1 Tracking study of muon acceleration with FFAGs S. Machida RAL/ASTeC 6 December, ffag/machida_ 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/2 . 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. C i : Enge coefficient.

10 Analytical modeling for non-scaling magnets (2) Up to G 20 and G 21 –Edge focusing Up to G 22 and G 23 –Octupole components of fringe fields Up to G 24 and G 25 –Dodecapole … Feed-down of multipole (octupole) has large effects when G 22 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 r^k 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. C i : 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 G 21. thin lens kick every 1 [mm]. We check Tune and time of flight in EMMA and 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 Berg’s 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. [1] J. S. Berg, “Longitudinal acceptance in linear non-scaling FFAGS.” TT T0T0

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 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  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 Initial ellipse ( ns, MeV) is tilted as about the same slope of separatrix. Less tilted (-50%). More tilted (+50%).

24 Distortion of longitudinal emittance Animation Less tilted (-50%). More tilted (+50%). Matched RF phase/2Pi Kinetic energy [GeV]

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  mm-mrad, normalized, zero vertical emittance. Errors of 0, 0.01, 0.02, 0.05 mm (rms), only one error seed. 0. mm0.01 mm 0.05 mm 0.02 mm Horizontal phase space (x, xp)

28 Resonance crossing with acceleration Horizontal is 5000  mm-mrad, normalized, zero vertical emittance. Errors of 0, 0.01, 0.02, 0.05 mm (rms), only one error seed mm0.05 mm Longitudinal phase space (phi, energy) Horizontal phase space (x, xp)

29 Dynamic aperture Transverse acceptance is limited by longitudinal motion, as well as resonance crossing ?

30 Parameters of EMMA a=1/12, b=1/5 (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 (0.09*2 , 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  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.  mm normalized) horizontal only vertical only horizontal and vertical

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. horizontal=2.56  horizontal=3.24 

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. vertical=4.90  vertical=6.40  2, 4 1, 3 2, 4

35 Time of flight for different amplitude Time of flight is calculated for each particle. Legend shows horizontal amplitude (  mm, normalized). Xp=0. Vertical amplitude is zero.

36 With increased “a” parameter. (meaning more voltage.) Increase the voltage twice as much (a=1/6). With a=1/6, all particles are accelerated, but trajectory in phase space still strongly depends on transverse amplitude. horizontal only a=1/12a=1/6 horizontal only

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

38 Parameters of GeV muon ring a=1/12, b=1/5 RF cavity every other cell. Voltage is MV per cavity. Harmonic number is 274 (about 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 10 3  mm-mrad, normalized.) Vertical amplitude is zero. Difference of ToF becomes smaller as accelerated

40 For GeV muon ring In GeV muon ring, particle with horizontal emittance of more than 36  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.