Design Issues for a Small Muon Cooling Demonstration Ring

1
Design Issues for a Small Muon Cooling
Demonstration Ring

• Steve Kahn
• Brookhaven National Lab
• Miami Muon Collider Workshop

2
Magnetic Model for the Mini Cooling Ring

• In this talk I will concentrate on the storage
  ring aspects of the Muon Cooling Ring. Harold
  described the cooling aspects yesterday.
• In this study we will use the four cell model
  that resulted from the last June meeting.
• The previous presentation in June discussed the
  six cell model. Figures from the six cell model
  are available if we need them.
• We chose the four cell model because we thought
  that it would better accommodate a ?15 cm
  aperture than the six cell model.
• We expect that this is true, since the ratio of
  the length of the magnet to the magnet aperture
  is greater.

3
Cell Geometry Description
Based on a Sketch from A. Garren
4
Dipole Ring Parameters
5
Using TOSCA

• Hard edge field calculations for the Garren-Kirk
  Weak Focusing Dipole Ring have shown promising
  results.
• It is essential to examine the ring using
  realistic fields that at least obey Maxwells
  equations.
• Tosca can supply fields from a coil and iron
  configuration.
• We can use the program to supply a field map that
  can be used by ICOOL and GEANT.
• Tosca itself can also track particles through the
  magnetic field that it generates.
• This allows us to avoid the descretization error
  that comes from field maps.

6
Tosca Model

• For the ease of calculation we initially modeled
  the dipole magnets by its coils only. This was
  for ease of calculations and is not way we would
  actually engineer the magnet if we actually built
  it.
• This permits the field to be calculated with
  Biot-Savart integration directly. No
  finite-element mesh is necessary if iron is not
  used.
• There are also limitations in the Tosca tracking.
  
• Tosca permits only 5000 steps. This limits the
  step size to 0.5 mm. This may limit the
  ultimate precision.

7
Tosca Modeling with Iron

• The need for iron
• Using an iron yoke to return the flux reduces
  stray fields in areas that we dont want field.
• It makes the field more uniform where we want the
  field.
• It gives more control over the quality of the
  field.
• This requires that we actually make a finite
  element mesh to solve Poissons equations.
• This has finally been done.
• Hopefully this has been done in a manner that is
  flexible enough for future changes that we will
  certainly have.

8
Four Cell Ring with Iron Flux Return
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Finding the Closed Orbit using Tosca

• We know that the closed orbit path must be in the
  xz plane and that it must have x0 at the
  x-axis from symmetry.
• We can launch test particles with different
  Xstart.
• The figures on the right show Xstart vs. ?x90 and
  Xstart vs ?x'90.
• Where ?x90 and ?x'90 are the variable differences
  after 90 advance.
• We find that the best starting values are
• Xstart55.03362 cm for ?x90
• Xstart55.05569 cm for ?x'90

Xstart vs. ?x90
Xstart vs. ?x'90
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Comparison of Closed Orbits with and without Iron
Coils Only
With Iron
Note the reverse curvature between magnets
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By Field Along the Closed Orbit Path
Coils onlyNo Iron
Coils plus Iron
Constant Hardedge Field

• Since coil only field has large negative field
  between the magnets, it must have larger field in
  the magnet to give the same integrated bend.

12
Using the Field Map

• We can produce a 3D field map from TOSCA.
• We could build a GEANT model around this field
  map.
• I will discuss this later.
• We have decided that we can provide a field to be
  used by ICOOL.
• ICOOL works in a beam coordinate system.
• We know the trajectory of the reference path in
  the global coordinate system.
• We can calculate the field and its derivatives
  along this path.
• We can describe the field everywhere from this.
  (This is just an application of Greens theorem)

13
Representation of the Field in a Curving
Coordinate System

• Chun-xi Wang has a magnetic field expansion
  formulism to represent the field in curved
  (Frenet-Serret) coordinate system.
• This formulism is available in ICOOL.
• Up-down symmetry kills off the an terms bs is
  zero since there is no solenoid component in the
  dipole magnets.
• The bn(s) are obtained by fitting
• to the field in the midplane orthogonal to the
  trajectory at s
• The field is obtained from a splining the field
  grid.

14
Fourier Expansion of bn(s)

• The bn(s) can be expanded with a Fourier series
• These Fourier coefficients can be fed to ICOOL to
  describe the field with the BSOL 4 option.
• We use the bn for n0 to 5.

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Low Order Harmonics along the Reference Path for
Field Map without Iron
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Higher Order Harmonics along the Reference Path
without Iron
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Low Order Field Harmonics along the Reference
Path with Iron Yokes
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(No Transcript)
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Filtered the Harmonic Plots Truncating the
Fourier Series at 50 terms
Note the field is plotted for One Cell, not Two
20
Filtered Higher Order Harmonics Truncated to 50
terms
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Some Comments about the Harmonic Distributions

• The effect of the focusing in y is contained in
  the B1(s) harmonic.
• This contains the edge focusing used in this
  ring.
• Notice that the B1 harmonic is larger for the no
  iron model.
• The iron model will have a larger ?y. We will
  discuss this later.
• The higher order harmonics show large amounts of
  high frequency noise.
• This appears to be related to the descretization
  of the mesh.
• It is partially, but not completely, removed by
  the truncation of the Fourier series, that will
  be used to describe each field harmonic.

22
Determining the Dynamic Aperture

• For the x-Px (y-Py) phase space we launch n
  tracks, each track starting 1 cm apart along the
  x (y) axis.
• The position in x-Px (y-Py) phase space is
  sampled after every cell.
• The stable orbits form ellipses the unstable
  ones have trajectories that are lost.
• A measure of the size of the stable phase space
  is the number of rings.

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Horizontal Dynamic Aperture (x vs. px)Without
Iron
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Vertical Dynamic Aperture (y vs. py)Without Iron
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Dynamic Aperture with the Iron Yoke
y vs. Py
x vs. Px
All harmonics
No sex and above
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Measure Dynamic ApertureCounting Rings
Old Table without Iron
New Table with Iron
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Calculating Transfer Matrices

• By launching particles on trajectories at small
  variations from the closed orbit in each of the
  transverse directions and observing the phase
  variables after a period we can obtain the
  associated transfer matrix.
• Particles were launched with
• ?x 1 mm
• ?x' 10 mr
• ?y 1 mm
• ?y' 10 mr

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90 Transfer Matrix

• This is the transfer matrix for transversing a
  quarter turn
• This should be compared to the 22 matrix to
  obtain the twiss variables

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Storage Ring Parameters

• The table below shows the Twiss Parameters as
  seen in ICOOL for both the realistic and hardedge
  models. These were calculated in a manner
  similar to those shown before
• Both ICOOL models look reasonably comparable to
  the original SYNCH and TOSCA models.
• This is extremely encouraging and says that the
  realistic fields do not significantly alter the
  lattice!
• Note that realistic on this transparency is coil
  only, no Iron!

30
A Comparison of the Lattice for the Cases with
and without Iron

• The ?x is similar between the three cases.
• ??y is different for the case with Iron. The
  larger??y for the iron case is consistent with a
  smaller B1 that we seen for iron previously. We
  do not have enough vertical focusing.
• Some attempt was made to scale the B1 term alone
  to see if one could reduce ?y. We could reduce it
  to 65 cm (which also reduced ?y to 101), but
  not much beyond that.

31
Field on the Vertical Symmetry Cross Plane
The figures show By on the symmetry plane for 3
cases where the horizontal extent of the magnet
aperture is varied. The vertical aperture is 30
cm. The field is shown at y0, 5, 10, 14 cm.
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Vertical Field for the Three Cases Compared
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Field on Mid-plane Section. Does the
half-aperture size affect the field shape?
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Top View of Muon Cooling Ring from Blackboard
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We Have a Possible Design of What This Ring Might
Look Like

• It should have a double containment.
• The inner containment would contain the beam
  channel and the interior part of the RF cavities
  and would contain the pressurized H2 gas.
• The RF would have thin windows which could be a
  grid of Be to provide a boundary condition.
• This containment would minimized the volume of H2
  gas required.
• This should impress the safety committee.
• The outer containment would be in liquid N2 at
  70K.
• This would provide the cryogenic temperature for
  the H2.
• The pole part of the magnet would be inside this
  containment.

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Section View Through the Magnet Vertical Mid-plane
37
More Containment

• The magnet flux return would be outside of the
  liquid N2 vessel.
• It would be a small increase in the reluctance
  path to jump the 3/8 stainless steel vessel
  wall.
• The warm copper coils could surround the flux
  return.
• This would permit simple water cooling for the
  coils.
• Some insulation separating the coils from the
  iron flux return might be necessary.

38
Section View Through the RF Cavity
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Kicker Scenarios

• A kicker to insert a muon beam with the desired
  size would require a more aggressive kicker than
  Palmer designed for the RFoFo ring. See table.
• It would require a similar stored energy.
• We would need a kicker rise time of 7 ns instead
  of 50 ns.
• This has got to be expensive, even if possible!

Before Kick
After Kick
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Proton Beam Insertion

• Insert a proton beam and let it interact.
• This is probably the most realistic scenario.
• We need to know the production spectrum for 400
  MeV protons. (Mars or Geant4)
• Will this produce sufficient ? to get the desired
  ? beam that we need.
• We need to verify this in Geant.
• We need to optimize where to inject protons to
  obtain the best muon beam at the desired radius
  and momentum.
• Select negative particles to reduce background.

41
Pion or Muon Beam Insertion

• Bring higher momentum ? or ? beam to outer edge
  of ring and let the beam loose energy by dE/dx
  loss until it is on orbit and on momentum.
• Romulus Godang is working on some version of
  this.
• These schemes need to be simulated to see if
  they can be made to work.

42
Field Maps

• We have 3D field maps of the field with and
  without iron ready for explotation.
• We have a Geant4 model of the 4 cell ring which
  will use these field maps.
• This is necessary for more than checking ICOOL
  which we all know is right.
• The errors in the field grow rapidly with the
  distance from the closed orbit. In the
  determination of the dynamic aperture it is not
  clear if we are measuring the breakdown in the
  parameterization of the of the field as an
  expansion around the closed orbit or the true
  aperture.
• The errors in Geant based on a field map should
  be more uniform over the aperture.

43
Geant4 Model of Cooling Ring Using Hardedge
Fields.
Muon with Reference momentum circulating in
Storage Ring Mode.
RF cavity
Detector Planes