COMPARISON OF COOLING CHANNELS FOR A NEUTRINO FACTORY
Eun-San Kim and Moohyun Yoon, PAL, POSTECH, Pohang, Kyungbuk, Korea, 790-784
Abstract
Ionization cooling is a preferred method for transverse
cooling of a muon beam in a neutrino factory. We design three different cooling channels, a DFOFO channel, an
alternating solenoid channel and a Super-FOFO channel,and show that the transverse beam emittance through these
cooling channels is cooled down by a factor of 7.7, 6.5 and6, respectively. The cooling channel design has been carried out by the code ICOOL. The transmission efficiency,transverse beam emittance, growth of longitudinal bunch
length, and momentum spread are investigated. It is shownthat these three cooling channels provide adequate cooling
performances, satisfying the so-called Palmer, Johnson andKeil (PJK) parameter for a neutrino factory.

1INTRODUCTION

A challenge in the design of a neutrino factory is the cooling of a muon beam. When muons are produced from a target, they occupy a large phase-space area. A successful design of a neutrino factory therefore requires that the transverse emittance of a muon beam after the capture, phaserotation, and buncher channels be sufficiently reduced sothat the muon beam can be accelerated efficiently in downstream sections. Furthermore, fast cooling is required because of relatively short lifetime of a muon beam. The lifetime of a muon is given by   = 2:197  10 6 E =m
seconds where E is the energy and m  is the rest mass ofa muon. A cooling method that can meet such a requirement is an ionization cooling [1]. In ionization cooling,
particles pass through a material medium and lose energy
through ionization interactions. The losses are parallel tothe particle motion and therefore both transverse and longitudinal momenta are lost. Only longitudinal momentumis then restored by acceleration through rf cavities. Thissequence results in a reduction of the angular spread of abeam, and therefore it leads to a decrease in the transverse
beam emittance. However, the random process of multiple
Coulomb scattering in material medium leads to increasingof beam emittance. When an absorber material is placedin a strong focusing field, the heating due to the multiplescattering can be minimized.
The equation for the normalized transverse emittance  nis given by [2]dn=dzEs21 dE n
12 dz E + 3 2E m c2 L ; R?

(1)

where z is the path length, E  is the muon beam energy,= v=c, LR is the radiation length of the absorber material, ? is the betatron function, and E s is the characteristic scattering energy which is approximately 13.6 MeV.

The normalized emittance  n is related to the usual unnormalized emittance  by  n = ? , where is the usual relativistic factor. In Eq. (1), the first term describes the cooling and the second term represents the heating due to themultiple scattering. Since the heating is proportional to ?
and 1=LR , small ? (i.e., strong focusing ) and a material
with high LR (i.e., a low-Z absorber ) need to be considered
in order to maximize the cooling. We consider a focusing
by solenoids, that is, the focusing by longitudinal magnetic
fields Bz . It has an advantage of focusing equally in bothtransverse planes. Solenoids can focus the beam to yieldsmall ? at absorber that are placed inside the magnet.
In this paper, we consider three other cooling channels, aDFOFO channel, an alternating channel and a Super-FOFO
channel [3]. The code ICOOL is employed for the cooling calculation. Engineering constraint in the design of thecooling channels is also investigated by considering for realistic physical parameters. Results from these simulations
may provide guidelines for optimal cooling systems. A design study of all the components of a neutrino factory byPalmer, Johnson and Keil (PJK) [4] requires that the cooling system result in normalized transverse and longitudinal emittances of about 1.5 mm-rad and 30 mm, respectively. Our goal is thus focused to obtain cooling systems
that meet the requirement by the PJK design.

2CHARACTERISTICS OF THE
COOLING CHANNELS

Input Beam The initial beam distribution of(x; Px ; y; Py ; z and Pz ) are generated by random Gaussian.
A correlation between the longitudinal momentum and thetransverse amplitude of the initial beam distribution is necessary to properly match the beam into the rf system. Thisis the only non-Gaussian modification in our initial beam
distribution. Correlation between longitudinal momentum
and transverse amplitude is given by relation of

x2 + y 2 P 2 + P 2
Pz = Pzi + a[ i 2 i + xi 2 yi ]:bPzi(2)

Here a and b are coefficients that determine the intensity ofcorrelation. P xi ; Pyi and Pzi is initial horizontal, verticaland longitudinal momenta, respectively. In our simulation
0.1 and 0.16 are used for a and b, respectively. Then thecorrelation is sufficiently strong to push up the longitudinal momemtum of the initial beam by 10%, increasing themomentum spread as well. We have the same initial correlations between longitudinal momentum and transverse
amplitude in all designed cooling channels.

2.1

DFOFO cooling channel

As can be seen in the design of a FOFO cooling channel [5],particle losses in the transverse direction were mainly occurred at the midpoint of rf cavities where beta function
was maximum. This requires a large radius of the rf window to reduce number of particle losses but this leads to alarge power deposited on the rf window. In order to reduce
particle losses in the transverse direction, we have reconfigured the FOFO lattice in such a way that the period of
FOFO magnetic fields is doubled. One cell geometry of ourdesigned cooling channel is shown in Figure 1. The minimum values of magnetic field B z occur at the midpoint of
the rf cavities and absorbers. The lattice is periodic andsolenoidal magnetic field sinusoidally varies on axis with
1.8 m period and 3.8 T amplitude on the z -axis. The beta
function in the DFOFO channel is minimum at midpointof rf cavities and therefore large acceptance channels canbe obtained. We note that the beam size has a minimum at
the midpoint of the rf cavities so that particle losses at theposition of rf window can be reduced.
The designed DFOFO cooling system is composed ofthree stage channels that have different thickness of aluminum window on both sides of the LH; First stage channel has 28 cells with window thickness of 650 m, second stage channel has 25 cells with window thickness of
400 m, and third stage channel has 47 cells with windowthickness of 300 m. Major parameters of the DFOFO
cooling channel are shown in Table 1. For the 3.8 T DF0F0
the relevant values are B =7.7 T at the coil, J =80 A=mm 2 ,R=0.625 m, giving BJR = 335 MPa.
Cooling Performance The initial beam distribution
before entering the cooling channel is generated by 12 mmrad transverse and 12 mm longitudinal emittances. A beamwith a longitudinal momentum of P z =212 MeV/c with rms
momentum spread of 13.2% is used.
In the designed DFOFO cooling channel the transverseemittance in both x and y planes is found to be reduced
to 13% of its initial value, while the longitudinal emittance
grows by a factor of 2. The overall six-dimensional emittance is reduced to 3.6% of its initial value. The cooling channel shows the particle transmission of 72%. Table I also shows the final beam parameters of rms beam
size x , rms bunch length  z and rms longitudinal momentum spread. The initial and final beam parameters for theDFOFO channels are shown in Table 2.

2.2

Alternating Solenoid Cooling Channel

In this channel, a lattice of solenoids with alternating direction is used to achieve the required focusing. Figure 2
shows configuration of alternating solenoid cooling channel with a cell length of 2.2 m. The cooling channel with
201.25 MHz rf cavities and 73 cells has a maximum magnetic fields of 3.4 T on the longitudinal axis and 6 T on
solenoid with current density of 58 A=mm 2 . The field falls
to zero and alternates in direction at the center of the rf cav-ities. The maximum and minimum values of magnetic fieldBz occur at close to the midpoint of absorber and rf cavities, respectively. The maximum value of the beta function
occurs at the midpoint of the rf cavities. The minimum
value occurs at close to the midpoint of absorbers. Accordingly, particles are lost more than in the DFOFO coolingchannel. The reason is that particle losses are mainly generated at the position of rf window where beam size is large.
The parameters of alternating cooling channel are shownin Table 1. For the 3.4 T alternating solenoid channel,
the relevant values are B =6 T at the coil, J =58 A=mm 2 ,R=0.805, giving BJR = 280 MPa.
Cooling Performance The initial beam distribution isgenerated by 11 mm-rad transverse and 16 mm longitudinal emittances. A beam with a longitudinal momentum ofPz =214 MeV/c with rms momentum spread of 9% is used.
The transverse emittance in both the x and y planes isreduced to 15% of its initial value, while the longitudinalemittance grows by a factor of 2. The overall normalized
six-dimensional emittance is reduced to 13% of its initialvalue. Table 2 shows the initial and final beam parameters
for the alternating cooling channels.

2.3

Super-FOFO Cooling Channel

The designed Super-FOFO cooling system is composed ofidentical 61 cells. The length of each cell is 2.6 m. One cellconfiguration of our designed cooling channel is shown inFigure 3. The minimum values of magnetic field B z occur at the midpoint of absorbers. The magnetic field increases from 3.7 T on z-axis to 6.7 T at the coils. The
lattice has magnetic fields that has a dig in the positions ofrf cavities and so acceptance could be extended. The beta
function in the Super-FOFO channel is minimum at midpoints of absorbers. Major parameters of the Super-FOFO
cooling channel are shown in Table 1. Our designed cooling channel satisfies the BJR condition in the solenoids.
For solenoids for the strong focusing in the Super-FOFOchannels, the relevant values are B =6.7 T, J =86 A=mm 2 ,R=0.41 m, giving BJR=236 MP a. For solenoids forthe coupling fields in the Super-FOFO channels, the relevant values are B =4 T, J =65 A=mm 2 , R=0.83 m, giving BJR=216 MP a. These BJR values in the SuperFOFO channles show lower requirements than the DFOFO
channel (335 MP a), alternating channel (280 MP a) and
FOFO channel (310 MP a).
Cooling Performance The initial beam distribution isgenerated by 9 mm-rad transverse and 15 mm longitudinal emittances. A beam with a longitudinal momentumof Pz =212 MeV/c with rms momentum spread of 8.5% isused.
In the Super-FOFO cooling channel the transverse emittance in both the x and y planes is found to be reduced to
17% of its initial value, while the longitudinal emittance
grows by a factor of 1.6. The overall six-dimensional emittance is reduced to 17% of its initial value. The coolingchannel shows the particle transmission of 54%. Without
the correlation, beam transimission is reduced from 54 %to 40%. Table 2 also shows the initial and final beam parameters for the Super-FOFO channels.
Table 1: Parameters of designed cooling channels
DFOFO Alternating SFOFO
frf (MHz)250
201.25201.25
rf gradient
15MV/m15MV/m15MV/mcell length
1.8m
2.2m
2.6m
magnetic field
3.8 T
3.4 T
3.7 Tchannel length
180 m
160 m
158 m
min. beta func.
35 cm
20 cm
30 cmmax. beta func.
70 cm
110 cm67 cm
Al radius(cm)20/16/92518Be thick.(m)125125125
Be radius(cm)19/15/13
20174

REFERENCES[1] A.N. Skrinsky, V.V. Parkhomchuk: Sov.J.Nucl. Phys. 12(1981) 3.
[2] D. Neuffer: Particle Accelerators 14 (1983) 75.
[3] E.-S. Kim and Moohyun : Jpn. J. Appl. Phus. Vol 40 (2001)
401.
[4] R. Palmer, C. Johnson and E. Keil: BNL-66971, CERN
SL/99-070, (1999).
[5] N.Holtkamp, et al., Submitted to Phys. Rev. ST: Acceleratorsand Beams (2000)

r[cm]solenoid

solenoid

62.5cm36 cm

RF20cmLH

Table 2: The intial and final beam parameters of the coolingchannels

RFLH LH

LH

1.8m

3.6m

Z [m]

Transmission(%)
Transverse emit.
(mm rad)Longitudinal emit.
(mm)
6D emittance(mrad)3 10 8

DFOFO
72
12.2/1.67

Alternating51
10.2/1.58

SFOFO
54
9/1.57

11.7/23.3

16/30

15/24

253/9.14

197/27130/22

9.2/12

10.2/14.4

10.2/14

Figure 1: One cell of the DFOFO cooling channel.

r[cm]solenoidsolenoid

80.5cm45 cm

Bunch length(cm)
Beam size(cm)Momen. spread(%)

RF20cm

4.4/1.6
13.2/19.7

3.9/1.1
9/25

LH

4.1/1.5
8.5/18.6

RFLH LH

LH

2.2m

4.4m

Z [m]

Figure 2: One cell of the alternating cooling channel.

3 CONCLUSION
In this paper, we have shown three ionization cooling channels based on different design principles. A DFOFO, analternating and a Super-FOFO cooling channels are designed and their cooling performances are calculated withthe code ICOOL. It is shown that cooling parameters ofthe PJK design for the neutrino factory can be achievedby these cooling channels. The purpose of our study is toobtain an efficient transverse cooling channels that satisfy
engineering constraint. Detailed simulation studies havebeen performed to obtain optimal parameters for each cooling channel. These studies will be valuable in selecting
the best design based on cooling performance, engineering
constraint and cost. The DFOFO cooling channel shows
relatively good particle transmission. The Super-FOFO
channel shows lower BJR values compared to the otherchannels.

r[cm]solenoidsolenoid

83cm
solenoidsolenoidsolenoid

41cm

RF20cm
LHRF
LH LH

LH

2.6m

5.2mZ [m]Figure 3: One cell of the Super-FOFO cooling channel.