Status of Low-  FOFO Snake for Final Stage of 6D Ionization Cooling

1. Status of Low- FOFO Snake for Final Stage of 6D Ionization Cooling Y. Alexahin (FNAL APC) Neutrino Factory & Muon Collider Collaboration Meeting, Oxford MS January 13-16, 2010

2. Goals 2 + – + – “cell” FOFO “snake” to achieve T < 0.5mm, L < 1mm requires:  Small -function at absorbers (at the solenoid center if the phase advance >180/cell  Transverse B-field to generate dispersion, can be created by tilting / displacing the solenoids or by additional dipole coils Nomenclature:  HFOFO = Helical FOFO channel of alternating solenoids (ASOL)  FOFO-xyz = FOFO with xyz resonance phase advance per focusing cell consisting of one solenoid + all other stuff (RF cavities etc.) Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

3. FOFO snake for final 6D cooling - original idea 3 + – Bz (T) 10*By (from Dec.2008 MCDW @ Jlab) Q1, Q2 p/100 z orbit lengthening x,y [cm] p/100 - no 2nd order chromaticity, - momentum compaction too small and of the wrong sign! z Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

4. Helical snake for final 6D cooling - the puzzle 4 By increasing B-field strength it is possible to get phase advance >180/cell and small -function at the solenoid center  much smaller emittance. Tune / period > odd_integer for resonant orbit excitation Puzzle: 2-cell period (planar snake), Q>1 6-cell period, Q>3 4-cell period, Q>3 6-cell period, Q>5 p < 0 significant over-focusing required p > 0 Naiive 1-dimensional considerations do not work for coupled motion: besides 180 phase advance of the normal mode there is also 180 rotation of the normal mode  the transverse field phase should change by either 360 or 0 / cell ! Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

5. Return to FOFO-180° 5 (from Dec.2009 MCDW @ BNL) Since there is no clear gain in going to 270° (just problems), more effort was applied to find solution for 180°. A superposed dipole field generated by additional coils can do the job. Such a field can not be obtained by displacing solenoids: Bz / BLS y x By / BLS Dy Dx Actually constant By works just as well! Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

6. FOFO-180° Momentum Acceptance 6 Q1 - 1 Q2 - 1 p/p0 p/p0 – 1 _min [cm] This design is not finished yet Search for a helical configuration is underway Emittances well below 0.5mm can be expected Attempts to equalize cooling rates of the transverse modes were unsuccessful so far p/p0 Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

7. Next Stop - HFOFO-240° 7 4/3 per cell  4  2 per 6-cell period: the resonance By harmonic is 4, but actually the 1rd harmonic is the most efficient! No big problem with the transverse mode cooling rate equalization: With solenoid pitch 1.4 mrad, constant quadrupole field of gradient 0.75T/m, 6mm LiH absorbers mode I II III tune 0.286+0.0024i 4.384+0.0049i 4.388+0.0048i _eq (mm) 0.74 0.38 0.40 But the momentum acceptance is too small: Q3 Q2 p/p0 Another difficulty: too high synchrotron tune with 6-cell period: Q1 ~ 0.3 Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

8. HFOFO-270° 8 Bz / Bz0 2.6 ! 1 2 4 3 r / Rin 3/2 per cell  3  2 per 4-cell period: the fundamental By harmonic is 1, the 3rd harmonic is working Original HFOFO-270° channel parameters (4-cell period): 800 MHz pillbox RF 2  8cm, Emax=32MV/m Solenoids: L=8cm, Rin=16cm, Rout=26cm, Inclination: 3mrad v+ h+ v- h- Absorbers: LiH, width (on-axis) 1.5cm, no wedge angle Total length of 4-cell period 1.12m Problems: Bzmax=18.5 T (on-axis) for p0=100MeV/c Limited dynamic momentum acceptance Large ratio Bz_coil/Bz_axis=2.6 ! B (V.Balbekov’s formulas used) Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

9. Choice of the Solenoid Geometry 9 Bz_coil/ Bz0 (left) and B3/Bz0 (right) vs coil inner radius Rin (cm). - Field on coil drops with smaller Rin, but nonlinearity grows. Bz_coil/ Bz0 (left) and B3/Bz0 (right) vs coil half-length L/2 (cm). - Both field on coil and nonlinearity decrease with L, but this leaves no room for RF! Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

10. Revised HFOFO-270° 10 600 MHz pillbox RF 2  8cm, Emax=17MV/m (on-axis) Solenoids: L=16cm, Rin=14.5cm, Rout=38cm (Bz_coil/Bz_axis=1.5) Inclination: 2.24mrad: v+, h+, v-, h- Absorbers: 2cm LH2 with 1mm LiH windows, no wedge angle Total length of 4-cell period 4  38cm = 1.52m Bz_axis=12.6T (Bz_coil=19T) for p0=90MeV/c With constant quadrupole field of gradient 1.5T/m mode I II III tune 0.191+0.0015i 3.254+0.0035i 3.288+0.0036i _eq (mm) 0.62 0.29 0.30 B (T)  (cm) y Bz 100Bx z (cm) x 100By z (cm) Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

11. Revised HFOFO-270° Momentum Acceptance 11  Static momentum acceptance is fine: (-9%,+27%), but the dynamic acceptance is only 25% (full width) – enough to accommodate the equilibrium momentum spread, but not the incoming beam  To increase dynamic acceptance one can: - reduce solenoid pitch angle (to reduce p) - reduce p0 (to increase 1/2) Both these methods are bad for longitudinal cooling –  Careful optimization is necessary Q3 Q2 p/p0 p0 = 80 p0 = 90 p/p0 - 1 Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

12. HFOFO-270° G4BL Simulations 12 x (cm) Precision check: periodic orbit red – MICCD, blue – G4BL z (cm) pz (MeV/c) y (cm) z (cm) z (cm) Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

13. HFOFO-270° G4BL Simulations 13 Phase portrait of initially Gaussian beam: blue – after 1st pass, red – after 16th pass (22.8m) px, py are mechanical momenta, p0=90MeV/c stochastics on, decays off px/p0 py/p0 (p - p0)/p0 x (cm) y (cm) t (ns) Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

14. HFOFO-270° G4BL Simulations 14 Fitted normalized emittances (final values 0.51mm, 0.36mm, 0.35mm), number of muons (N16/N1=908/5222=17.4%) and Palmer’s 6D cooling efficiency 1N Q6 2,3N pass # N pass # Total 6D cooling efficiency is just Q6=2: 50% cooling 50% shaving pass # Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

15. Summary & Outlook 15  FOFO-180 promises large momentum acceptance, DA and small emittances. However, no way to equalize the transverse mode cooling rates has been found yet. Still, there are some possibilities to explore.  HFOFO-270 can provide emittances T ~ 0.35mm, L ~0.5mm with LH2 absorbers, but the transmission is poor. It can be (probably) improved with nonlinear shape of the absorbers reducing ionization losses for low-momentum muons.  If both efforts fail, retreat below 180: the allowed maximum field of 19T (on the coil) will be achieved with smaller cell length  the increase in min and emittances will be moderate Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010

16. Plans (from MCDW09) 16  Search for the final stage HFOFO configuration (by IPAC10)  Front end with HFOFO (by April 2010 IDS meeting)  Determination of the number of stages and configuration for each stage (by the next MCDW)  Optimization of each stage parameters (2010-2012)  End-to-end simulation with matching between the stages (by 2012) Low- HFOFO Status - Y. Alexahin NFMCC Meeting Oxford, MS, January 14 2010