1. Resonance crossing and error tolerances Shinji Machida KEK FFAG05 at Fermilab, April 3-7, 2005

2. Purpose To simulate non-scaling muon FFAG, electron FFAG from 10 to 20 MeV is proposed. There are mainly two issues to be studied. –Acceleration without an RF bucket. –Fast crossing of integer and half-integer resonances. 6D full tracking is performed to study resonance crossing and error tolerances in an electron FFAG. –Alignment –Gradient –Kinematic errors are included

3. Lattice model Trbojevic and Courant lattice at TRIUMF workshop. –C=15 m, N=45. Each magnet is split into 10 thin lens elements. –Tune –Lattice functions –Momentum dependent path length are reasonably reproduced. Trbojevic at TRIUMF FFAG2004

4. Error source Alignment errors –Specified +-100%. (0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1 mm) –Distribution is uniform. –Excite integer resonances. Gradient errors –Specified +-100%. (0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1 %) –Distribution is uniform. –Excite half-integer resonances.

5. Resonance crossing and its speed During acceleration –Integer resonance crossing 6 times (H), 7 times (V) –Half-integer crossing 12 times (H), 14 times (V) Crossing speed can be varied with different RF frequency and voltage. Tune variation in a cycle RF freq. [GHz]1.50.7470.3150.1570.07870.03950.0197 RF volt. [MV]2.51.250.50.250.1250.0640.032 turns5102550100200400 Horizontal tune:13.6 - 7.9 Vertical tune:9.2 - 2.2

6. Crossing speed in a cycle dp/dt is almost constant. -> crossing speed becomes small at higher momentum. Longitudinal phase space Tune variation in a cycle dp/p vs. time

7. Animations Resonance crossing in phase space (vertical only). Tune spread due to chromaticity. eFFAG_seed2/kb_rmax005_ellipse eFFAG_seed2/kq_rmax0005_ellipse

8. Alignment errors (seed#1) Initial emittance = 100  mm-mrad (norm.)

9. Gradient errors (seed#1) Initial emittance = 100  mm-mrad (norm.)

10. Alignment errors, 5 turns

11. Alignment errors, 50 turns

12. Alignment errors, 400 turns

13. Alignment errors

14. Gradient errors, 5 turns

15. Gradient errors, 50 turns

16. Gradient errors, 400 turns

17. Gradient errors

18. Summary Resonance crossing is demonstrated in phase space. Additional distortion due to finite chromaticity (tune spread) is inevitable. Error tolerances depend on crossing speed (and criterion). 550400 alignment 100  m50  m20  m gradient1%0.5%0.1% (+-100%, uniform distribution.) (growth is less than twice.)