Particle Diffusion inFFT Space ChargeMethod

J. Holmes

May, 2013

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Case Studies• ORBIT was run for the following case:

– 2D direct force FFT space charge solver.

– Lattice:• Straight uniform focusing channel.• Phase advance of 2π in 50 meters ->(call 50 meters one turn and Qx = Qy = 1.0

bare tunes.

– Uniform coasting KV distribution• 5×1013 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.

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KV distribution and Incoherent Tunes

• Matched KV distribution maintains uniformity over time.

• Incoherent tune footprint also remains constant.

• Tune depression ~ 1/3.

X-Y distribution X-Y Tunes

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Particle Profiles are Preserved

• Profiles for 100K particles after 1000 turns with no space charge and space charge with various grids.

X profiles – 1000 Turns Y Profiles – 1000 Turns

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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.

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Beam Moment Evolution• Beam centroid a few microns off center, except for symmetrized

beam

• Second moments for matched and mismatched cases.

First Moment <X> Second Moment <ΔX2>

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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 andspace charge with various grids.

256x256 grid and variousnumbers of particles.

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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

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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

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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, 64x64Symmetrized

400K Particles, 256x256Symmetrized

11 Managed by UT-Battellefor 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, 64x64Symmetrized

1M Particles, 256x256Symmetrized

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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 statisticalAlphax = Alphay differ

Lattice and statisticalBetax = Betay differ