will radiate 10,000x more power than a single electronA similar result holds for synchrotron radiation.

The speed at which the sine moves is the phase velocity

ckphase

v

The group velocity is kg

v

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 ynecessaril vkg

However, in a vacuum, vgroup= c

CHERENKOV RADIATION

Remember the definitions of “Phase Velocity” and “Group Velocity” for EM waves:

Group and Phase Velocities

Plasma Effects See Rybicki & Lightman, Chapter 8

Recall that when we found the wave solutions to the wave equations, we setthe dielectric constant = 1

But in a plasma, you can have dielectric constant, and no “sources” – solveWave equations and get E&M waves

Define plasma frequency

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p2 =

4πne2

m

Then the wave solutions for E & B have

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k = c−1 ω2 −ωp2

ω2 =ωp2 + k 2c 2

Instead of

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=ck

Phase velocity:

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vph ≡ω

k=c

nr

Where nr = index of refraction

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nr = 1−ωp

2

c 2

Group velocity:

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vgr =∂ω

∂k= c 1−

ωp2

ω2

Is always < c

Cherenkov Radiation:

If a charged particle moves faster than the speed of light in the plasma, thenthe 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 with11,000 photomultiplier tubes to detect the Cherenkov radiation from neutrinos, includingsolar 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