Muon Coalescing 101 Chuck Ankenbrandt Chandra Bhat Milorad Popovic Fermilab NFMCC Meeting @ IIT March 14, 2006.

Muon Coalescing 101

Chuck Ankenbrandt

Chandra Bhat

Milorad Popovic

Fermilab

NFMCC Meeting @ IIT March 14, 2006

Context for this talk • Suppose that some day:• A proton driver based on an 8-GeV linac exists;• High-int. muon beams are available with low emittances in all three planes;• The proton driver linac can be used to accelerate both protons and muons.

• Achieving low emittances requires parametric resonance ionization cooling. However, Slava Derbenev has found that PIC doesn’t work well for very intense bunches because of space-charge tune shifts. That led Rolland Johnson and Slava to develop scenarios that produce a large number of less intense bunches.

• In one specific scenario, each of ten equally spaced proton bunches produces a train of sixteen equally spaced muon bunches. That works as is for a neutrino factory; however, to achieve high luminosity in a collider, it is highly desirable to combine the bunches.

• That in turn led them to ask the question addressed in this talk: How can muon bunches be combined to enhance the luminosity of a muon collider?

General combining considerations

• Combining ought to be done after accelerating to high energy, where space charge is not a problem and adiabatic damping of beam sizes provides room to operate. At high energy, momentum-dependent path lengths work better than velocity differences for combining bunches.

• There are two bunch-combining techniques presently used operationally for protons at Fermilab: slip-stacking and coalescing. The specific implementations used for protons are much too slow for muons. The approach described here is a fast form of coalescing. Fast coalescing ignores slow niceties, so reducing the dilution of longitudinal emittance is a major consideration.

First-Order Ring Physics1. Muon Decays in Rings

cLdk eBp eBfRp

eBf

pRC

22

)(2972

TeslaBfmc

Bfce

C

Ln dkdk

Decay length

So the number of turns to decay is given by

mcp where f is the fill factor

First-Order Ring Physics2. Space Charge

22

3

B

Nr

n

o

for Gaussian bunch

p

Numbers: Compare p /

9/

m

mrr popo

40/ nnp

20

1/ pNN

3/

p

p BB CB

2

80

1)()/()( 222

pp m

m

at the same energy

First-Order Ring Physics3. Slippage

p

p

f

f

where

and it’s easier to use

p

eBfL

p

p

R

Ln ttc

22

20

20

22

11

t

012

Here,

p

p

R

R

t

2

1

p

pRRC

t

2

22

Nc, number of turns to coalesce=R

L

20

p

Lm

n

n t

dk

c

20

eBp

eBRfp

Where Lo=half-length of bunch train

Assuming momentumspread is constant

Schematic of the LINAC and Coalescing Ring

Coalescing Ring

20 GeV Muon

LINAC

Bunch train

with 1.3GHz structure

Bunch LE~ 0.03 eVs

dE~ 20 MeV

vernier LINAC

Muon Coalescing RingThe following parameters are assumed for the Coalescing Ring:

Injection beam : 1.3GHz bunch structure   # of bunches/train = 17   Ring Radius = 52.33m; Revolution period= 1.09s   Energy of the muon = 20 GeV (gamma = 189.4)   gamma_t of the ring = 4

If we assume Ring-Radius/rho (i.e., fill factor) = 2, then B-Field = 2.54T     (This  field seems to be reasonable)      h for the coalescing cavity = 42, 84   Number of trains/injection = less than 37 (assuming ~100ns for injection/extraction)   RF voltage for the coalescing cavity = 1.9 MV (h=42) = 0.38 MV (h=84) fsy ~ 5.75E3Hz   Tsy/4 = 43.5us   Number of turns in the ring ~40

Constraints:Muon mean-life = 2.2us (rest frame)Muon mean-life in  lab = 418us for 20 GeV beamTime (90% survival) = 43.8us

Radius=52.3m

Injection extraction

Initial Simulation Results

• Three scenarios in a 20 GeV ring for up to 37 groups of 17 bunches of 1.3GHz

• Scenario1: rf cavities in the ring takes 54 s

• Scenario2: vernier linac takes about 46-54 s

• Scenario3: vernier linac and rf cavities in the ring takes about 38 s

1st ScenarioMuon Bunch train from the LINAC

Muon Bunch train in the coalescing bucketT=0 sec

dE~ 20 MeV

Muon Bunch train in the coalescing bucketT= 31.6 sec

Muon Bunch train in the coalescing bucketT= 54 sec

dE~ 200 MeVBunch Length~ 1.5ns

2nd Scenario• A vernier-linac to give a tilt in the

Longitudinal Phase-space

Muon Bunches after pre-linac

•And next inject the beam into the Coalescing Ring

Bunch train before the special

purpose pre-linac

Muon Bunch train in the Coalescing RingT=0 sec

Muon Bunch train in the Coalescing RingT=46 sec

dE~ 100 MeVBunch Length~ 4ns

Muon Bunch train in the Coalescing RingT=71 sec

dE~ 60 MeVBunch Length~ 3ns

Muon Bunch train in the Coalescing RingT=0 sec

3rd Scenario

Muon Bunch train in the Coalescing RingT=38 sec

dE~ 200 MeVBunch Length~ 1.5ns

Summary and conclusions

• Fast coalescing requires:• Short muon bunch trains (less than half the

distance between proton bunches)• A large momentum ‘ramp’ across each train• Small transition gamma (weak focusing lattices?)• Large radial acceptance in the ring• The energy ramp can be generated with a vernier linac

and/or with rf cavities in the ring.• Coalescing leads to multiple constraints (on ring

circumferences, bunch spacings, rf frequencies, etc.)• Longitudinal emittance dilution is a concern.• Of course, global optimization is required.

Mindset and motivation• We are much more likely to get a proton driver if it can be designed

and sited in such a way that it provides a versatile multistage upgrade path to transform existing facilities into sources of megawatt-class proton beams (as well as being an ILC testbed).

• We are much more likely to get a proton driver, a stopping muon program, a neutrino factory, and a muon collider if we can maintain synergy among all of them. In particular, the path to a neutrino factory should not diverge from the path to a muon collider.

• Even though a neutrino factory might be implemented with only modest muon cooling, early achievement of extreme muon cooling would have several important advantages:

• Muons could be accelerated in the proton driver linac;• The rest of the neutrino factory (except cooling) would be

easier to implement;• The path from the neutrino factory to the muon collider

would be much easier.