Accelrator Betatron Description

Betatron

Maria Kazachenko

Physics department

Montana State University

What is betatron?Any sufficiently advanced

technology is indistinguishable

from magic.

Arthur C. Clarke

Used in• Nuclear reactions• X-ray sources in medicine• Possible solar flare mechanism

Newe- acceleration with EM induction

Before: fast e- - only in cosmic rays

CR source Energy

Supernova 1014 eV

Sun 105 eV

Milky Way 108 eV

Betatron 108 eV

Donald Kerst; e- accelerator; 1940

Particle accelerator that uses the electric field induced by a varying magnetic field to accelerate electrons to high speeds in a circular orbit.

Introduction

Outline

Methods of electrons acceleration (historically)

1. Van de Graaf high voltage generator (E=const, B=const)

too big

2. Linear accelerator (E changes, B=const)

too long

3. Circular accelerator (E changes, B=const)

relativistic effects

4. Betatron accelerator (B changes, vortex E)

How it works?

Magnetic field distribution

Equilibrium orbit and stability

Electron injection

5. Conclusion

Before a betatron

KE V

elKE ( ) 3.2RaC MeV

elKE 5 20

10RaMeV kg

R meters

Particle acceleration in electric field

Nature: Beta-radioactive materials;

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

Is it possible to get 5 MeV KE without using

5 MV potential?

Use multiple acceleration with lower

potential?

Why do we need to accelerate particles?

To measure smth small requires smth smaller

De Broglie and wave-particle dualism

h

p

Disadvantage: single acceleration, size

Linear Accelerator

e

-1000 V +1000 V -1000 V+1000 V

e

-1000 V +1000 V -1000 V +1000 V

e

-1000 V +1000 V -1000 V+1000 V-1000 V +1000 V -1000 V +1000 V

e

0 V +1000 V +2000 V +3000 V

e e e

To get KE=106eV, we need 1000 V not 106V.

If 1000 plates, KE=1000*Vsingle_pair=106eV

KE=3000 eV

e

Linear Accelerator

e e

Sloan and Kots got mercury ions accelerated up to 2.85 MeV; 1.85 meter linac

36 electrodes

n

n

LT const

v

Could be ~1 km, easily!

High voltage ion source

Acceleratingplates

Source of radiofrequency (RF)

Vacuum chamber

Target

X-rays

710 10f Hz MHz

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

Proton 50-100MeV Electron 25 KeV

Impossible to accelerate electrons in cyclotron up to several million of eV

Why can’t we use cyclotron to accelerate electrons?

Time period2 m

T= ( )c

f vq B

t T

2

Tt

0

2 2

2 2

1

1 1

mm T

v vc c

( )m m v

E- acceleration with EM induction

e- rotating in a circle in magnetic field BAfter one revolution Ekin increases by

- How can we make e- rotate in a circle? - Using special configuration of magnetic field.

0 00 0

1

1= = ( )

2 2 r

e e

t t

m v c qBr P

qB B c

dP d qF q E q P P

dt r c dt c

0r const Br P ; if

E

0rr 0

r 0

KE=dU 2 ;

dU 20 , 5 ;

r E

V r cm

t=0.001 seconds, S=290 km, 18.5 MeV, 925.000 revolutions

Basic principle of how the betatron works

0 00

0

2 r

2 rt t

qP

c

qP

c

qB

Pc

020 2t t

qB P B r

r c

2t

t

BB

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

tB BS

tB

2t

t

BB

0

Bt

SB

r max 0

2 2 20

max

P

2

qrB

q r BKE

m

Stability of motion on the equilibrium orbit

Is motion on the equilibrium orbit stable? S=300 kilometers!!! T=1/1000 sec

2

c

m n

F

qvB qvAF =

c cr

n

AB

r

mv

r

1. Radial stability

2. Axial stability

Barrel-type magnetic field lines

Lorentz force deflects electrons back to the median plane.

unstable stable

center edgeB B

How to realize the initial condition in practice?

First betatron. Electron injection.

0 0 0 0rq

P mv Bc

•Ausserordentlichhochgeschwindigkeitelektronenentwickelndenschwerarbeitsbeigollitron German for "extraordinarily high-speed electron generator".

B=B(t) => very short time when B~B0

“Betatron”

Summary

Instrument Shape Electric field

Magnetic field

Electron energy,

MeV

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

Synchrotron torus variable variable 10.000

Betatron in use (in the past)

1. Fast electrons in particle physics2. X-rays (radiation oncology)

Best e--accelerators now

1. Large electron-positron collider – 8*104 MeV

2. International Linear Collider, 106 MeV

Questions?

Syncrotron radiation

222 15

2 2 20 max

21.3*10

3rad

q KE fE qE W B KE

mc m c H

Magnetic mirrorA magnetic mirror is a magnetic field configuration where the field strength changes

when moving along a field line.

Adiabatic invariantsFor periodic motion, the adiabatic invariants are the action integrals taken over period of the motion.

pdq0

dB constdt

First adiabatic invariantMagnetic moment cons-nin time-dependent B(cyclotron motion)

Second adiabatic invariant(longitudinal motion) ||J mv ds const

Particle Trapping

2 2

||

||min

22

2

sin;

20 _

sin ( )sin 2 ;

sin

perpendicperpendic

R

R

mvB

B Bbounce back

constB B

B

B

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 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

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 at which

to at which b(t)=bm

The change in the trap length l with time changes from l(0)=1 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:

Gyrosynchrotron Radiation