Muon Coalescing 101

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

2. 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?

3. 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.

4. First-Order Ring Physics • Muon Decays in Rings Decay length where f is the fill factor So the number of turns to decay is given by

5. First-Order Ring Physics 2. Space Charge Numbers: Compare at the same energy for Gaussian bunch

6. First-Order Ring Physics 3. Slippage where Here, and it’s easier to use Where Lo=half-length of bunch train Nc, number of turns to coalesce= Assuming momentum spread is constant

7. Coalescing Ring 20 GeV Muon LINAC Schematic of the LINAC and Coalescing Ring vernier LINAC Bunch train with 1.3GHz structure Bunch LE~ 0.03 eVs dE~ 20 MeV

8. Injection extraction Radius=52.3m Muon Coalescing Ring The following parameters are assumed for the Coalescing Ring: Injection beam : 1.3GHz bunch structure   # of bunches/train = 17 Ring Radius = 52.33m; Revolution period= 1.09s   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 beam Time (90% survival) = 43.8us

9. 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

10. 1st Scenario Muon Bunch train from the LINAC

11. Muon Bunch train in the coalescing bucket T=0 sec dE~ 20 MeV

12. Muon Bunch train in the coalescing bucket T= 31.6 sec

13. Muon Bunch train in the coalescing bucket T= 54 sec dE~ 200 MeV Bunch Length~ 1.5ns

14. 2nd Scenario • A vernier-linac to give a tilt in the Longitudinal Phase-space Bunch train before the special purpose pre-linac Muon Bunches after pre-linac • And next inject the beam into the Coalescing Ring

15. Muon Bunch train in the Coalescing Ring T=0 sec

16. Muon Bunch train in the Coalescing Ring T=46 sec dE~ 100 MeV Bunch Length~ 4ns

17. Muon Bunch train in the Coalescing Ring T=71 sec dE~ 60 MeV Bunch Length~ 3ns

18. 3rd Scenario Muon Bunch train in the Coalescing Ring T=0 sec

19. Muon Bunch train in the Coalescing Ring T=38 sec dE~ 200 MeV Bunch Length~ 1.5ns

20. 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.

21. 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.