1. Parametric Resonance Ionization Cooling of Muons
Alex Bogacz*
in collaboration with
Kevin Beard*, Slava Derbenev* and Rol Johnson
*Jefferson Laboratory
 Muons Inc.
7-th International Workshop on Neutrino Factories and Superbeams, LNF Frascati, June 21, 2005
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

2. Overview
Final transverse ionization cooling - Parametric resonance enhancement
Resonant transport channels - lattice prototypes
quadrupole based
solenoid based
Transverse beam dynamics in the cooling channel – tracking studies
‘soft-edge’ solenoid
linear transfer matrix
nonlinear corrections (in tracking)
thin ‘ideal’ absorber model
Chromatic aberrations - compensation of the detuning effects with RF
tracking studies
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

3. Transverse parametric resonance cooling
Transport channel (between consecutive absorbers) designed to replenish large angular component, x’, sector of the phase-space, ‘mined’ by ionization cooling process.
Parametric resonance in an oscillating system - perturbing frequency is equal to the harmonic of the characteristic (resonant) frequency of the system, e.g half-integer resonance
Normal elliptical motion of a particle’s transverse coordinate in phase space becomes hyperbolic – resulting beam emittance has a wide spread in x’ and narrow spread in x – sector of the phase-space where ionization cooling is most effective
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

4. Transfer matrix of a periodic resonant lattice
Symplectic transfer matrix, M(s), for a beamline (in x or y)
Lattice period can be designed in such a way that sin  = 0   = n , n = 1, 2.
Coordinate and angle are uncoupled - resulting beam emittance has a wide spread in x’ and narrow spread in x.
x x’ = const
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

5. Symmetrized double cell (Dfx = 3p = Dfy)
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

6. Angular ‘shearing’ of the transverse phase-space
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

7. 4-cell resonant channel
absorber
Uniform triplet lattice resonantly perturbed by a singlet
Absorber placed half way between triplets
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

8. Solenoid cell (Dfx = p = Dfy)
c L[cm]= B[kG]= Aperture[cm]=10
c L[cm]= B[kG]= Aperture[cm]=10
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

9. ‘soft-edge’ solenoid model
Zero aperture solenoid - ideal linear solenoid transfer matrix:
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

10. ‘soft-edge’ solenoid – edge effect
Non-zero aperture - correction due to the finite length of the edge :
It decreases the solenoid total focusing – via the effective length of:
It introduces axially symmetric edge focusing at each solenoid end:
axially symmetric quadrupole
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

11. ‘soft-edge’ solenoid – nonlinear effects
Nonlinear focusing term DF ~ O(r2) follows from the scalar potential:
Scalar potential in a solenoid
Solenoid B-fields
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

12. ‘soft-edge’ solenoid – nonlinear effects
In tracking simulations the first nonlinear focusing term, DF ~ O(r2) is also included:
Nonliner focusing at r = 20 cm for 1 m long solenoid with 25 cm aperture radius
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

13. Thin absorber with re-acceleration
Ionization cooling due to energy loss (-Dp) in a thin absorber followed by immediate re-acceleration (Dp) can be described as:
The corresponding canonical transfer matrix can be written as
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

14. Final cooling – initial beam parameters
p = 287 MeV/c
after helical cooling channel
erms
normalized emittance: ex/ey
mmmrad 30
longitudinal emittance: el
(el = sDp sz/mmc)
momentum spread: sDp/p
bunch length: sz mm
0.8
0.01
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

15. Solenoid cell, with absorber – 6D tracking
each absorber: Dp/p = 0.05
4 cm Be Dp = 14 MeV/c
detuning effect - the momentum-dependent betatron frequency causes off- momentum particles to be out of resonance with the focusing lattice
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

16. Solenoid cell, with absorber and RF– 6D tracking
each absorber: Dp/p = 0.05
4 cm Be Dp = 14 MeV/c
by choosing suitable synchrotron motion parameters, the resonance condition can be maintained
synchrotron phase advance of 2p/8 per cell – two RF cavities at zero-crossing (cavity gradient: 17.3 MeV/m at 400 MHz)
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

17. Chromatic aberration compensation with RF– cells 1-4 top plots: NO RF, bottom plots: with the RF
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

18. Chromatic aberration compensation with RF– cells 5-8 top plots: NO RF, bottom plots: with the RF
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

19. Solenoid cell – G4BL view
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

20. Summary Present status… Future work…
Prototype PIC lattices - quadrupole and solenoid channels
Lattices with absorbers studied via transfer matrix code
Beam dynamics studies vie multi-particle tracking
Building and testing G4BL tools
Chromatic aberration compensation with synchrotron motion
Proof-of-principle transport code tracking (solenoid triplet channel)
Future work…
G4beamline simulation of a qudrupole/solenoid channel with absorbers
G4beamline simulation with absorbers followed by RF cavities - include multiple scattering and energy straggling effects
Emittance calculation - implementation of ecalc9
Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004