Accelerator Physics Weak Focusing. S. A. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage Jefferson Lab Old Dominion University Lecture 2

Transcription

1 Accelerator Physics Weak Focusing S. A. Bogacz, G. A. Krafft, S. DeSilva, R. Gamage Jefferson Lab Old Dominion University Lecture 2

2 Betatrons 25 MeV electron accelerator with its inventor: Don Kerst. The earliest electron accelerators for medical uses were betatrons.

3 300 MeV ~ 1949

4 Electromagnetic Induction Faraday s Law: Differential Form of Maxwell Equation S E Faraday s Law: Integral Form Faraday s Law of Induction S B t B E ds ds t d E dl dt S 2 RE B

5 Transformer

6 Betatron as a Transformer In the betatron the electron beam itself is the secondary winding of the transformer. Energy transferred directly to the electrons Radial Equilibrium Energy Gain Equation d dt 2 RE B R c eb / m d dt ee 2 mc c

7 Betatron condition To get radial stability in the electron beam orbit (i.e., the orbit radius does not change during acceleration and electrons relativistic), need mc R const eb d ec 1 db B 2 2 dt mc 2R dt 2R B R 2 2 B R B r This last expression is sometimes called the betatron two for one condition. The energy increase from the flux change is ec er 2 B B r R 2 Rmc mc

8 Two for one in pictures Bz r B r R R 2 B r rdr B R 2 B r R R 2 2 B z z 0 1 B r R z B B 2 2 R 2

9 Transverse Beam Stability Ensured by proper shaping of the magnetic field in the betatron

10 Relativistic Equations of Motion Standard Cylindrical Coordinates x dv dt z r ˆ ˆr y d v ˆ v ˆ rr v dt dv q v d B 0!! dt m dt x r cos y r sin rˆ cos xˆ sin yˆ ˆ sin xˆ cos yˆ v v rˆ r v v ˆ r r z zˆ r x y drˆ / dt ˆ d ˆ / dt rˆ

11 Cylindrical Equations of Motion In components q m 2 r r v B rbz r q q r 2r v B zb rb m m q m r z q v q z B r B m z m Zero th order solution r t constant R t constant 0 0 t t z t 0 r 0

12 Magnetic Field Near Orbit Get cyclotron frequency again, as should qb r R, z 0 0 z m Magnetic field near equilibrium orbit 0 B r B, ˆ z B r z B ˆ ˆ 0z r R r r R z r r Br Bz zrˆ zzˆ z z Bz Br Bz B 0 B 0, Br 0 0 r z z c

13 Field Index Magnetic Field completely specified by its z-component on the mid-plane B z B r, z B ˆ ˆ ˆ 0z r R z zr r Power Law model for fall-off B r, z 0 B R / r z The constant n describing the falloff is called the field index nb 0 B r, z B ˆ ˆ ˆ 0z r R z zr R 0 n

14 Linearized Equations of Motion Assume particle orbit close to or nearby the unperturbed orbit r t r t R t t t z t z t nb0 nb0 0 Bz B0 r Br z R R 2 r r 2R c q c r B R B R nb r R B 0 c 0 0 c c 0 0m R 0 R 2 r r R r const c c c q nb 0 2 z Rc z nc z 0m R c

15 Weak Focusing For small deviations from the unperturbed circular orbit the transverse deviations solve the (driven!) harmonic oscillator equations r 1 n r const R / 2 2 c c c z n z 2 c The small deviations oscillate with a frequency n 1/2 Ω c in the vertical direction and (1 n) 1/2 Ω c in the radial direction. Focusing by magnetic field shaping of this sort is called Weak Focusing. This method was the primary method of focusing in accelerators up until the mid 1950s, and is still occasionally used today. 0 0

16 Stability of Transverse Oscillations For long term stability, the field index must satisfy 0n 1 because only then do the transverse oscillations remain bounded for all time. Because transverse oscillations in accelerators were theoretically studied by Kerst and Serber (Physical Review, 60, 53 (1941)) for the first time in betatrons, transverse oscillations in accelerators are known generically as betatron oscillations. Typically n was about 0.6 in betatrons.

17 Physical Source of Focusing 0 n n 1 B r changes sign as go through mid-plane. B z weaker as r increases Bending on a circular orbit is naturally focusing in the bend direction (why?!), and accounts for the 1 in 1 n. Magnetic field gradient that causes focusing in z causes defocusing in r, essentially because Bz / r Br / z. For n > 1, the defocusing wins out.

18 First Look at Dispersion Newton s Prism Experiment prism red screen violet Dispersion units: m Bend Magnet as Energy Spectrometer position sensitive material High energy p x D p p x p Bend magnet Low energy

19 Radial Equilibrium Linearized Dispersion for Betatron R c p eb / m eb pp R R B B RB RB RB e p n RB RB 1 n RB e p 1 n R D R radial p R 1 n 0 0 0

20 Evaluate the constant r 1 n r const R / 2 2 c c c 0 For a time independent solution r R (orbit at larger radius) 1 n R const R / 2 2 c c c p p const R R R 1 p c c c p General Betatron Oscillation equations p 2 2 r 1 n cr cr p z n z 2 c 0

21 No Longitudinal Focusing p R c r cr 1 p 2 0 p 1 t 0 c c 2 c p 0 R dt R p ct c dt 2 p 0 1 n Speed Change Path increase from displaced orbit

22 Classical Microtron: Veksler (1945) l 6 Extraction l 5 l 4 l 3 l 2 Magnetic Field l 1 y RF Cavity x 2 1

23 For the geometry given Basic Principles d 2 dt v x 2 d( mv) e E v B dt d( mv x) ev ybz dt d( mv y ) evxbz dt 2 c v 0 For each orbit, separately, and exactly x d 2 dt v y 2 2 c v 0 v x ( t) vx0 cos( ct) v y ( t) vx0 sin( ct) y x t v x0 ( ) sin c c t y t v v ct x0 x0 ( ) cos c c

24 Non-relativistic cyclotron frequency: Relativistic cyclotron frequency: 2 f eb / m c c eb / m z Bend radius of each orbit is: 0, v / c / l x l c c In a conventional cyclotron, the particles move in a circular orbit that grows in size with energy, but where the relatively heavy particles stay in resonance with the RF, which drives the accelerating DEEs at the non-relativistic cyclotron frequency. By contrast, a microtron uses the other side of the cyclotron frequency formula. The cyclotron frequency decreases, proportional to energy, and the beam orbit radius increases in each orbit by precisely the amount which leads to arrival of the particles in the succeeding orbits precisely in phase.

25 Microtron Resonance Condition Must have that the bunch pattern repeat in time. This condition is only possible if the time it takes to go around each orbit is precisely an integral number of RF periods 1 f f c RF f f c RF First Orbit For classical microtron assume can inject so that 1 1 c Each Subsequent Orbit f f RF fc 1 f RF

26 Parameter Choices The energy gain in each pass must be identical for this resonance to be achieved, because once f c /f RF is chosen, Δγ is fixed. Because the energy gain of non-relativistic ions from an RF cavity IS energy dependent, there is no way (presently!) to make a classical microtron for ions. For the same reason, in electron microtrons one would like the electrons close to relativistic after the first acceleration step. Concern about injection conditions which, as here in the microtron case, will be a recurring theme in examples! f c / f B / B RF z 0 B 0 2 mc e B T 1.07kG@10cm Notice that this field strength is NOT state-of-the-art, and that one normally chooses the magnetic field to be around this value. High frequency RF is expensive too!

27 Δγ lower Classical Microtron Possibilities f f c RF Assumption: Beam injected at low energy and energy gain is the same for each pass,, 1, 1 1/2 1/3 1/4,, 1,,, 1,,, 1, 2, 1, 2, 1 3, 1, 3/2, 1/2 4, 1, 4/3, 1/3 5, 1, 5/4, 1/4 3, 2, 3, 2 4, 2, 2, 1 5, 2, 5/3, 2/3 6, 2, 3/2, 1/2 4, 3, 4, 3 5, 3, 5/2, 3/2 6, 3, 2, 1 7, 3, 7/4, 3/4 5, 4, 5, 4 6, 4, 3, 2 7, 4, 7/3, 4/3 8, 4, 2, 1 B bigger

28 For same microtron magnet, no advantage to higher n; RF is more expensive because energy per pass needs to be higher Extraction Magnetic Field y RF Cavity x 3 2

29 Going along diagonal changes frequency To deal with lower frequencies, go up the diagonal Extraction Magnetic Field y RF Cavity x 4 2

30 Phase Stability Invented independently by Veksler (for microtrons!) and McMillan V c (t) s ( l1) / frf 2 s f RF t t 1/ f RF Electrons arriving EARLY get more energy, have a longer path, and arrive later on the next pass. Extremely important discovery in accelerator physics. McMillan used same idea to design first electron synchrotron.

31 Generic Modern Synchrotron Focusing RF Acceleration Bending Spokes are user stations for this X-ray ring source

32 Synchrotron Phase Stability Edwin McMillan discovered phase stability independently of Veksler and used the idea to design first large electron synchrotron. V c (t) s h / frf 2 s f RF t t 1/ f RF h Lf / c RF Harmonic number: # of RF oscillations in a revolution

33 Transition Energy Beam energy where speed increment effect balances path length change effect on accelerator revolution frequency. Revolution frequency independent of beam energy to linear order. We will calculate in a few weeks Below Transistion Energy: Particles arriving EARLY get less acceleration and speed increment, and arrive later, with repect to the center of the bunch, on the next pass. Applies to heavy particle synchrotrons during first part of acceleration when the beam is non-relativistic and accelerations still produce velocity changes. Above Transistion Energy: Particles arriving EARLY get more energy, have a longer path, and arrive later on the next pass. Applies for electron synchrotrons and heavy particle synchrotrons when approach relativistic velocities. As seen before, Microtrons operate here.

34 Ed McMillan Vacuum chamber for electron synchrotron being packed for shipment to Smithsonian

35 Full Electron Synchrotron

36 GE Electron Synchrotron Elder, F. R.; Gurewitsch, A. M.; Langmuir, R. V.; Pollock, H. C., "Radiation from Electrons in a Synchrotron" (1947) Physical Review, vol. 71, Issue 11, pp

37 Cosmotron (First GeV Accelerator)

38 BNL Cosmotron and Shielding

39 Cosmotron Magnet

40 Cosmotron People

41

42 Bevatron Designed to discover the antiproton; Largest Weak Focusing Synchrotron

43 Strong Focusing Betatron oscillation work has showed us that, apart from bend plane focusing, a shaped field that focuses in one transverse direction, defocuses in the other Question: is it possible to develop a system that focuses in both directions simultaneously? Strong focusing: alternate the signs of focusing and defocusing: get net focusing!! Order doesn t matter

44 Linear Magnetic Lenses: Quadrupoles Source: Danfysik Web site

45 Comment on Strong Focusing One main advantage of strong focusing. In weak focusing machines, n < 1 for stability. Therefore, the fall-off distance, or field gradient cannot be too high. There is no such limit for strong focusing. n 1 is now allowed, leading to large field gradients and relatively short focal length magnetic lenses. This tighter focusing is what allows smaller beam sizes. Focusing gradients now limited only by magnet construction issues (pole magnetic field limits).

46 Weak vs. Strong Benders

47 First Strong-Focusing Synchrotron Cornell 1 GeV Electron Synchrotron (LEPP-AP Home Page)

48 Alternating Gradient Synchrotron (AGS)

49 CERN PS 25 GeV Proton Synchrotron

50 CERN SPS Eventually 400 GeV protons and antiprotons

51 FNAL First TeV-scale accelerator; Large Superconducting Benders

52 LEP Tunnel (Now LHC!) Empty LHC

53 Storage Rings Some modern accelerators are designed not to accelerate much at all, but to store beams for long periods of time that can be usefully used by experimental users. Colliders for High Energy Physics. Accelerated beamaccelerated beam collisions are much more energetic than accelerated beam-target collisions. To get to the highest beam energy for a given acceleration system design a collider Electron storage rings for X-ray production: circulating electrons emit synchrotron radiation for a wide variety of experimental purposes.

54 Princeton-Stanford Collider

55 SPEAR Eventually became leading synchrotron radiation machine

56 Cornell 10 GeV ES and CESR

57 SLAC s PEP II B-factory

58 ALADDIN at Univ. of Wisconsin

59 VUV Ring at NSLS VUV ring uncovered

60 Berkeley s ALS

61 Argonne APS

62 ESRF