1. Particle dynamics in electron FFAG Shinji Machida KEK FFAG04, October 13-16, 2004

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 with Simpsons to study particle dynamics in an electron FFAG. –Simpsons: Thin lens tracking code with acceleration. The independent variable is time.

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. Study items Acceleration without an RF bucket. –Beam parameters required. Allowable initial phase spread final and momentum spread at extraction. Transverse acceptance (“dynamic aperture”). Resonance crossing. –Integer resonance with misalignment. –Half-integer resonance with gradient errors.

5. Longitudinal phase space Reference parameters: frequency 1.5 GHz total voltage 56 kV x 45 cell Particles with initial phase of 0 to 150 deg. (0 to -0.08 m) are accelerated. However, momentum spread at extraction becomes large.

6. If we can put particles with only from 40 to 60 deg and choose proper RF voltage, momentum spread can be reduced.

7. Transverse acceptance Initial amplitude of 0, 1, 2, 5, 10 mm. horizontalvertical Large amplitude particle, ~10 mm, cannot be accelerated to the final momentum. 10mm 5mm 1mm 2mm 0mm

8. Integer resonance crossing Alignment errors as well as crossing speed are parameters. –Alignment errors: 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1mm (+- 100%, uniform). –Crossing speed: nominal to 5 times slower. Crossing speed –Instead of nominal operation parameter: 1.5 GHz, 2.5 MV total, 315 MHz, 0.500 MV, 25 turns 747 MHz, 1.25 MV, 10 turns 1.5 GHz, 2.50 MV, 5 turns –Observer betatron oscillations and acceleration.

9. Freq=1.49 GHz (h=76), 2.5 MV, 5 turns 0mm 0.01mm 0.02mm

10. Freq=1.49 GHz (h=76), 2.5 MV, 5 turns 0.05mm 0.1mm 0.2mm 0.5mm 1mm

11. Freq=1.49 GHz (h=76), 2.5 MV, 5 turns Freq=0.747 GHz (h=38), 1.25 MV, 10 turns Freq=0.315 GHz (h=16), 0.5 MV, 25 turns Alignment errors is fixed: +-0.05mm (100%)

12. Tolerance (alignment) If an acceleration is done within 5 turns, alignment errors of +-0.05mm (or +-0.1mm) is ok. However, when crossing speed is reduced, for example 10 turns, alignment errors of +-0.05mm give significant blow up.

13. Half-integer resonance crossing Gradient errors as well as crossing speed are parameters. –Gradient errors: 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, 5 % –Crossing speed: nominal to 80 times slower. Crossing speed –Instead of nominal operation parameter: 1.5 GHz, 2.5 MV total, 19.7 MHz, 0.032 MV, 400 turns 78.7 MHz, 0.125 MV, 100 turns 157 MHz, 0.250 MV, 50 turns 315 MHz, 0.500 MV, 25 turns 747 MHz, 1.25 MV, 10 turns 1.5 GHz, 2.50 MV, 5 turns –Observer betatron oscillations and acceleration.

14. Freq=19.7 MHz (h=1), 0.032 MV, 400 turns 0.02% 0.05% 0.1% 0.2% 0.5%

15. Freq=78.7 MHz (h=4), 0.125 MV, 100 turns 0.05% 0.1% 0.2% 0.5% 1%

16. Freq=157 MHz (h=8), 0.25 MV, 50 turns 0.05% 0.1% 0.2% 0.5% 1%

17. Freq=0.315 GHz (h=16), 0.5MV, 25 turns 0.1% 0.2% 0.5% 1% 2%

18. Freq=0.747 GHz (h=38), 1.25 MV, 10 turns 0.1% 0.2% 0.5% 1% 2%

19. Freq=1.49 GHz (h=76), 2.5 MV, 5 turns 0.2% 0.5% 1% 2% 5%

20. Tolerance (gradient) General rule: “rate/dk^2 is constant”, may be applicable. Gradient error has to be less than 0.5%.

21. Summary Initial beam should be well controlled. –Phase space spread. –Transverse emittance. The following is preliminary numbers. –Alignment errors should be less than 0.05mm if longer acceleration (<10 turns) will be studied. –Gradient errors have to be less than 0.5%. General rule of “rate/dk1^2 is constant”, may be applicable.