1. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010 1 Update on Parametric-resonance Ionization Cooling (PIC) V.S. Morozov Old Dominion University V. Ivanov, R.P. Johnson, M. Neubauer Muons, Inc. A. Afanasev Hampton University and Muons, Inc. A. S. Bogacz, Y.S. Derbenev Thomas Jefferson National Accelerator Facility K. Yonehara Fermi National Accelerator Laboratory Muons, Inc.

2. 2 PIC Concept 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 Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

3. Varying 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 – one free oscillations’ period low-integer multiple of the other - / + = 1 or 2 – dispersion magnitude oscillation period D factor of 2 shorter than + Required features can be produced by epicyclic magnetic field configuration – solenoid with two superimposed different-period transverse helical fields ▫ uniform smoothly-varying fringe-field-free configuration 3 PIC Requirements Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

4. 4 Epicyclic Channel Two transverse helical fields with wave numbers k 1 and k 2 Equation of motion Analytic solution under approximation k c = const ( p z = const ) Dispersion function containing two oscillating terms Condition for dispersion to periodically return to zero Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

5. 5 Approach to Designing Correlated Optics Consider second helix perturbation Adjust desired free-oscillation period ratio - / + = 1 or 2 in primary helix By choosing wave number k 2 of second helix, set dispersion oscillation period D = |2  /(k 2 -k 1 )| such that + / D = 2 Adjust strength of second helix to create oscillating dispersion Iteratively adjust - / + and + / D by changing helices’ parameters until correlated optics is achieved Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

6. 6 Single Helix Equilibrium condition Orbit stability condition Betatron tunes For given r = Q + /Q -, one can solve for ∂b/∂a if Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

7. No solution for Q + = Q - Two solution regions for Q + = 2Q - –|  | << 1, -2 < q < -1, B 2 /B 1 ~ 1 –|  | >> 1 Choose:  = -5.4 q = -1.54 B sol = 2 T b d = -0.154 T b q = 0.065 T/m k c = 32.9 m k = -61.0 m Q - = 0.464, Q + = 0.929 B 2 /B 1 ~ 0.04 7 Adjusting Betatron Tunes Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

8. Using one-period linear transformation matrix Track particle over many periods and take Fourier transform of coordinate vector component 8 Determining Betatron Tunes Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

9. 9 Finding Periodic Orbit No exact analytic solution in case of two helices Stable periodic orbit does not always exit Begin with single helix where stable periodic orbit is knows to exist Use one or combination of the following to find periodic orbit when second helix is present – Adiabatically increase strength of second helix while tracking orbit – Use “friction” force making particle trajectory converge to periodic orbit – Increase second helix’s strength from zero in steps finding periodic orbit iteratively on each step Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

10. 10 Dispersion in Epicyclic Channel Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010 Second helix strength | D | = const | D | oscillates not reaching 0| D | oscillates reaching 0

11. 11 Periodic Orbit in Epicyclic Channel Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

12. Consider: no solenoid, two helices of equal strengths with equal-magnitude and opposite-sign wave numbers Field periodic with = 2  / k, Vertical field only at any point in horizontal plane Periodic orbit lies in horizontal plane 12 Another Option for PIC Channel Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

13. More conventional orbital dynamics problem Horizontal and vertical motion uncoupled Magnetic structure accommodates both muon charges Transverse motion stable in both dimensions Dispersion has oscillatory behavior 13 Periodic Orbit and Dispersion Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010

14. Eliminates scattering on pressurizing gas and cavity wall while enhancing accelerating voltage 14 RF Cavity Concept for EPIC & REMEX Muons, Inc. The 2010 NFMCC Collaboration Meeting University of Mississippi, January 13-16, 2010 Open CellEPIC/REMEX Cell Closed Pillbox Cell BEAM Beryllium grids Irises Be wedge Thermal stabilizer