# Presentation: 493061

## Design Issues for a Small Muon Cooling Demonstration Ring

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

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Cell Geometry Description

- Based on a Sketch from A. Garren

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Dipole Ring Parameters

| Parameter | Value |
| --- | --- |
| Reference Momentum | 172.1 MeV/c |
| Number of Half-Cells | 4 |
| Bend Angle per Cell | 90° |
| Ring Circumference | 3.8 m |
| Number of RF cavities | 4 |
| RF Gradient | 40 MV/m |
| Absorber | Pressurized H2 |
| Hardedge Dipole Field | 1.8 T |
| Straight Length per Half-Cell | 405cm |
| Dipole Radius of Curvature | 31.8 cm |

*Dipole Ring Parameters*

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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 Maxwell’s 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Tosca Modeling with Iron

- The need for iron
- Using an iron yoke to return the flux reduces stray fields in areas that we don’t 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 Poisson’s 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Four Cell Ring with Iron Flux Return

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Finding the Closed Orbit using Tosca

- We know that the *closed orbit* path must be in the *xz plane* and that it must have *x’=0 * at the *x-axis  *from symmetry.
- We can launch test particles with different *X**start*.
- 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
- Xstart=55.03362 cm for x90
- Xstart=55.05569 cm for x'90

- Xstart vs. x90

- Xstart vs. x'90

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Comparison of Closed Orbits with and without Iron

- With Iron

- Coils Only

- Note the reverse curvature between magnets

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## By Field Along the Closed Orbit Path

- Coils only—No 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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 Green’s theorem)

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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 *a**n* terms;  *b**s* is zero since there is no solenoid component in the dipole magnets.
- The *b*n(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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Fourier Expansion of bn(s)

- The *b**n**(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 *b**n** *for* n=0 *to* 5.*

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Low Order Harmonics along the Reference Path for Field Map without Iron

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Higher Order Harmonics along the Reference Path without Iron

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Low Order Field Harmonics along the Reference Path with Iron Yokes

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Slide 18

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Filtered the Harmonic Plots:  Truncating the Fourier Series at 50 terms

- Note: the field is plotted for One Cell, not Two

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Filtered Higher Order Harmonics– Truncated to 50 terms

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Horizontal Dynamic Aperture (x vs. px)Without Iron

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Vertical Dynamic Aperture (y vs. py)Without Iron

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Dynamic Aperture with the Iron Yoke

**x vs. P****x**

**y vs. P****y**

**All harmonics**

**No sex and above**

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Measure Dynamic Aperture:Counting Rings

**New Table with Iron**

**Old Table without Iron**

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 90° Transfer Matrix

- This is the transfer matrix for transversing a quarter turn:
- This should be compared to the 2×2 matrix to obtain the twiss variables:

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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!*

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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 y=0, 5, 10, 14 cm.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Vertical Field for the Three Cases Compared

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Field on Mid-plane Section.  Does the half-aperture size affect the field shape?

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Top View of Muon Cooling Ring from Blackboard

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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 *H**2** *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 *H**2** *gas required.
- This should impress the safety committee.
- The outer containment would be in liquid *N**2  *at 70°K*.*
- This would provide the cryogenic temperature for the *H**2*.
- The pole part of the magnet would be inside this containment.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Section View Through the Magnet Vertical Mid-plane

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Section View Through the RF Cavity

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## 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.

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004

## Geant4 Model of Cooling Ring Using Hardedge Fields.

- Muon with Reference momentum circulating in *Storage Ring Mode.*

- RF cavity

- Detector Planes

- S. Kahn -- Muon Cooling Ring Workshop

- Dec 14, 2004