1. 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 D. Cline

2. June 21, 2005 S. Kahn -- Small Muon Cooling Ring2 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring3 Demonstration Ring Design Parameters Pressurized H 2 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.

4. June 21, 2005 S. Kahn -- Small Muon Cooling Ring4 Sketch of One Cell of the Cooling Ring Cavity Magnet

5. June 21, 2005 S. Kahn -- Small Muon Cooling Ring5 1.6 m

6. June 21, 2005 S. Kahn -- Small Muon Cooling Ring6

7. June 21, 2005 S. Kahn -- Small Muon Cooling Ring7 Cooling with the Hard-edge Model Scalable Ring: The ring operates on the 3 rd Harmonic Observe cooling with a Merit factor of 20 (without decays).

8. June 21, 2005 S. Kahn -- Small Muon Cooling Ring8 Comparison of Closed Orbits with and without Iron With Iron Coils Only Note the reverse curvature between magnets

9. June 21, 2005 S. Kahn -- Small Muon Cooling Ring9 B y 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring10 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring11 Cooling Results Using the Shaped Iron Poles AdmittanceEquilibrium XX 17.2 mm5.8 mm YY 3.5 mm2.1 mm ZZ 18.0 mm5.0 mm

12. June 21, 2005 S. Kahn -- Small Muon Cooling Ring12

13. June 21, 2005 S. Kahn -- Small Muon Cooling Ring13 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring14 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring15 Backup Slides

16. June 21, 2005 S. Kahn -- Small Muon Cooling Ring16 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring17 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 H 2 gas) which are beyond the needs of a demonstration.

18. June 21, 2005 S. Kahn -- Small Muon Cooling Ring18 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring19 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 B0B0 B1B1 EdgeInside Magnet Reverse Field between Magnets

20. June 21, 2005 S. Kahn -- Small Muon Cooling Ring20 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 magnet Less edge focusing No return flux

21. June 21, 2005 S. Kahn -- Small Muon Cooling Ring21 Vertical Field for the Three Cases Compared The dynamic aperture that we see appears to be limited by the field quality. The figure shows B y 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.

22. June 21, 2005 S. Kahn -- Small Muon Cooling Ring22 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring23 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.

24. June 21, 2005 S. Kahn -- Small Muon Cooling Ring24 Fourier Expansion of b n (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. where

25. June 21, 2005 S. Kahn -- Small Muon Cooling Ring25 Determining the Dynamic Aperture For the x-P x (y-P y ) phase space we launch n tracks, each track starting 1 cm apart along the x (y) axis. The position in x-P x (y-P y ) 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring26 Horizontal Dynamic Aperture (x vs. p x ) Without Iron My Hardedge model Realistic Field Model Realistic Model with no sex

27. June 21, 2005 S. Kahn -- Small Muon Cooling Ring27 Vertical Dynamic Aperture (y vs. p y ) Without Iron My Hardedge Model Realistic Field Model Realistic field w/ no sex

28. June 21, 2005 S. Kahn -- Small Muon Cooling Ring28 Dynamic Aperture with the Iron Yoke without Pole Shaping x vs. P x y vs. P y All harmonics No sex and above

29. June 21, 2005 S. Kahn -- Small Muon Cooling Ring29 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. June 21, 2005 S. Kahn -- Small Muon Cooling Ring30 Summarizing Dynamic Aperture Plots Below the size of the dynamic aperture is measured by counting rings. Each ring represents 1 cm spatial aperture.

31. June 21, 2005 S. Kahn -- Small Muon Cooling Ring31 Cooling Results Using the Shaped Iron Poles

32. June 21, 2005 S. Kahn -- Small Muon Cooling Ring32 How Well Does This Magnet Reproduce the SYNCH Lattice Description? 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.