Accelerator Physics. G. A. Krafft Jefferson Lab Old Dominion University Lecture 2

Transcription

1 Accelerator Physics G. A. Krafft Jefferson Lab Old Dominion University Lecture 2

2 Four-vectors Four-vector transformation under z boost Lorentz Transformation v ' v v v v ' 1 1 ' v v 2 2 v ' v v Important example: Four-velocity. Note that interval d 2 1 dt Lorentz invariant. So the following is a 4-vector cu dct dx dy dz,,, c 1, x, y, z d d d d

3 4-Momentum Single particle mechanics must be defined in terms of Fourmomentum p mcu mc 1,,, x y z Norms, which must be Lorentz invariant, are u u 1, p p mc What happens to Newton s Law? But need a Four-force on the RHS!!! dp d F F ma dp / dt

4 Electromagnetic (Lorentz Force) Non-relativistic F q E v B Relativistic Generalization (ν summation implied) Electromagnetic Field F F qf u 0 E E E x y z E 0 cb cb x z y E cb 0 cb y z x E cb cb z y x 0

5 Relativistic Mechanics in E-M Field Energy Exchange Equation (Note: no magnetic field!) d dt qe v 2 mc Relativistic Lorentz Force Equation (you verify in HW!) d mv dt q E v B

6 Methods of Acceleration Acceleration by Static Electric Fields (DC) Acceleration Cockcroft-Walton van de Graaf Accelerators Limited by voltage breakdowns to potentials of under a million volts in 1930, and presently to potentials of tens of millions of volts (in modern van de Graaf accelerators). Not enough to do nuclear physics at the time. Radio Frequency (RF) Acceleration Main means to accelerate in most present day accelerators because one can get to MV in a meter these days. Reason: alternating fields don t cause breakdown (if you are careful!) until much higher field levels than DC. Ideas started with Ising and Wideröe

7 Cockcroft-Walton Proton Source at Fermilab, Beam Energy 750 kev

8 van de Graaf Accelerator Brookhaven Tandem van de Graaf ~ 15 MV Generator Tandem trick multiplies the output energy

9 Ising s Linac Idea Prinzip einer Methode zur Herstellung von Kanalstrahlen hoher Voltzahl (in German), Arkiv för matematik o. fysik, 18, Nr. 30, 1-4 (1924).

10 Drift Tube Linac Proposal Idea Shown in Wideröe Thesis

11 Wideröe Thesis Experiment Über ein neues Prinzip zur Herstellung hoher Spannungen, Archiv für Elektrotechnik 21, 387 (1928) (On a new principle for the production of higher voltages)

12 Sloan-Lawrence Heavy Ion Linac The Production of Heavy High Speed Ions without the Use of High Voltages David H. Sloan and Ernest O. Lawrence Phys. Rev. 38, 2021 (1931)

13 Alvarez Drift Tube Linac The first large proton drift tube linac built by Luis Alvarez and Panofsky after WW II

14 Earnest Orlando Lawrence

15 Germ of Idea* *Stated in E. O. Lawrence Nobel Lecture

16 Lawrence s Question Can you re-use the same accelerating gap many times? F ma qv B B 2 2 d x qb x 2 v v 0 2 y 2 c x 2 2 d y qb y 2 v v 0 2 x 2 c y v dt m dt d v dt m dt d 2 2 v v qb v v v v x y x y y x 0 dt m v v t v x y t d is a constant of the motion gap

17 Cyclotron Frequency v t v cos t ; v t v sin t x 0 c y 0 c v v x t x sin t ; y t y cos ct c 0 c The radius of the oscillation r = v 0 /Ω c is proportional to the velocity after the gap. Therefore, the particle takes the same amount of time to come around to the gap, independent of the actual particle energy!!!! (only in the non-relativistic approximation). Establish a resonance (equality!) between RF frequency and particle transverse oscillation frequency, also known as the Cyclotron Frequency f f /2 rf c c qb 2 m c

18 What Correspond to Drift Tubes? Dee s!

19 U. S. Patent Diagram

20 Magnet for 27 Inch Cyclotron (LHS)

21 Lawrence and His Boys

22 And Then!

23 Beam Extracted from a Cyclotron Radiation Laboratory 60 Inch Cyclotron, circa 1939

24 88 Inch Cyclotron at Berkeley Lab

25 Relativistic Corrections When include relativistic effects (you ll see in the HW!) the effective mass to compute the oscillation frequency is the relativistic mass γm f c c /2 qb 2 m where γ is Einstein s relativistic γ, most usefully expressed as E E E mc E E E mc 2 tot 0 kin kin m particle rest mass, E kin particle kinetic energy

26 Cyclotrons for Radiation Therapy

27 Bragg Peak