The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

Study of Stochastic Cooling in the Accumulator Cooler RingatMUSES ProjectNaohito INABE, Kouichi YOSHIDA, Masanori WAKASUGI and Takeshi KATAYAMA*The Institute of Physical and Chemical Research (RIKEN)
Wako-shi, Saitama 351-01, Japan,*Center for Nuclear Study, Graduate School of Science, Univ. of Tokyo,Tanashi, Tokyo 188, Japan

Abstract
We studied availability of a stochastic coolingfor a cooling accumulation of RI beam in the A CR.
In the ACR fast cooling is essential for an RI beamwith large emittance and momentum spreadbecause the RI has an intrinsic life time. In
general the stochastic cooling is useful to obtain afast cooling time in a region where emittance andmomentum spread are large.
From calculations
under some conditions, we could get a very shortcooling time ( < 0.1 s) and confirm that thestochastic cooling is available with cooling of theRI beam in the ACR.

1.

Introduction

conditions.
In this paper we will mention
calculations and results. Availabilityofstochastic cooling will be also discussed.
2.

Transversal cooling

The transversal cooling time of the stochastic
cooling is given by[4],whereW ; band-width of a feed back system
N ; total number of particles
LlJ..L ; phase advance between pickup andkickergV j3p~k, g is gain of a feed back systemand ~P and ~k is ~ values of pickup andkicker
M+U, M is mixing factor and U is noiseF;factor given, (noise of system) divided(beam noise per Schottoky band)
Using parameters of a beam and a system, g isgiven
g1 ;One of purposes for the accumulator cooler ring[1] (ACR) at the MUSES project[2] is cooling of anion beam including a radio-isotope (RI) beam
produced by projectile-fragmentation in the BigRIPS. Due to production mechanism the RI beam
has large emittance (Ex ; horizontal, Ey; vertical)
and momentum spread (Llp/p). Moreover, due tomulti.:.tum injection, Ex of the beam injected into theACR is very large. In the ACR initial values of Ex,
Ey and Llp/p are 125 1t mm*mrad, 10 1t mm*mradand ±0.15 %. Requirement of cooling in the ACR isto have a very. fast cooling time for the RI beamwith large Ex, Ey and Llp/p because the RI has an
intrinsic life time.
Stochastic cooling[3] is one of useful methods
for cooling of the ion beam. In particular, a fastcooling time is obtained by the stochastic coolingfor in a region of large £x, Ey and Llp/p as theinitial beam in the ACR. On the other hand, in aregion of small Ex, Ey and Llp/p, the cooling time ofthe stochastic cooling is longer than that ofelectron cooling. In this way the stochastic coolingwill be used as pre-cooling that reduces inftiallarge £x, Ey and Llp/p to about 0.1 times. After precooling the electron cooling will be used to makefinal small values of Ex, Ey and Llp/p.
We calculated transverse and longitudinalcooling times of the stochastic. cooling in several(2)

whereq;
f0 ;
zp ;

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charge of beam
revolution frequency
coupling impedance between beamand pickup
fk;
efficiency of kicker
Gamp;
gain of amplifier~;
Lorentz parameter of beam
E;energy of beam
From (1) and (2), we can get an optimum gain(Gamp_opt) to get a minimum cooling time as shown in(3)sinLlJ..Lw ~E_ tGamp o p--- .
(M+U) N (qe ffoZpfk(3)

The 11 thSymposium on Accelerator Science and Technology, Harima Science Garden City, 1.997

Output power of an amplifier for Gamp_opt is given---------+----------i--------···i···········{···········{···········{······· .

10~

ll;

--------+---------:-----------j----------·j-----------j------- --j----------

1

where, k is Boatsmann constant and T and Tn aretemperatures of the amplifier and a pickup system
Using eq. (4) weincluding a pre-amplifier.
calculated P's for 11 Be4 + (400 MeV /u), 39Ca20+ (300MeV /u), 132Sn50+ (200 MeV /u) and 200Pb80+ (100MeV /u) with Ap/p= ±0.15 % and Ex=100 1tmm*mrad. In the calculation we assumed Tn=20 K,
W=2 GHz and Zp=lOO n. The other parameters ofthe ACR are shown in ref [1]. Results are shown inFig. 1 in term of dependence of number of particles.
As shown in Fig. 1, the obtained P's are very large
especially for small number of particles as RI
beam. From the result, we studied the coolingtimes under limited power of the amplifier (Plim).

~j.........

0.1

1"1 ......... 1"; ......... 1": .........1": ......1:........... 1": ...... ..

;V,: ~Ffl~! ]:11ee4+

0.00011000

10 4l:

.:

10 6

10 5

10 7

10 8:!
1010

10 9

NI.IT'IberofPartk:lesFig. 2 Transversal cooling time.

3. Longitudinal cooling

200 Pb80+ :: '~j=+~Llll:~10---------+---------+----------f---:3--~-------·f---------+--------

1 o4......... ) ........... ......... ) ...........\ .....

51000~,: ----------~--------~i!~iii!
······----~-----------~-----------l---·····--·~---·······-~------~: ········

100.................................

We also studied longitudinal cooling time by anotch filter method[5] which cooling is done in a
Similar to theregion of dispersion free.
transversal cooling, the cooling time of thelongitudinal cooling is limited by output power ofthe amplifier. In the power-limited system thecooling time is given by the sam.e equation as eq.
(5). For the longitudinal cooling 'topt is given.l ...........l..................... J~_:_'topt=(N )(1 + W 2 h+u(o))
12W~JW

10~~~~~~~~~~~~~~
1010
10 910 810 7
10 610 5104
1000
Number of Particles

where W c is average frequency of the amplifierand U(O) is give byFig. 1 Output power of an amplifier.
In a power-limited system the cooling time isu(o) =given_1_ = 1 Gamp_lim (2 _ Gamp_lim)
Gamp_opt't!im 'topt Gamp_opt(6)

w {_fu__§_) ur(1 + 12W~
KWc AEo2}-12(7)where, AE is spread of energy and(5)where, subscript "lim" means the limited systemand 'topt is a minimum cooling time obtained fromeqs. (1), (2) and (3). In eq. (5) Gopt_lim has a value
obtained under the condition P=Plim in eq. (4).
Under the assumption of Plim = 10 kW wecalculated the cooling times for several beams
with several number of particles. In Fig. 2 theresults are shown. As shown in Fig. 2, theobtained cooling time is less than 0.1 s for particle
number of 103 -106 as RI beam. In particular for
beam with a high charge the cooling time is very
fast.

'Y )
1(-11y __1j(Ytl 1+ywhere y is Lorentz parameter and 'Yt is transition y.
Gamp_opt and P for the longitudinal cooling are givenby(8)1Gamp opt =.L.A__1_(1+_w_)-l Eo
2 (qef NW c 12Wc IKZpf~ 1+u{0}-P=lk(T+Tn)WGirup op{1+-1-)
u(o)
3(9)

where A is a mass number of the beam. The otherparameters which don't explain in eqs. (6) - (9) arethe same as the transverse cooling.

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We calculated particle number dependence ofthe cooling time. Kinds of particles and conditions
are the same as those used in the transversalcooling. In Fig. 3 results are shown. For number ofparticles < 107 the obtained cooling times are lessthan 0.1 s. In particular the cooling time ofparticle with high charge is very fast. Thesetendencies are similar to the transversal cooling.

'·',~~~,, ::~~~~;~~:-~L~:~
50

100150200

lp(O)Fig. 5 ZP dependence of cooling time. Here,
number of particles is lOs.

4. Availability of stochastic coolingNumber of Particles

Fig. 3 Longitudinal cooling time.
We also calculated cooling times for several
values of temperatures of a pickup system (Tn) andcoupling impedances (Zp)· Fig. 4 and 5 show Tn andZP dependences. In both calculations we assumed
number of particles lOs. For calculation of Tndependence we assumed ZP = 100 Q and for that of~dependence, Tn = 20 K. As shown in Fig. 4 the Tn
dependence is weak. Cooling times of highly
charged beams are shorter than 0.01 s even in 100
K. It is due to good signal-to-noise ratio because
voltage induced by the highly charged beam at
the pickup is much larger than that by noise. As
shown in Fig. 5 ZP dependence is strong especiallyfor region below 50 Q. To obtain 't < 0.1 s ZP isrequired zp > 50 Q.

l0.1'==J~;~~~~~~~~=-:~~~~:;o=·=~·=·~~

f'"' ·-~~~=-+~~-~-r~~.:.r==~:'''l----jo.oo 1 o'---'-~..__,_zo~~~4-'--o~~-'6Lo~~_Lao~~_J1oo
Temperature (K]Fig. 4 Tn dependence of cooling time. Here,
number of particles is lOs.

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From calculations in sec. 2 and 3 we obtained
fast cooling times in almost whole region. This
indicates that the stochastic cooling is available
with the cooling of the RI beam in the ACR.
One of the important things to construct asystem of the stochastic cooling in the ACR is tomake a pickup system with large coupling
impedance. The large coupling impedance is also
important to reduce a needed output gain of theamplifier because a voltage from pickup itself islarge. On the other hand it is not necessary for thepickup system to be very low temperature. This
makes construction of the system to be easy.
On the base of the calculation we are designing
the pickup system with high coupling impedance.
References[1] K. Ohtomo et al., Proc. of 17th PAC,
Vancouver, May 1997 to be published.
[2] T. Katayama et al., Proc. of STORI 96,
September 1996, Bernkastel, Germany and tobe published Nuclear Physics A.
[3] S. van der Meer, CERN report,
CERN /ISR-PO /72-31(1978).
[4] D. A. Goldberg et al., LBL report, LBL25924 (1989).
[5] T. Katayama et al., Particle Accelerators
21 (1987) 99.