C. Dimopoulou

A. Dolinskii, T. Katayama, D. Möhl, F. Nolden, C. Peschke, M. Steck, L. Thorndahl

Simulations of stochastic cooling of antiprotons in the collector ring CR

TUIOB02 @ COOL'11Alushta, Ukraine, September 2011

Antiprotons3 GeV, 108 ions

Rare isotopes740 MeV/u, 109 ions

δp/p (rms) εh,v (rms)

π mm mrad

δp/p (rms) εh,v (rms)

π mm mrad

Before cooling 0.35 % 45 0.2 % 45

After cooling 0.05 % (*) 1.25 (*) 0.025 % 0.125

Phase space reduction 9x103 1x106

Cooling down time ≤ 9 s ≤ 1 s

Cycle time 10 s 1.5 s

Required performance of CR stochastic cooling Short bunch of hot secondary beam from production target into the CR After bunch rotation and adiabatic debunching the δp/p is low enough to apply stochastic cooling Fast 3D stochastic cooling required to profit from production rate of secondary beams

(*) 20% lower (if possible) for HESR as accumulator ring (instead of RESR)

C. Dimopoulou, COOL'11

Overview of the CR stochastic cooling systems

Systems in frequency band 1-2 GHz

Pickup Kicker pbars RIBs Method

PH KH hor. hor., final stage

difference PU

PV KV vert. vert., final stage

difference PU

PH+PV KH+KV long. long., final stage

Sum PU + notch filter

PP KH ----- hor. + long.,first stage

Palmer: difference PUat high D

PP KV ---- vert., first stage

difference PU

System in frequency band 2-4 GHz (future option)

P2-4 K2-4 long. -------- Sum PU + notch filter

Main issue for pbars: increase ratio

noise thermal

)Q ( signalSchottky 2

C. Dimopoulou, COOL'11

1925-2011

Principle of betatron cooling & basic ingredients

Phase advance PU-K ≈ 900

Coherent term= cooling force x undesired mixing (PUK)

Diffusion= heating from Schottky noise (desired mixing (KPU)) + from thermal noise

''rms'' theory (analytical model)( Fokker-Planck equation for )JNtJ ),(

High amplification needed, electronic gain ~ 10 7 (140 dB)

System gain g = PU response x Electronic gain x K response ~ 10-2

C. Dimopoulou, COOL'11

UMggBN

W

dt

d

22211

Phase advance PU-K ≈ 900

High amplification needed, electronic gain ~ 10 7 (140 dB)

Good cooling for overlapping Schottky bands i.e. M=1 and low ratio thermal noise/Schottky signal U

B and M depend in a contradictory way on the spread ΔT/T= - Δf/f ~ - ηring/pk Δp/p of the beam particles, they vary during momentum cooling

In reality: choose ηring/pk for a compromise between B and M

To cool all the particles within the initial momentum distribution B ≥ 0

Principle of betatron cooling & basic ingredients

C. Dimopoulou, COOL'11

UMggBN

W

dt

d

22211

,

EDDF

Et ns

ENtE ),(

Principle of momentum cooling with notch filter

Fokker-Planck equation (solved with CERN code)

The response of the notch filter

provides the cooling force,induces extra undesired mixing

Coherent term= cooling force x undesired mixing (PUK)

System gain G = PU response x Filter response x Electronic gain x K response

p

δpi π π

i ep

pm e

i

sin1

2H 0f/f 2

notch

well-separated Schottky bands M>1B ≥ 0 for increased undesired mixing

very small |η|≈1%i.e. ring almost @ γtr

C. Dimopoulou, COOL'11

Features and developments for the 1-2 GHz system

Optical notch filter (< 40 dB deep notches within 1-2 GHz )

PU/Kicker tank consists of 2 plates (up+down or left+right) with 64 electrodes/plate PH/KH=PV/KV rotated by 900

Plunging of PU electrodes i.e. moving closer to beam during cooling No plunging of KI electrodes

Slotline PU electrodes at 20-30 KCryogenic low-noise preamplifiers at 80 K (open option of preamplifiers in UHV at 20 K)Kickers at 300 K

Effective noise temperature at preamplifier input Teff =73 K

Beam

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1.0 1.2 1.4 1.6 1.8 2.00.6

0.7

0.8

0.9

1.0

1.1

1.2

f(GHz)0 10 20 30 40 50 60

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

ypu

(mm)

Longitudinal PU/K impedance, sensitivity, PU plunging

pk ZZ 4

S(f))()f(y)f,( c ySZZ pp

Relative measurements on prototype PU:

,2

rms bI

PZ p

p

,2

rms

kk P

UZ b

HFSS simulations, absolute values:

circuit convention:

)f((y))f(y)f,( c SSZZ kk 11.25)f( cpZ at yPU= ±60 mm

Simplify:

yslope1)( yS

75.37)f( cpZ at yPU= ±20 mm

S(f)

Plunging of PU electrodes:factor 1.8 in sensitivity (3.4 in Zp)from yPU=±60 mm ±20 mm

C. Dimopoulou, COOL'11

Input parameters & requirementsCR Circumference 3 GeV antiprotons

221.45 mβ=0.9712, γ=4.197, rev. frequency f0=1.315 MHz

Ring slip factor η, slip factor PU-K ηpk

Distance PU-K/circumference-0.011, -0.033 0.378

Beam intensityInitial rms momentum spread Initial rms emittance εh,v

108

3.5 10-3, Gaussian/parabolic 45 π mm mrad

System bandwith 1-2 GHz

Number of PU, K (longitudinal cooling)Number of PU, K (transverse cooling)

128, 128 64, 64

PU, Kicker impedance at midband 1.5 GHz

PU/K sensitivity S(y)=1+slope* yPU/K sensitivity vs. frequency S(f)

no plunging considered, PU electrodes at ± 60 mm11.25 Ohm, 45 Ohm slope= 24.5 m-1

Effective temperature for thermal noise 73 K

ideal, infinitely deep notch filter + 900 phase shifter

Total installed power at kickers (limited by funding, can be upgraded)

4.8 kW

Goal: Cool longitudinally from σp/p= 3.5 10-3 4 10-4 in 9 sSimultaneous transverse cooling from εh,v = 45 ≈ 1 π mm mrad

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Momentum cooling: Cooling force and diffusion G|| = 150 dB (3.2 107); t=10 s

t=0, 2.5, 5, 7.5 and 10 s

Coherent term: • linear notch filter response around Δp/p=0 cooling force• momentum acceptance of system (undesired mixing ≥ 0) > total initial Δp/p Cooling of all particles

Schottky noise dominates long. cooling time ~ N

Notch filter cuts thermal noise around all harmonics

C. Dimopoulou, COOL'11

It deforms the cooling force and suppresses Schottky noise within the distribution,cooling loop is stable (Nyquist plot)

),,(1

),G( ),G(

tEmS

EmEm

),,(),G( )()(n n),,( kp tEmBTFEmmZmZtEmS kp

-*

*

*

PV

20 dE

E

dE/d

d

df e),,(

E

i

EmtEmB

Feedback by the beam included:

Momentum cooling: Feedback by the beam

G|| = 150 dB (3.2 107); t=10 s

C. Dimopoulou, COOL'11

Optimization:For a given signal/noise ratio there is a gain so as to reach the desired σp/p in the desired time.Lower gain leads to lower σp/p but cooling takes longer.

Momentum cooling: Results

For ultimate σp/p : increase signal/noise by plunging the PU electrodes during cooling

C. Dimopoulou, COOL'11

Required installed power = 4 Pmax ( to account for signal fluctuations)

),( ),()(n ef22

1)(

2p

20s tEmEGmZtP

m Ep

Total cw power in bandwidth at kicker: Pmax = Ps(t=0)+ Pn

, decreases as σp/p shrinksSchottky

2||effn kT

4

1GWP filtered thermal

Betatron cooling rate: details

)(2cos)(

t

p

pxmtB pkpkc

)(

1)(

tpp

mtM

c

)(

1

)(

1

2

slope)( 0

20

2 tmZW

f

nfNe

TktU

m ppp

effB

)()(

|)sin(|)()( pk

tUtM

tBtgopt

Optimum gain

UMggBN

W

dt

d

2pk |)sin(|2

211

C. Dimopoulou, COOL'11

long

t

ini

ep

pt

p

p

)(

Simultaneous notch filter momentum cooling ONAnsatz from Fokker-Planck results at dB 150|| G

Interplay between betatron & momentum cooling

Beyond power limits...cw Pmax= 950 W !

For precise treatment, feedback by the beam must be included

dB 141 initial )( cmG

Betatron cooling: First results

Reached εh = 4 π mm mrad in 9 s

M dominates the heating at all t: M~10 U in principle, need long cooling at very low gain (plunging helps only at the end)

Initially:

, 2.1U ! 11M and grows...

C. Dimopoulou, COOL'11

0 2 4 6 8 100

5

10

15

20

25

30

35

40

45

h

p/p

rms p/

p (

10-4),

rms h

(m

m m

rad)

t(s)

Conclusions I

Pbar filter momentum cooling from σp/p= 3.5 10-3 4 10-4 in 9 s is possible in the 1-2 GHz band:

• with a gain around 150 dB (3.2 107),• required max. installed power ~ 2.6 kW (cw ~0.7 kW),• assuming unplunged PU electrodes (conservative case), plunging expected to help reaching lower σp/p,• feedback by the beam not negligible but loop stable.

The design η=-0.011 of CR is optimum for both 1-2 and 2-4 GHz bands (undesired mixing)

C. Dimopoulou, COOL'11

Conclusions II

Preliminary results show that betatron cooling is possible

• with separately optimized simultaneous filter momentum cooling (150 dB,~2.6 kW),

• down to εrms ~ 4 π mm mrad within 9 s,

• with an electronic gain at midband around 140 dB (107),

• with max. required installed power ~ 4 kW (cw ~1 kW) per plane h/v i.e. beyond the foreseen available power,

• assuming unplunged electrodes.

As expected, betatron cooling suffers from large desired mixing M (required by filter momentum cooling) dominating the diffusion at all t. Way out: slow-down momentum cooling in the beginning

C. Dimopoulou, COOL'11

Outlook

Include feedback by the beam into betatron cooling model

Time-optimization of momentum and betatron cooling together, distribution of available power accordingly, e.g.,

• Initially, slower filter cooling to help the betatron cooling, then inversely to reach ultimate emittances and momentum spread.• Apply initially time-of-flight and later notch filter momentum cooling, with simultaneous betatron cooling.

Include plunging of PU electrodes, expected to reduce diffusion by factors 4-9, especially transversally

Additional filter momentum cooling in the 2-4 GHz band, study handshake between 2 bands

C. Dimopoulou, COOL'11

Circumference 221.45 m

Max. magnetic rigidity 13 Tm

Antiprotons Rare Isotopes Isochronous Mode

 

Max. particle number 108 109 1-108  

Kinetic energy 3 GeV 740 MeV/u 790 MeV/u  

Velocity v 0.971 c 0.830 c 0.840 c  

Lorentz γ 4.20 1.79 1.84  

Transition γT 3.85 2.82 1.67 – 1.84  

Frequency slip factor η -0.011 0.186 0  

Betatron tunes Qx and Qy 4.28, 4.84 3.19, 3.71 2.23, 4.64  

Revolution frequency 1.315 MHz 1.124 MHz 1.137 MHz  

CR Parameters

,

EDDF

Et ns

),,(1

),G(Re )()(n n f e 2),( kp

20 tEmS

EmmZmZtEF

mkp

2

k20effn ),,(1

),G( )(n f Tk

2

1),(

tEmS

EmmZtED

mk

2

kp030

2s ),,(1

),G(

1 )()(n n

11 f e),(

tEmS

Em

mmZmZEtED

mkp

Fokker-Planck & Formulae I

cohEF 0f incohns EDDD 20f

2

1

ENtE ),(

[D.Möhl et al., Physics Reports 58 (1980)]

),G(Re )()(n n f e 2),( kp20 EmmZmZtEF

mkp

2k

20effn ),G( )(n f Tk

2

1),( EmmZtED

mk

2kp0

30

2s ),G(

1 )()(n n

11 f e),( Em

mmZmZEtED

mkp

Fokker-Planck & Formulae II

),(H ),( )( notch||

u EmieEmGEmG

0

)( 2 )(

p

EpxE pku