1. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 1 Bunch Coalescing in a Helical Channel* Cary Yoshikawa Chuck Ankenbrandt Dave Neuffer Katsuya Yonehara *Work supported by Muons, Inc. DOE SBIR grant DE-SC00002739 (MAP-doc-4302)

2. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 2 Outline  Introduction  Motivation/advantages  Muon Collider Layout (where bunch coalescing fits in)  Number of bunches to merge  Initial Conditions  Helical Bunch Coalescing subsystem 1)Acceleration to desired initial energy 2)Creation of linear energy-time correlation 3)Drift 4)Capture in single RF bucket 5)Deceleration and longitudinal cooling  Simplification of the subsystem that creates linear energy-time correlation  Summary and Future

3. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 3 Motivation/advantages The primary advantages to coalesce bunches in a helical channel are: Shorter bunching distances can be achieved than those of a straight channel. Table 1: Coalescing distances and decay losses in straight and helical channels for muon bunches that have initial energy-distance relationship of dE/d(ct) = 4.09 MeV/m. Slip factor for the helical channel is η HC = η/(β 3 γ) = 0.43. L z = (β 3 γ m μ )/ (η/(dE/d(ct))| bunches ) is the longitudinal bunching distance in the straight channel. It provides a natural match out of and into the Helical Cooling Channel, although the HCC theory is being exploited to extend its applicability to match between other cooling systems with different dynamics. We have just been awarded a phase I that addresses this (Complete Muon Collider Cooling Channel Design and Simulations). KE (MeV) P (MeV/c) Straight: η=(1/γ T 2 – 1/γ 2 ) L z,straight (m) L z,HC (m) L HC (m) Decay loss in a straight (%) Decay loss in a helical channel (%) 100176-0.2641206085107.4 120199-0.2191746085136.6 150233-0.1712776085175.7 175260-0.1423856085215.1 200287-0.1195176085254.6 225313-0.1026746085294.3 250340-0.0888586085333.9

4. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 4 Muon Collider Layout (where bunch coalescing fits in) "R&D Proposal for the National Muon Accelerator Program", August 27, 2010 Helical Cooler(MB) Helical Cooler(SB) Helical Bunch Merger We wish to coalesce bunches in a helical channel to minimize the decay losses (distance) between the upstream multi-bunch (MB) and downstream single-bunch (SB) coolers. Helical Cooler (MB) = Helical Cooler Multi-Bunch Helical Cooler (SB) = Helical Cooler Single-Bunch (The MB and SB coolers can be different from an HCC if necessary.)

5. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 5 Number of bunches to merge Front end Scenario Drift, Buncher, Rotator Length Rf VoltagesFull length ( w 75m cooling)  + /p (  t <0.03,  L <0.3m)  - /p (  t <0.03,  L <0.3m) Core bunches, N B, all  - /p IDS/NF80.6, 33, 42m0  9, 12, 15230m0.1050.11620/0.107 N=1055.3, 31.5, 330  12, 15, 182050.1290.14316/0.141 N=847.8, 35.5, 27 m0  15, 18, 201800.1240.13613/0.123 Table 1: Comparison of muon source front end systems. Shorter bunch train systems capture more muons Capture both μ + and μ - Require higher gradient rf Longitudinal bunch size increased

6. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 6 Initial Conditions Katsuya has a 6D HCC cooler that works well at a kinetic energy of 120 MeV (cf. next slide). We will use his final beam distributions as our initial beam distributions.

7. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 7 HCC Parameters for 200 MHz case 2/28/11 - 3/04/117 Zbb’bzνEκλεμεμ εTεT εLεL ε 6D unitmTT/mTGHzMV/ m mmm rad mmmm 3 Chann el length @ ref RF p ⊥ /p z Trans- missio n RMS normalized 01.021238900 11001.2-0.21-4.20.2161.0 0.751.94.39.4 2911.8-0.42-6.00.4161.00.70.620.861.80.99 3244.2-2.29-14.00.8161.00.30.380.341.10.07 MAP Winter Meeting 2011, Design study of HCC, K. Yonehara HCC reported by K. Yonehara at the 2011 Winter MAP Meeting @ JLAB Assumed emittances at start of coalescing subsystem Target emittances at end of coalescing subsystem in this study, although downstream 6D cooler for a single bunch is yet to be designed.

8. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 8 All manipulations were performed with constant HCC parameters that resulted in a slip factor : η HC = 0.43 = η/(β 3 γ) = (1/ βγ)(Δτ/τ)/(ΔE/E) which has dynamics representative of Katsuya’s HCC. Values of other parameters used throughout Helical Bunch Coalescing subsystem: λ = longitudinal spatial period = 1 m r ref = 16 cm, κ = pitch = 1 B sol (z-axis) = 5.7 T, B sol (on ref) = 5.0 T = 0.72 T, = -1.2 T/m Helical Bunch Coalescing subsystem Helical Channel Parameters

9. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 9 Helical Bunch Coalescing subsystem 1: Acceleration to desired initial energy  Shifts energy of all bunches up equally.  Bunch coalescing works better at higher energies, as will be illustrated on the next slide.  Bunches are accelerated from KE~120 MeV (p=~200 MeV/c) out of Katsuya’s HCC to KE=200 MeV (p=287 MeV/c), while emittances are assumed to remain constant.  This acceleration was not simulated, but its motivation is demonstrated on the next slide.

10. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 10 Helical Bunch Coalescing subsystem 200 MeV central energy choice time (nsec) KE (MeV) z = 0 m z = 60 m z = 40 m z = 20 m Free drift in helical channel w/ η = 0.43.

11. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 11 Helical Bunch Coalescing subsystem 2: Creation of linear energy-time correlation  40m of low gradient, variable frequency RF that puts different bunches at different energies.  V’ max = 1 MV/m  Frequencies determined by zero crossing at center reference bunch and 30º phase at 4 bunches away from the reference.  204.17 MHz < f < 271.84 MHz We’ll simplify later.

12. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 12 z = 40 m chosen for end of linear E(t) creation z = 20 m z end (acceleration) = z start (linear E(t) creation) = z = 0 m z = 50 m Helical Bunch Coalescing subsystem 2: Creation of linear energy-time correlation

13. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 13 Helical Bunch Coalescing subsystem 3: Drift  60m of drift in the same helical channel w/o RF to coalesce the bunches together.  Distance to coalesce bunches is determined by the slope |dE/dt| of the bunches and the slip factor η: where dE/d(ct) = 4.09 MeV/m η HC = 0.43

14. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 14 z = 40+0 m z = 40+20 m z = 40+40 m z = 40+60 m end of drift; 13 bunches time (nsec) KE (MeV) Helical Bunch Coalescing subsystem 3: Drift

15. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 15 z = 40+0 m z = 40+60 m end of drift; 13 bunches z = 40+60 m end of drift; 11 bunches z = 40+60 m end of drift; 9 bunches σ rms (t) = 1.329 ns σ rms (t) = 0.8473 ns σ rms (t) = 0.7912 ns time (nsec) KE (MeV) Helical Bunch Coalescing subsystem 3: Drift

16. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 16 Helical Bunch Coalescing subsystem 4: Capture in single RF bucket  5m of RF to capture several bunches into a single RF bucket.  f = 200 MHz in vacuum  V’max = 10 MV/m

17. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 17 Helical Bunch Coalescing subsystem 4: Capture in single RF bucket RF capture into a single bunch from 13 initial bunches. Bunches at the end of the drift section at z = 40(E-t correlation build-up) + 60(drift) = 100 m and start of RF capture are shown in (a). Longitudinal dynamics of muons at 5 m into RF capture is shown in (b) using same time scale as (a). Zoomed in views are shown in (c) and (d), where only muons in the separatrix are shown in (d).

18. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 18 Helical Bunch Coalescing subsystem 4: Capture in single RF bucket 94.2%(1225/1300) 99.7%(897/900) 98.4%(1082/1100)

19. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 19 Helical Bunch Coalescing subsystem 4: Capture in single RF bucket How do emittances of the captured single bunch compare to acceptance of existing HCC designs? What follows are rms emittances. So, if the rms emittance of the single bunch is the same or smaller than that of an HCC, it is considered to be within the acceptance.

20. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 20 HCC ε T (325 MHz) ≤ 20.4 mm Helical Bunch Coalescing subsystem 4: Capture in single RF bucket HCC ε T (200 MHz) ≤ 21 mm (Emittance values are rms.)

21. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 21 HCC ε L (325 MHz, 27 MV/m) ≤ 42.8 mm HCC ε L (200 MHz, 16 MV/m) ≤ 23 mm Helical Bunch Coalescing subsystem 4: Capture in single RF bucket (Emittance values are rms.)

22. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 22 Helical Bunch Coalescing subsystem 5: Deceleration and longitudinal cooling Properties of and operational needs on the single bunch: The single muon bunch will need to be decelerated to the appropriate energy of the 2 nd 6-D cooler. ε T is likely to be smaller than the 2 nd 6-D cooler acceptance, so emittance exchange can be utilized if necessary to reduce ε L.

23. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 23 A design that addresses the above issues is to implement radial wedges, a gas absorber, and phase the RF to zero crossing at the single bunch center. The gas absorber and zero RF phase maximizes RF bucket size. Emittance exchange in gas is enhanced by the radial wedges to expedite reduction of ε L at expense of growth of ε T, where increase in ε T is tolerable. Helical Bunch Coalescing subsystem 5: Deceleration and longitudinal cooling

24. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 24 Simplification of the subsystem that creates linear energy- time correlation Configuration Original~50% Fill Factor~25% Fill Factor RF Field Fill Factor (%)95%49%24.5% Cavity Length (cm)1025 Number of Cavities4008040 V’max (MV/m)1.02.24.4 The most complicated section in the bunch coalescing subsystem is the portion that creates the linear energy-time correlation via a large number of RF cavities, each having its own frequency, being only 10 cm long, and having nearly 100% fill factor along z. The following simplifications are considered:

25. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 25 Simplification of the subsystem that creates linear energy- time correlation At end of energy-time correlation creation, longitudinal dynamics visually appear identical between original and simplifications.

26. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 26 Simplification of the subsystem that creates linear energy- time correlation Efficiency as measured by muons in single bunch separatrix. ~4%

27. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 27 Summary & Future We have demonstrated a working example of the main components of a helical channel bunch coalescing subsystem. A 6-D simulation with realistic beam dynamics has been performed. Simplifications of the most challenging portion (creation of energy-time correlation) have a modest impact on performance. Future: Design and simulate the portion that decelerates and longitudinally cools the single bunch to fit a second 6-D cooler. Determine/estimate impact of space charge. Consider possible increase in acceptance by increasing λ = 1.0m to 1.6m to allow more bunches to be merged, although 10 seems to be enough. Further optimize the parameters of the bunch coalescing subsystem. Possibly re-optimize upstream and downstream 6D coolers to better match the bunch coalescing subsystem.

28. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 28 Back up

29. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 29 325 MHz Parameter list Z±Δr ±Δp/ p bb’bzνκλNμNμ εTεT εLεL ε 6D unitmcm%TT/mTGHzmmm rad mmmm 3 Channe l length Full Width Full width @ ref RF 1015221.3-0.5-4.20.3251.0 38820.442.812900 2408101.3-0.5-4.20.3251.0 3755.9719.7415.9 3497101.4-0.6-4.80.3251.00.93544.0115.010.8 412932.51.7-0.8-5.20.3251.00.83271.024.82.0 52191.71.82.6-2.0-8.50.651.00.53270.582.13.2 62431.61.33.2-3.1-9.80.651.00.43270.421.30.14 72731.3 4.3-5.6-14.10.651.00.33270.321.00.08 83031.21.14.3-5.6-14.11.31.00.33270.341.10.07 12/02/0929MC Design workshop @BNL, K. Yonehara (HCC reported by Katsuya Yonehara at the Dec. 2009 MCDW @ BNL) start of merger end of merger

30. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 30 Simplification of the subsystem that creates linear energy- time correlation HCC ε T (325 MHz) ≤ 20.4 mm HCC ε T (200 MHz) ≤ 21 mm As expected, there are effectively no differences in ε T due to modifications that affect the longitudinal dynamics.

31. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 31 Simplification of the subsystem that creates linear energy- time correlation HCC ε L (325 MHz, 27 MV/m) ≤ 42.8 mm HCC ε L (200 MHz, 16 MV/m) ≤ 23 mm Longitudinal emittance growth for 4.4 MV/m (~25% FF) and 2.2 MV/m (~50% FF) are similar for 9 and 11 bunches, but has no impact for 13 bunches within the ~3% statistical error of this preliminary study.

32. Muons, Inc. 3/5/2012MAP Collab. Meeting at SLAC Cary Y. Yoshikawa 32 Compare RF power consumption in 200 and 325 MHz base HCCs 2/28/11 - 3/04/1132 νEL cavity R cavity Dissipatio n P peak Stored EDissipation P ave GHzMV/mcm MW/mJ/mkW/m RF 10 cavities/λRep rate = 15 Hz 0.2161057.143.331317.9 0.416728.623.278.23.4 0.816414.314.719.50.76 0.325271035.365.434113.1 0.6527717.736.085.22.6 1.32748.823.221.30.58 In STP condition Tried to find 20 kW/m @ 77 Kelvin of cooling power Need a special cooling system MAP Winter Meeting 2011, Design study of HCC, K. Yonehara (HCC reported by Katsuya at the 2011 Winter MAP Meeting @ JLAB)