1. Particle Diffusion in FFT Space Charge Method J. Holmes May, 2013

2. 2Managed by UT-Battelle for the U.S. Department of Energy Case Studies ORBIT was run for the following case: – 2D direct force FFT space charge solver. – Lattice: Straight uniform focusing channel. Phase advance of 2 π in 50 meters ->(call 50 meters one turn and Q x = Q y = 1.0 bare tunes. – Uniform coasting KV distribution 5×10 13 protons per 50 meter length. Emittances εx = εy = 1.0 mm-mr Vary some parameters: – Number of macroparticles – Space charge grid size – Symmetrization of the macroparticle distribution Focus on individual particles starting at different values of x.

3. 3Managed by UT-Battelle for the U.S. Department of Energy KV distribution and Incoherent Tunes Matched KV distribution maintains uniformity over time. Incoherent tune footprint also remains constant. Tune depression ~ 1/3. X-Y distributionX-Y Tunes

4. 4Managed by UT-Battelle for the U.S. Department of Energy Particle Profiles are Preserved Profiles for 100K particles after 1000 turns with no space charge and space charge with various grids. X profiles – 1000 TurnsY Profiles – 1000 Turns

5. 5Managed by UT-Battelle for the U.S. Department of Energy Particle Profiles are Preserved X and Y profiles for 1M particles after 1000 turns with space charge with various grids. These profiles are much smoother than those with only 100K particles.

6. 6Managed by UT-Battelle for the U.S. Department of Energy Beam Moment Evolution Beam centroid a few microns off center, except for symmetrized beam Second moments for matched and mismatched cases. First Moment Second Moment

7. 7Managed by UT-Battelle for the U.S. Department of Energy RMS Emittances Remain Constant… For fine mesh and low number of particles, RMS emittances increased over 1000 turns. As the grid was refined and particle numbers increased, the emittances became constant. 100K particles, no space charge and space charge with various grids. 256x256 grid and various numbers of particles.

8. 8Managed by UT-Battelle for the U.S. Department of Energy Diffusion of Reference Particles – 100K Particles Emittance evolution of 3 particles: – X0 = 0 mm, y0 = 0 mm – X0 = 3 mm, y0 = 3 mm – X0 = 6 mm, y0 = 6 mm 100K Particles, 64x64100K Particles, 256x256

9. 9Managed by UT-Battelle for the U.S. Department of Energy Diffusion of Reference Particles – 333K Particles Emittance evolution of 3 particles: – X0 = 0 mm, y0 = 0 mm – X0 = 3 mm, y0 = 3 mm – X0 = 6 mm, y0 = 6 mm 333K Particles, 64x64333K Particles, 256x256

10. 10Managed by UT-Battelle for the U.S. Department of Energy Diffusion of Reference Particles – 400K Particles, Symmetrized Emittance evolution of 3 particles: – X0 = 0 mm, y0 = 0 mm – X0 = 3 mm, y0 = 3 mm – X0 = 6 mm, y0 = 6 mm 400K Particles, 64x64 Symmetrized 400K Particles, 256x256 Symmetrized

11. 11Managed by UT-Battelle for the U.S. Department of Energy Diffusion of Reference Particles – 1M Particles Emittance evolution of 3 particles: – X0 = 0 mm, y0 = 0 mm – X0 = 3 mm, y0 = 3 mm – X0 = 6 mm, y0 = 6 mm 1M Particles, 64x64 Symmetrized 1M Particles, 256x256 Symmetrized

12. 12Managed by UT-Battelle for the U.S. Department of Energy Why aren’t emittances constant? Because of space charge, the lattice and statistical Twiss parameters differ. The lattice parameters are used in the emittance calculations, but the statistical parameters should be used. Lattice and statistical Alphax = Alphay differ Lattice and statistical Betax = Betay differ