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

Transcription

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

2 Course Outline Course Content Introduction to Accelerators and Short Historical Overview Basic Units and Definitions Lorentz Force Linear and Circular Accelerators Particle Motion in EM Fields Linear Beam Dynamics Periodic Systems Magnetic Multipoles Nonlinear Perturbations Coupled Motion

3 Synchrotron Radiation Radiation Power and Distribution Insertion Devices X-ray Sources Free Electron Lasers Technical Components Particle Acceleration Cavities and RF Systems Spin and Spin Manipulation Collective Effects Particle Distributions Vlasov Equation Self-consistent Fields

4 Landau Damping Beam-Beam Effects Relaxation Phenomena Radiation Damping Toushek effect/ibs Beam Cooling

5 Energy Units When a particle is accelerated, i.e., its energy is changed by an electromagnetic field, it must have fallen through an Electric Field (we show later by very general arguments that Magnetic Fields cannot change particle energy). For electrostatic accelerating fields the energy change is E q q q charge, Φ, the electrostatic potentials before and after the motion through the electric field. Therefore, particle energy can be conveniently expressed in units of the equivalent electrostatic potential change needed to accelerate the particle to the given energy. Definition: 1 ev, or 1 electron volt, is the energy acquired by 1 electron falling through a one volt potential difference. a b

6 Energy Units ev = C 1 V = J MeV = 10 ev = J To convert rest mass to ev use Einstein relation electron,0 0 = mc where m is the rest mass. For electrons E E kg 310 m/sec J = MeV Recent best fit value MeV

7 Some Needed Relativity Accelerator Physics is Applied Special Relativity Following Maxwell Equations, which exhibit this symmetry, assume all Laws of Physics must be of form to guarantee the invariance of the space-time interval ct ' x' y' z ' ct x y z Coordinate transformations that leave interval unchanged are the usual rotations and Lorentz Transformations, e.g. the z boost ct ' ct z x' y' x y z ' z ct

8 Relativistic Factors Following Einstein define the relativistic factors v c = v c Easy way to accomplish task of defining a Relativistic Mechanics: write all laws of physics in terms of 4-vectors and 4- tensors, i.e., quantities that transform under Lorentz transformations in the same way as the coordinate differentials.

9 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

10 4-Momentum Single particle mechanics must be defined in terms of Fourmomentum 1,,, /,,, p mcu mc E c p p p x y z x y z Norms, which must be Lorentz invariant, are (g μν =g μν =(1,-1,-1,-1)) u u 1, p p mc What happens to Newton s Law? But need a Four-force on the RHS!!! dp d F ma dp / dt F

11 Electromagnetic Field Described by the Four-vector potential A, ca, ca, ca x y z Field Tensor F A A 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

12 Electromagnetic (Lorentz Force) Non-relativistic F q E v B Relativistic Generalization (ν summation implied) F qf u Electromagnetic Field with lower second index 0 Ex Ey Ez Ex 0 cbz cb y F F g Ey cbz 0 cbx Ez cby cbx 0

13 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

14 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

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

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

17 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).

18 Drift Tube Linac Proposal Idea Shown in Wideröe Thesis

19 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)

20 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)

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

22 Earnest Orlando Lawrence

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

24 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

25 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

26 What Correspond to Drift Tubes? Dee s!

27 U. S. Patent Diagram

28 Magnet for 27 Inch Cyclotron (LHS)

29 Lawrence and His Boys

30 And Then!

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

32 88 Inch Cyclotron at Berkeley Lab

33 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

34 Cyclotrons for Radiation Therapy

35 Bragg Peak

36 gg Lorentz Group: Linear Lie Group The group of Lorentz Transformations is a Linear (Matrix) Lie group. It can therefore be analyzed by standard methods. First find the generators G e I G t t g g G g gg 0 gg gg General anti-symmetrical matrix yields 0 x y z 0 x y z x 0 z y x 0 z y G y z 0 x y z 0 x z y x 0 z y x 0 t

37 Six Fundamental Generators Rotations S 1,2,3 Boosts K 1,2,3 Generators ,, ,,

38 Commutation Relations (S 1,2,3 ) 2 and (K 1,2,3 ) 2 are straightforward to compute (ε S) 3 = ε S and (ε K) 3 = ε K for any unit three-vector ε Commutation Relations are S, S S S, K K K, K S i j ijk k i j ijk k i j ijk k Arbitrary Lorentz transformation connected to the identity is G S K expg du mc qf u u exp qf / mc u 0 d Electric Fields give boosts Magnetic Fields give rotations