1. Dejan Trbojevic Non-Scaling Fixed Gradient FFAG Optimization and Proton Therapy Accelerator from 25 – 250 MeV FFAG04@KEK 12 – 16, October 2004 CONTENT:   Again: Why is the Dispersion Action H important?   Optimization:   For the Muon Acceleration the priority is the path length.   Would not be bad also to get small orbit offsets?   We have to be in the “STABLE” region for tunes in the whole acceleration region for dp/p.   Well let’s get the shortest circumference if possible?   The vertical tune should be at the end of acceleration larger than 0.1.   Proton Therapy Accelerator – how to get it right?

2. When did we start? FFAG Lattice Without Opposite Bends  The first publication was from the Montauk workshop on September 30, 1999: Trbojevic, D., Courant, E. D., and Garren, A., FFAG Lattice Without Opposite Bends, Colliders and Collider Physics at the highest Energies, AIP CONFERENCE PROCEEDINGS, Volume 530, Montauk, New York 1999, pp. 333-338, American Institute of Physics, Melville, New York, 2000, Editor B.J. King. “FFAG lattice without opposite bends”,  Trbojevic, D., “FFAG lattice without opposite bends”, KEK Workshop on FFAG Synchrotrons, October 11, 2000.  Accelerator physics seminar talk at Brookhaven National Laboratory: Dejan Trbojevic, : ”Fixed Field Alternating Gradient Lattice (FFAG) without Opposite Bends”.  Accelerator physics seminar talk at Brookhaven National Laboratory: Dejan Trbojevic, December 14, 2000: ”Fixed Field Alternating Gradient Lattice (FFAG) without Opposite Bends”.  Muon Collaboration Meeting at Berkeley,. Dejan Trbojevic: “Some taught about recirculator”.  Muon Collaboration Meeting at Berkeley, February 2, 2001. Dejan Trbojevic: “Some taught about recirculator”.  Collaboration Meeting Neutrino Factory at Brookhaven National Laboratory  Trbojevic, D., Courant, E., Garren, A. “Fixed field alternating gradient lattice design without opposite bends”. Eighth European Particle Accelerator Conf. (EPAC’02), Paris, France, June 3-7, 2002, pgs. 1199-1202 (2002) BNL-69007. “FFAG LATTICE FOR MUON ACCELERATION WITH DISTRIBUTED RF”,  PAC2003, Portland, Oregon, May 16, 2003, “FFAG LATTICE FOR MUON ACCELERATION WITH DISTRIBUTED RF”, D. Trbojevic, J.S. Berg, M.Blaskiewicz, E.D. Courant, R. Palmer, BNL, Upton, New York, A.A. Garren, LBL, Berkeley, California, USA.  FFAG update at the KEK workshop July 8, 2003.

3. Required Range of Energies (or  p/p) Aperture limitation is defined by the maximum value of the DISPERSION function:  x < +/- 30 mm if the 0.5 <  p/p < 1.5 then: D x < 60 mm Why is the Dispersion Action H Relevant? The normalized dispersion amplitude Corresponds to the 1/2 !!! The basic idea has remained the same:  x = D x  p/p < 30 mm

4. Basic Dispersion function definitions: The “dispersion action” H function:

5. Combined function magnet: dispersion and “twiss” functions

6. How to obtain the minimum of :

7. Conditions for the minimum of the function for the combined function magnet:

8. The major conclusions from previously compiled analytical formulas for different lattices like FODO, doublet, triplet, double bend achromat, triple bend achromat etc. comparisons are listed bellow: The FODO cell H function and conditions for the minimum of the average value through the cell are: The dispersion function in the FODO cell: a minimum at  ~140 o

9. The minimum emittance lattice: The minimum emittance lattice requires reduction of the function H:The minimum emittance lattice requires reduction of the function H: –The normalized dispersion amplitude corresponds to the 1/2 –Conditions are for the minimum of the betatron function  x and dispersion function D x to have small values at the middle of the dipole (combined function dipole makes it even smaller).  min  Ld  15 D xmin =  Ld/24 

10. 5 NSLS reduction of the emittance: 10 times

11. Comparison between fodo, doublet, and triplet lattices:

13. Lattice parameter dependence on the H function:

14. Maximum orbit offsets and path length arround the ring dependence on the H function:

15. Maximum closed orbit offsets for a series of muon acceleration lattices made from conditions for the minimum of

16. Stability range of betatron tunes for Muon acceleration lattices developed by the minimum condition:

17. Proton Therapy non-scaling Fixed Gradient 25 MeV – 250 MeV accelerator  Required Parameters  Magnet properties  Lattice parameters at reference momentum p o  Optimization of the lattice: tunes and orbit offsets  Lattice properties during acceleration

18. Proton Therapy non-scaling Fixed Gradient 25 MeV – 250 MeV accelerator B  0.7272814 2.4321837

19. Proton Therapy non-scaling Fixed Gradient 25 MeV – 250 MeV accelerator