Particle dynamics in electron FFAG

Shinji Machida

KEK

FFAG04, October 13-16, 2004

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 ind

ependent variable is time.

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 atTRIUMF FFAG2004

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.

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.

If we can put particles with only from 40 to 60 degand choose proper RF voltage, momentum spreadcan be reduced.

Transverse acceptance

Initial amplitude of 0, 1, 2, 5, 10 mm.

horizontal vertical

Large amplitude particle, ~10 mm, cannot beaccelerated to the final momentum.

10mm

5mm1mm

2mm

0mm

Integer resonance crossing• Alignment errors as well as crossing speed are param

eters.– 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 tot

al,• 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.

Freq=1.49 GHz (h=76), 2.5 MV, 5 turns

0mm

0.01mm

0.02mm

Freq=1.49 GHz (h=76), 2.5 MV, 5 turns

0.05mm

0.1mm

0.2mm

0.5mm

1mm

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 isfixed: +-0.05mm (100%)

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.

Half-integer resonance crossing• Gradient errors as well as crossing speed are parame

ters.– 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 tot

al,• 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.

Freq=19.7 MHz (h=1), 0.032 MV, 400 turns

0.02%

0.05%

0.1%

0.2%

0.5%

Freq=78.7 MHz (h=4), 0.125 MV, 100 turns

0.05%

0.1%

0.2%

0.5%

1%

Freq=157 MHz (h=8), 0.25 MV, 50 turns

0.05%

0.1%

0.2%

0.5%

1%

Freq=0.315 GHz (h=16), 0.5MV, 25 turns

0.1%

0.2%

0.5%

1%

2%

Freq=0.747 GHz (h=38), 1.25 MV, 10 turns

0.1%

0.2%

0.5%

1%

2%

Freq=1.49 GHz (h=76), 2.5 MV, 5 turns

0.2%

0.5%

1%

2%

5%

Tolerance (gradient)

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

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 accel

eration (<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.