Hybrid macroparticle algorithm for modeling space charge

A space charge algorithm is constructed that is a hybrid between envelope and multiparticle models. The transverse dynamics is simplified by tracking the transverse envelope of each macroparticle. The equations of motion are derived from the Hamiltonian-fluid formulation of the Vlasov Poisson system. A novel electrostatic self-field solver is derived that solves directly for the self-consistent on-axis potential and linear defocusing force, including longitudinal and beam pipe boundary conditions. The implementation of the algorithm, hyper3d, is presented. It is tested against the particle-in-cell code warp, using various configurations of a periodic focusing structure with rf cavities. The required number of macroparticles is reduced substantially compared to standard particle-tracking codes. This model is well adapted to cases where the transverse dynamics is linear and where the details of the longitudinal dynamics are the principal point of interest; an example is where a dc beam is converted to a bunched beam.

Figure 1

Root mean square transverse beam size over the first five FODO lattice periods for different codes.

Figure 2

The 2 rms beam envelope, over the bunching FODO lattice periods for different codes. The bunchers are centered at the locations indicated by the vertical lines.

Figure 3

The longitudinal phase space portrait measured at the final view screen for the warp code (above) and hyper3d (below) with 100,000 macroparticles. The color indicates the normalized particle density.

Figure 4

The relative growth of the 4 rms emittance of a beam in a long focusing channel simulated in hyper3d.

Figure 5

The single-threaded performance of the hyper3d code, measured as the average time in seconds per time step, for different numbers of grid points and macroparticles. The computation time scales linearly with the number of macroparticles and slightly more than quadratically with the number of grid points.