1. 11 th Int. Workshop on Neutrino Factories, Superbeams & Beta Beams, Fermilab/IIT, July 20-25, 2009 Helical FOFO Snake Update Y. Alexahin (FNAL APC)

2. Basic Idea 2 absorbersRF cavitiesalternating solenoids z [cm] B x  50 B [T] BzBz B y  50 x, y [cm] x z [cm] y x y The idea: create rotating B  field by periodically tilting solenoids, e.g. with 6- solenoid period. Periodic orbits for μ+ and μ- look exactly the same, just shifted by a half period (3 solenoids). With tune Q  >1 (per period) r  D>0  muons with higher momentum make a longer path  longitudinal cooling achieved even with planar absorbers Periodic orbit for p=200MeV/c HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

3. Why double cavities? 3 solenoids Double cavity increases packing factor w/o reduction in the transit factor Hopefully the breakdown RF gradient will not be much reduced due to magnetic field with such configuration HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

4. Beta-functions & tunes 4 The best results with 7mrad pitch angle, no absorber wedge angle: modeIIIIII tune 1.239+0.012i1.279+0.007i0.181+0.002i  _eq (mm)3.24.56.9 ImQ=0.007  cooling rate d log  / dz = 2  2  /L ImQ = 1/70m Bmax=2.3T  j=58A/mm^2, Itot=4.4MA/solenoid There is difficulty in equalization of damping rates of the transverse modes yy xx at the absorber locations ~ 70cm - compare with MICE’s ~ 45cm Solenoids: L=24cm, Rin=60cm, Rout=92cm, pitch 7mrad, B z max=2.35T (p=200MeV/c) RF: 200 MHz pillbox 2x36cm, Emax=16MV/m Absorbers: 15cm LH2 planar HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

5. Tracking with MICCD 5 z-v 0 t x [cm]y [cm] pp P y /p 0 blue - initial, red – after 25 periods (153m) “True” action variables of 1771 particles evenly distributed in tetrahedron (J I + J II )/2.6 + J III /4 < 1 (cm) Phases chosen at random No decay or stochastic processes Courant-Snyder invariant =2J, to compare with normalized emittance multiply by  2:  CSImax  10cm or 2.2  for   N =2cm canonical momenta (include vector-potential), p 0 =200MeV/c P x /p 0 HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

6. Tune “repulsion” from integer resonance 6 no absorber, no RF p/200 orbit length/L 0 Q I, II Nice surprise: Large 2nd order chromaticity due to nonlinear field components keeps both tunes from crossing the integer ! p/200 - in contrast to classical HCC with homogeneous absorber where to ensure longitudinal damping. Momentum compaction factor: HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

7. Excursion - Longitudinal Hamiltonian 7 pp  -  0 (rad) Contour plot of the longitudinal Hamiltonian for =0.014 - kinematic nonlinearity limits momentum acceptance, not insufficient RF bucket depth. where = orbit length/L 0 Kinetic energy in quasi-static approximation =0.014 K pp =0.007 U RF Longitudinal separatrix reaches maximum at  0.007 HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

8. Tracking with G4BL 8 blue - initial (the same ensemble), red – after 20 periods (122m) p y /p 0 (p-p 0 )/p 0 p x /p 0 x [cm] y [cm] t [ns] Stochastic processes on, but no decays mechanical momenta, p 0 =200MeV/c HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

9. Analysis of G4BL Tracking Data 9 Normal mode emittances (normalized) vs period # (  _eq = 0.32, 0.45, 0.69) G4BL stochs. on MICCD N/N 0  2N G4BL stochs. off  N [cm]  1N  3N no decays!  With decays transmission over 20 periods (122.4m) is > 85%  Initial/final  N [cm]=1.7, 1.8, 2.1 / 0.37, 0.7, 1.1  6D cooling efficiency (following R.Palmer’s definition) Q 6  20  multiply  m by  0  0 Recipe for emittance calculation:  compute normal mode amplitudes a m for each muon using lattice eigenvectors  subtract average values  find distribution in action variables J m =| a m | 2  fit with exponential function Transmission vs period # HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

10. Equalization of transverse modes damping rates  R. Palmer proposed to add ~ constant solenoidal field to better mix the transverse modes and equalize their damping rates.  This can be achieved by powering e.g. negative solenoids with slightly lower current.  With just (I + - I - )/ I + = 1.6% : modeIIIIII tune 1.211+0.0100i1.301+0.0108i0.196+0.0003i  _eq (mm)3.83.236.5 (fast transverse cooling w/o longitudinal blowup was the intent): 10 yy z xx Unfortunately, there is no appreciable gain in equilibrium emittance due to large  -wave excited with unequal field in + and - solenoids. Transmission also suffers (red vs. blue) - computed without stochastic effects and decay N/N0N/N0 period # HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

11. Damping Rate Equalization with Quadrupole Field 11 HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009 Most efficient “equalizer” is constant quadrupole field superimposed on the main solenidal field. With a gradient of just 0.057 T/m: modeIIIIII tune 1.240+0.0088i1.286+0.0087i0.178+0.0036i  _eq (mm)3.93.84.7 (w/o quad was3.24.56.9) z  y (cm) Dy (cm) z Dx (cm)  x (cm)  The sum of betas smaller - no increase in transverse emittance (despite smaller long. emittance)  50% stronger long. damping due to larger dispersion (smaller solenoid pitch required)  The effect does not depend on quad tilt (and polarity) as well as on the muon sign - the channel is still good for both!

12. G4BL Simulations with Quad “Equalizer” 12 HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009 1.71, 1.79, 2.20 / 0.554, 0.584, 1.006 1.66, 1.80, 2.05 / 0.367, 0.701, 1.096 N/N0N/N0 period #  2N  3N  1N  2N  3N with quad There is not so much reduction in final emittances due to the damping rate equalization as increase in the initial values due to changes in nonlinear transformation from “ true ” normal forms (Here emittances in cm) w/o quad with quad w/o quad: Initial / final emittances (cm) with quad: w/o quad

13. Absorber Container and RF Cavity Walls 13 HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009 A MODERN AIRCRAFT ALUMINUM ALLOY (thanks to Don Summers) Aluminum Alloy Yield Tensile Radiation Name Composition Density Strength Strength Length % by weight g/cc ksi ksi cm 300K 300K 20K 6061-T6 1.0Mg.6Si.3Cu.2Cr 2.70 40 45 68 8.86 2090-T81 2.7Cu 2.2Li.12Zr 2.59 74 82 120 9.18 Supposedly 2090-T81 walls can be made 0.1mm thick (as compared to 0.18mm of 6061-T6 in MICE). Three cases considered below: No container nor cavity walls (original lattice) Two 0.1mm 2090-T81 container walls / absorber, no safety walls nor cavity walls Two 0.1mm 2090-T81 container walls / absorber + three 0.38mm Be walls / two RF cavities Q 1 Q 2 Q 3  1N (mm)  2N (mm)  3N (mm) E RF (MV/m) No walls1.239+0.012i 1.279+0.007i 0.181+0.0025i 3.2 4.5 6.9 16.0 Abs. walls 1.241+0.012i 1.281+0.007i 0.185+0.0026i 3.7 5.1 7.5 16.3 + RF walls1.243+0.013i 1.286+0.007i 0.191+0.0033i 4.0 5.8 7.3 17.5 Though the effect of the walls is tolerable (<30% increase in the equilibrium emittance) it would be worth while to look for a lighter material for the absorber walls (Be?)

14. G4BL Simulations with Walls 14 HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009 Final emittances for the cases 1 and 3:  1N (mm)  2N (mm)  3N (mm)  6D (mm^3) No walls3.677.0110.96 218.2 With all walls3.637.7710.22 212.9 - Strange result, may be due to a faster cooling with 9% increase in losses and the RF gradient period #  1N  2N  3N N/N0N/N0 with walls w/o walls

15. Summary & Outlook 15  Tilting solenoids of ASOL channel (Dave ’ s nomenclature) increases momentum acceptance and provides some longitudinal cooling with only mild deterioration of transverse acceptance and cooling (look for the optimum!)  Transverse damping rates can be equalized by adding quadrupole field without compromising transverse or momentum acceptance  HFOFO with 200MHz double RF gives  T < 6mm,  L ~10mm  Another 200MHz section with single RF can give  T ~ 3mm,  L ~10mm  Combine HFOFO cooling with Daves bunching / rotation  Higher phase advance per cell  small betas @ absorbers  Better options for the absorber and RF cavity windows? HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009