Lecture title: Electron Cooling. Physics 598ACC, 2007 Summer Term, UIUC Accelerators, Theory and Practice. July 16, 2007 Sergei Nagaitsev

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1 Lecture title: Electron Cooling Physics 598ACC, 007 Summer Term, UIUC Accelerators, Theory and Practice July 16, 007 Sergei Nagaitsev Fermi Fermi National National Accelerator Accelerator Laboratory Laboratory

2 Summary Beam temperature, emittance; Liouville s theorem Beam cooling; Non-Liouvillean processes Electron cooling theory Electron cooling: practical implementations In my lecture I will use the Gaussian units: e = 4.8e-10 G.U., c = 3e10 cm/s

3 Beam sources The simplest conceptual model of an electron beam source is a planar diode. Cathodes typically operate at 1400K (kt 0.1 ev) An effective work function is about 1.6 ev Ion sources are generally more complex. The ions are typically extracted from the plasma of a gas discharge. 3

4 Beam temperature The electrons in the tail of the Fermi-Dirac distribution inside a cathode and the ions in the plasma source have a Maxwellian velocity distribution given by ( m v ) x + vy + vz f ( vx, vy, vz ) = f0 exp kt 7 For electrons: vrms = kt 6 10 T e [ cm/s] The thermal velocity spread of electrons (or ions) remains present in the beam at any distance downstream of the source. However, it may increase or decrease as will be discussed later. Let s assume that the beam propagates in the z-direction 4

5 Beam acceleration Suppose that the initial state, which we will denote by i, corresponds to the distribution emerging from the particle source: ( ( ) m v ) = i fi ( vi ) f0 vx, vy exp kti where v i, T i stand for initial v z, Tz After acceleration the distribution will be: ( ) m v f v 0 f f ( v f ) = f0 vx, vy exp, where v f = vi + v ktf and mv 0 = ev0 = ( ) E k 0 5

6 Beam acceleration (cont Beam acceleration (cont d) d) The new temperature: Finally, we have 6 [ ] ( ) f f f v m v kt = ( ) 0 0 v v v v v i i f + = + = ( ) = + = v v v v v v v v i i i f ( ) ( ) = v v v v v v v i i i f [ ] ( ) k i i i f E kt v v v m kt 4 4 0

7 Longitudinal cooling due to acceleration kt f ( kt ) The cooling effect predicted by this equation is very dramatic. Take, for example, an electron beam with an initial temperature of kt = 0.1 ev. After acceleration to Ek = 10 kev, the longitudinal temperature drops, while transverse does not change: -7 kt f 6 10 ev Relativistic expression: i E k kt f ( kt ) i β γ mc 7

8 Beam density effect on long. temperature Let s estimate the mean potential energy of electrons after acceleration: E 1/3 p Ce n = Cmc where C is a numeric constant (~ ), n particle density, r e classical electron radius (.8e-13 cm) Example: 1-Amp electron beam at 10 kev, beam radius a = 1-cm: Ib 8-3 n = cm πa βce Thus, E p C 10-4 ev!!! After acceleration, the long. temperature dramatically drops, but then starts increasing due to electron-electron interactions. The rate of increase depends on beam density and other factors r e n 1/3 8

9 Emittance Typically, T i is much smaller than the kinetic energy after acceleration. Thus, each particle has a velocity only slightly different from v 0. In the lab. frame a particle velocity has angles w.r.t. the z -direction: vx v θ x x y = θ y v 0 y = Let s define the rms emittance as: ε ( ) x x xx 1/ x = Similarly, for y Emittance is a measure of the beam s phase-space area (or volume) v 0 9

10 Emittance (cont d) Electron beam: 1-cm radius (a), uniform density, kinetic energy E k = 10 kev, emitted from a cathode with kt i = 0.1 ev 1/ 1/ a kt (*home work problem) i a kti ε rms = = v0 m β mc 1/ a kti Relativistic expression: ε rms = βγ mc Let s define the normalized emittance: ε n = βγε rms This quantity is independent of beam energy. For the above example ε =. μm n 10

11 Liouville s theorem 11

12 The original paper: J. de Math., p. 34,

13 Liouville s paper 13

14 From Wikipedia In physics, Liouville's theorem, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase-space distribution function is constant along the trajectories of the system - that is that the density of system points in the vicinity of a given system point travelling through phase-space is constant with time. In the simple case of a non-relativistic particle moving in Euclidean space under a force field F with coordinates x and momenta p, Liouville's theorem can be written as 14

15 In simple words No system of magnets or electro-magnetic fields can reduce the phase-space density of beam The density of particles in phase-space, or the phase-space volume occupied by a given number of particles remains invariant under Hamiltonian forces. 15

16 Consequence of Liouville s theorem While the volume in phase space remains constant, the shape generally does not. In fact, nonlinearities (aberrations) in the field configurations may cause considerable distortions (filamentations) Examples: spherical aberrations; synchrotron motion 16

17 Emittance and Liouville s theorem Normalized emittance is an approximate (but very convenient) measure of the phase-space volume. In an ideal linear accelerator with a perfect vacuum and no interactions between particles the emittance should remain constant. However 17

18 Why cool beams? Particle accelerators, by imparting high energies to charged particles, create a beam in a state with a virtually limitless reservoir of energy in one (longitudinal) degree of freedom. This energy can couple (randomly and coherently) to other degrees of freedom by various processes, such as: scattering; improper bending and focusing; interaction with environment (e.g. vacuum chamber) radiation; Normally, it is necessary to keep energy spreads in the transverse degrees of freedom at 10-4 of the average longitudinal energy. 18

19 What is beam cooling? Cooling is a reduction in the phase space occupied by the beam (of same number of particles). Equivalently, cooling is a reduction in the random motion of the beam. 19

20 Examples of NON-COOLING Beam scrapping (removing particles with higher amplitudes) is NOT cooling. Expanding the beam transversely would lower its transverse temperature. This is NOT cooling. Coupling between degrees of freedom may lead to exchange of energies. If the dynamics is Hamiltonian, this is NOT cooling. We carefully reserve the term beam cooling for non-hamiltonian processes where Liouville s theorem is violated; i.e., where there is an increase in phase-space density. 0

21 Need for cooling Injection help: stacking, accumulation, phasespace manipulation etc. Antiparticle production: accumulation of many pulses of antiparticles Internal fixed target: emittance grows from target scattering Colliding beams: beam-beam effects, residual gas scattering Precise Energy Resolution: narrow states, threshold production 1

22 Stochastic cooling Synchrotron radiation Electron cooling Types of beam cooling Laser cooling (of certain ion beams) Ionization cooling (not yet tested) A technique for rapid transverse cooling in a straight transport line has yet to be found.

23 How does electron cooling work? The velocity of the electrons is made equal to the average velocity of the ions. The ions undergo Coulomb scattering in the electron gas and lose energy, which is transferred from the ions to the co-streaming electrons until some thermal equilibrium is attained. Electron Gun Electron Collector Electron beam Storage ring Ion beam 1-5% of the ring circumference 3

24 Binary collisions Let s consider the simplest case of a single stationary Coulomb center, interacting with a monochromatic beam of particles, moving in z-direction. Consider only binary collisions. The trajectory is a hyperbola, such that, e e where ρ α β is the 90-degree impact = mu parameter θ tan = ρ ρ 4

25 The friction force Let s now find the friction force, F, acting on the scattering center, α. From symmetry F can only be along z-direction u d β F = m u z u dt β Scattering is elastic only direction of u changes! 5

26 The friction force The z-projection of particle velosity: β θ ρ Δu z = u sin = u ρ + ρ Thus, the force F F = u u mδu β z udσ = The integral diverge logarithmically n β Let s introduce a cut-off, the maximum impact parameter ρ max >> ρ Then ρmax ρdρ ρ max Λ ln ρ + ρ 0 ρ u u ( 4πmnβu ρ ) 0 ρdρ ρ + ρ 6

27 Finally, we obtain F The friction force 4π m = eα eβ nβ where Λ is the so-called Coulomb log Truncating the integral is not a trick. There is always some cut-off (i.e. beam radius or Debye shielding) Debye shielding (homework problem): At impact parameters greater than Debye radius, D, the coulomb potential decays exponentially [ ] D 740 cm T n [ ev] Λ [ ] -3 cm u u 3 7

28 Moving foil analogy Consider electrons as being represented by a foil moving with the average velocity of the ion beam. Ions moving faster (slower) than the foil (electrons) will penetrate it and will lose energy along the direction of their momentum (de/dx losses) during each passage until all the momentum components in the moving frame are diminished. p v p Foil r F * * * * 4πn ( re mc ) = E = * mv p * represents rest frame Λ r v v * p * p 8

29 Friction force (cont d) For an arbitrary electron distribution function (for example): r 1 v v ex ey vez f ( v = e) exp 3/ (π ) σ exσ eyσ ez σ ex σ ey σ ez r F a moving ion experiences the following force: r r v 4 p e 3 b( Vp ) = 4π mere c nebη Λ r r f ( ve) d ve A 3 Vp ve Drag rate, MeV/c per hour Momentum deviation, MeV/c r V r r V r V p p r ve r v e 3 f r ( v e ) d 3 r v e 9

30 Was invented by G.I. Budker (INP, Novosibirsk) as a way to increase luminosity of p-p and p-pbar colliders. Electron cooling First mentioned at Symp. Intern. sur les anneaux de collisions á electrons et positrons, Saclay, 1966: Status report of works on storage rings at Novosibirsk First publication: Soviet Atomic Energy, Vol., May 1967 An effective method of damping particle oscillations in proton and antiproton storage rings 30

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34 Gerard K. O Neill O ( ) 199) Was a professor of physics at Princeton University ( ). He invented and developed the technology of storage rings for the first colliding-beam experiment at Stanford. He served as an adviser to NASA. He also founded the Space Studies Institute. 34

35 The First Report 3 / e e p i e 4 e p Maxwellian m T Am T n e m m Z A = π τ Budker s formula

36 First Cooling Demonstration Electron cooling was first tested in 1974 with 68 MeV protons at NAP-M storage ring at INP(Novosibirsk). 36

37 IVth All-Soviet Conference on Part. Accelerators, Moscow, First experimental success and first report on electron cooling of protons in NAP-M : E p 50 MeV Ip 50 μa E e 37 kev Ie 0.1 A φ p_equilibrium 1 mm τ cool 3 sec - in full agreement with Budker s theory (classic plasma formulae). G.I.Budker, Ya.S.Derbenev, N.S.Dikansky, V.I.Kudelainen, I.N.Meshkov, V.V.Parkhomchuk,D.V.Pestrikov, B.N.Sukhina, A.N.Skrinsky, First experiments on electron cooling, in Proc. of IVth All-Soviet Conference on Part. Accel., v., p.30, 1975; IEEE Trans. Nucl. Sci., NS- (1975) 093; Part. Accelerators 7 (1976)197; Rus. Atomic Energy 40 (1976)

38 1975 Unexpected results after e-cooler improvement Provisional text not revised by CERN Translation Service NUCLEAR PHYSICS INSTITUTE SIBERIAN BRANCH OF USSR ACADEMY OF SCIENCE PS/DL/Note 76-5 October 1976 Preprint N.P.I G.I. Budker, A.F. Bulyshev, N.S. Dikansky, V.I. Kononov, V.I. Kudelainen, I.N. Meshkov, V.V. Parkhomchuk, D.V. Pestrikov, A.N. Skrinsky, B.N. Sukhina NEW EXPERIMANTAL RESULTS OF ELECTRON COOLING Presented to the All Union High Energy Accelerator Conference, Moscow, October 1976 (Translated at CERN by O. Barbalat) * * * * * 38

39 Improvements: B-field homogeneity in the cooling section about 10-4, electron energy stability better than As result - the betatron oscillation damping time is inversely proportional to the electron current and for a current of 0.8 A it amounts to 83 ms (proton energy of 65 MeV) -much shorter of The Budker s numbers! What a puzzle! 39

40 Flattened electron velocity distribution T Puzzle explained = β T γ The role magnetic field magnetized cooling Cathode mc e - B Interaction of an ion with a Larmor circle p ± p + F e - 40

41 The role of the solenoidal field The solenoidal magnetic field allows to combine strong focusing with the requirement (for efficient cooling) of low electron transverse temperature in the cooling interaction region; Cooling rates with a "strongly" magnetized electron beam are ultimately determined by the electron longitudinal energy spread only, which can be made much smaller than the transverse one. 41

42 Effects of a strong magnetic field It is obvious that the influence of magnetic field becomes important when the duration of an ionelectron interaction exceeds the Larmor oscillation period: vr ρ > ω L mcv eh If the ion velocity is small compared to the electron velocity spread and the impact parameter is large, the ion practically interacts with a Larmor circle moving only along the magnetic field. This might lead to an order of magnitude enhancement in the cooling force. r 4

43 Friction force for various degrees of magnetization Friction force From: Electron cooling: physics and prospective applications, by V.V. Parkhomchuk and A.N. Skrinsky, Rep. Prog. Phys. 54 (1991), p v e ve v i 43

44 First cooler rings Europe , Initial Cooling Experiment at CERN M.Bell, J.Chaney, H.Herr, F.Krienen, S. van der Meer, D.Moehl, G.Petrucci, H.Poth, C.Rubbia NIM 190 (1981) 37 USA , Electron Cooling Experiment at Fermilab T.Ellison, W.Kells, V.Kerner, P.McIntyre, F.Mills, L.Oleksiuk, A.Ruggiero, IEEE Trans. Nucl. Sci., NS-30 (1983) 370; 44

45 Stochastic cooling Stochastic cooling was invented by Simon van der Meer in 197 and first tested in 1975 at CERN in ICE (initial cooling experiment) ring with 46 MeV protons. FNAL

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48 Fred Mills, one of the Fermilab physicists working on the electron cooling tests in 1980, writes: 48

49 Fermilab s legacy: Cool Before Drinking... 49

50 Second Generation of Cooler Storage Rings IUCF Cooler Ring (Bloomington,IN, USA) 1988 Test Storage Ring (MPI, Heidelberg, Germany) 1988 Low Energy Antiproton Ring (CERN) 1989 TARN-II ( Test Accumulator Ring for NUMATRON, Tokyo University, Japan) 1989 CELSIUS (Uppsala University, Sweden) 1990 Experimental Storage Ring (GSI, Darmstadt, Germany) Electron Cooling - Nagaitsev 50

51 Second Generation of Cooler Storage Rings 199 COoler-SYnchrotron (FZ Juelich, Germany) 199 CryRing (MSI, Stockholm, Sweden) 1993 ASTRID (Aarhus University, Denmark) SIS (GSI, Darmstadt, Germany) Heavy Ion Medical ACcelerator (NIRS, Japan) 000 Antiproton Decelarator (CERN) 00 Electrostatic cooler storage ring at KEK (KEK, Tsukuba, Japan) Electron Cooling - Nagaitsev 51

52 ESR Electron Cooler at GSI (Darmstadt) Electron Cooling - Nagaitsev 5

53 005 Recycler Electron Cooler (Fermilab, USA) 006 Two cooler rings complex (IMP, Lanzhou, China) Low Energy Ion Ring (CERN) 0?? TARN-II renovated (RIKEN, Japan) 005 S-LSR : Solid magnet Laser equipped cooler Storage Ring (Kyoto University, Japan) Electron Cooling - Nagaitsev 53

54 Electron cooling applications Particle physics with electron cooled protons, deuterons and antiprotons : Antiproton physics => LEAR π-meson physics => IUCF, COSY, CELSIUS First antihydrogen generation in-flight => => LEAR (stochastic cooling) Antihydrogen generation in traps => => AD => ATHENA and ATRAP Electron Cooling - Nagaitsev 54

55 Electron cooling applications Nuclear physics Studies ofradioactive nuclei and rare isotopes, exotic nuclei states (like bare nuclei decay, etc.) => => ESR High precision mass spectroscopy => ESR New stage of experiments in atomic and molecular physics => => TSR, CryRing, ASTRID Electron Cooling - Nagaitsev 55

56 Mass-spectrometry spectrometry of radioactive nuclei at GSI Electron Cooling - Nagaitsev 56

57 Mass spectrometry of excited nuclear states Electron Cooling - Nagaitsev 57

58 Detection of single cooled ion at GSI Electron Cooling - Nagaitsev 58

59 Energy loss by the beam due to residual gas after electron cooling is turned OFF Electron Cooling - Nagaitsev 59

60 Ordering effects in coasting and bunched ion beams CRYRING, Manne Siegbahn Lab Phase transition to a 1D crystal (string). Min. dist. btw. ions 4 mm Electron Cooling - Nagaitsev 60

61 Most recent addition to the electron cooling family a 300- kv cooler at IMP, Lanzhou Built and commissioned by Budker INP Has implemented several novel ideas: Electrostatic electron beam bends Variable-profile electron beam Extra-uniform (1ppm) guiding magnetic field Electron Cooling - Nagaitsev 61

62 Budker INP design E e =300 kev 1 - electron gun; - main gun solenoid ; 4 - electrostatic deflectors; 5 - toroidal solenoid; 6 - main solenoid; 7 - collector; 8 - collector solenoid; 11 - main HV rectifier; 1 - collector cooling system. 1 m Electron Cooling - Nagaitsev 6

63 Schematic of Fermilab s Electron Cooler 63

64 Electron beam parameters Electron kinetic energy 4.34 MeV Absolute precision of energy 0.3 % Energy ripple 10-4 Beam current 0.5 A DC Duty factor (averaged over 8 h) 95 % Electron angles in the cooling section (averaged over time, beam cross section, and cooling section length), rms 0. mrad Electron Cooling - Nagaitsev 64

65 Where do we get antiprotons? The Antiproton Source is made up of three parts. The first is the Target: Fermilab creates antiprotons by striking a nickel target with protons. Second is the Debuncher Ring: This triangular shaped ring captures the antiprotons coming off of the target. The third is the Accumulator: This is the storage ring for the antiprotons. Recently, we have added another ring, the Recycler, for additional antiproton storage. Electron Cooling - Nagaitsev 65

66 Recycler Main Injector Recycler The Recycler is a fixedmomentum (8.9 GeV/c), permanent-magnet antiproton storage ring. Main Injector The Main Injector is a rapidlycycling, proton synchrotron. Every seconds it delivers 10 GeV protons to a pbar production target. It also delivers beam to a number of fixed target experiments. Electron Cooling - Nagaitsev 66

67 Electron cooling system setup at Fermilab Pelletron (MI-31 building) Cooling section solenoids (MI-30 straight section) 67

68 Electron cooling: long. drag rate For an antiproton with zero transverse velocity, electron beam: 500 ma, 3.5-mm radius, 00 ev rms energy spread and 00 μrad rms angular spread Drag rate, MeV/c per hour Linear approx. F p λp Momentum deviation, MeV/c F Non-magnetized cooling force model + e p 3 ( c ) Λ f ( v ) d ve = 4πne m re f ( v e ) = 1 e exp v v e v v e ( ) π 3 / σ σ σ σ σ = βγθ e c σ n // e δe βγmc JCS = γeβc v p 3 v e Lab frame quantities 68

69 Cooling force Experimental measurements Two experimental techniques, both requiring small amount of pbars, coasting (i.e. no RF) with narrow momentum distribution and small transverse emittances Diffusion measurement For small deviation cooling force (linear part) Reach equilibrium with ecool Turn off ecool and measure diffusion rate Voltage jump measurement For momentum deviation > MeV/c Reach equilibrium with ecool Instantaneously change electron beam energy Follow pbar momentum distribution evolution 69

70 1 Cooling rate for small amplitudes For small momentum deviations (< 1 MeV) the cooling force is linear: F - λp. The distribution function in momentum is close to being Gaussian. Dist. function, arb. units rms spread 0.17 MeV/c σ D rms = λ Momentum deviation, MeV/c 5x10 10 pbars, mm mrad (n, 95%) Cooled by 500 ma electron beam on axis 70

71 Cooling OFF-ON ON Momentum spread (rms, MeV/c) and electron current (A) By turning the electron cooling OFF and ON again one can determine both the diffusion and cooling rates Cooling OFF: Fit Electron beam current, Amps Time, sec 5x10 10 pbars, mm mrad (n, 95%) σ () t = + Dt σ 0 Cooling ON: D σ () t = σ 0 exp( λt) + λ D.5 MeV /hr λ 43 hr -1 D λ 71

72 Drag force measurements: electron energy jump by + kev Momentum distribution (log scale) Evolution of the weighted average of the pbar momentum distribution function 1. MeV per division 0.0 Momentum deviation [MeV/c] MeV/hr Time [mn] Beam emittance was measured by Schottky: 1.5 μm (n, 95%). In the cooling section this corresponds to a 0.9 mm radius (rms), electron current 500 ma Cooling force is in reasonable agreement with predictions 7

73 Conclusions Beam cooling is widely used, but each cooling technique has a limited range of applicability Low energy electron cooling (for ions below 500 MeV/nucleon) found many excellent experimental applications in nuclear and atomic physics. Fermilab now has a world-record operational electron cooling system Since the end of August 005, electron cooling is being used on (almost) every Tevatron shot Electron cooling allowed for the latest advances in the Tevatron luminosity 73