Beam Cooling - Aarhus Universitet · Beam Cooling • Introduction to cooling, temperature, phase...
Transcript of Beam Cooling - Aarhus Universitet · Beam Cooling • Introduction to cooling, temperature, phase...
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