Betatron Maria Kazachenko Physics department Montana State

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• Betatron
• Maria Kazachenko
• Physics department
• Montana State University

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What is betatron?
Any sufficiently advanced technology is
indistinguishable from magic. Arthur C. Clarke
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Introduction
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Outline

• Methods of electrons acceleration (historically)
• Van de Graaf high voltage generator (Econst,
  Bconst)
• too big
• 2. Linear accelerator (E changes, Bconst)
• too long
• 3. Circular accelerator (E changes, Bconst)
• relativistic effects
• 4. Betatron accelerator (B changes, vortex E)
• How it works?
• Magnetic field distribution
• Equilibrium orbit and stability
• Electron injection
• 5. Conclusion

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Before a betatron
Why do we need to accelerate particles? To
measure smth small requires smth smaller De
Broglie and wave-particle dualism
Particle acceleration in electric field
Nature Beta-radioactive materials

• Human
• vacuum tube
• electron gun
• Van de Graaf generator

Disadvantage single acceleration, size
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Linear Accelerator
3000 V
1000 V
2000 V
0 V
KE3000 eV
To get KE106eV, we need 1000 V not 106V. If
1000 plates, KE1000Vsingle_pair106eV
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Linear Accelerator
X-rays
e
High voltage ion source
Accelerating plates
Source of radio frequency (RF)
Vacuum chamber
Target
Sloan and Kots got mercury ions accelerated up to
2.85 MeV 1.85 meter linac 36 electrodes
Could be 1 km, easily!
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An Early Circular Accelerator

• In 1929, Ernest Lawrence developed the first
  circular accelerator
• This cyclotron was only 4 inches in diameter, and
  contained two D-shaped magnets separated by a
  small gap
• An oscillating voltage created an electric field
  across the small gap, which accelerated the
  particles as they went around the accelerator

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Why cant we use cyclotron to accelerate
electrons?

• Time period

Proton
50-100MeV Electron 25 KeV
Impossible to accelerate electrons in
cyclotron up to several million of eV
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E- acceleration with EM induction
e- rotating in a circle in magnetic field B After
one revolution Ekin increases by
t0.001 seconds, S290 km, 18.5 MeV, 925.000
revolutions
- How can we make e- rotate in a circle? - Using
special configuration of magnetic field.
if
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Basic principle of how the betatron works
Conclusion Electron will have circular motion
of constant radius if the half of the average of
the magnetic field within the circle is equal to
the value of magnetic field on the orbit.
Special B (r) distribution
Time evolution of the magnetic field
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Stability of motion on the equilibrium orbit
Is motion on the equilibrium orbit stable? S300
kilometers!!! T1/1000 sec
1. Radial stability
stable
unstable
2. Axial stability Barrel-type magnetic field
lines Lorentz force deflects electrons back to
the median plane.
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First betatron. Electron injection.

• Ausserordentlichhochgeschwindigkeitelektronenentwi
  ckelndenschwerarbeitsbeigollitron
• German for "extraordinarily high-speed electron
  generator".

Betatron
How to realize the initial condition in practice?
BB(t) gt very short time when BB0
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Summary

• Betatron in use (in the past)
• Fast electrons in particle physics
• X-rays (radiation oncology)
• Best e--accelerators now
• Large electron-positron collider 8104 MeV
• International Linear Collider, 106 MeV

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Questions?
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• Syncrotron radiation

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Magnetic mirror

• A magnetic mirror is a magnetic field
  configuration where the field strength changes
  when moving along a field line.

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Adiabatic invariants

• For periodic motion, the adiabatic invariants are
  the action integrals taken over period of the
  motion.

First adiabatic invariant Magnetic moment
cons-n in time-dependent B (cyclotron motion)
Second adiabatic invariant (longitudinal motion)
Particle Trapping
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Magnetic mirror magnetic field
configuration where the field strength changes
when moving along a field line, as a result
charged particles bounce back from the high field
region. Fermi acceleration Decrease of the
field line length provides the first-order Fermi
acceleration Betatron acceleration Compression
of the magnetic field lines provides betatron
acceleration
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Particle Acceleration in a Collapsing Trap
A magnetic trap between the Super-Hot
Turbulent-Current Layer (SHTCL) and a Fast
Oblique Colisionless Shock (FOCS) above magnetic
obstacle (MO) Particles are captured into a
collapsing magnetic trap where they accelerate
further to high energies. Apart from the
First-Order Fermi acceleration the authors have
suggested taking into account the betatron effect
in collapsing traps, i.e. an increase in the
transverse momentum as the trap contracts. Main
idea of the paper to develop a trap model
in which both Fermi and betatron accelerations
are at work, compare efficiencies, pitch-angle
distributions, total kinetic energy of trapped
electrons.
Ref. Somov, B.V. and Kosugi, T., ApJ, 485, 859,
1997
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• The formation of a trap. Its contraction.
• Particle acceleration

Electron energy in the magnetic reconnection
region (RR) increases from a coronal thermal
energy of 0.1 keV at least to an energy of 10keV.
Each magnetic flux tube is a trap since BmgtB0.
Particle injection is impulsive, i.e. electrons
fall into trap at the initial time and
subsequently either precipitate into the loss
cone or become trapped, acquiring additional
energy. Due to motion from RR to chromosphere,
the length of the trap decreases gt particles
energy in a trap increases
due to Fermi mechanism. When magnetic trap
contracts transversely, particles are accelerated
by betatron mechanism.
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• Transverse contraction
  changes from at which
  
• to at which b(t)bm
• The change in the trap length l with time
  changes from l(0)1 to l0 or to some
  residual trap length.
• Longitudinal invariant
• Transverse invariant
• As a result
• When two mechanisms act, the pitch angle is
• The nonrelativistic KE
• Pitch angle when particle falls into loss cone
• Kinetic energy at the escape time

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Gyrosynchrotron Radiation