Dynamics of transverse and longitudinal ionization cooling of muon beams
Chun-xi Wang and Kwang-Je Kimy
Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439
yand University of Chicago, 5270 South Ellis Avenue, Chicago, Illinois 60637
Abstract
Ionization cooling is the best-known cooling mechanismthat may reduce the muon beam emittance, before most of
the muons decay, to a useful level for the envisioned muon
colliders and neutrino factories. We review progress in understanding the linear beam dynamics of both transverseand longitudinal cooling.

1INTRODUCTION

In order to reduce the transverse and longitudinal emittances of a muon beam for neutrino factories and muon
colliders, ionization cooling channels are being developed[1, 2, 3]. For transverse cooling, promising designs consistof continuous straight solenoids to provide strong transverse focusing, liquid hydrogen absorbers at minimum
beta locations to provide ionization energy loss, and lowfrequency rf accelerating field between the absorbers toprovide longitudinal acceleration and focusing. For longitudinal cooling, various emittance exchange schemes havebeen proposed.
To understand the basic beam dynamics of ionization
cooling and to establish a theoretical framework for the design of cooling channels, second-order beam-moment andenvelope equations in solenoidal ionization cooling channels are being studied. We adopt this approach because(a) the second moments of beam phase-space distribution
contain most of the important beam properties such as rms
beam size and angular divergence, and (b) the evolution
equations of second moments close on themselves for linear dynamics that often dominate beam evolution.
Beam-moment equations have been developed in severalpapers for ionization cooling over the past several years[4, 5, 6]. We further developed the beam-moment study
emphasizing analytical solutions. There are 10 independent second moments for the two transverse degrees of
freedom and 21 for the transverse and longitudinal combined. In general, it is nontrivial to analytically solve sucha large number of moment equations. Despite the difficulties, significant progress has been made recently. The first
analytical solution was found for the transverse cooling ofa cylindrical symmetric beam in a straight solenoidal cooling channel [7]. Then transverse cooling of a general beam
was solved [8]. To achieve these, it is important to choose
the right parameterization to characterize those moments so This work was supported in part by the U.S. Department of Energy,
Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38;
and by grants from the Illinois Board of Higher Education; the Illinois Department of Commerce and Community Affairs; and the National ScienceFoundation.

that it provides convenient descriptions of beam properties
and evolution. Symmetries and associated invariants play akey role in the choice of parameters [9].
Lately our attention has been focused on the dynamics
of longitudinal cooling in a bent-solenoid channel. Although the final solution has not been worked out, progresshas been made, especially on the dispersion in a bentsolenoid channel that is desired for exchanging the longitudinal emittance to the transverse emittance for cooling. In
this short report, we briefly outline the existing theory andprogress underway. Readers are encouraged to check out
our earlier review [10] as well as more comprehensive andupdated presentations.

2 TRANSVERSE COOLING DYNAMICS
The magnetic field in a solenoid is completely determinedby the on-axis field strength B s()
1dB (s)B(s; x) = B(s)es 2 ds x + nonlinear terms: (1)Here es is the unit vector in the s-direction and x is thetransverse displacement. Because of the longitudinal magnetic field, particles with transverse momentum will gyratearound the axis. Thus a particle’s motion in the lab frame isfully coupled between the two transverse degrees of freedom and looks rather complicated. Fortunately, beam dynamics is dramatically simplified in the Larmor rotating
frame that rotating at 1/2 the cyclotron frequency about theaxis. In the Larmor frame, the two transverse motions aredecoupled and each degree of freedom can be treated withstandard Courant-Snyder theory. Because of the cylindrical symmetry of a solenoidal channel, beam dynamics inthe two subspaces are identical. Thus only one set of Twiss
parameters ( , , ) is needed to describe the lattice properties, and they satisfy the familiar equations
20; 0 2;; (2)^^^^ = 2^ ^ = ^ ^ ^ = 1+ ^^with periodic boundary conditions. Here the solenoid lattice is characterized by  s
eB s = p s 2 , where ps isthe longitudinal momentum.
Due to the cylindrical symmetry, there are four linearly
independent quadratic invariants in a solenoidal channel:
xPy yPx plus
the canonical angular momentum L z
three Courant-Snyder-type invariants. Expressed in termsof phase-space coordinates fx, Px , y, Py g in the Larmor
frame, the Courant-Snyder invarants readIx
s x2
s xPx
s Px2 ;
(3)( )=( ( ) 2 )=~ ~ ~ ~= ^( )~ + 2^( )~ ~ + ^( ) ~Iy = ^(s)~y2 + 2^(s)~yP~y + ^(s)P~y2 ;(4)~~x~P + y~Px ^ ~ ~Ixy = ^(s) x~ y~ + 2^(s) y2 + (s)Px Py : (5)= (2)21p

(X~ )4De

y Ix +x Iy 2xy Ixy LLz
22(x y 2xy L =4); (6)and the 4D emittance is4D  det  = (x y 2xy L2=4)2 :(7)Here the four parameters L,  x , y , xy are the beam angular momentum and emittances that can be written asL  hLz i = hx~P~y ihy~P~x i;q1
x  hx~2 ihP~x2 i hx~P~x i2 = hIx i;
2y (8)(9)

hy~2 ihP~y2 i hy~P~y i2 = 12 hIy i;qsxy ~(10)~hx~y~ihP~x P~y i h x~Py +2 y~Px i2 = 12 hIxy i: (11)For an empty solenoidal channel, these four beam parameters are conserved quantities. To respect the cylindricalsymmetry, it is often convenient to use  sx y =and ax y = , the rotationally symmetric and asymmetric parts of the transverse emittances.
When absorbers and rf fields are applied, ionization
cooling takes place. Assuming the reference particle is kept
at a constant momentum p s , based on moment equations,the four beam parameters evolve as [7, 8]=(=( + )2)2d s
1 ^ s=Lds L^ 1da=  a ; ddsxy =  xy :ds









^

+ 0 ; (12)(13)Here, the ionization dumping is characterized by(s) =1 dE

ps v ds loss

= p1s dp
dsloss;(14)

where dE=dsjloss is a positive quantity representing the energy loss per unit length in the absorber material, and v isthe muon’s speed. The multiple-scattering heating is characterized by(s) =

13:6 MeV 2 1 ;ps v

LRad(15)

where LRad is the radiation length of the material. The
beam evolution equations have been solved analytically.
Thus the linear dynamics of transverse cooling is solved
analytically.
The transverse cooling behavior in a periodic coolingchannel can be summarized as: 1) incoming beam quickly
matches into a periodic cooling channel via filamentationand particle loss; 2) the asymmetric part of the matchedbeam will be cooled away exponentially without heating;3) the symmetric emittance and angular momentum will becooled by ionization energy loss while heated by multiple
scattering; and 4) the final equilibrium state is a round beamwith only symmetric emittance determined by the balanceof cooling and heating (may have net angular momentum
depending on channel design). The effects of longitudinal dynamics on the transverse cooling have not been addressed yet. Figure 1 compares the analytical calculationand tracking simulations (including all effects) for a periodic solenoidal cooling channel.
9000

normalized transverse emittance

The last invariant is unique to the cylindrically symmetricsolenoidal channels.
Using these four linearly independent quadratic invariants, the equilibrium Gaussian distribution can be written
asDPGeant
ICOOL
Theory

8000
7000
6000
5000
4000
3000
2000
1000
02040

6080

meter

100

120

140

Figure 1: Transverse cooling calculated analytically, withICOOL, and with DPGeant. (Simulation courtesy of P. Lebrun; reproduced with permission from APS, PRL, 84.)3 FORMULAS FOR BETA FUNCTION
AND ORBIT STABILITY
A major task in transverse cooling channel design is togenerate a focusing lattice that has small beta function
and large momentum acceptance [11] using achievablesolenoid fields. Since continuous focusing is necessary tocontain the divergent muon beam, the solenoid magnets
cannot be treated as a few simple lenses. To facilitate lattice design, analytical formulas for beta function and orbit
stability are derived in ref. [12]. The lattice properties arewithdetermined by the Hills equation x 00 K s x
K s  s 2 . For a periodic solenoidal channel, the onaxis field B s varies continuously with period L. It is natural to use the Fourier coefficients f# n g of the normalized
L 2 K L & to characfocusing strength function # & s
terize the solenoid field. Here & L is the normalized
position. The beta function can be calculated with+ () =0()= ( )()( )==( )#p " X
1 <[# ei2ns=L ]L sin(#0 )n(s) =  p# sin  1 ++ :n2 #00n=1(16)Here  is the one-period phase advance that can be calcu
= . And is the trace of the one-period
lated viatransfer matrix that can be calculated withcos =  2

p 1 j# j2
 sin #0  Xn#0 ) +2p#0 n=1 #0 n2 +    :p

 = 2 cos((17)
The orbit stability can be determined by the well-known
criteria j j < . Higher-order expressions of and are2

available in ref. [12]. Using these formulas one can quickly
estimate the basic properties of a solenoidal channel fromthe Fourier coefficients of its focusing function. Furthermore, insight can be gained from these analytical expressions. For example, the form of Eq. (16) already suggests afew general strategies for obtaining a small s : 1) shorter
period; 2) stronger field (resulting in larger # 0 and higherorder passband); 3) larger phase-advance term
; and 4)
larger harmonics to cancel the unity term within the brackets of Eq. (16).

()
sin4

LONGITUDINAL COOLINGDYNAMICSLongitudinal cooling is much harder to achieve. The ionization process itself does not effectively cool the beam
momentum spread because the energy-loss rate is not sensitive to beam momentum except for very low-energy muons.
Despite numerous proposals and simulation studies, there
are no solid demonstration of a working scheme. A promising option is the “emittance exchange” scheme: introducing dispersion to spatially separate muons of different energies and then using wedge absorbers to discriminatively
cool them. Our present focus is to upgrade an effective
transverse cooling channel such that both longitudinal andtransverse degrees of freedom can be cooled via emittanceexchange.
To generate dispersion in a solenoid channel, thestraightforward approach is to superimpose dipole fields
with the solenoid field and make the solenoids bend along
the curved reference orbit determined by the dipole field.
Neither the solenoid nor the dipole field can be treated aspiecewise constant blocks, and the lattice consists of rather
complicated combined function magnets. Since the mainsolenoid field continuously rotates the beam and tends tomake the beam rotationally symmetric, it may be advantageous to have symmetric focusing. To achieve this, gradient dipoles (with field index n=1/2) could be used. 1In terms of the Hamiltonian, the bent-solenoid model under investigation can be written as

1 p2 + p2  + 1 (s)2 x2 + y2 + (s)Lz (18)2 x y 2xÆx2
1 k(s) x2 y2 + 1 Æ2+v(s)z2 ;+(s) 2(s)2 22

H =where fz; Æ g are the canonical variables for the longitudinal phase space,  s is the bending radius, k s is thequadrupole strength, and s is the longitudinal focusing strength. The first line corresponds to the straightsolenoidal channel that has been solved for the transverse
cooling. For a gradient dipole with symmetric focusing,= s 2 k s k s , i.e., the quadrupole component is=  s 2. The total
tied to the bending radius as k s

()

1 ()

( )= ( )v( )()()=12()

1 R. Palmer first suggested the possibility of replacing some of the
weak solenoids of a transverse cooling channel with gradient dipoles tokeep the required focusing while introducing the dipole field.

( ) = (s) 2 +1=2(s)2. Thefocusing strength becomes K s
Hamiltonian reduces to
1 2 2 12
22 px + py + 2 K (s) x + y + (s)Lz (19)xÆ 1  2+
Æ +v(s)z 2 :(s) 2~ x, D~ y ) of thisIn the Larmor frame, the dispersions ( DH =system have been worked out to satisfy= cos[(s()s)] ;(20)
sin[(s)] ;(21)
D~ y00 + K (s)D~ y =(s)Rss)ds is the rotating angle of the Larmor
where (s) = 0 (D~ x00 + K (s)D~ xframe. These equations lay the foundation for the dispersion design of a symmetrically focused bent-solenoid channel. Note that the r.h.s. driving term is not independent of
the focusing strength K s . This makes it difficult to finda dispersion solution. Nonetheless, example solutions havebeen demonstrated.
Using dispersion functions, the transverse and longitudinal motions can be decoupled. This paves the way to analytically solve the beam dynamics. Moment equations forthe bent-solenoid channel have also been developed. Effort is being made to reduce the 21 moment equations toa much smaller number of envelope equations. Hopefully,
analytical solutions for the longitudinal cooling can be obtained in the near future. There is still much to learn about
longitudinal cooling.

()

5REFERENCES[1] C. Ankenbradt et al., Phys. Rev. ST Accel. Beams 2, 081001(1999).
[2] “A Feasibility Study of a Neutrino Source Based on a Muon
Storage Ring,” edited by N. Holtkamp and D. Finley, 2000.
[3] D. Neuffer, “+ –  COLLIDERS,” CERN 99-12 (1999).
[4] C.M. Celata and A.M. Sessler, Proceedings of the 1998 European Particle Accelerator Conference, 1058 (1998).
[5] B.A. Shadwick and J.S. Wurtele, Proceedings of the 1999
Particle Accelerator Conference, 1716 (1999).
[6] G. Penn and J.S. Wurtele, PRL 85(4) 764, 2000.
[7] Kwang-Je Kim and Chun-xi Wang, PRL 85(4) 760, 2000.
[8] Chun-xi Wang and Kwang-Je Kim, “Transverse ionization
cooling of unmatched beam in solenoidal focusing channels,” to be published.
[9] Chun-xi Wang and Kwang-Je Kim, “Beam parameterizationand invariants in a periodic solenoidal channel,” to be published.
[10] Kwang-Je Kim and Chun-xi Wang, “Progress in the Linear Beam Dynamics Study of Ionization Cooling Channel,”
Proceeding of Neutrino Factory Conference 2000.
[11] V. Balbekov, MUCOOL Note 98 (2000).
[12] Chun-xi Wang and Kwang-Je Kim, “Recursive solution forbeam dynamics of periodic focusing channels,” PRE 64(5),2001.