FODO-based Linear Quadrupole PreCooler

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FODO-based Linear Quadrupole PreCooler

• M. Berz, D. Errede, C. Johnstone, K. Makino D.
  Neuffer, and A. Van Ginneken, A. Tollestrup

WG1, July 3 NuFACT02 Imperial College,
London July 1-6, 2002
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Contents

• Basics
• What emittances are required for a neutrino
  factory?
• Equilibrium emittances in cooling channels
• General staging remarks for cooling

3
What emittances drive the cooling in a neutrino
factory

• Storage ring design ?n(rms)
• pjk 50-GeV design 1.5 mm rad
• Fermi 50-GeV design 3.2 mm-rad
• RLAs
• Study I 1.1 mm-rad
• Study II lt2 mm-rad
• Required cooled emittance is 1-2 mm-rad for
  U.S. baseline acceleration scenario
• For Japanese FFAG scenario, emittance requirement
  is 1 cm-rad, almost an order of magnitude larger

4
The equilibrium emittance is given by ?N,min
??(14 MeV)? (??m?LR.d??ds)
where ??is the transverse beta function at the
absorber, ? the relativistic velocity, m? the
mass of the muon, LR the radiation length of the
absorber material, and dE/ds the energy lost in
the absorber.
To achieve a minimum equilibrium emittance of 1.7
mm-rad _at_200 MeV/c , ?? has to be 0.4 m. This
is to be compared to the rms emittance after
capture of 20 mm-rad, indicating a factor of 10
requirement in transverse cooling for the U.S.
Scenario (factor of 2 for Japanese acceleration
scenario.
5
Staging the cooling

• Clearly a factor of 10 decrease in transverse
  emittance is difficult using a single cooling
  structure
• large emittances, low betas large apertures in
  elements (?max ? 1/ ??)
• how would you effectively stage an order of
  magnitude in cooling
• Look at the cooling dynamics

6
Full simulation transverse cooling along Quad
channel
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Observations

• Cooling rates are constant until the
• 1.5 x ?equilibrium
• lowering ?? does not significantly increase the
  cooling rate
• Conversely, raising the initial uncooled
  emittance
• ?? can be scaled upwards also
• very significant for cost and optical design of
  cooling at ultra-large emittances after capture.

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A staging strategy

• Emittance-dependent
• Factor of 2 cooling/plane per stage
• ?n rms (mm-rad) 20 ? 10 ? 5 ? 2.5
• ??(m) 1.5 ? 0.75 ? 0.4
• Factor 2 2 2
• practical limit for ?? of 0.4 m
  FFAG scenario

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Choice of optical cooling structure

• With the much large ?? one can start thinking
• nonsolenoidal structures, i.e. quadrupoles
• removing components from apertures of elements,
• hopefully smaller apertures lower costs
• nonsuperconducting elements

10
Quadrupole Cooling Channels General
Considerations

• Longitudinal and and Transverse Acceptance of
  Channel
• Compare acceptances of different optical
  structures
• Insertion of Absorber
• Location of minimal beam size, both planes
• Calculate equilibrium emittance limit
• Physical Limitations
• Quadrupole aperture and length constraints
• Available space between magnets
• Match to emittance-exchange channel
• Can same optical structure be used for emittance
  exchange
• not required for FFAG scenario

11
Optical Structures for Ultra-large Emittance Beams

• FODO cell
• Simple-lens/Alternating Gradient systems have the
  largest combined transverse and chromatic
  acceptance of any optical structure for
  light/magnetic optics.
• Generally, acceptance is limited only by the
  physical apertures of the components
• Before proposing a channel--
• What constraints should be imposed on quadrupole
  design?

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Quadrupole Design

• Aperture vs. length role of fringe fields
• Maximum poletip field
• Design Constraints
• Maximum aperture Magnet Length
• Normal Conducting Quads Poletip Field lt 2T
• Separated Components rf, in particular removed
  from component apertures increasing acceptance
  and decreasing component cost

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Consequences of hoice of Optical Structure

• FODO
• Simplest alternating focussing and defocussing
  (in one transverse plane) lenses
• A minimum in beta or beam size cannot be achieved
  simultaneously in both planes
• Doublet or Triplet quadrupole system
• 2 or 3 consecutive, alternating focussing and
  defocussing quadrupoles
• required to form simultaneous low beta points in
  both planes (interaction regions of colliders,
  for example)

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Acceptance of Quadrupole Channels

• Transverse Acceptance
• Using only linear elements (quadrupoles and/or
  dipoles), the transverse dynamic aperture is
  normally larger than the physical aperture
  (unless a strong resonance is encountered in a
  long series of cooling cells).
• Practically, FODO and doublet/triplet quadrupole
  channels have transverse acceptances or apertures
  limited only by poletip strength for a given
  gradient (?8T for superconducting quads and ?2T
  for normal conducting).
• Longitudinal Acceptance
• The FODO cell is a simple lens system and has the
  largest chromatic acceptance of any
  quadrupole-based structure. Lattices based on
  FODO cells have been designed which transmit up
  to a factor of 4 change in momentum.
• Doublet/triplet-based quadrupole structures are
  momentum limited to approximately ?5 deviation
  from the central momentum of the channel.
• Logistics
• Because the FODO cell cannot achieve a minimum
  beta point in both planes, its valid application
  is just after capture and phase rotation, where
  the transverse and longitudinal emittances are
  very large.
• Conversely, the limited momentum acceptance of
  the triplet/doublet quadrupole channels restrict
  their implementation to after emittance exchange
  has occurred.
• The rest of this talk will discuss
  the FODO cooling channel only

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Construction of FODO Quad Cooling Cell

• 1/2
  
  1/2
• abs F rf D
  rf F rf D abs
• COOLING CELL PHYSICAL PARAMETERS
• Quad Length 0.6 m
• Quad bore 0.6 m
• Poletip Field 1 T
• Interquad space 0.4 - 0.5 m
• Absorber length 0.35 m
• RF cavity length 0.4 - 0.7 m
• Total cooling cell length 2m (rf extending into
  magnet)
• 4m (separated rf)
• For applications further upstream at larger
  emittances, this channel can support a 0.8 m
  bore, 0.8 m long quadrupole with no intervening
  drift and without matching to the channel
  described here.

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Acceptance in the Presence of Fringe Fields

• Outside the physical 60 cm aperture after
  applying a known fringe field model
• Stable in the presence of fringe fields over a
  momentum range, dp/p -22 to gt100
• Although it is outside the physical aperture,
  fringe fields are the limiting effect on the
  dynamic aperture

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Fringe-field components

• Origin of longitudinal Defocussing quad with
• field components in beam envelope for the
• fringe fields quatoed acceptance of
  the channel
• Strong only on diagonal BUT Beam envelope
  quickly
• near poletips deviated from the diagonal
  in a FODO
• channel

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Enhanced physical apertures in a quad

• Star-shaped vacuum chambers can be used in the
  FODO channel quads effectively increasing their
  acceptance this is done in the Fermilab MI for
  injection/extraction.
• However, star chambers have not been assumed in
  the following simulations.

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FODO LATTICE AND INSERTION OF ABSORBER

• LATTICE FUNCTIONS OF A FODO CELL FOR A QUAD
  COOLING CHANNEL, P0200 MeV/c
• In a FODO cell, the combined minimum for ?x and
  ?y is at their crossing point, halfway between
  quadrupoles.
• The achievable ?transverse in the absorber is
  about 1.6 m, or a factor of 4 larger than the
  average transverse ? in the 2.75 m SFOFO channel
  (0.4 m).
• This channel can a full transverse normalized
  emittance of 2-2.5? for ? 20 mm-rad at a p0 of
  200 MeV/c.

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Longitudinal Acceptance and Lattice Stability at
Different Momenta

• LATTICE FUNCTIONS for 155, 245, and 300 MeV/c,
  clockwise.
• ?average at the absorber ranges from 1.57 to 1.90
  at 155 and 245 MeV/c, respectively, which
  represents the momentum range of the 2.75 m SFOFO
  channel.
• ?max is 3.90 m _at_155 MeV/c and 3.19 m _at_245 MeV./c
• The acceptance reach of this channel is clearly
  larger than 300 MeV/c i.e. ?avg is still only
  2.3 m. and ?max is 3.5 m

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Fodo Cell Properties as a Function of Momentum
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Longitudinal Acceptance and Lattice Stability at
Different Momenta
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Acceptance of the Linear Quad Channel

• With this stability what is the acceptance of the
  linear quad channel?
• More importantly, how does it compare with the
  upstream SFOFO channels?

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Effective Cooling Range

• BUT
• as we know ?? at the absorber is increasing with
  momentum, but so is the normalized acceptance
• so what is the real cooling range in energy?

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• The equilibrium emittance is given by
• ?N,min ??(14 MeV)?
• (??m?LR.d??ds)
• where ??is the transverse beta function at
  the absorber, ? the relativistic velocity, m? the
  mass of the muon, LR the radiation length of the
  absorber material, and dE/ds the energy lost in
  the absorber.
• Then, one cools when
• ?N,min/ ?? ? constant
• or
• if the unnormalized acceptance is constant vs.
  momentum
• ?? / ?? ? constant

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This implies

• Cooling occurs from 150 MeV/c to past 400 MeV/c
  in this channel

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Preliminary Tracking Studies of the FODO quad
cooling channel

• Tracking Studies were performed
• with full nonlinear terms
• with/without quadrupole fringe fields (different
  models)
• with multiple scattering (windows absorber)
• dE/dx as a function of energy
• full energy loss function including straggling
  and spin
• 200 MHz sinusoidal rf

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Preliminary Tracking Studies of the FODO quad
cooling channel-details

• Tracking Studies were performed
• with full nonlinear terms
• with/without quadrupole fringe fields (different
  models)
• with multiple scattering (windows absorber)
• full straggling implemented (windows absorber)?
• Preliminary estimates of cooling were obtained
  by
• 1. determining the invariant phase ellipse of
  the quad channel
• 2. tracking in 1 cm steps along the x axis to
  determine the dynamic aperture of the channel
• 3. particles were then launched on the outer
  stable invariant ellipse at various x,x
  coordinates and on one inner phase ellipse near
  the calculated equilibrium emittance
• 4. particle positions were plotted for 10 cells,
  but at increasing distance down the length of the
  cooling channel (cells 21-30, 31-40, 101-110, for
  example) until the cooling converged.
• 5. the rough cooling factor was obtained by
  comparing the outermost stable ellipse with the
  final ellipse which clearly contained the
  majority of the particles.

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Quad Cooling Channel Simulationby COSY Infinity
4m Cell

• Muons (180MeV/c to 245MeV/c)
• Magnetic Quadrupoles (k2.88)
• Liquid H Absorber -dE/dx -12MeV/35cm
• Cavities Energy gain 12MeV/Cell to
  compensate the loss in the absorber

K. Makino Emittance Exchange
Workshop at LBNL, October 3-19, 2001
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Tracking the Quad Cooling CellsMomentum 220
MeV/c, Starting from x2cm,4cm,,30cm, for 100
Cells (a) Without Cooling (b) With Cooling (no
scattering)(c) With Cooling and Scattering

(d) Pseudo-Invariant Ellipses with Cooling
(damping factor corrected)
(a)
(b)
(c)
(d)
K. Makino Emittance Exchange
Workshop at LBNL, October 3-19, 2001
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Tracking the Quad Cooling Cells with
ScatteringMomentum 220 MeV/c, Starting from
x10cm, 15cm Pseudo-Invariant Ellipses(a)
Initial Ellipses (b) for 11-20 Cells (c) for
31-40 Cells (d) for 101-110 Cells
(a)
(b)
(c)
(d)
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Full Simulation includes

• dE/dx as a function of energy
• dE/dx curves have now been loaded as a function
  of energy into COSY
• i.e. energy lost in each absorber depends on the
  particles energy.
• Straggling
• A. Van Ginnekens energy loss function has been
  interfaced to COSY.
• Low energy tail of the straggling function is
  believed to be an important
• loss mechanism in the early channels AND the
  spin of the particle
• affects the average energy of the distribution.
  In this simulation the
• energy of the reference particle is calculated
  with full straggling and spin.
• The energy loss of other particles are calculated
  relative to this
• reference particle following the dE/dx curve.
• rf bucket
• A sinusoidal 200-MHz rf waveform has been
  implemented assuming a
• gradient of 10MV/m.

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Straggling Energy distribution after hydrogen
absorber
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Straggling Energy distribution after Al window
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Generation of particle energy distribution after
absorber
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Quantifying the Cooling Description of a Merit
Factor

• TRANSVERSE
• Load an elliptical distribution in x,x which
  corresponds to the stable phase ellipses
• of a quad channel without cooling. The exact
  shape will be a Gaussian whos rms
• width is 1/2 - 1/2.5 the half aperture of the
  quadrupoles, which is 30cm (?max 3.1 m,
• _at_p200MeV/c). An initial to final rms ratio can
  be calculated for the transverse
• cooling factor.
• LONGITUDINAL
• Two cases will be loaded one with little
  longitudinal loss, and one filling the
• bucket for comparison.
• Minimal bucket loss
• ?E 12 MeV rms bunch length 7.5 cm
• ?s 60 ? ? 54 for a 3? distribution
• Maximul bucket loss
• ?E 24 MeV rms bunch length 15 cm
• ?s 60 ? ? 108 for a 3? distribution
• Longitudinal and transverse losses will form the
  overall transmission factor.
• The product of the transmission and cooling
  factors combined with the decay
• losses will provide the merit factor for this
  emittance range.

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Full simulation transverse cooling along channel
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Tracking of particles launched along the diagonal
in x,y
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Goals Achieved with a the FODO-based Quad Cooling
Channel

• Cool transversely by a factor of 2 in the
  emittance of each plane from an full normalized
  emittance of about 80-125 mm-rad to 30-40 mm-rad
  in each plane.
• Inject cleanly (without matching) into an
  emittance exchange channel using the same
  FODO-based optical cell.
• Recool transversely with the same channel.
• After this channel, inject into more
  sophisticated cooling channels such as solenoidal
  ones to cool to the final required emittances
• This is the only cooling needed for an FFAG
  scenario (emittance exchange is not needed)

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Future Design Studies

• Move the central energy of the channel to the
  minimum in the total strength of the reheating
  terms (300-400 MeV/c) by increasing the poletip
  strength of the quadrupoles (they are still
  normal conducting I.e. lt2T).
• Although this, in principle, halves the momentum
  spread, this quad channel will still accept more
  the -22-100 total momentum bite for effective
  cooling due to its naturally high longitudinal
  acceptance.
• This implies a momentum bile of 270 MeV/c - 700
  MeV/c
• Compare cooling rates, final emittances, and
  losses at the higher momenta
• Load exact particle distributions from
  target/capture/buncher string
• Investigate superconducting rf and increased
  absorber lengths. (The quadrupole end fields fall
  much more quickly than solenoidal ones
• and if quads are normal conducting, mirror
  plates can be used)
• Evaluate other absorbers He?, LiH (rotating drum
  target)