Tracking study of muon acceleration with FFAGs

1
Tracking study of muon acceleration with FFAGs

• S. Machida
• RAL/ASTeC
• 6 December, 2005
• http//hadron.kek.jp/machida/doc/nufact/
• ffag/machida_20051206.ppt pdf

2
Contents

• Code development
• Example of scaling and non-scaling muon rings
• Beam dynamics study

3
Code development

4
Tracking philosophy

• Combination of Teapot, Simpsons, and PTC.
• All the elements are thin lens like Teapot.
• Time as the independent variable like Simpsons.
• We keep track of absolute time (of flight.)
• Separation of orbit from magnet geometry 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 modeled in the same platform.
• At the moment, analytical model instead.

5
Four criteria of Berg for correct tracking

• No TPS.
• Geometry is separated from orbit.
• Modeling of end fields
• Initial matching

6
Why another code?

• Tracking codes always have some approximations.
• I would like to know exactly what approximations
  are taken.
• Fields in sector bend
• Modeling of end fields
• Can be optimized for FFAG.
• Superperiodicity is large and a ring consists of
  simple cell.
• Scaling and non-scaling can be compared with same
  kinds of approximations (for ISS).

7
Lattice geometry for non-scaling doublet

• First, all magnets center are placed on a circle
  whose radius is circumference/2p.
• Shift QD outward to obtain net kick angle at QD.
  The magnitude was chosen such that time of flight
  at the minimum and maximum momenta becomes equal.
• Rotate QF counterclockwise to make the axis of QF
  parallel to line E-F.

8
Integration method

• Kick and drift
• Magnetic fields including fringe region is split
  into thin lenses.
• When a particle reach one of thin lenses, Bx, By,
  and Bz are obtained analytically or 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.

9
Analytical modeling for non-scaling magnet (1)

• Shifted quadrupole
• 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.
• Ci Enge coefficient.

10
Analytical modeling for non-scaling magnets (2)

• Up to G20 and G21
• Edge focusing
• Up to G22 and G23
• Octupole components of fringe fields
• Up to G24 and G25
• Dodecapole
• Feed-down of multipole (octupole) has large
  effects when G22 and higher order is included.
• It is not clear if it is real or numerical
  defects due to subtraction of two large numbers.

11
Analytical modeling for scaling magnet

• rk type magnet
• Soft edge model with Enge type fall off.
• Scalar potential in cylindrical coordinates.
• where
• s distance from hard edge.
• g scaling parameter of the order of
  gap.
• Ci Enge coefficient.

12
Acceleration

• At the center of long straight, longitudinal
  momentum is increased.
• RF acceleration at every other cells. Can be any
  place.

13
Scaling and non-scaling machines

14
Check of the code in non-scaling FFAG

• With the following parameters
• B fields expansion up to G21.
• thin lens kick every 1 mm.
• We check
• Tune and time of flight in EMMA and 10-20 GeV
  ring.
• Serpentine curve.

15
Closed orbits

• Iteration gives closed orbits.

Whole view.
One cell.
16
Tune and time of flight of EMMA
- Good agreement with Bergs results.
17
Choice of longitudinal parameters
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.
DT
T0
1 J. S. Berg, Longitudinal acceptance in
linear non-scaling FFAGS.
18
Serpentine curve

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

19
Tune, ToF and displacement of 10-20 GeV muon ring

20
Check of the code in 0.3 to 1 GeV scaling FFAG

• Closed orbit in horizontal direction.

21
Beam dynamics study- EMMA (electron model) as an
example -

22
Study items

• Distortion of longitudinal emittance.
• With zero initial transverse amplitude.
• Dynamic aperture without acceleration.
• At injection energy.
• Alignment errors of 0, 0.01, 0.02, 0.05 mm (rms),
  but with only one error seed.
• Resonance crossing with acceleration.
• 1000 and 5000 p mm-mrad, normalized
• Strength of linear resonance is independent of
  particle amplitude.
• Higher order resonance becomes significant with
  larger particle amplitude.
• Alignment errors of 0, 0.01, 0.02, 0.05 mm (rms),
  but with only one error seed.

23
Distortion of longitudinal emittance
Less tilted (-50).
Initial ellipse (-0.125 ns, -0.25 MeV) is
tilted as about the same slope of separatrix.
More tilted (50).
24
Distortion of longitudinal emittance
Less tilted (-50).
Animation
Matched
Kinetic energy GeV
More tilted (50).
RF phase/2Pi
25
Dynamic aperture without acceleration.

• Without acceleration. Kinetic energy is 10 MeV.
  16 turns.
• Errors of 0, 0.01, 0.02 mm (rms), with only one
  error seed.

26
Dynamic aperture without acceleration.

• Without acceleration. Kinetic energy is 10 MeV.
  16 turns.
• Errors of 0.02 mm (rms), with only one error seed.

27
Resonance crossing with acceleration

• Horizontal is 1000 p mm-mrad, normalized, zero
  vertical emittance.
• Errors of 0, 0.01, 0.02, 0.05 mm (rms), only one
  error seed.

Horizontal phase space (x, xp)
0. mm
0.01 mm
0.02 mm
0.05 mm
28
Resonance crossing with acceleration

• Horizontal is 5000 p mm-mrad, normalized, zero
  vertical emittance.
• Errors of 0, 0.01, 0.02, 0.05 mm (rms), only one
  error seed.

Horizontal phase space (x, xp)
0.01 mm
0.05 mm
Longitudinal phase space (phi, energy)
29
Dynamic aperture

• Transverse acceptance is limited by longitudinal
  motion, as well as resonance crossing ?

30
Parameters of EMMA

• a1/12, b1/5 (it was 1/4 before.)
• RF cavity every other cell.
• Voltage is 32.8 kV per cavity.
• Harmonic number is 68 (about 1.3 GHz.)
• Acceleration is completed in 673 cells.

31
Acceleration with finite transverse emittance

• 11 particles starting from the same longitudinal
  coordinates (0.092p, 10 MeV).
• Transverse amplitude is different (0, 0.04, 0.16,
  0.36, 0.64, 1.00, 1.44, 1.96, 2.56, 3.24, 4.0 p
  mm normalized).
• For vertical only, we choose (0, 0.1, 0.4, 0.9,
  1.6, 2.5, 3.6, 4.9, 6.4, 8.1, 10. p mm
  normalized)

horizontal only
vertical only
4.9
2.56
6.4
0
3.24
0
4.0
8.1
horizontal and vertical
1.44
1.96
0
2.56
32
Acceleration with finite transverse emittance

• Initial phase dependence
• 4 particles have same amplitude, but different
  phase.
• Time of flight for each particles.
• Measure time to finish one revolution for each
  accelerating particle.
• With increased a parameter. (meaning more
  voltage.)

33
Initial phase dependence

• Serpentine curve for different initial phase
  (horizontal only).
• Not so much deference among particle with
  different initial phase.
• Particle 1 is always gain less energy.

horizontal2.56 p
horizontal3.24 p
4
3
4 3
2 1
2 1
34
Initial phase dependence

• Serpentine curve for different initial phase
  (vertical only).
• As one expects, particles 1 has the same curve as
  3, and 2 has the same curve as 4 in vertical
  plane.

vertical4.90 p
vertical6.40 p
2, 4
2, 4
1, 3
1, 3
35
Time of flight for different amplitude

• Time of flight is calculated for each particle.
• Legend shows horizontal amplitude (p mm,
  normalized). Xp0.
• Vertical amplitude is zero.

36
With increased a parameter. (meaning more
voltage.)

• Increase the voltage twice as much (a1/6).
• With a1/6, all particles are accelerated, but
  trajectory in phase space still strongly depends
  on transverse amplitude.

horizontal only
horizontal only
a1/12
a1/6
37
Acceleration with finite transverse emittance

• A particle with horizontal emittance of more than
  4 p mm is not accelerated. (When vertical
  emittance is zero.)
• When both horizontal and vertical emittance are
  finite, a particle of more than 2.56 p mm is not
  accelerated.
• Effects of finite vertical emittance is smaller
  than horizontal.

38
Parameters of 10 - 20 GeV muon ring

• a1/12, b1/5
• RF cavity every other cell.
• Voltage is 16.246 MV per cavity.
• Harmonic number is 274 (about 200.5 MHz.)
• Acceleration is completed in 16 turns (1344
  cells.)

39
Acceleration with finite transverse emittance

• Horizontal amplitude are
• (0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 x 103 p
  mm-mrad, normalized.)
• Vertical amplitude is zero.
• Difference of ToF becomes smaller as accelerated.

0
25
36
40
For 10 - 20 GeV muon ring

• In 10-20 GeV muon ring, particle with horizontal
  emittance of more than 36 p mm is not
  accelerated. (When vertical emittance is zero.)
• Effects of finite transverse amplitude is less
  than that of EMMA simply because of smaller
  physical emittance.

41
Summary

• Tracking code is made to study FFAG dynamics.