1. Simulation of Particle Trajectories for RIKEN Rare-RI Ring Nishina Center, RIKEN SUZUKI Hiroshi Nov. / 11 / 2011

2. RIKEN Rare RI Ring Rare-RI Ring BigRIPS (production, separation for rare RI) Determine the mass of unstable nuclei supplied by BigRIPS with a precision of the order of 10 -6 Beam SHARAQ (used as injection beam line) RIKEN Rare-RI Ring is an isochronous one with an order of 10 -6. Unstable nuclei turn 2000 times in the ring. To determine the mass of the nuclei, TOFs (revolution time) are measured.

3. Isochronous magnetic field Isochronous magnetic field with an order of 10 -6 ． –In One section (60 degrees), there are 4 magnets. –All magnets are rectangular ones. –The field of the inner two magnets are flat in radial direction. –The field of the outer two magnets have n-value and higher components. To design the isochronous ring, the simulation program for TOFs (and the trajectories) of the particles should be precise with an order of 10 -6 to 10 -7. One section of RIKEN Rare RI Ring

4. Simulation Program Program consists of 4 parts, written in C Language. –main.c : Parameters for the desirable ring are set. –Ring.c : Each element of the ring is controlled and the particles are rotated 2000 times along their orbits. –ST.c : represents the straight sections. –DM.c : represents the magnet sections including fringe region. Geometrical Tracking is used. –In the region with magnetic field (inside of the magnet and fringe region), the orbit is divided every 5 mm, to achieve the precise determination of the trajectories.  One section (60 degrees) is divided into =1400 segments. 4 th order Runge=Kutta method is used. One section (60 degrees)

5. Field distribution 3 kinds of the field distribution were used for the simulation. 1) Cut off : very simple approximation. 2) Continuous : fringe shape is represented with Enge function. Enge coefficients were deduced from fitting of the TOSCA simulation. 3) Discrete (Mesh) data : deduced from TOSCA simulation Most precise, but takes much time. 2) Continuous Distribution Fringe field (TOSCA calculation) Inside of magnets a=0.3522, b=-1.4023, c=-0.3483, d=-0.2031 Z=z/2D, 2D is a gap of the dipoles Enge Function

6. 3) Discrete (Mesh) Data  Interpolation of the data i) Choose 16 field-data around the objective point (In this figure, only x-z plane (mid-plane) is shown.) ii) Calculate the spline interpolation along the beam axis (shown with 4 red boxes). Then deduce 4 field-strengths at red points beside the objective point. For deducing the calculation time, these spline interpolations should be calculated before the trajectory calculations. iii)Field at the Objective point is calculated from 4 red data with spline interpolation. z (beam axis) x Interpolation of the field data

7. Segment Number Optimization The convergence of TOF with the number of the segments in one section (60 degrees) was checked. –The continuous distribution was used. –When the segment is more than 1400, the TOF calculation achieves to an order of 10 -7 precision (absolute value).  One section is divided into 1400 segments (size is 5 mm). Vertical axis: –In the case of relative value (the difference from the TOF of the central momentum particle), 10 -6 precision is achieved when segment number is more than 300 step.

8. Data-Mesh Size Optimization The convergence of TOF with the mesh size was checked. Beam-axis direction –When the mesh size is less than 5 mm, the TOF calculation achieves to an order of 10 -7 precision.  The mesh size is 5 mm in beam-axis direction (one section has 1400 data- mesh). Radial direction –When the mesh size is less than 15 mm, the TOF calculation achieves to an order of 10 -7 precision.  Mesh size in radial direction is also 5 mm. Beam-axis direction Radial direction

9. Results of Simulation (Ideal Case) Trajectories for +0.5, 0, -0.5% particles (10 turns) Trajectory –The trajectories of particles which have different momentum remain parallel in the ring (The momentum dispersion remains 6.7 cm/%). Isochronisity –In one section, outer 2 magnets have a slope of the magnetic field. –Its shape is written as B(x) = B 0 * (1+0.330*x+0.092*x 2 ) Difference of the TOF In the ideal calculation, isochronisity with 10 -6 precision can be reached with 2 nd order correction.

10. Isochronous region Horizontal direction –The TOF of the center-orbit particle is the longest. –The isochronous region for 10 -6 precision is 100  mm mrad. Vertical direction –The TOF of the center-orbit particle is the shortest. –The isochronous region for 10 -6 precision is 25  mm mrad. The isochronous region is enough for the emittance of the beam through the injection line (SHARAQ beam line). Horizontal direction Vertical direction Isochronous region

11. Design of the Pole Tip Designing of the pole tip is in progress. –Attach sims to make the slope of the field for outer 2 magnets in one section. –The isochronisity with an order of 10 -5. –For more isochronisity, we are going to use the trim coils. Shape of the pole tip for the outer magnets isochronisity

12. Summary To design the isochronous field of RIKEN Rare RI Ring, a high precision (10 -6 ) beam-optics simulation has been developed. A geometrical tracking is adopted.  For that purpose, one section of the ring (60 degrees) is divided into 1400 segments (5-mm step).  Data mesh in beam-axis direction is 5 mm.  Data mesh in radial direction is also 5 mm. In the ideal case, –The field slope for isochronisity in radial direction is B(x) = B 0 * (1+0.330*x+0.092*x 2 ) –The region of isochronisity with 10 -6 … Horizontal ： 100π mm mrad Vertical ： 25π mm mrad The designing of the pole tip is now in progress.

13. Principle Principle of the mass measurement –Compare the revolution time of reference nuclei (well-known mass) and objective nuclei (unknown mass). (velocity  is also needed.) Nuclei (A/Q)velocityRigidityPeriod (revolution time) m0/qm0/q  0 B0B0 T0T0 m/qm/q  B0B0 T, where

14. Appendix Y.Yamaguchi et.al. NIMB 266 (2008) 4575-4578

15. Features of the ring Dipoles only (Cyclotron-like storage ring) –Wide momentum acceptance (±0.5%) –Parallel beam (Equilibrium orbits) –Isochronous ring Isochronous magnetic field –n-value (1 st order) and trim coils (higher order) for outer two dipole magnets Only one objective nuclei is required for the measurement. –Production rate with SRC cyclotron and BigRIPS beam line 78 Ni ： ~2cps 138 Sn : ~1cps (U 238 beam @ 1000pnA) 1 section of the Rare RI ring (60 deg)