1. 1 EPIC SIMULATIONS V.S. Morozov, Y.S. Derbenev Thomas Jefferson National Accelerator Facility A. Afanasev Hampton University R.P. Johnson Muons, Inc. Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

2. Outline Concept of Parametric-resonance Ionization Cooling (PIC) PIC linear optics requirements Epicyclic twin-helix channel for PIC – Magnetic optics design – Possible practical implementation G4beamline simulations of twin-helix channel – Cooling with wedge absorbers followed by regions of static electric field – Effect on the orbit and its compensation – Timing of rf cavities – Cooling with wedge absorber and rf cavities Conclusions and future plans 2 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

3. Parametric resonance induced in muon cooling channel Muon beam naturally focused with period of free oscillations Wedge-shaped absorber plates combined with energy-restoring RF cavities placed at focal points (assuming aberrations corrected) – Ionization cooling maintains constant angular spread – Parametric resonance causes strong beam size reduction – Emittance exchange at wedge absorbers produces longitudinal cooling Resulting equilibrium transverse emittances are an order of magnitude smaller than in conventional ionization cooling 3 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc. PIC Concept

4. Resonant dynamics: angular spread grows while beam size shrinks Absorbers keep angular spread finite Absorbers Optics to restore parallel beam envelope Beam envelope without absorbers PIC Principle 4 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

5. Equilibrium angular spread and beam size at absorber Equilibrium emittance (a factor of improvement) w Absorber platesParametric resonance lenses PIC Schematic 5 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

6. PIC Channel Optics Requirements 6 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc. Horizontal free oscillations’ period x equal to or low-integer multiple of vertical free oscillations’ period y Oscillating dispersion – small at absorbers to minimize energy straggling – non-zero at absorbers for emittance exchange – large between focal points for compensating chromatic and spherical aberrations  Correlated optics: correlated values of x, y and dispersion period D – x = n y = m D, e.g. x = 2 y = 4 D or x = 2 y = 2 D Fringe-field-free design

7. Practical fringe-field-free approach Periodic solutions of source-free Maxwell equations in vacuum Harmonic of order n given by Total field Helical Harmonics 7 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

8. Consider two dipole helical harmonics ( n = 1 ) of equal strengths with equal- magnitude and opposite-sign wave numbers k 1 = - k 2 = 2  / Field periodic with = 2  / k, Vertical field only in horizontal plane Twin Helix 8 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

9. Vertical field only in horizontal plane  Periodic orbit in horizontal plane Horizontal and vertical motion uncoupled Region of stable transverse motion in both planes Periodic Orbit and Dispersion 9 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

10. Superimpose straight quad to redistribute horizontal and vertical focusing D =  x = 2 y = 4  x = 0.25, y = 0.5 Down side: cannot satisfy correlated optics conditions for both charges Adjusting Correlated Optics 10 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

11. Dispersion: Chromaticity: Scaling pattern: Dispersion and Chromaticity 11 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

12. Layer of positive-helicity helical conductors with cos  azimuthal current dependence Layer of negative-helicity helical conductors Normal quad Possible Practical Implementation 12 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

13. Layer of positive-tilted loops with cos z longitudinal current dependence Layer of negatively-tilted loops Normal quad Adopt existing technology? Possible Practical Implementation 13 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

14. G4beamline Simulations 14 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

15. Going InComing Out No absorber and no RF 10 5 100 MeV/c  - through 100 periods of “twin helix” with correlated optics Initially parallel beam uniformly distributed with 10  10 cm square Dynamical Aperture Test 15 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

16. 2 cm thick Be wedge absorber with 0.3 thickness gradient 1 m helix period, absorbers placed every 2 periods ( x = 0.25 ) at points with 3 cm dispersion for appropriate distribution of cooling decrements Timing of rf cavities is not straightforward, absorbers are followed by regions of static electric field adjusted to compensate energy loss of 2 cm Be Energy recovery regions are short (2 cm) to decouple from transit time effects and reduce optics perturbation In practice, as much space as possible should be taken up by absorbers and rf Absorber / Energy Recovery Model 16 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

17. 10 3 200 MeV/c muons, uniform  x =  y =  2 mm,  x =  y =  50 mrad,  p/p =  2.7% Beam started along the “unperturbed” periodic orbit, however, the orbit has changed due to absorbers / energy recovery, this causes initial mismatch Stochastic processes are off Beam Cooling Simulation 17 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

18. Periodic momentum changes Longitudinal Cooling 18 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

19. Particle tracked over many periods until cooling makes it converge to new periodic orbit Particle observed at the same point within 2 period (~  2 in front of each absorber) New Periodic Orbit 19 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

20. Cooling Process in Phase Space 20 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc. Starting point New periodic orbit

21. Conceptually different picture when electric field is adjusted to restore the original momentum corresponding to correlated optics Near Correlated Optics 21 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc. Steady oscillations established

22. Particle still cools but the phase space splits into two islands Stable resonance? Phase Space View 22 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

23. Similar picture at 250 MeV/c Electric field is tuned to give periodic momentum close to 250 MeV/c Correlated Optics at 250 MeV/c 23 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

24. Phase Space View 24 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

25. Blue: trajectory with magnetic field only Red: trajectory with absorbers / energy recovery Periodic Trajectory in Phase Space 25 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

26. Reference particle becomes unstable, rf cavities’ timing set manually,  s = 30  The trajectory is not perfect but exhibits the same characteristic behavior Tracking Single Particle with RF Cavities 26 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

27. Phase Space View 27 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

28. 10 3 200 MeV/c muons, uniform  x =  y =  6 mm,  x =  y =  50 mrad,  p/p =  2.7%,  t =  0.04 ns Stochastic processes are off Beam Cooling with Absorbers / RF 28 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

29. Longitudinal Cooling 29 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.

30. Cooling channel’s correlated optics is well-understood The basic model with wedge absorbers and rf cavities is in place Cooling simulations initiated Next steps – Phase space dynamics with absorbers and rf in the correlated optics case needs to be understood – Induce parametric resonance probably using lumped quadrupoles – Turn stochastic processes on – Look into aberration compensation, there is a well-understood approach to correcting at least chromatic aberrations – Compare final emittances in case of conventional ionization cooling and PIC Conclusions and Future Plans 30 Operated by JSA for the U.S. Department of Energy Muon Accelerator Program - Winter Meeting, March 1, 2011 Muons, Inc.