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Transcript
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 ) qB 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= qE 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