June 21, 2005 S. Kahn -- Small Muon Cooling Ring1 Small Muon Ring for a Cooling Demonstration Steve Kahn For S. Kahn, H. Kirk, A. Garren, F. Mills and

June 21, 2005 S. Kahn -- Small Muon Cooling Ring

1

Small Muon Ring for a Cooling Demonstration

Steve KahnFor

S. Kahn, H. Kirk, A. Garren, F. Mills and D. Cline


2

The Need for Muon Cooling Muon beams will be needed for Neutrino Factories and eventually

Muon Colliders. Because muons mostly come from the decay of pions and the

pion capture process generally produces a large phase space, reduction of phase space is required.

This is particularly necessary for a muon collider which requires cooling by ~10-5. Neutrino Factories would benefit by a reduction of muon phase space by ~10-2.

The approach is to use ionization cooling to reduce the phase space. Specifically, we want

A compact ring with edge focusing dipole magnets. The beam enclosure filled with high-pressure hydrogen gas to

serve as the energy loss absorber. RF cavities to restore the longitudinal energy loss.


3

Demonstration Ring Design Parameters

Pressurized H2 gas filled ring. The gas is the absorber.

40 Atm @ 300 K 10 Atm @ 77 K

Four Weak focusing dipoles Dipoles use edge focusing. Iron yokes for flux return.

Nominal dipole field of 1.8 T 2.3 T would be possible with

Vanadium Permendur. RF cavities in the drift region

between magnets to replace energy loss in gas.

Using 201.25 MHz cavities. 10 MV/m gradient.

Parameter Value Dipole Field 1.8 T

Number of Cells 4 Reference Momentum 172.12 MeV/c Ring Circumference 3.81 m

X Aperture 20 cm Y Aperture 10 cm

Pz Acceptance 10 MeV/c Minimum X 38 cm Maximum X 92 cm Minimum Y 54 cm Maximum Y 66 cm

Hydrogen Gas Pressure 40 Atm @ 300º K RF Gradient 10 MV/m

RF Frequency 201.25 MHz Total RF Length 1.2 m Total Orbit Turns 100

Table 1: Parameters that describe the muon cooling ring.


4

Sketch of One Cell of the Cooling Ring

Cavity

Magnet

rho=31.8

gamma=22.5

Xc=22.5 cm

Yc=22.5 cm

25 cm

Cavity

Cavity


5

1.6 m


6


7

Cooling with the Hard-edge Model

rp Scalable Ring:

The ring operates on the 3rd Harmonic

Observe cooling with a Merit factor of 20 (without decays).


8

Comparison of Closed Orbits with and without Iron

With IronCoils Only

Note the reverse curvature between magnets


9

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.


10

Pole Shaping to Improve the Horizontal Aperture

We have shaped the poles of the magnet to make the field on the symmetry plane field more uniform.

This improves the horizontal aperture.


11

Cooling Results Using the Shaped Iron Poles

Admittance Equilibrium

X17.2 mm 5.8 mm

Y3.5 mm 2.1 mm

Z18.0 mm 5.0 mm


12


13

Injection into This Demonstration Ring

Kicker Scenarios: In order to kick the entire beam into the ring on orbit would

require ~10 kjoules in 7 ns! We can kick in different parts of the beam and add them

together in analysis. This still would require a substantial kicker.

Proton beam Insertion. Pions produced in the gas would decay to muons. We need to study if enough muons are produced and captured by this method.

Muon (or pion) beam insertion. Bring in higher momentum (or ) to the outer edge of the ring and let them lose energy by dE/dx loss until it is captured on orbit and momentum by the rf.

This seems like the most promising approach at this moment. Simulation studies of this scheme need to be done.


14

Conclusions

We have just successfully finished a phase I SBIR and have submitted a phase II SBIR. The phase I SBIR was to do a feasibility

study of this cooling ring which we have succeeded in doing.

The phase II SBIR will be to do an engineering study of this ring and to build a principle component such as a magnet.


15

Backup Slides


16

Why a Small Cooling Ring? We would like to pursue a small muon cooling ring as a

demonstration experiment that we can afford to build. We have set as a goal to design a ring that should cost

about $5M. We are not sure that this precise cost will be achieved but we are choosing this as a guide.

We would like to obtain enough cooling so that it would be quite clear that we have actually cooled the beam.

We have just successfully finished a phase I SBIR and have submitted a phase II SBIR.

The phase I SBIR was to do a feasibility study of this cooling ring which we have succeeded in doing.

The phase II SBIR will be to do an engineering study of this ring and to build a principle component such as a magnet.


17

Beam Dynamics Simulations

The storage ring was designed using SYNCH. ICOOL was used for particle tracking through the lattice taking into

account energy loss in the absorbers and energy recovery from the rf cavities.

A merit factor is used as a measurement of the cooling performance:

In this study decaying muons will be excluded for this merit factor since this is a demonstration and they can be corrected for.

Studies for some cooling ring variations have shown merit factors as high as 400.

The high merit factor solutions typically require aggressive parameters (~5T magnets, 45 MV/m rf or 100 bar H2 gas) which are beyond the needs of a demonstration.

F

i

F

i

F

i

z

z

y

y

x

xonTransmissiMerit


18

A Realistic Field Description Using Tosca The hard edge cooling simulations for the small cooling

ring have shown promising results yielding demonstrable cooling.

It is essential to examine the ring using fields from magnets that can actually be built. (This is more than merely obeying Maxwell’s equations.)

Tosca can supply fields from a coil and iron configuration. Tosca can provide field maps that can be used by ICOOL

and GEANT for tracking. Tosca can also track particles through the field it

generates. We can actually find the closed orbit in Tosca itself.

Since the closed orbit is in the x-z plane we need merely find the track which has x’=0 and x=0 after one turn. This is a 1 parameter minimization.


19

History of Magnet Models That Have Been Used for This Cooling Ring: Coils Only Model

Advantages. Simple model gives a good

description of the field. Lattice parameters are

consistent with Synch Model. Edge focusing works well. Essentially no quadrupole

component inside magnet. Maximum field not constrained by

iron. Can design to higher fields. Disadvantages.

Spray flux over all space. Can insert entire cooling ring

into iron box for external shielding.

Need structural support system that is not ferro-magnetic.

Field Harmonics Along Path

B0 B1Edge Inside

MagnetReverse Field between Magnets


20

Magnets with Iron Yokes Iron magnets with flat poles.

We see very little reverse field between magnets.

We see less edge focusing than expected. We see a non-zero quadrupole component

present in the center of the magnet. Consequently the lattice parameters

obtained from this magnet do not agree with the design values from Synch.

Also we did not get as large a dynamic aperture as we had hoped.

I will discuss dynamic aperture later. This was the situation last December at Miami

There was a positive note. We did see cooling if we turned off the random processes. (Multiple scattering, straggling).

Field Harmonics along the Path Non-zero quad

inside magnetLess edge focusing

No return flux


21

Vertical Field for the Three Cases Compared The dynamic aperture that

we see appears to be limited by the field quality.

The figure shows By at y=0 in the vertical symmetry plane for three pole widths.

We would like a good field over 25 cm.

We have limited physical space due to RF cavity constraints.

The hard edge has a perfectly uniform transverse field profile by definition. This gives it a large dynamic aperture.

Comparison of Symmetry Plane Fields

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Radius, m

By, t

esla

+-25

-25,+40

+-40


22

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.

The limitation of this technique is that the field errors grow with radial distance from the reference path.

This could limit our computation of the dynamic aperture.


23

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.

nny xsbsxB )(),(


24

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 n=0 to 5.

T

sikN

knkn ecb

1

0,

T

sik

T

nnk esbT

c )(1

0

, where


25

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.


26

Horizontal Dynamic Aperture (x vs. px)Without Iron

My Hardedge model

Realistic Field Model Realistic Model with no sex


27

Vertical Dynamic Aperture (y vs. py)Without Iron

My Hardedge Model

Realistic Field ModelRealistic field w/ no sex


28

Dynamic Aperture with the Iron Yokewithout Pole Shaping

x vs. Pxy vs. Py

All harmonics

No sex and above


29

Dynamic Aperture with a Shaped Pole

Shaping the pole significantly increases the horizontal dynamic aperture.

It does not affect the vertical dynamic aperture. This can be done only by

Reducing the distance between magnets. This would also reduce the space allowed for the RF. We can’t afford to do that.

Reducing the vertical magnet aperture (gap). We will reduce it from 15 cm to 10 cm.


30

Summarizing Dynamic Aperture Plots

Below the size of the dynamic aperture is measured by counting rings. Each ring represents 1 cm spatial aperture.

Case x Px y Py x Px No High Order

y Py

No High Order

Hard Edge 12 14

Coils Only 5 4 8 7

Iron Flat Poles 6 5 7 5

Iron Shaped Poles 13 7


31

Cooling Results Using the Shaped Iron Poles


32

How Well Does This Magnet Reproduce the SYNCH Lattice Description?

Min Beta vs Quadrupole

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 1.2 1.4 1.6 1.8 2 2.2 2.4

Quadrupole Factor

Be

ta,

m Beta X

Beta Y

Synch Nominal Added Quad

x 99.88º 117.58º 101.68º

x 37.85 cm 27.01 cm 32.91 cm

x 0 -0.00315 -0.0039

y 92.63º 68.11º 94.95º

y 56.89 cm 81.96 cm 62.70º

y 0 0.00092 0.00180

Phase Advance vs Quad Factor

0

20

40

60

80

100

120

140

1 1.2 1.4 1.6 1.8 2 2.2 2.4

Quadrupole Factor

Mu

, d

eg

ree

s

Mu X

Mu Y

•The lattice parameters determined from transfer matrix for this storage ring with the nominal iron magnets has moved from the original SYNCH values.

•Adding quadrupole does seem to bring the lattice parameters closer to the SYNCH values

•So far we have merely scaled the quadrupole component. We need to modify the poles to include this quadrupole. This is in progress.

Documents

June 21, 2005 S. Kahn -- Small Muon Cooling Ring1 Small Muon Ring for a Cooling Demonstration Steve Kahn For S. Kahn, H. Kirk, A. Garren, F. Mills and