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Lecture title: Electron CoolingLecture title: Electron Cooling

Physics 598ACC, 2007 Summer Term, UIUCPhysics 598ACC, 2007 Summer Term, UIUCAccelerators, Theory and PracticeAccelerators, Theory and Practice

July 16, 2007Sergei Nagaitsev

Fermi National Accelerator LaboratoryFermi National Accelerator Laboratory

SummarySummary

Beam temperature, emittance; Liouville’s theoremBeam cooling; Non-Liouvillean processesElectron cooling theoryElectron cooling: practical implementations

In my lecture I will use the Gaussian units:e = 4.8e-10 G.U., c = 3e10 cm/s

S. Nagaitsev – Electron Cooling 2

Beam sourcesBeam sources

The simplest conceptual model of an electron beam source is a planar diode.

Cathodes typically operate at 1400K (kT ≈ 0.12 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.


Beam temperatureBeam 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

For electrons: 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


( )⎥⎥⎦

⎤

⎢⎢⎣

⎡ ++−=

kTvvvm

fvvvf zyxzyx 2

exp),,(222

0

[ ]cm/s106 7erms TkTv ⋅≈=

Beam accelerationBeam acceleration

Suppose that the initial state, which we will denote by i, corresponds to the distribution emerging from the particle source:

where stand for initial After acceleration the distribution will be:

( ) ( )


⎥⎦

⎤⎢⎣

⎡−=

i

iyxii kT

vmvvfvf2

exp,)(2

0

ii Tv , zz Tv ,

( ) ( ) 20

2220

2

0 where,2

exp,)( vvvkT

vvmvvfvf if

f

fyxff +=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=

kEeVmv== 0

20

2 and

Beam acceleration (contBeam acceleration (cont’’d)d)

The new temperature:

Finally, we have


[ ]( )22fff vvmkT −=

( ) 20

220

22 vvvvv iif +=+=

( )2

40

4

20

220

220

22...

81

211 ⎟

⎟⎠

⎞⎜⎜⎝

⎛+−+=⎟

⎠⎞⎜

⎝⎛ +=

vv

vvvvvv ii

if

( ) ( ) ...41

41

40

4

20

22

220

2+−++=

vv

vvvvv ii

if

[ ] ( )k

iiif E

kTvvvmkT

24

2224

20

≈⎟⎠⎞⎜

⎝⎛ −≈

Longitudinal Longitudinal ““coolingcooling”” due to accelerationdue to acceleration

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:Relativistic expression:


( )k

if E

kTkT2

2

≈

eV106 -7×≈fkT

( )222

2

mckTkT i

f γβ≈

Beam density effect on long. temperatureBeam density effect on long. temperature

Let’s estimate the mean potential energy of electrons after acceleration:

where C is a numeric constant (~ 2), n – particle density, re – classical electron radius (2.8e-13 cm)Example: 1-Amp electron beam at 10 keV, beam radius a = 1-cm:

Thus, Ep ≈ 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

3/123/12 nrCmcnCeE ep =≈


3-82 cm 103.4 ⋅≈=

ceaIn b

βπ

EmittanceEmittance

Typically, Ti is much smaller than the kinetic energy after acceleration. Thus, each particle has a velocity only slightly different from v0. In the lab. frame a particle velocity has angles w.r.t. the z -direction:

Let’s define the rms emittance as:

Similarly, for y…Emittance is a measure of the beam’s phase-space area (or volume)


0vvx x

x =′≡θ0v

vy y

y =′≡θ

( ) 2/1222 xxxxx ′−′=ε

EmittanceEmittance (cont(cont’’d)d)

Electron beam: 1-cm radius (a), uniform density, kinetic energy Ek = 10 keV, emitted from a cathode with kTi = 0.1 eV

(*home work problem)

Relativistic expression:Let’s define the normalized emittance:

This quantity is independent of beam energy.

For the above example


2/1

2

2/1

0 22⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

mckTa

mkT

va ii

rms βε

2/1

22⎟⎠⎞

⎜⎝⎛=

mckTa i

rms βγε

rmsn βγεε =

m 2.2 με =n

LiouvilleLiouville’’ss theoremtheorem


The original paper: J. de Math., p. 342, 1838The original paper: J. de Math., p. 342, 1838


LiouvilleLiouville’’ss paperpaper


From From WikipediaWikipediaIn 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 travellingthrough 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


In simple wordsIn 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.


Consequence of Consequence of LiouvilleLiouville’’ss theoremtheorem

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


EmittanceEmittance and and LiouvilleLiouville’’ss theoremtheorem

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…


Why cool beams?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.


What is beam cooling?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.


Examples of NONExamples of NON--COOLINGCOOLING

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’stheorem is violated; i.e., where there is an increase in phase-space density.


Need for coolingNeed for cooling

Injection help: stacking, accumulation, phase-space manipulation etc.Antiparticle production: accumulation of many pulses of antiparticlesInternal fixed target: emittance grows from

target scatteringColliding beams: beam-beam effects, residual gas scatteringPrecise Energy Resolution: narrow states, threshold production


Types of beam coolingTypes of beam cooling

• Stochastic cooling• Synchrotron radiation• Electron 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?How does electron cooling work?

Storage ring

ElectronGun

ElectronCollector

1-5% of the ringcircumference

Electron beam

Ion beam

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.

S. Nagaitsev – Electron Cooling

Binary collisionsBinary 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,where is the 90-degree impact

parameter


ρρθ ⊥=⎟

⎠⎞

⎜⎝⎛

2tan

2muee βαρ =⊥

The The ““frictionfriction”” forceforce

Let’s now find the friction force, F, acting on the scattering center, α.From symmetry F can only be along z-direction

Scattering is elastic – only direction of u changes!


∑−=β

βzu

dtdm

uuF

The friction forceThe friction force

The z-projection of particle velosity:

Thus, the force F

The integral diverge logarithmicallyLet’s introduce a “cut-off”, the maximum impact parameterThen


22

22 2

2sin2

ρρρθβ

+−=−=Δ

⊥

⊥uuuz

( )∫∫∞

⊥⊥ +

=Δ−=0

22224

ρρρρρπσ ββ

β dumnuuudnum

uuF z

⊥>> ρρmax

∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛≅

+≡Λ

⊥⊥

max

0

max22 ln

ρ

ρρ

ρρρρd

The friction forceThe friction force

Finally, we obtain

where Λ is the so-called Coulomb logTruncating 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


3224

uunee

mF Λ= ββα

π

[ ] [ ][ ]3-cmn

eVcm740 TD ≈

28

Moving foil analogyMoving foil analogy

p

*

*

2*

22**** )(4

p

p

p

e

vv

mvmcrnEF

rr

⋅π

−=−∇=Λvp

Foil

* represents rest frame

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.


Friction force (contFriction force (cont’’d)d)

For an arbitrary electron distribution function (for example):

a moving ion experiences the following force:


⎟⎟⎠

⎞⎜⎜⎝

⎛−−−= 2

2

2

2

2

2

2/3 222exp

)2(1)(

ez

ez

ey

ey

ex

ex

ezeyexe

vvvvfσσσσσσπ

r

eeep

epee

ep

epebeepb vdvf

vVvV

AvdvfvVvV

ncrmVF rrrr

rrr

rr

rrrr

33

33

42 )(||

)(||

4)( ∫∫ −

−⋅≡

−

−⋅Λ⋅⋅−= ηπ

-25

-20

-15

-10

-5

0

5

10

15

20

25

-8 -6 -4 -2 0 2 4 6 8

Momentum deviation, MeV/c

Dra

g ra

te, M

eV/c

per

hou

r

30

Electron coolingElectron cooling

Was invented by G.I. Budker (INP, Novosibirsk) as a way to increase luminosity of p-p and p-pbarcolliders.

First mentioned at Symp. Intern. sur les anneaux de collisions áelectrons et positrons, Saclay, 1966: “Status report of works on storage rings at Novosibirsk”

First publication: Soviet Atomic Energy, Vol. 22, May 1967 ”An effective method of damping particle oscillations in proton and antiproton storage rings”


31S. Nagaitsev – Electron Cooling

34

Gerard K. OGerard K. O’’Neill (1927Neill (1927--1992)1992)

Was a professor of physics at Princeton University (1965-1985). 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.



1966 – The First Report

2/3

e

e

p

i

e4

ep2Maxwellian m

TAmT

ne

mm

ZA

283

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⋅⋅=

πτ

Budker’s formula

36

First Cooling DemonstrationFirst Cooling Demonstration

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



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.2, p.302, 1975; IEEE Trans. Nucl. Sci., NS-22 (1975) 2093; Part. Accelerators 7 (1976)197; Rus. Atomic Energy 40 (1976) 49.

τcool 3 sec - in full agreement with Budker’s theory (classic plasma formulae).

IVth All-Soviet Conference on Part. Accelerators, Moscow, 1974

1974 - First experimental success and first report on electron cooling of protons in NAP-M :

Ep 50 MeV Ip 50 μAEe 37 keV Ie 0.1 Aφp_equilibrium 1 mm


1975 – Unexpected results after e-cooler improvement

Provisional textnot revised by CERNTranslation Service

NUCLEAR PHYSICS INSTITUTE SIBERIAN BRANCH OF USSR ACADEMY OF SCIENCE

PS/DL/Note 76-25October 1976

Preprint N.P.I. 76-32G.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)* * * * *


As result - “… the betatron oscillation damping time is inversely proportional to the electron current

electron energy stability – better than 10-5.

and for a current of 0.8 A it amounts to 83 ms (proton energy of 65 MeV) –

What a puzzle!-much shorter of “The Budker’s numbers!”

Improvements: B-field homogeneity in the cooling section – about 10-4,

“Flattened” electron velocity distribution

The role magnetic field – magnetized cooling

Puzzle explainedPuzzle explained……


222

2

|| mc

TT Cathode

γβ=

B

e-

p±

e-

p+

Interaction of an ion with

“a Larmor circle”

F


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.

Effects of a strong magnetic fieldEffects of a strong magnetic field


• It is obvious that the influence of magnetic field becomes important when the duration of an ion-electron interaction exceeds the Larmor oscillation period:

• 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.

eHmcvv r

L

r ≡>ω

ρ

Friction force for various degrees of magnetizationFriction force for various degrees of magnetization


Frictionforce

From: “Electron cooling: physics andprospective applications”, by V.V. Parkhomchuk and A.N. Skrinsky, Rep. Prog. Phys. 54 (1991), p. 919.

vive⊥ve||

44

First cooler rings

Europe – 1977 – 79, Initial Cooling Experiment at CERNM.Bell, J.Chaney, H.Herr, F.Krienen, S. van der Meer, D.Moehl, G.Petrucci, H.Poth, C.Rubbia– NIM 190 (1981) 237

USA – 1979 – 82, Electron Cooling Experiment at FermilabT.Ellison, W.Kells, V.Kerner, P.McIntyre, F.Mills, L.Oleksiuk, A.Ruggiero, IEEE Trans. Nucl. Sci., NS-30 (1983) 2370;


45

Stochastic coolingStochastic cooling

Stochastic cooling was invented by Simon van derMeer in 1972 and first tested in 1975 at CERN in ICE (initial cooling experiment) ring with 46 MeVprotons. FNAL - 1980.


48

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


49

FermilabFermilab’’ss legacy: Cool Before Drinking...legacy: Cool Before Drinking...


Electron Cooling - Nagaitsev 50

Second Generation of Cooler Storage Rings

1988 - IUCF Cooler Ring (Bloomington,IN, USA)

1988 – Low Energy Antiproton Ring (CERN)

1988 – Test Storage Ring (MPI, Heidelberg, Germany)

1989 – TARN-II (“Test Accumulator Ring for NUMATRON, Tokyo University, Japan)

1990 – Experimental Storage Ring (GSI, Darmstadt, Germany)

1989 – CELSIUS (Uppsala University, Sweden)


Second Generation of Cooler Storage Rings

1993 – ASTRID (Aarhus University, Denmark)

1992 – COoler-SYnchrotron (FZ Juelich, Germany)

2000 - Heavy Ion Medical ACcelerator(NIRS, Japan)

1998 - SIS (GSI, Darmstadt, Germany)

1992 – CryRing (MSI, Stockholm, Sweden)

2000 – Antiproton Decelarator (CERN)

2002 – Electrostatic cooler storage ring at KEK (KEK, Tsukuba, Japan)


ESR Electron Cooler at GSI (Darmstadt)ESR Electron Cooler at GSI (Darmstadt)


20?? –TARN-II–renovated (RIKEN, Japan)

2006 – Two cooler rings complex (IMP, Lanzhou, China)

2006 - Low Energy Ion Ring (CERN)

2005 – Recycler Electron Cooler (Fermilab, USA)

2005 – S-LSR : Solid magnet Laser equipped cooler Storage Ring (Kyoto University, Japan)


Electron cooling applicationsElectron cooling applications

Antiproton physics => LEAR

Antihydrogen generation in traps =>=> AD => ATHENA and ATRAP

Particle physics with “electron cooled” protons, deuterons and antiprotons :

π-meson physics => IUCF, COSY, CELSIUS

First antihydrogen generation in-flight =>=> LEAR (stochastic cooling)


Electron cooling applicationsElectron cooling applications

Studies of radioactive nuclei and rare isotopes, exotic nuclei states (like bare nuclei decay, etc.) =>

=> ESR

Nuclear physics

High precision mass spectroscopy => ESR

New stage of experiments in atomic and molecular physics =>

=> TSR, CryRing, ASTRID


MassMass--spectrometry of radioactive nuclei at GSIspectrometry of radioactive nuclei at GSI


Mass spectrometry of excited nuclear statesMass spectrometry of excited nuclear states


Detection of single cooled ion at GSIDetection of single cooled ion at GSI


Energy loss by the beam due to residual gas Energy loss by the beam due to residual gas after electron cooling is turned OFFafter electron cooling is turned OFF


Ordering effects in coasting and bunched ion Ordering effects in coasting and bunched ion beamsbeams

CRYRING, Manne Siegbahn LabPhase transition to a 1D crystal(string). Min. dist. btw. ions 4

mm


Most recent addition to the electron cooling family Most recent addition to the electron cooling family –– a 300a 300--kV cooler at IMP, LanzhoukV cooler at IMP, Lanzhou

Built and commissioned by Budker INPHas implemented several novel ideas:

Electrostatic electron beam bendsVariable-profile electron beamExtra-uniform (1ppm) guiding magnetic field


1 - electron gun; 2- main “gun solenoid”; 4 - electrostatic deflectors; 5 - toroidal solenoid; 6 - main solenoid; 7 - collector; 8 - collector solenoid;

11 - main HV rectifier; 12 - collector cooling system.

1 m

Ee=300 keVBudker INP design

63

Schematic of Schematic of FermilabFermilab’’ss Electron CoolerElectron Cooler



Electron beam parametersElectron 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 ≤ 0.2 mrad


Where do we get Where do we get antiprotonsantiprotons??

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.


Recycler Recycler –– Main InjectorMain Injector

The Recycler is a fixed-momentum (8.9 GeV/c), permanent-magnet antiproton storage ring.

The Main Injector is a rapidly-cycling, proton synchrotron. Every 1.5-3 seconds it delivers 120 GeV protons to a pbar production target. It also delivers beam to a number of fixed target experiments.

Recycler

Main Injector

67

Electron cooling system setup at Electron cooling system setup at FermilabFermilab

Pelletron(MI-31 building)

Cooling section solenoids

(MI-30 straight section)


68

-25

-20

-15

-10

-5

0

5

10

15

20

25

-8 -6 -4 -2 0 2 4 6 8


Dra

g ra

te, M

eV/c

per

hou

rElectron cooling: long. drag rateElectron cooling: long. drag rate

For an antiproton with zero transverse velocity, electron beam: 500 mA, 3.5-mm radius, 200 eV rmsenergy spread and 200 μrad rms angular spread

Non-magnetized cooling force model

( ) ( ) e

pe

peeee fcrmn v

vv

vvvF 3

3

22 d4−

−Λ= ∫

+∞

∞−

π

( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡−−=

⊥

⊥

⊥2||

2||

2

2

||22/3 22

exp2

1)(σσσσπee

evv

f v

ceβγθσ =⊥

mcE

βγδσ ≈//

Lab frame quantities

ceJn CS

e βγ=

Linear approx.

pFp λ−≈


69

Cooling force Cooling force –– Experimental measurementsExperimental measurementsTwo 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 > 2 MeV/c• Reach equilibrium with ecool• Instantaneously change electron beam energy• Follow pbar momentum distribution evolution

70

0.0001

0.001

0.01

0.1

1

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0


Dis

t. fu

nctio

n, a

rb. u

nits

For small momentum deviations (< 1 MeV) the cooling force is linear: F ≈ - λp. The distribution function in momentum is close to being Gaussian.

Cooling rate for small amplitudesCooling rate for small amplitudes

rms spread 0.17 MeV/c

λ

σ2

2 Drms =

5x1010 pbars, 2 mm mrad (n, 95%)Cooled by 500 mA electron beam on axis

71

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 200 400 600 800 1000 1200

Time, sec

Mom

entu

m sp

read

(rm

s, M

eV/c

)an

d el

ectr

on c

urre

nt (A

)

By turning the electron cooling OFF and ON again one can determine both the diffusion and cooling rates

Cooling OFFCooling OFF--ONON

Electron beamcurrent, Amps

Cooling OFF:

Cooling ON:

D ≈ 2.5 MeV2/hrλ ≈ 43 hr -1

Fit ( ) Dtt += 2

0σσ

( ) ( )

λλ

λσσ

22exp

220

DtDt +−⎟⎠⎞

⎜⎝⎛ −=

5x1010 pbars, 2 mm mrad (n, 95%)

72

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [mn]M

omen

tum

dev

iatio

n [M

eV/c

]

Drag force measurements: electron energy jump by +2 keVDrag force measurements: electron energy jump by +2 keV

Momentum distribution (log scale)

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

1.2 MeV per division

21 MeV/hr

Evolution of the weighted average of the pbar momentum distribution function

Cooling force is in reasonable agreement with predictionsCooling force is in reasonable agreement with predictions


ConclusionsConclusions

Beam cooling is widely used, but each cooling technique has a limited range of applicabilityLow 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 operationalelectron cooling system

Since the end of August 2005, electron cooling is being used on (almost) every Tevatron shotElectron cooling allowed for the latest advances in the Tevatron luminosity

lecture title: electron cooling - illinoisweb.hep.uiuc.edu/home/derrede/598acc/e cooling...

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