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Betatron
Maria Kazachenko
Physics department
Montana State University
What is betatron?
Any sufficiently
advanced
technology is
indistinguishable
from magic.
Arthur C. Clarke
Introduction
Donald Kerst; eaccelerator; 1940
New
e- acceleration with EM induction
Used in
•
Nuclear reactions
•
X-ray sources in medicine
•
Possible solar flare mechanism
Particle accelerator that uses the electric field
induced by a varying magnetic field to accelerate
electrons to high speeds in a circular orbit.
Before: fast e- - only in cosmic rays
CR source
Energy
Supernova
1014 eV
Sun
105 eV
Milky Way
108 eV
Betatron
108 eV
Outline
Methods of electrons acceleration (historically)
1.
2.
3.
4.
5.
Van de Graaf high voltage generator (E=const, B=const)
too big
Linear accelerator (E changes, B=const)
too long
Circular accelerator (E changes, B=const)
relativistic effects
Betatron accelerator (B changes, vortex E)
How it works?
Magnetic field distribution
Equilibrium orbit and stability
Electron injection
Conclusion
Before a betatron
Why do we need to accelerate particles?
To measure smth small requires smth smaller
De Broglie and wave-particle dualism

h
p
Particle acceleration in electric field
KE  V
Nature:
Is it possible to get
5 MeV KE without using
5 MV potential?
Beta-radioactive materials;
KEel ( RaC )  3.2MeV
Human:
• vacuum tube
• electron gun
• Van de Graaf generator
Use multiple
acceleration with lower
potential?
KEel  5MeV  20kg Ra
R  10meters
Disadvantage: single acceleration, size
Linear Accelerator
e
0V
e
+1000 V
e
+2000 V
e
+3000 V
KE=3000 eV
e
-1000 V
+1000
+1000
-1000 V
V
V
e
+1000
V
-1000
-1000V
V
+1000
V
e
-1000
VV
+1000
+1000V
V
-1000
To get KE=106eV, we need 1000 V not 106V.
If 1000 plates, KE=1000*Vsingle_pair=106eV
e
+1000
V
-1000
+1000
V
-1000VV
Linear Accelerator
X-rays
e
e
High voltage
ion source
Accelerating
plates
Vacuum
chamber
Source of radio
frequency (RF)
Target
Ln
 T  const
vn
Sloan and Kots got mercury ions accelerated up to 2.85 MeV;
1.85 meter linac
36 electrodes
f  107 Hz  10MHz
Could be ~1 km, easily!
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
Why can’t we use cyclotron to accelerate
electrons?
T=
Time period
m
m0
v2
1 2
c
m  m (v )
T
t
2
2  m  c
 f (v )
qB
T 
Proton
Electron
1
v2
1 2
c
50-100MeV
25 KeV
Impossible to accelerate electrons in cyclotron up to
several million of eV
t T
E- acceleration with EM induction
e-
rotating in a circle in magnetic field B
After one revolution Ekin increases by
KE=dU r  2 r0 E;
dU r  20V , r0  5cm;
E
r0
t=0.001 seconds, S=290 km, 18.5 MeV, 925.000 revolutions
- How can we make e- rotate in a circle?
- Using special configuration of magnetic field.
me ve c 1
qB 
 P
; if r  r0  const  P  B
qB
B
c
dP
1
d
q
F=
 qE  q

 Pt  P0 =
( t   0 )
dt
2 r0c dt
2 r0c
r
Basic principle of how the betatron works

q
B  2  Pt 
Bt r0
 r0
2c
qB 
P
c
q
P0 
0
2 r0 c
Pt 
q
t
2 r0 c
 Bt 
Bt
2
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
Bt 
Bt
Bt
SB
Bt 
Time evolution of the magnetic field
SB
r0
Pmax  qrB0
Bt
2
KEmax
q 2 r 2 B02

2m
Stability of motion on the equilibrium orbit
Is motion on the equilibrium orbit stable? S=300 kilometers!!! T=1/1000 sec
1. Radial stability
A
B n
r
mv 2
Fc 
r
qvB qvA
Fm =
 n
c
cr
2. Axial stability
Barrel-type magnetic field lines
Bcenter  Bedge
Lorentz force deflects electrons back
to the median plane.
unstable
stable
First betatron. Electron injection.
•Ausserordentlichhochgeschwindigkeitelektronenentwickelndenschwerarbeitsbeigollitron
“Betatron”
German for "extraordinarily high-speed electron generator".
How to realize the initial
condition in practice?
P0  mv0 
q
B0 r0
c
B=B(t) => very short time
when B~B0
Summary
Instrument
Shape
Electric
field
Magneti
c field
Electron
energy,
MeV
Betatron in use (in the
past)
1. Fast electrons in particle
physics
2. X-rays (radiation
oncology)
Van de Graaf
generator
linear
constant
constant
25
Linear
accelerator
linear
variable
constant
2.85
(50.000)
Cyclotron
circle
variable
constant
0.025
Betatron
torus
constant
variable
300
1. Large electron-positron
collider – 8*104 MeV
Synchrotron
torus
variable
variable
10.000
2. International Linear
Collider, 106 MeV
Best e--accelerators now
Questions?
Syncrotron radiation
E  qE  Wrad
2
2  q   KE  2
f
15
  2 
B  KE  1.3*10
2 
2
3  mc   m0 c 
H max
2
Magnetic mirror
A magnetic mirror is a magnetic field configuration where the field strength changes
when moving along a field line.
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
 pdq
d
B
 0    const
dt
J   mv||ds  const
mv 2 sin 2   perpendic


; B   perpendic   || 
2B
B
 ||min  0  bounce _ back

sin  (  )
sin 2 
2 ;
  const 

B
BR
B
sin 2  
BR
2
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
Particle Acceleration
in a Collapsing Trap
A magnetic trap between the Super-Hot TurbulentCurrent 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
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
Bm>B0.
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 =>
particles energy in a trap increases
due to Fermi mechanism. When magnetic trap contracts transversely,
particles are accelerated by betatron mechanism.
Transverse contraction
changes from
to
at which b(t)=bm
The change in the trap length l with time
to l=0 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:
at which
changes from l(0)=1
Gyrosynchrotron Radiation