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

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

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

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

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

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

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

Solenoid cell (Dfx = p = Dfy) c1 L[cm]=130 B[kG]=32.4 Aperture[cm]=10 c2 L[cm]=80 B[kG]=-34.1 Aperture[cm]=10 Alex Bogacz, Workshop on Muon Collider Simulations, Miami Beach, FL December 15, 2004

‘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

‘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

‘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

‘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

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

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

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

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

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

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

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

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