1

Tracking code development for FFAGs

S. Machida

ASTeC/RAL

21 October, 2005

http://hadron.kek.jp/~machida/doc/nufact/

ffag/machida_211005.ppt & pdf

2

Tracking philosophy

• Combination of Teapot, Simpsons, and PTC.– All the elements are thin lens like Teapot.– Time as the independent variable like Simpsons.– Separation of orbit from magnet location like PTC.– No implementation of polymorphism unlike PTC.

• Read B fields map as an external data file.– Scaling as well as non-scaling (semi-scaling) FFAGs are mo

deled in the same platform.

• No name at the moment.

3

Lattice geometry

• First, all magnets’ center are placed on a circle whose radius is 0.370x42/2.

• Opening angle “a” is 0.265/0.370x2/42 and “b” is 0.105/0.370x2/42.

• Shift QD by 0.03416 [m] outward to obtain net kick angle at QD. The magnitude was chosen such that time of flight at 10 and 20 [MeV] becomes equal.

• Rotate QF counterclockwise to make the axis of QF parallel to line E-F.

4

Quadrupole modeling

• Soft edge model with Enge type fall off.– Scalar potential in cylindrical coordinates.

where

and

s: distance from hard edge.

g: scaling parameter of the order of gap, g=0.011 [m].

Ci: Enge coefficient, C0=0.1455, C1=2.2670, C2=-0.6395,

C3=1.1558, C4=C5=0. [same as Meot’s numbers]

€

P2 r,θ,z( ) =r2 sin2θ

2G2,0 z( ) +G2,2 z( )r2 + ⋅⋅⋅[ ]

€

G2,2k z( ) = −1( )k 2

4 k k! 2 + k( )!

d2kG2,0 z( )dz2k

€

G2,0 z( ) =G0

1+ exp Cizi

i= 0

5

∑ ⎛ ⎝ ⎜ ⎞

⎠ ⎟

€

z =s

g

5

Integration method

• Kick and drift– Quadrupole including fringe region is split into thin lenses.– When a particle reach one of thin lenses, Bx, By, and Bz are

interpolated using pre-calculated data at neighboring four grid points.

– Lorentz force is applied and direction of the momentum is changed.

– Between thin lenses, a particle goes straight.

6

Closed orbits

• Iteration gives closed orbits.

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

x [m]

10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV

2.7

2.6

2.5

2.4

2.3

2.2

0.50.40.30.20.10.0

x [m]

10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV

Whole view. One cell.

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Acceleration

• At the center of long straight, longitudinal momentum is increased.

• RF acceleration at every other cells. Can be any place.

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Check of the code

With the following parametersB fields expansion up to r2.

thin lens kick every 1 [mm].

We checkTune and time of flight in EMMA.

Serpentine curve.

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Tune and time of flight

0.4

0.3

0.2

0.1

0.0

201816141210kinetic energy [MeV]

horizontal vertical

52.00

51.98

51.96

51.94

51.92

51.90

51.88

51.86

201816141210kinetic energy [MeV]

- Tune and time of flight are calculated from particle tracking. Tune: FFT of 1024 turns data. Time of flight: total path length of polygon divided by c.

- Good agreement with Berg’s results.

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Choice of longitudinal parameters

€

a =qV

ω ⋅ΔT ⋅ΔE

€

b =T0

ΔT

€

ΔT = 0.10 [ns] from tracking result (previous page).

If , [kV] per cell (x2 per cavity).

We chose according to a reference by Berg [1].

Then, .

RF frequency is

Hz.

€

a =1

12

€

V =16.4

€

b =1

4

52.00

51.98

51.96

51.94

51.92

51.90

51.88

51.86

201816141210kinetic energy [MeV]

[1] J. S. Berg, “Longitudinal acceptance in linear non-scaling FFAGS.”

ΔTT0€

T0 = 0.025

€

frf =h

Trev=

68

51.905 ×10−9

=1.310 ×109

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Serpentine curve

• With 337 passages of RF cavity (674 cells), a particle is accelerated from 10 to 20 [MeV].

20x10-3

18

16

14

12

10

0.50.40.30.20.10.0RF phase/2Pi

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Study item 1: gutter acceleration with finite transverse emittance.

• Without any lattice errors.• Initial longitudinal emittance: +-0.9 deg. * +-0.1 %• Initial transverse emittance: 0.1 mm (rms, normalized)• Gaussian with 2 sigma cut. Number of macro particle is 100.

20x10-3

18

16

14

12

10

0.50.40.30.20.10.0RF phase/2Pi

20x10-3

18

16

14

12

10

0.50.40.30.20.10.0RF phase/2Pi

transverse emittance is zero. transverse emittance is 0.1 mm.

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Study item 1: gutter acceleration with finite transverse emittance (cont.).

• 6 particles starting from the same longitudinal coordinates (0.24*2, 10MeV).

• Transverse amplitude is different (0, 0.02, 0.08, 0.18, 0.32, 0.5 mm).

• This explains longitudinal emittance distortion (previous page) due to transverse amplitude.

20x10-3

18

16

14

12

10

0.50.40.30.20.10.0RF phase/2Pi

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Summary

• New tracking code is being developed to study FFAG dynamics.

• Longitudinal emittance distortion due to finite transverse emittance is observed.