Epicyclic Helical Channel for Parametricresonance Ionization Cooling PIC

1
Epicyclic Helical Channel for Parametric-resonanc
e 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

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

3
PIC Schematic

• Equilibrium angular spread and beam size at
  absorber
• Equilibrium emittance
• (a factor of
  improvement)

4
PIC Requirements

• 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
• normal-mode oscillations periods must be
  low-integer multiples of dispersion magnitude
  oscillation period ?1,2 / ?D 1 or 2
• 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

5
Simplified PIC Model

• Two transverse helical fields with wave numbers
  k1 and k2
• Equation of motion
• Cyclotron wave number
• Analytic solution under approximation kc const
  (pz const)

6
Oscillating Dispersion

• Dispersion function containing two oscillating
  terms
• Condition for dispersion to periodically return
  to zero
• k1? k2? kc/2 ? k gives B1? 9B2 and elliptic
  orbit
• Dispersion function

7
Oscillating Dispersion (Cont.)
p?p?p
absorbers
aberration correction
8
EPIC Based on HCC

• ? ? pT /pz 1 gives more complicated picture
• Since B2 ltlt B1 , consider secondary helix
  perturbation
• Start with single-periodic Helical Cooling
  Channel (HCC)
• exact analytic solution
• well-studied in simulations
• orbit stability conditions

9
HCC Stability Region Normal-Mode Tunes

• In usual HCC sgn(k) sgn(kc), ? gt 0
• In EPIC channel sgn(k) ?sgn(kc), ? lt 0

10
Adiabatic Turn On of Secondary Helix
11
Effective Force Approach

• Introduce effective friction force
• total energy conserved while phase volume is not
• analogous to cooling
• all trajectories converge towards periodic orbit
• aids in finding periodic orbit when analytic
  solution is not available

12
EPIC Dispersion in Realistic Fields
G4BeamLine p 250 12.5 MeV/c low ? for primary
helix
Matlab p 100 1 MeV/c ? 1 for primary helix
13
Possible Technical Implementations

• Solenoid with direct superposition of required
  helical harmonics
• Adopt procedure developed for HCC by Kashikhin et
  al.
• circular current loops centered on elliptical
  helix (different from periodic orbit)

14
Incorporating RF in 2-period HS

• Hydrogen-pressurized RF cavity
• Be wedge included in the walls replaces grids
  used with HPRF
• many of required technical solutions already
  exist (using ceramic ring to adjust frequency,
  small-diameter power feed through)
• or Tapered no-RF cooling blocks with RF sections
  in between
• Transverse-longitudinal coupling is minimal

15
Recent Progress

• Found magnetic field configuration giving
  required dispersion
• uniform
• smoothly-varying
• fringe-field free
• Showed preliminary demonstration of needed
  features
• Developed technique to identify periodic orbit
• straightforward to find normal-mode tunes
  numerically using single-period orbital transfer
  matrix
• Came up with ideas for technical implementation
• Have ideas how to incorporate RF
• Started preliminary G4BeamLine simulations

16
Work Plan

• Numerical studies of orbital motion in epicyclic
  (double-periodic) channel
• Use developed tools to determine stability region
  around periodic orbit
• Determine characteristic parameters such as
  normal-mode tunes
• Learn to control parameters of motion with
  quadrupole and higher-order magnetic fields
  components
• Identify configuration and strengths of sextupole
  and octupole field components for aberration
  corrections
• Demonstrate in full-scale G4BeamLine simulations
• compensation of chromatic and spherical
  aberrations
• cooling and emittance exchange with wedge
  absorbers
• Explore possible technical solutions
• Study implementation of RF cavities as part of
  PIC channel design taking into account scattering
  in pressurizing gas if needed
• Investigate space-charge limitations

Muon Collider Design Workshop BNL, December 1-3,
2009
16