Beam Beam CoolingCooling

• Introduction to cooling, temperature, phasespace and Liouville

• Stochastic cooling• Electron cooling• Laser cooling• Radiation damping• Ionisation and other cooling

Søren Pape Møller, Aarhus University, Denmark

cooling is not:collimation (loss of particles)adiabatic damping

Beam Beam coolingcooling

• Since beamcooling is ”slow”, it is mostlyused in storage rings;

• however, ionisation and ”stochasticcooling” has been or will be used for e.g. for muons

What is cooling? What is Temperature?

IntroductionIntroduction

⟩⟨= ⊥⊥2

//21

//23 )( vmTk r

v is the velocity relative to the reference particlemoving with the average ion velocity. (internaltemperature)Temperature is a measure of the disorderedmotion.Cooling is hence a reduction of the temperature,i.e. of the disordered motion.Clearly, other parts of the total system willget warmer.

My old thermodynamics teacher

• How do you measure the temperature of an ant?

What is cooling? What is Temperature?

IntroductionIntroduction

⟩⟨= ⊥⊥2

//21

//23 )( vmTk r

v is the velocity relative to the reference particlemoving with the average ion velocity.Temperature is a measure of the disordered motion.

In an accelerator

⎟⎟⎠

⎞⎜⎜⎝

⎛⟩⟨

+⟩⟨

=

⟩Δ⟨=

⊥VH

McT

ppMcT

ββεγβ

β

11

/

222

222//

WhyWhy beam beam coolingcooling??

WhyWhy beam beam coolingcooling??Improve beam quality• beam size, emittance• energy spread• intensity of beam, accumulation, stacking• lifetime of beam

Counteract degradation of beam qualitydue to interaction of beam particles with• other particles (intrabeam scattering)• rest-gas (internal targets)• non-ideal fields, resonances, instabilities

injection errors

StackingStacking by by coolingcooling

septum

x

y

StackingStacking by by coolingcooling 22

0 100 200 300 400 500 6000

50

100

150

200

curr

ent [

mA

]

time [sec]

ASTRID SR source:~200 mA accumulatedfrom many injectionsof ~5 mA

Fermilab antiprotonaccumulatorstacking for 1 hour

108/2 sec pbar at 8 GeV

injection stack tail core

shutter

Liouville formelt

konstant0/dtd

kræfter vekonservatifor Liouville gtetsligninKontinuite

etc.friktion somkræfter eHamiltonsk ikkeer ogpotential,et fra udledeskan der del, eHamiltonskeller vekonservati

dener /hvor ,/ /siger ligninger Hamiltons),,(nktion Hamiltonfu

impulser e tilhørendde og F.eks.nktionenLagrangefuer ),,(hvor ,/

,r koordinate ekonjugered såkaldte med faserumlt dimensiona 6 ipartikler som beskrivessystemet kan ker vekselvirikke epartiklern Hvis

faserum.lt dimensiona 6 i beskrivespartikler af bestående systemEt

=⇒=

⇒∂∂

−=

∂∂∂∂=+∂−∂=

−≡=

−≡∂∂=

∑

∑

ρρ

ρρ

i

i

ii

zyx

ii

pQ

dtd

Q

qHpHqQqHpLqptpqHH

,p,ppx,y,zUTtqqLqLp

p, qN

NN

&&

&

&

rr

PhasePhase spacespace and and LiouvilleLiouvilleLiouville: For hamiltonian systems, the phase space density

is constant (when measured along a trajectory)The phase space volume (emittance) is conserved

q

p

∫ = constantpdq

Quadrupole focusing

x’

x

Often the two transverse and the longitudinal degrees of freedomare decoupled

PhasePhase spacespace, , LiouvilleLiouville and and coolingcooling

Liouvilles theorem means that cooling is not possiblefor Hamiltonian systems, that is systems with forcesthat can be derived from potentials.In addition particles cannot be injected into alreadyfilled areas of Phase space.All you can do is to change the form of phase space.

However, with velocity-dependent forcesdrag, friction (dissipative) forces

electron, radiation, Laser, ionisation cooling

cooling is indeed possible!!

CoffeeCoffee, , creamcream, , LiouvilleLiouville and and StochasticStochastic coolingcooling

StochasticStochastic coolingcooling principleprinciple

q

p

q

p

macroscopicemittance

Maxwells demon

StochasticStochastic coolingcoolingLiouville: Cooling is not possible with electromagnetic forcesdeflecting the particles (continous fluid, og N=∞).When single particles can be observed, and a correspondingcorrection applied, cooling is possible!This is the secret of stochastic cooling!

pick-up kickerN=108, σ=5mmσ/√N=0.5μm

In reality W< ∞

StochasticStochastic coolingcooling

g

xn

<x>s

g(<x>s+ xn)

transversepick-up transverse

kicker

Cooling timeτ≈N/W

t

t

Ts=1/2W

W bandwidth

pulse atpick-up

pulse atkicker

WTN

TTNN S

S 2==

StochasticStochastic coolingcooling exerciseexercise1) Ask for 5 random numbers with <x>=0 and σ=12) Find actual <x> (in general <x>≠0)3) Subtract error in mean to restore mean to zero4) Calculate new σ5) Goto 1)6) Watch σ as function of time7) What is the cooling time?8) Include electronical noise!

0 5 10 15 200.01

0.1

1

σ

turns

N=20 N=5

CoolingCooling TimeTime

pick-up kicker pick-up

small goodmixing mixing

)(1 timecooling optimum

1)/(1gain optimum

)(2

11

υτ

υ

υτ

+Γ=

<+Γ=

⎥⎦⎤

⎢⎣⎡ +Γ−=

NW

g

gN

gWmixing

noise/signal-ratio

CoolingCooling time 2time 2

τ ∝ NDecrease gain as cooling proceedsGood mixing, Γ = 1, by designing storage ring soη=∂(ΔT/T)/∂(Δp/p) is large. However small mixing PU→KLarge bandwidth (W> GHz, Ns~10-3N)Weak dependence on energyZ dependence in υ

)(1 timecooling optimum

1)/(1 gain optimum

)(2

11

υτ

υ

υτ

+Γ=

<+Γ=

⎥⎦⎤

⎢⎣⎡ +Γ−=

NW

g

gN

gW

StochasticStochastic coolingcooling

Betatron cooling: 2 systems (hor. and vert.)dist. PU → kicker = odd number of λ/4

Momentum cooling:acc. gap instead of transverse kicker

(i) PU in high-dispersion region Δx/x=D Δp/p(ii) detect Δf/f=η Δp/p and correct Δp/p

Stochastic cooling facilities:ISR (1977), ICE, AA, AC, LEAR, AD @ CERNFermilabTARNCOSY, GSI

Stochastic Cooling

FNAL antiproton sourceFNAL antiproton source

Fermilab antiprotonaccumulatorstacking for 1 hour

108/2 sec pbar at 8 GeV

injection stack tail core

shutter

Δp/p (Δf/f)

StochasticStochastic CoolingCooling

AD at CERN

Δp/p (Δf/f)

Electron Electron coolingcoolingIons

e-gun (Te~0.2 eV) collector

~108/cm3

Laboratory frame )( frame particle Iv

Electron Electron CoolingCooling

Invented by Budker in Novosibirsk

NAP-M ringat INP, 1974,68 MeV p

magnets:solenoidstoroids

MSL in Stockholm

MSL electron cooler

Cryostat

Electron gun

Collector

Vacuum chamber

Interaction region

Magnet winding (NC)

Cryopump

Ion beam

Support frame NEG pump

Correction dipole*

Superconducting solenoid

Return yoke

ASTRID electron cooler

Electron Electron coolingcooling 22

Initially

ieeI

iI

eI

TvmvMT

vv

≡>>≡

≥

22

22

21

21

Finally

rmse

rmseI

rmsI

fe

fI

vMmv

Mmvv

TT

431

heating no

2 ≈=≡

=

I

IIIC

I

eePFrms

eI

vvvvL

mvenZF

vvfvv

=−=

=>

ˆ ˆ4

)()( )(

2

42π

δ

Electron Electron coolingcooling drag forcedrag force

Ie

C

eee

ePFrms

eI

vTmL

mZneF

TmvT

mvfvv

2/324

22/3

324

)2/exp(2

)( )(

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−⎟⎟⎠

⎞⎜⎜⎝

⎛=<

π

π

Electron Electron coolingcooling timetime

nLeZMm 1

42

2

ηγτ = { PFrms

eIe

PFrmseI

PFI

vvmT

vvv

)(

)( )(2/3

223

341

<⎟⎠⎞

⎜⎝⎛

>

π

π

Typically τ~tens of seconds ( Z=1)

FMv

dtdv

vII

I

=≡−1

1τ

Electron Electron coolingcooling

Electron cooling at AD

LEAR, ICE, AD @ CERN

CRYRING, CELSIUS

TSR, COSY, SIS, ESR

IUCF, Fermilab

TARN, ..

ASTRID

Mod. to simple Mod. to simple descriptiondescription1) Flattened distribution due to acceleration

↓⇒≈ //// 0 τT

2) )(0)(0 3−

⊥⊥⊥ ∝↓→≈→∞=≠ IvTBB ττ

In practice B=∞ only for distantcollisions

VirtuesVirtues of electron of electron coolingcoolingVersatile cooling techniqueLongitudinal and transverse coolingCooling times τ≈0.1-1secA/Q2

T// << 0.1 eVT⊥ ≈ 0.1 eVin addition: adiabatic expansion T // ∝ B

Laser Laser coolingcooling

V

γ

V + dV

a) b)

V + dV + dV’

c)

V + NdV

d)

Ions

Laser)cos1(' Doppler

:recoilIon

0 θ

λ

cvhvhv

chvhqMvr

−=

=== h

Kick from one photonabsorption-emission

1s2p

1s2s

5485Å(2.3 eV) τ=42ns

meV12recoil single :limit UltimateeV 0.2 :m 2in change

152m/ :m 2in s10½

Ø3mm)in s)(1mW spontaneoud(stimulate saturationAt meV12 :energyin Change

/ :momentumin Change

1-7

=

=⋅=Γ=

==Δ=Δ

=Δ

vrr

vpEhp λ

100 keV Li+

Laser Laser coolingcoolingF(v)

v

Laser Laser coolingcooling in in ASTRIDASTRID

Laser Laser coolingcooling

Virtues of laser cooling:

Laser cooling is fast

However:

Only effective for longitudinal cooling

Not versatile: Li+, Be+, Mg+, …

Radiation Radiation dampingdampingIn principle: any charged particlein practise: only electrons/positronssince τ≈E/(U0/T0)

RF

Vertical betatron coolingHorisontal: dispersion!Longitudinal: finite energy quanta

(for details,see lectures by L. Rivkin)

Ionisation Ionisation coolingcoolingSlowing down in matterFriction forceNot hadrons due to large inelastic cross sectionNot electrons due to short radiation length

Can only be used for in μ in μ-collider/ ν-factory

ν’s produced by decaying μ’sμ’s produced from decaying π’sπ’s produced by p’s on target

Since μ’s do not live forever (τ0=2.2 μs)cooling has to be fast (even when γ is large: τ= γτ0.

Also emittances are very large!

vF −∝

Ionisation Ionisation coolingcooling principleprincipleTransverse cooling:muons lose energy by dE/dx and longitudinal momentum isrestored by RF

To minimize heating from Coulomb scattering:Small β⊥ (high-field solenoids)Large LR (low-Z absorber): Liquid H2

Ionisation Ionisation energyenergy coolingcoolingIonisation energy cooling using a wedge and dispersion

R F

possiblepossible νν--factoryfactory at CERNat CERN

MICE at RALMICE at RAL

OtherOther coolingcooling methodsmethods

Stimulated radiation cooling

Radiative cooling

ConclusionsConclusions

high

N⋅10-8 s

low

high

all

Stochastic

low

10-10-2 s

any

medium0.01<ββ <0.1

ions

Electron

any

10-6 s

any

any

muons

Ionisation

low

10-4-10-5 s

any

any (butDoppler)

some ions

Laser

any

>10-3 s

any

very highγ>100

e-/e+

Radiation

Favouredbeam temperature

Coolingtime

Favouredbeam intensity

Favouredbeam velocity

Species