Epicyclic Helical Cooling Channels

1
Epicyclic Helical Cooling Channels

• Andrei Afanasev,
• Muons, Inc/Hampton U
• Low-Emittance Muon Collider Workshop
• Fermilab, June 8-12, 2009

2
Generating Variable DispersionModified
Helical Solenoid

• This is a demo of a beamline that satisfies
  conditions for alternating dispersion function
  needed for Parametric-resonance Ionization
  Cooling (PIC)
• Detailed simulations for the proposed beamline
  are in progress

3
PIC Concept

• Muon beam ionization cooling is a key element in
  designing next-generation high-luminosity muon
  colliders
• To reach high luminosity without excessively
  large muon intensities, it was proposed to
  combine ionization cooling with techniques using
  parametric resonance (Derbenev, Johnson, COOL2005
  presentation)
• A half-integer resonance is induced such that
  normal elliptical motion of x-x phase space
  becomes hyperbolic, with particles moving to
  smaller x and larger x at the channel focal
  points
• Thin absorbers placed at the focal points of the
  channel then cool the angular divergence of the
  beam by the usual ionization cooling mechanism
  where each absorber is followed by RF cavities

4
PIC Concept (cont.)
Comparison of particle motion at periodic
locations along the beam trajectory in transverse
phase space
Ordinary oscilations vs Parametric resonance
Conceptual diagram of a beam cooling channel in
which hyperbolic trajectories are generated in
transverse phase space by perturbing the beam at
the betatron frequency
5
PIC Challenges

• Large beam sizes, angles, fringe field effects
• Need to compensate for chromatic and spherical
  aberrations
• Requires the regions with large dispersion
• Absorbers for ionization cooling have to be
  located in the region of small dispersion to
  reduce straggling impact
• Suggested solution (Derbenev, LEMC08 AA,
  Derbenev, Johnson, EPAC08)
• Design of a cooling channel characterized by
  alternating dispersion and stability provided
• by a (modified) magnetic field of a solenoid
  avoid fringe fields by introducing a continuous
  periodic helical-type structure

6
Helical Cooling Channel (HCC)

• Proposed to use for 6D muon cooling with
  homogeneous absorber, see for details
• Derbenev, Johnson, Phys.Rev.ST Accel.Beams 8,
  041002 (2005)
• Under development by Muons, Inc.
• Y. Derbenev and R. Johnson, Advances in
  Parametric Resonance Ionization Cooling, ID 3151
  - WEPP149, EPAC08 Proceedings
• V. Kashikhin et al., Design Studies of Magnet
  Systems for Muon Helical Cooling Channels, ID
  3138 - WEPD015, EPAC08 Proceedings

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Helical Solenoid

• Helical Solenoid geometry and flux density for
  the Helical Solenoid magnet

8
Straight (not helical) Solenoid

• BzB0, BxBy0
• Dispersion 0
• Transverse-momentum
• cooling only

XY-plane
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Helical Solenoid

• Derbenev, Johnson, Phys.Rev.ST Accel.Beams 8,
  041002 (2005) proposed for 6D cooling of muons
• Dispersion const
• Orbit is constrained by a ring (adiabatic case)
• Periodic orbit is a circle for a particular
  choice of initial conditions(non-adiabatic case)

XY-plane
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Our ProposalEpicyclic Helical Solenoid

• Superimposed transverse magnetic fields with two
  spatial periods
• Variable dispersion function

XY-plane
k1-2k2 B12B2 EXAMPLE
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Epicyclic Helical Solenoid (cont.)

• Wave numbers k1,k2 may be opposite-sign or
  same-sign
• Beam contained (in the adiabatic limit) by
• Epitrochoid (same sign) - planets trajectory in
  Ptolemys system
• Hypotrochoid (opposite-sign)- special case is an
  ellipse

k12k2 B12B2
k1-k2 B12B2
k1-2k2 B12B2
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Particle Trajectory in SolenoidDouble-Periodic
Transverse Field
In the adiabatic limit k1-k2ltltkc
transverse-plane trajectory is bound by an
ellipse
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Periodic Orbits in EHS
,
.

• Equation of motion
• b1, b2 - transverse helical fields with spatial
  periods described by wave-number parameters k1
  and k2,
• Cyclotron wave number
• Approximation pzconst gives analytic solution
  for eqs. of motion (Note it is a particular
  solution of a non-homogeneous Diff.Eq. in
  practice choose initial conditions on particle
  kinematics such that general solution of
  corresponding homogeneous equation does not
  contribute)

.
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Oscillating Dispersion
.

• The dispersion function for such an orbit is a
  superposition of two oscillating terms,
• The dispersion function has nodes if
• For example, k1-k2kc/2, then B29B1 , solution
  for the orbit is
• Dispersion function is periodic and sign-changing
• Numerical analysis with Mathematica (with no
  pzconst approximation) gives close results

15
Stability of Reference Orbit

• Following DerbenevJohnson (2005)

• Periodic reference orbit solution exists for
  sgn(kc)/-sgn(k), resulting in different
  stability regions
• Same-sign solution was considered for HCC so far
  (stable for kappa1)
• Opposite-sign solution (for larger transverse
  helical field component) is needed to obtain
  oscillating dispersiongt lower-kappa region
  (0.5) is stable

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Transverse-plane Trajectory in EHS
B1?0, B20 (HS) ? B1?0, B2?0
(Epicyclic HS)
p?p?p
k1-k2kc/2 k1-k2/2kc/4

• Change of momentum from nominal shows regions of
  zero dispersion
• and maximum dispersion
• Zero dispersion points Locations of plates for
  ionization cooling
• Maximum dispersion Correction for aberrations

17
Periodic orbit in Epicyclic HS
HS
Epicyclic HS

• Conditions for the periodic orbit need to be
  satisfied

-Derbenev, Johnson, PRST (2005) gt AA, Derbenev
(2008)
Numerical analysis (using Mathematica for
tracking) Transverse-plane periodic orbit is
elliptic at low kappa (0.1-0.2), but deviates
from ellipse at large kappa (gt1)
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G4Beamline Tracking

• G4BeamLine simulation of muon trajectories in the
  epicyclic helical solenoid field in lab (left
  plot) and its projection on the transverse plane
  (right plot).
• Muon momentum1005 MeV/c muons with the
  parameters chosen as follows k1-k2 -kc/2,
  Bz7T, B1/B29. Example using simplified magnetic
  fields.

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G4Beamline Tracking (cont)

• G4BeamLine simulation of muon beam transport in
  EHS using realistic magnetic fields.
• Muon momentum250 MeV/c, momentum spread 25
  MeV/c (top plot) and 12.5 MeV/c (bottom plot).
• Transverse-to-longitudinal momentum ratio 0.15
  on a minor axis.

p250/-25 MeV/c
p250/-12.5 MeV/c
20
Designing Epicyclic Helical Channel

• Solenoiddirect superposition of transverse
  helical fields, each having a selected spatial
  period
• Or modify procedure by V. Kashikhin and
  collaborators for single- periodic HCC V.
  Kashikhin et al., Design Studies of Magnet
  Systems for Muon Helical Cooling Channels, ID
  3138 - WEPD015, EPAC08 Proceedings
• Magnetic field provided by a sequence of
• parallel circular current loops with centers
• located on a helix
• (Epicyclic) modification
• Circular current loops are centered
• along the parametric curves.
• The simplest case will be an ellipse (in
  transverse plane)
• Calculations with Mathematica validated this
  concept (AA)

21
Solution with Kashikhin Method
Bz
Field generated by a series of circular current
loops with centers displaced following an
ellipse in transverse plane Maximum displacement
(from a circle) is 1/9
Bx,y
Major difference from Helical Solenoid reference
trajectory has a different eccentricity compared
to the line that guides centers of current loops
22
Implementing in PIC

• Plan to develop an epicyclic helical solenoid as
  part of PIC cooling scheme and Reverse-Emittance
  Exchange
• Design concept similar to a single-period Helical
  Solenoid
• But needs more periods per channel to develop
  parametric resonance
• Elliptic option looks the simplest
• But ellipses have different eccentricity for
  (transverse) reference orbits vs centers of
  magnetic loops that form an epicyclic Helical
  Solenoid
• Detailed theory, numerical analysis and
  simulations are in order AA, Derbenev
  TheoryNumerical analaysis
• K. Yonehara, V. Ivanov, G4BL simulations