Initial consideration on the low-  section of muon linac

1. Initial consideration on the low- section of muon linac 4th meeting on muon g-2/EDM experiment Jan. 30, 2009 Masanori Ikegami KEK

2. Outline • Assumptions • Space-charge effect? • Two possible options • Summary

3. Assumptions • Muon generation with • 40,000 muons per bunch • Isotropic momentum spread of 3 keV/c • RMS size of 2.5 mm • RMS pulse length of 3 psec • Ideal initial acceleration to =0.08 • Simply adding the longitudinal momentum keeping the RMS beam size, RMS pulse length, and the momentum spread. =0.08 corresponds to 3 MeV for proton.

4. Kinetic parameters MUONPROTON Mass [MeV/c2] 105.7 938.3  0.08 0.08  1.0032 1.0032  0.0803 0.0803 p0 [MeV/c] 8.48 75.3 Ek [MeV] 0.340 3.02

5. Transverse distribution assuming Analogously,

6. Longitudinal distribution with being the longitudinal position of the design particle. with being the arrival time of the design particle to a certain position. assuming

7. Longitudinal distribution (cont.) We here introduce for the longitudinal coordinate with the fundamental frequency. We assume We also introduce with being the kinetic energy of the design particle.

8. Comparison with J-PARC MEBT x’ [mrad] E [keV] MUON MUON 0.36 0.24 -2.5 2.5 -0.35 0.35 x [mm] -0.36  [deg] -0.24 0.084 keV•deg  0.025 mm•mrad 0.89 mm•mrad J-PARC J-PARC x’ [mrad] E [keV] 2.6 16 -1.1 1.1 -5.8 5.8 x [mm]  [deg] 93 keV•deg  3.15 mm•mrad -16 -2.6 2.9 mm•mrad

9. Envelope equation Envelope equation for an axial symmetric bunched beam is; ; distance along the beam line ; external focusing strength ; unnormalized rms emittance

10. Envelope equation (cont.) ; generalized perveance for a bunched beam ; number of particle per bunch ; classical radius of the particle m for proton m for muon

11. Envelope equation (cont.) with ; peak current, ; bunch frequency, ; elementary electric charge For J-PARC linac 30 mA operation, For muon linac 103 difference in

12. Envelope equation (cont.) ; form factor ; aspect ratio in the beam frame For  > 1,  ~ 40 in muon linac@=0.08

13. Beam envelope for drift (=0.08) Red: w/ space charge Blue: w/o space charge Red: w/ space charge Blue: w/o space charge The beam envelope for drift (without external focusing) Obtained by integrating the envelope equation No acceleration Drift for 12 m or 0.5 sec

14. Two options • Option 1: No longitudinal focusing with low frequency • s is set to 0 deg (on crest). • The frequency is chosen to be 324 MHz or lower. 3 ps  0.36 deg @324 MHz  210-5 deviation in energy gain • Optional transverse focusing • Option 2: With longitudinal focusing matched to the injected beam • The focusing strength is chosen to be matched with the pulse width and momentum spread to avoid emittance growth from the nonlinear nature of the RF force. • The transverse focusing is introduced, at least, to cancel the RF defocusing force.

15. Transverse focusing • In both options, transverse beam manipulation before and after the accelerating section may be effective to ease the tolerance for the good field region. • The increased transverse momentum spread in the accelerating section should be addressed. Quad’s Quad’s beam Accelerating section x’    x

16. Option 1: No longitudinal focusing with low frequency

17. Instability 10-4 10-4 Num. of cells Num. of cells Non-relativistic approximation. Start with 0.36 deg deviation from the crest. Might be feasible with small number of cells.

18. Tentative plan for Option 1 Plan 1-A: DTL without DTQ • = 0.08 to 0.7 Energy gain: 42 MeV Total length: 21 m Number of cells: 58 Time of travel: 0.18 s Comparable to J-PARC DTL It may be possible to increase assuming a short pulse.

19. Tentative plan for Option 1 (cont.) Plan 1-B: Higher gradient DTL without DTQ • = 0.08 to 0.7 Energy gain: 42 MeV Total length: 10.5 m Number of cells: 29 Time of travel: 0.09 s is increased assuming a short pulse.

20. Issues for Option 1 • Instability • Small number of cells is required. • Effect from abrupt cell  variation? • Small margin for the momentum spread specification • Tolerance for RF phase and amplitude • Transverse field uniformity • Low shunt impedance at high- side • Unable to introduce frequency jump ZT2 ~ 8M/m @=0.7 extrapolated from J-PARC SDTL • Long total length • RF defocusing from RF errors

21. Option 2: With longitudinal focusing matched to the injected beam

22. Matching condition • We adopt the smooth approximation (or assume continuous focusing in both transverse and longitudinal). • We find adequate external focusing strength to make the initial beam widths the equilibrium (matched beam widths). • The external focusing strength can be found by solving the following equations.

23. Matching condition (cont.) Neglecting the space-charge term, Then, Similarly, ; Phase advance per meter for betatron oscillation ; Phase advance per meter for synchrotron oscillation

24. Matching condition (cont.) We here assume acceleration up to  = 0.7. To keep , As , we should satisfy the following condition, At  = 0.7,

25. Matching condition (cont.) To keep , As , we should satisfy the following condition, or At  = 0.7,

26. Longitudinal focusing force ; Average accelerating field ; Synchronize phase ; Transit time factor ; RF wave length ; Rest mass

27. RF defocusing ; Focusing strength from quadrupole magnets ; Strength of RF defocusing We need to introduce transverse focusing force to cancel the RF defocusing.

28. Case study 1 Comparable to J-PARC DTL for  = 0.08 or for  = 0.7 Conventional DTL can provide sufficient longitudinal focusing at low- side, but not at high- side.

29. Case study 1 (cont.) Ratio Red: Requirement Blue: Conventional DTL Required It may be possible to increase assuming a short pulse.

30. Case study 1 (cont.) Another possible way to vary is to adjust .

31. Tentative plan for Option 2 Plan 2-A: 324 MHz E0-ramped DTL • = 0.08 to 0.7 • Energy gain: 42 MeV • Total length: ~14 m • Number of cells: ~40 • Time of travel: 0.12 s The longitudinal focusing is adjusted by ramping . Transverse focusing can be provided with DTQ’s. may be too high for high- side.

32. Case study 2 Higher frequency option for  = 0.08 or for  = 0.7 648 MHz DTL can provide sufficient longitudinal focusing at high- side with reasonable .

33. Case study 2 (cont.) Ratio Red: Requirement Blue: Conventional DTL Required It may be possible to increase assuming a short pulse.

34. Case study 2 (cont.) Another possible way to vary is to adjust .

35. Tentative plan for Option 2 (cont.) Plan 2-B: 648 MHz s-ramped DTL • = 0.08 to 0.7 • Energy gain: 42 MeV • Total length: ~17 m • Number of cells: ~94 • Time of travel: 0.14 s It is disadvantageous to ramp E0 to obtain required ks0 with higher frequency. It may be possible to adopt higher gradient to shorten total length.

36. Issues for Option 2 • Feasibility of precise s- or E0- ramping • Susceptible to the longitudinal parameter of the injected beam • Nonlinearity of the RF force in s-ramping scheme • Low shunt impedance at high- side • Frequency jump? • Long total length • Transverse focusing to cancel RF defocusing • DTQ or external quad?

37. Other possibilities • To match the longitudinal focusing at the injection, but gradually introduce mismatch in the downstream portion without s- or E0- ramping • Does the beam distribution adiabatically change? • To manipulate the longitudinal beam parameter with buncher and debuncher before and after the accelerating section. buncher debuncher beam Accelerating section z’    z

38. Summary • Low- section for muon linac are considered assuming ideal acceleration up to =0.08. • Space-charge effects seem to be negligible. • Two options are proposed:Option 1 No longitudinal acceleration Option 2 With longitudinal acceleration matched to the injected beam