HIGH FIELD - LOW ENERGY MUON IONIZATION COOLING CHANNEL

Hisham Kamal Sayed and Robert B. PalmerBrookhaven National Laboratory, Upton, NY, USA

David NeufferFermi National Laboratory, Batavia, IL, USA

(Dated: July 23, 2015)

Muon beams are generated with large transverse and longitudinal emittances. In order to achievethe low emittances required by a muon collider, within the short lifetime of the muons, ionizationcooling is required. Cooling schemes have been developed to reduce the muon beam 6D emittancesto ≈ 300 µm-rad in transverse and ≈ 1-1.5 mm in longitudinal dimensions. The transverse emittancehas to be further reduced to ≈ 50-25 µm-rad with an upper limit on the longitudinal emittance of ≈76 mm in order to meet the high-energy muon collider luminosity requirements. earlier studies of thetransverse cooling of low energy muon beams in high field magnets showed a promising performance,but did not include transverse or longitudinal matching between the stages. In this study we presentthe first complete design of the high field - low energy ionization cooling channel with transverseand longitudinal matching. The channel design was based on strong focusing solenoids with fieldsof 25-30 T and low momentum muon beam starting from 135 MeV/c that decreases gradually. Thecooling channel design presented here is the first to reach micron scale emittance. We present thechannel’s optimized design parameters including the focusing solenoid fields, absorber parametersand the transverse and longitudinal matching.

I. INTRODUCTION

The muon collider offers significant advantages over anelectron machine as a lepton collider. Muons, with a mass200 times heavier then electrons, have an increased cross-section for Higgs production, by ≈ four orders of mag-nitudes [1]. More importantly, a muon collider does notsuffer from some of the difficulties associated with highenergy electrons, e.g. synchrotron radiation and beam-strahlung. This allows us to use multiturn acceleratorswith much lower cost and size.

However, producing and accelerating high intensitymuon beams is a challenging task. Muon beams are pro-duced from pion beam decay. Pions are generated bycollision of multi-GeV proton beam with a target. Theresulting pion beam has large transverse and longitudinalemittances, which require reduction in all six phase-spacedimensions to fit within the longitudinal and transverseacceptances of the muon accelerator. In addition to thesechallenges, muons have a very small lifetime of 2.2 µs,which requires fast and aggressive cooling and accelera-tion.

Muon colliders were first proposed by Budker in 1969,and later more detailed design concepts were proposedin [2–4]. Muon accelerators must deliver high luminos-ity muon beams with transverse emittances less than 50µm-rad for high-energy muon collider rings. Reductionof the initial large 6D transverse emittance of muon beamis achievable only by ionization cooling, where muon mo-menta are reduced in all directions as they pass throughthe absorber material. Absorbers are followed by sets ofRF cavities which restore only the longitudinal compo-nent of the momentum. This process cools only the trans-verse emittance directly, but when combined with emit-tance exchange it also reduces longitudinal emittance,

providing 6D phase space cooling of the muon beam.

A complete design concept of a high intensity muoncollider from production to collision was given in [5]. Thedesign has five major components; the high power targetfor pion production [6], the RF bunching, the 6D cooling,the fast acceleration, and finally the injection into thecollider ring. The Muon Collider design starts with theMuon Front End, consisting of the pion production targetand the bunching channel [6–9] that yields bunch trainsof both muon signs.

The Muon Front End is followed by the 6D coolinglattices which reduce the longitudinal emittance until itbecomes possible to merge the trains into single bunches,one of each sign. Further cooling in all dimensions isapplied to the single bunches in the following 6D coolinglattices.

A complete scheme of muon cooling for a Muon Col-lider was presented in [5]. In this scheme muons of bothsigns are cooled transversely in a linear channel usingLiH absorbers in a periodic 2.8 T solenoid lattice withRF cavities for acceleration. Following this stage, a 6Dcooling channel which uses wedge shaped liquid hydro-gen absorbers, tilted solenoids for focusing and disper-sion generation, and RF cavities for acceleration. Sincethe collider luminosity is proportional to N2, where N isthe number of muons per bunch, it is desirable to haverelatively few bunches with larger N . A bunch mergingchannel would therefore follow the initial 6D cooling [5].

A second 6D cooling channel follows the bunch mergingchannel. The second 6D cooling channel will have muchstronger solenoid focusing and should reduce the muonbeam emittance to εT = 280 µm−rad and εL = 1.5 mm.

Several 6D ionization cooling channel designs havebeen proposed [10–13]. Simultaneous transverse and lon-gitudinal cooling is achieved by using wedge-shaped ab-

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sorber materials or a differential path length in an ab-sorber region with non-vanishing dispersion.

In 6D cooling channels particles with different mo-menta will experience differing path lengths inside theabsorber due to the dispersion, leading to reduction inthe longitudinal emittance. These 6D ionization coolingchannels operate with muon beams of central momen-tum ≈ 220 MeV/c and cannot provide cooling beyondεT ≈ 280 µm− rad and εL ≈ 1.5 mm [11–13].

The final stage of the muon collider ionization coolingis a transverse only cooling channel [14]. It utilizes liquidhydrogen absorbers inside strong focusing solenoids. Thereduction in the transverse emittance takes place at theexpense of heating the longitudinal emittance.

The final transverse cooling channel for the muon col-lider was proposed in [5, 14, 15]. The design proposedcooling low-energy muons with liquid H2 absorbers in30-50 T solenoids. The reported design concept showedthat a final emittance of 25 µm is achievable, assumingideal matching between the stages and lowering the beamenergy down to 5 MeV.

The focus of this work is to present a complete designof the high field - low energy transverse cooling chan-nel for use in the final cooling segment of the the MuonCollider. The design reported here is based on a 25-30T focusing solenoids. The design includes complete andrealistic transverse and longitudinal matching betweenthe stages. The maximum field strength in the channelwas limited to 30 T to reduce the cost and adhere to thecurrent technological limitations of high field magnets.

The channel works in a momentum regime that startsat 135 MeV/c and decreases gradually. At such low mo-menta, the longitudinal emittance rises from the heatingeffect of the negative slope of energy loss versus energy.Starting with initial emittances achieved by 6D coolingchannels, we show that the channel design can achievetransverse emittances of 55 µm-rad within a longitudi-nal acceptance of 76 mm-rad. This allows the use of thischannel design in Muon collider final cooling stages.

In the following we first discuss the physics and motiva-tion of ionization cooling at low energy (Section II). Wethen review and the discuss feasibility and parameters ofhigh field magnets (Section III). Following that we givethe details of our final cooling channel design (SectionIV), followed by simulation and tracking studies (SectionV), establishinging the cooling channel performance.

II. IONIZATION COOLING AT LOWENERGIES

Inside the absorber material, the reduction in beammomenta providing transverse cooling is opposed by themultiple Coulomb scattering. Multiple Coulomb scatter-ing in the absorber material limits the ionization coolingand acts as a heating element. Strong focusing fields atthe absorber location help in minimizing the longutidnalheating effect. Equ. 1 gives the rate of change of the nor-

malized transverse emittance inside an absorber material[16]

FIG. 1. Analytical estimate of the minimum equilibrium emit-tance achieved in Liquid hydrogen absorber using focusingsolenoid fields of 20, 30, 40 T. Reducing the muon beam ki-netic energy and using strong focusing field can achieve verysmall emittance values down to 25 µm-rad.

dεNds

= − εNβ2E

dE

ds+

β⊥E2s

2β3mc2LRE(1)

Where β⊥(≈ 2P/0.3B) is the transverse betatron func-tion of the beam, β = v/c, LR is the radiation length ofthe absorber material, and Es = 13.6 MeV.

The energy loss in material dE/ds is given by theBethe-Bloch formula [17]

dE

ds= 4π NA ρ r2e mec

2Z

A[

1

β2ln(

2mec2γ2β2

I(Z))− 1− δ

2β2]

(2)Where NA is Avogadro’s number, A,Z are the atomic

weight and number of the absorber material of densityρ. and me and re are the electron mass and classicalradius respectively. The density effect factor δ ≈ 0. Thematerial ionization energy I(Z) in eV can be estimatedby [16]

I(Z) = 16Z0.9 (3)

When both the ionization cooling and the multiplescattering heating in the absorber material come to equi-librium, the normalized transverse emittance of the beamcan not be reduced beyond the value of the equilibriumemittance given by

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TABLE I. High field Magnet Parameters

Magnet Length [m] Inner radius [m] Coil Thickness [m] Current Density I/A [A/mm2]

0.317 0.025 0.029 164.260.337 0.055 0.041 142.430.375 0.098 0.056 125.880.433 0.157 0.067 119.070.503 0.228 0.120 85.990.869 0.355 0.089 39.600.868 0.454 0.104 44.300.992 0.575 0.252 38.60

εequ,N =β⊥E

2s

2βmc2LR(dE/ds). (4)

Which gives the minimum achievable emittance in anyabsorber material in a simpler form

εmin,N ∝E

BLR(dE/ds)(5)

For a given absorber material εmin,N depends on thebeam energy E and the focusing strength B. Reduc-ing the beam energy gradually in the cooling channelwith strong focusing fields can provide a reduction inthe transverse emittance to values in the range of 60-25 µm-rad. Muon ionization cooling to emittances be-low 50µm-rad can be achieved with liquid hydrogen ab-sorbers placed in high field solenoids which have smallbores.

The disadvantage of the ionization cooling of low en-ergy muons is the increase in the longitudinal emittance.At low energies the increase of the energy loss with re-duced energy heats the beam longitudinally. The changeof the RMS energy spread of the beam during ionizationcooling is given by

dσ2E

ds= −2

∂(dE/ds)

∂Eσ2E +

d〈∆E2RMS〉ds

(6)

The second term of the last equation represents therandom fluctuations in the particle energy loss, whichis a heating effect. In the long-pathlength Gaussian-distribution limit, this term is given in (MeV cm)2/g by[16]

d〈∆E2RMS〉ds

≈ 0.157 ρZ

Aγ2(1− β2

2) (7)

As the muon beam transverse emittance is reduced, thelongitudinal emittance increases accordingly. We aim tolimit the longitudinal heating to within the longitudinalacceptance of the muon acceleration chain. Acceleration

to high energy reduces the relative momentum spreadδP/P and improves the longitudinal acceptance.

Fig. 1 shows an analytical estimation of the minimumequilibrium emittance achieved in the Liquid hydrogenabsorber using focusing solenoid fields of 20, 30, 40 T.By reducing the muon beam kinetic energy and usingstrong focusing fields one can achieve very small emit-tance values down to 25 µm-rad.

FIG. 2. A set of eight superconducting coaxial coils providinga peak field of 50 T. The inner radius of the smallest coils is0.025 m.

III. HIGH FIELD MAGNETS

In this section we will briefly describe two possiblehigh-field magnets that can be used in the cooling chan-nel. A high field magnet was proposed in [18], it hasa set of eight coaxial superconducting solenoid coils. Itwas designed to deliver peak fields up to 50 T. The coilsparameters and currents are given in Table I and theschematic of the magnet is shown in Fig. 2. In order toprovide peak fields ranging from 30 to 25 T, the currentdensities are scaled accordingly. These coils were used inthe simulation of the cooling channel.

Another variety of high field magnet design, which is anall superconducting 45 T solenoid, was presented in [19].The conceptual design of the 45 Tesla solenoid is based

4

FIG. 3. Schematic of the elements of one ionization cooling stage. Each stage starts with strong coaxial focusing coils whichenclose the LH2 absorber, followed by matching coils, energy-phase rotation RF cavities, and acceleration RF cavities.

on Bi-2223 HTS tape, and the magnet operates at 4.2 Kto take advantage of the high current carrying capacityat that temperature. An outer Nb3Sn shell surroundsthe HTS solenoid.

Another 45 T hybrid magnet composed of a 33.5 T re-sistive magnet nested in an 11.5 T outsert is currentlyoperating at The National High Field Magnet Labora-tory ”NHFML” [20]. The disadvantage of this magnet isits large power consumption, and such large power con-sumption might not suit our needs. We would like to notethat a 27 T solenoid using an 8 T YBCO insert at 4Khas recently been successfully tested at the same facility.

The magnet design details do not affect the coolingchannel performance. We have used a scaled-down ver-sion of the 50 T magnet design in our simulations. Amore simple superconducting magnet design with 25-30T peak field will deliver the same ionization cooling per-formance.

IV. CHANNEL DESIGN

The high-field low-energy cooling channel is composedof 16 stages in a straight section. Each stage consists offive major components: A set of coaxial strong focusingsolenoid coils, liquid hydrogen absorber, two matchingsections, longitudinal phase space rotation RF cavities,and acceleration RF cavities. Figure 3 shows a schematicof a single stage. A schematic of all of the 16 stages isshown in Figure 4.

The muon beam momentum is reduced gradually alongthe channel to reduce the value of the equilibrium emit-tance and achieve the desired cooling. As a result eachcooling stage has its own set of unique design parametersoptimized for the momentum band at which the stage

operates.

A. Individual stage design

Each stage contains a cylindrically shaped liquid hy-drogen absorber placed inside the innermost coil of thehigh field magnet where β⊥ has a minimum value. Fig.6 shows the β⊥ inside the absorber for each stage. Suchabsorbers are feasible [21]. The strong focusing solenoidcoils are preceded and followed by a set of matching coilsto match the beam transport from a 3.5 T constant fo-cusing field to the peak 30-25 T value while mitigatingchromatic effects.

Due to the fact that the beam mean momentum be-fore passing through the absorber is different from thatafter exiting the absorber; the matching coils are asym-metric. The matching is done by propagating a set ofβ⊥-functions for different momenta through the absorbermaterial to account for the energy loss and increased en-ergy spread. The muon momenta were chosen to rep-resent the momentum band of the muon beam at eachstage. An automated optimization algorithm was usedto match the β⊥-functions in and out of the absorber[22].

The matching segment has no RF cavities and wasutilized to develop an energy-time correlation requiredfor the energy-phase rotation. The length of the driftspace between the absorber and the first RF cavity con-trols the amount of energy-phase rotation and hence thebunch length and energy spread after rotation. The driftlength was optimized for each individual stage to reducethe energy spread and achieve the required cooling in thefollowing stage. The first set of RF cavities after the driftis set to have zero phase to rotate the longitudinal phase

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FIG. 4. A layout schematic of 16 stages of the cooling channel. The colored boxes in the top represent the cooling stages (adetailed schematic of one stage is show in Fig. 3.). The bottom figures show a sample of the on-axis field of the strong focusingsolenoid; the shaded areas show the corresponding absorbers lengths.

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3 3.5 4

B [

T]

Z [m]

PART I

(a) First quarter: Long absorbers. Limit thefield integral to limit the transverse -

longitudinal coupling and limit unnecessaryincrease in σt.

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3 3.5 4

B [

T]

Z [m]

PART II

(b) Second quarter: Long absorbers butrelatively smaller transverse amplitudes and

already longer bunch length.

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3 3.5 4

B [

T]

Z [m]

PART III

(c) Third quarter: Medium absorber thickness.Larger energy spread will lead to unwanted

chromatic effects.

0

5

10

15

20

25

30

35

0 0.5 1 1.5 2 2.5 3 3.5 4

B [

T]

Z [m]

PART III

(d) Fourth quarter: Small absorber thickness.Very small transverse amplitudes.

FIG. 5. On-axis field of four of the focusing solenoids used in the ionization cooling channel.

space of the muon beam. Another set of RF cavities fol-lowing the phase rotation section are used to acceleratethe muon beam to the required energy for the followingcooling stage. The accelerating RF cavities frequenciesand phases are optimized to match the bunch length atevery stage.

Each cooling stage performance was optimized using

automatic optimization algorithm based on tracking of alarge statistical ensemble of particles. The optimizationincluded the transverse and longitudinal matching be-tween the stages. The optimization objectives were setto maximize cooling and minimize particle losses.

A set of field flip matches are inserted between thestages to limit the accumulation of the canonical angu-

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0.016

0.018

0.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0 2 4 6 8 10 12 14 16

β [

m]

Stage

FIG. 6. Transverse Twiss β-function at the center of strongfocusing solenoid in each stage of the 16 stages.

lar momentum [15]. The matching was performed byminimizing the transverse β-beat for particles within therequired momentum band. The field flip locations aregiven in Table II.

B. Design variation between the stages

The initial energy, energy spread, and absorber length

for every stage are optimized to minimize−dε‖dε⊥

. Thebeam energy is gradually reduced and the bunch lengthincreases gradually along the channel. As a result eachstage has its own set of matching coils, focusing coils, RFcavities frequencies, which are different from the otherstages.

The selection of the on-axis field at the absorbers lo-cations and the match in and out of the strong focusingportion has a major impact on the cooling performance ofthe channel. We adopted four differing focusing solenoidsets which were used throughout the channel accordingto the value of the transverse beam emittance and en-ergy spread. The strength of the focusing field and thematching coils are optimized for each stage individually.

At the initial stages where the transverse emittanceis large 300 µm-rad we would like to avoid the longi-tudinal emittance growth from the amplitude dependenttransit time effect [6]. The field integral is minimized toprovide the required focusing for reducing the εequ andmatch the length of the absorber. The peak magneticfield is designed to have a short decline to the transportfield value of 3.5 T. In the later stages the energy of thebeam is further decreased and hence the growth of theenergy spread per unit absorber length is larger. Shorterabsorbers are used and the cooling will require a strongerfield of 30 T.

In the first four stages (first quarter of the channel)

LH2 absorbers are long which requires long section ofthe peak field. It is desired to limit the field integralto avoid the unnecessary increase in the bunch lengthdue to transverse - longitudinal coupling in the high fieldsolenoids [6]. The on-axis field profile for each stage wasoptimized to meet these two requirements. See Fig. 5(a).

In the second quarter (stages 5-8), LH2 absorbers havea moderate length and the beam has relatively smallertransverse amplitudes and already long bunch length.The on-axis field in this case will have slightly more adi-abatic match to the lower field and slightly smaller peakfield length. See Fig. 5(b).

In the third and forth quarters (stages 9-16), LH2 ab-sorbers have medium thickness. The beam has slightlylarger energy spread which will lead to unwanted chro-matic effects. A more adiabatic match to the peak fieldwas required to compensate for the chromatic effects andhas a small impact on the already long bunch length. SeeFig. 5(c) and 5(d).

As the bunch goes through the energy phase rotationsection in each stage it gets longer, we start the simu-lation of the channel with σt=5 cm and the final bunchlength is 1.8 m. The RF frequencies were chosen to keepthe bunch length σct < λ/20. Table II shows RF pa-rameters used in every stage. The gradients assumedmaximum surface fields ≈

√f , and assuming reentrant

vacuum cavities with surface to accelerating gradients∝ f0.75. Only the last accelerating stage required aninduction linac.

In the first stage a set of 325 MHz RF cavities are usedand are placed inside the 3.5 T transport coils. In thelater stages as the bunch gets longer, which mandatesthe use of lower RF frequencies. The size of the RFcavities increases, and the cell structure is switched touse smaller coils inside the RF field. The bunch lengthand the energy spread at different stages are show in Fig.7.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16

Stage

cστ [m]

100 σΕ

FIG. 7. Bunch length and energy spread across the channel.

7

TABLE II. Parameters of the high-field low-energy cooling channel.

Stage P Energy Spread σE LH2 Thickness Drift Length RF Length RF Frequency Field Flip

N MeV/c MeV cm m m MHz

1 135.0 2.29 65 0.434 2.25 325 Yes2 130.0 2.48 60 0.459 2.25 250 Yes3 129.0 2.78 60 0.450 2.5 220 No4 129.0 3.10 59 0.458 2.5 201 No5 122.0 3.60 57 1.629 5.0 201 Yes6 124.0 4.90 53 2.22 4.5 180 No7 116.0 3.40 42 2.21 3.25 150 No8 111.0 3.90 40 2.0 3.5 150 No9 106.0 3.50 40 3.13 5.0 125 Yes10 98.0 3.07 35 3.13 5.0 120 No11 89.4 3.11 20 3.12 5.0 110 No12 87.9 2.76 20 3.1 8.0 100 No13 85.9 2.67 20 3.0 7.5 100 Yes14 79.7 3.08 15 2.7 7.0 70 No15 71.1 4.0 15 2.6 6.0 50 No16 71.0 3.80 13 2.5 6.0 20 No17 70.0 3.80 10 – – 20 –

V. SIMULATION OF THE COOLINGCHANNEL

The high field - low energy cooling channel was simu-lated using the G4BEAMLINE code [23], which is basedon GEANT4 [24] physics libraries. The magnetic field ofthe solenoids was computed using a realistic configura-tion of coils and their current settings.

The RF cavities were modeled as cylindrical pillboxesrunning in the TM010 mode. A reference particle wasused to determine each RF cavity’s relative phase. Thereference particle is a muon that is used as a clock todetermine the entry times for each cavity in the channel.The reference particle is allowed to lose energy in ab-sorber and gain energy during RF acceleration, so thatit travels with the bunch.

G4BEAMLINE uses GEANT4 physics model QGSP-BERT, and it includes all the relevant physical processesfor simulating the low energy beam interaction with ab-sorber material. This includes energy loss, straggling,multiple scattering and muon decays.

In this simulations we used a Gaussian input beamwith normalized transverse emittance ε⊥ of 300 µm−radand longitudinal emittance εL of 1.5 mm. The initial mo-mentum distribution was generated with an average lon-gitudinal momentum of 135.0 MeV/c. Tracking was per-formed using an initial sample of 107 particles matchedand injected into the first ionization cooling stage.

0

50

100

150

200

250

300

0 20 40 60 80 100 120 140

Z [m]

εt [µm]εL [mm]

Transmission [percent]

FIG. 8. Transverse and longitudinal emittance in the channel,along with the transmission which includes muon decay.

VI. OPTIMIZATION OF THE CHANNELPERFORMANCE

Each stage performance was optimized automaticallyusing multi-layered parallel optimization algorithms [9,25]. The optimization was based on tracking of themuon beam through the whole cooling channel. The op-timization was performed to improve both the transverse

8

and the longitudinal motion as both are coupled duringthe cooling and matching. The optimization parame-ters included the focusing coils, matching coils, absorberlengths, drift lengths from the absorber to first energy-phase rotation cavity and RF frequencies/phases. Theoptimized channel parameters are given in Table II. Thetransverse and longitudinal emittances evolution throughthe channel and the muon transmission are shown inFig.8. The channel was able to cool the muon beamfrom 300 µm-rad down to 55 µm-rad. The longitudinalemittance of the beam after cooling is 76 mm.

VII. SUMMARY

A high luminosity muon collider requires a very smalltransverse beam emittance. The muon beam, which isinitially produced with alarge transverse emittance, hasto go through a series of 6D ionization cooling channelsto reduce the emittance down to 300 µm-rad. The 6Dcooling channels are limited due to the compact natureof those channels and the limited equilibrium emittancevalues they can achieve while operating at momentum of200 MeV/c with a maximum focusing strength of ≈ 15T. Further reduction in emittance requires reducing theoperating energy and using much stronger fields; fields ofthe order of 25-30 T.

In this work we presented the first complete channelwith real matching of a high field - low energy cooling.The maximum peak field considered in this study waslimited to the current state of the art magnets i.e. 30T and the lowest beam momentum was 70 MeV/c. Thedesign parameters of the cooling channel were discussedin detail.

The channel shows a promising performance as the fi-nal cooling stage for the muon collider. More work willbe required to further reduce the transverse emittanceto the 25 µm-rad desired for a very high-luminosity andhigh-energy muon collider.

It is worth mentioning that the channel performanceis not dependent on the magnet design or the absorberouter geometry. Any magnet design that delivers the onaxis fields discussed in section IV will deliver the requiredcooling performance.

VIII. ACKNOWLEDGMENT

Authors are grateful to H. Kirk, S. Kahn, M. Palmer,R. Ryne, D. Stratakis, and R. Weggel for useful discus-sions. This research used resources of the National En-ergy Research Scientific Computing Center, which is sup-ported by the Office of Science of the U.S. Departmentof Energy under Contract No. DE-AC02-05CH11231.

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