Ionization Cooling of Muons in Linear RF System G. E. Penn and J. S. Wurtele, Department of Physics, UC Berkeley and Center for Beam Physics, LBNL,Berkeley CA 94720 USAAbstract
Cooling of muon beams is a crucial component of either amuon collider or a neutrino factory. Design studies of cooling channels predominantly use particle tracking codes.
Alternatively, one may use moment equations [1], which
incorporate the basic aspects of ionization cooling: energy
loss and scattering in material, acceleration by radio frequency (rf) cavities, and focussing in solenoid magnets.
The extension of this model to include the impact of engineering constraints (apertures and windows) and beam
structure on cooling channel is discussed. The predictionsof the moment analysis are compared with particle tracking
simulations.

cles within a six-dimensional phase space volume) is not
as good, but is still encouraging.
Ionization cooling [8] is the method of choice for compressing phase space for muon beams. It has the advantagethat it can cool on a timescale fast compared to the muon
lifetime. One difficulty with ionization cooling is that, withtypical parameters, longitudinal emittance tends to growduring transverse cooling because of the unfavorable slopeof the particle energy loss as a function of particle momentum. Thus some form of longitudinal cooling using dispersive regions is desirable. One option is to pass the beamthrough a helical wiggler channel, which results in a continuous dispersion that can be combined with wedge-shaped
absorbers to cut down the energy spread of the beam, at theexpense of increasing transverse emittance.

1 INTRODUCTION
Interest in developing intense muon beams has been growing in recent years, both for a muon collider[2] and for aneutrino factory[3, 4]. The scenario envisioned for producing muon beams has pions produced through the bombardment of a target by energetic protons. The pions thendecay into muons This is an expensive process that results
in a diffuse muon beam with a large energy spread. Themuons are captured through a combination of drift and induction linac or low-frequency rf longitudinal phase rotation sections. The beam is then bunched into rf buckets andinjected into the cooling channel. The beam has a low sixdimensional phase space density—few muons are in the acceptance of the downstream system. Without cooling, thelow number of useful muons per incident proton would result in a highly inefficient system and would require unrealistic proton driver powers.
The cooling channel is, typically, a series of solenoidal
magnets interspersed with absorber and rf sections. Suchcooling channels have novel designs and considerations,and are currently simulated with single-particle tracking
codes such as GEANT[5] and ICOOL[6]. We report thedevelopment [1] of a model, similar to the Courant-Snyder
formalism [7] for quadrupole focusing, that can describe
muon cooling using the beam moments. In principal thisshould straightforwardly allow for a much faster design
process. In practice, however, some care must be taken
to allow for beam scraping at apertures, higher-order correlations between transverse amplitude and longitudinal velocity, and the fact that the longitudinal phase space fills
up the rf bucket (and, therefore, is not well-described by aGaussian). Results give very good agreement for the transverse emittance. The level of agreement between the actual
figure of merit (which is not emittance, but rather parti Work supported by the USDOE Division of High-Energy Physics

22.1

FORMALISM

Single-Particle Motion

The magnetic field inside of a cylindrically symmetric~ = r  [A (r; z )^e ], where to lowsolenoid is given by B
est order in radius A  ' rBz (r = 0; z )=2. The equations of motion in the Larmor frame, neglecting stochas1 dP00z 0
2
tic effects, are XR+ qEvP XR +  XR = YR P ds and2z 0
XR P1 dP
YR00 + qEvP YR +  YR =ds ; where XR =x cos ' y sin ', YR = x sin ' + y cos ', and '0 = qPAz r 'qB(z) , Ez is the rf electric field, and dP=ds is the rate
2Pzof momentum loss in absorbers. The right hand side terms
are sources of canonical angular momentum, Lcanon =xPy yPx + qrA ; and do not appear in quadrupole focussing systems. The solutions are parametrized
by the bep
tatron function and phase as X R = A1cos(1 ),ppand YR = A2p cos( 2 ); in the absence of rf orabsorbers, the transverse amplitudes A1 and A2 are constant and correspond to the Courant-Snyder invariants. Thebetatron function equation is

2 p p ( p )2 + 4 p2 2 4 = 0;
00

0(1)

where 0 = 1= p and 2 (z ) is the linearized focussing
term. The angular momentum Lcanon ' Pz A1A2 sin(21). The evolutionof the longitudinal
momentum is con
sistent with P z2 1 + (x0 )2 + (y0 )2 = P 2.

2.2

Transverse Moment Equations

For a cylindrically symmetric beam, the distribution shouldin general be a function of the angular momentum L canon
and the combined amplitude A 21 + A22 . The equations ofmotion for the lowest-order beam moments is obtained including the effect of multiple scatter as given by S Fraction in Transverse Aperture Cut

Transverse EmittanceεN (π m rad)

0.014
0.012

ICOOL Simulation

0.010

Moments Equations

0.008
0.006
0.004
0.002
0.000
025

50

75

100

1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0125Figure 1: Comparison of the moment code and the ICOOLparticle tracking code. Both incorporate engineering constraints such as scattering in the finite thickness rf and absorber windows and apertures.

(d=ds)hx 2i ' (13:6 MeV=P v)2 =LR . This expression
0makes use of a Gaussian fit to the Moliere model of multiple scatter.
In the limit where transverse fields are linear with radiusand coupling to longitudinal motion is weak:

PS
1 dP+
N2mc
Pz ds2SqEz= 2 + v P  2Pmc
z z
NmcqB
 (  L)Pz NPzS=+ 2(  L)  2Pmc
N1dP L P SL =
(2);
Pz ds
N 2mcwhere 2mcLN = hLcanon i. The averaged equation for Pz?

?

?

0?

?

?

0?

?

?

?

?

0?

1.0

εN/εCUT

1.5

2.0

Phase Space Cuts
trans < 15 mm, long < 150 mm
0.70?

0?

0.5 cooling

Figure 2: Fraction of a beam within a phase space cut plotted as a function of the ratio of the beam emittance to thecut amplitude. The beam is assumed to have a Gaussiantransverse phase space distribution. The initial and finalvalues for a typical cooling channel is shown. Simulatedperformance is about half the number shown due to particles spilling out of the longitudinal rf bucket and scraping.

Population in 6D cuts

0before0.0Z (m)N =after cooling

0.60
0.50
0.40
0.30
0.20

ICOOL Simulation

0.10

Moments Equations

0.00
02550 Z (m) 75

100

125

isdPqBz
hPz i ' qE+
mcN (  L) :vz
ds
Pz
00?

(3)

The longitudinal emittance tends to grow from a varietyof effects including nonlinearities in the rf bucket, Landaustraggling in material, and the differential rate of energy
loss in material (slower particles tend to lose more energy
in the relevant energy range). For a well-bunched beam
having a large longitudinal emittance, with typical values
being 10 – 30 mm, the dominant effect is the slope of thed v dP  :energy loss curve, with the result that  0L ' v1 dP
ds L

2.3

Amplitude Correlations

For beams having a large transverse emittance, the relative longitudinal motion of a particle is strongly depen-Figure 3: Performance of a cooling lattice. The fraction ofparticles in the phase space acceptance of the downstream
accelerator complex is plotted as a function of propagation
distance in the cooling channel.

dent on transverse amplitude. This results in a significant non-linear correlation being required in order for thebeam to be matched into a given rf bucket. This effect on
the longitudinal dynamics can be considered without reevaluating the transverse motion of the particles. The average forward velocity
 of a particle over many betatron oscillations is vz ' v 1 (A21 + A22)=4p + Lcanon 
=2Pz .
For cases where  = 0 and L = 0, the momentum distribution for a matched beam can be described in terms

2.4

Simulations with Phase Space Cuts

The figure of merit used for the cooling channel is the number of particles propagating within the 6D phase space acceptance of the downstream systems. This consists of independent cuts on longitudinal and transverse amplitude,
where the transverse cut is taken to be Pz A2? =mc <0:015 m rad, and the longitudinal phase space area istaken to be 0.15  m. The typical value for transverse amplitude is hP z A2? =mci ' 4N . Considering the transverse
cuts alone, only 13% of the beam lies within the aperturecut at the beginning of the cooling channel where  N '0:012 m rad. At the end of cooling, when N ' 0:002 m
rad, almost 89% of the beam survives the transverse aperture cuts (Fig. 2). This compression of phase space is offset
by beam scraping, particle decays, and longitudinal emittance growth (by  a factor of 2) which result in losses.
The longitudinal cut roughly corresponds to the size of therf bucket.
In the moments equations, the effect of radial apertures
is mimicked by applying an appropriate cut in transverse
amplitude, and treating the beam as a Gaussian distribution with a cutoff in the tails. Scraping leads to a reductionin the number of particles and in emittance. In Figures 1and 3, results for a SFOFO cooling channel similar to thatused in the BNL “Feasibility Study 2” are shown for an idealized beam, and compare well with predictions from themoments equations. The moments equations are slightlyoptimistic, mostly because of difficulty in exactly matching the beam to the rf bunching.

2.5

Phase Space Cutstrans < 15 mm, long < 150 mm
0.70
Population in 6D cutsof a single parameter CP as P ' P0 1 + CP A2? + ÆP ,where P0 is the nominal momentum and ÆP is a stochastic term. For beams having large transverse emittance,
the non-Gaussian correlation increases the total momentumspread, with P2 = 02 + 8m2 c2 CP2 2N ; the added term is oforder of P2 ? at the point of tightest focus.

0.60
0.50
0.40
0.30
0.20

ICOOL Simulation

0.10Moments Equations

0.00
02550 Z (m) 75

100125Figure 4: Emittance exchange in a bunch beam is demonstrated using an idealized beam with a low initial emittance.
The net six-dimensional phase (not shown) exhibits some
growth due to the lack of rf to maintain bunching in thisexample.
nels, so the beam can be cooled without spilling particles
out of the rf bucket.

3CONCLUSIONSA paraxial theory has been developed for transverse beam
moments in a ionization cooling lattice. This theory is similar in form to the Courant-Snyder formalism for quadrupole
focussing systems. Good predictions for the beam emittance are obtained. The inclusion of apertures and longitudinal rf buckets leads to solutions to the moment equations that have the main features of the results of tracking codes, but currently overpredict, by a factor of order
20%, the number of muons in the six dimensional phasespace acceptance. Extensions of the model to include sixdimensional phase space dynamics are underway.

Emittance Exchange with helical wigglers

Inside helical wiggler channels, particles undergo helicalmotion with an equilibrium radius that depends on particle
energy. Close to resonace between Larmor motion and therotating dipole field, where P = qB s w =2, the resulting
energy dispersion can become quite large. Here, Bs is thesolenoid field strength, and  w is the wiggler period. Using
wedge-shaped absorbers, this dispersion can lead to emittance exchange, reducing the energy spread of the beam at
the expense of increasing transverse emittance.
In Figure 4 a basic example is shown, for a 300 MeV/cbeam having a small initial emittance  N ' 0:002mrad.
The solenoid field is a constant 3.2 T, while the rotating
dipole field with a 2.5 m period is increased with a tanhramp to a peak value of 0.2 T. A series of LiH wedges
reduce the energy spread nearly in half. Current work is
aimed at developing an emittance exchange section whichworks for higher emittance beams; another goal is to incorporate emittance exchange into transverse cooling chan-4

REFERENCES[1] G. Penn, J.S. Wurtele, Phys. Rev. Lett. 85, 764 (2000).
[2] Charles M. Ankenbrandt et al. (Muon Collider Collaboration), Phys. Rev. ST Accel. Beams 2, 081001 (1999).
[3] S. Geer, Phys. Rev. D57, 6989 (1998).
[4] Proceedings of the ICFA/ECFA Workshop “Neutrino Factories based on Muon Storage Rings”, Lyon; to be published inNucl. Instrum. Methods A.
[5] GEANT4 Collaboration, CERN/LHCC 98-44, “GEANT4:
An object-oriented toolkit for simulation in HEP”; see also
http://wwwinfo.cern.ch/asd/geant4/geant4.html.
[6] R.C. Fernow, in Proceedings of the 1999 Particle Accelerator
Conference, NY, p. 3020.
[7] E.D. Courant and H.S. Synder, Ann. Phys. 3, 1 (1958).
[8] A.N. Skrinsky and V.V. Parkhomchuk, Sov. J. Part. Nucl. 12,
223 (1981).