TimeReversed Particle Simulations In GPT or There And

Time-Reversed Particle Simulations In GPT (or “There And Back Again”) Simon Jolly Imperial College FETS Meeting, 12/10/05

Time-reversed Simulations • GPT only has capacity to run time forwards in simulations. • To make comparisons with “downstream” emittance measurements, need to find a way of running time backwards. • Create “reverse” simulation by making divergence negative ie. all angles are inverted: w Is this a realistic assumption to make? w Does it produce realistic results?

Backwards Simulations • “Time-reversed” (backwards) technique tested in the following way: w Create beam and track forwards 300 mm; w Invert transverse velocity (angle) of each particle and reverse longitudinal profile: equivalent to a reflection in X-Y plane; w Re-insert “reversed” beam into GPT and track forward another 300 mm. w “Reverse” beam a second time and compare to original model at t=0.

Simulation Parameters • 2 different beam models used: w “Parallel” beam - circular, uniform beam; xrms = yrms = 5 mm, x’ = y’ = 0, z = 0, 35 ke. V, 60 m. A, 100% SC, E = 0, 10, 000 particles. w Gaussian beam - xrms = yrms = 1. 6 mm, x’rms = y’rms = 1. 7 mrad, x, rms = y, rms = 8. 3 x 10 -3 mm mrad, z = 0, 35 ke. V, 60 m. A, 100% SC, E = 0, 10, 000 particles. • 2 different space charge models used: 2 Dline and tree 2 D (“reverse” simulation tests SC model accuracy).

Parallel/Gaussian X-Y Profiles Parallel beam Gaussian beam

Parallel Beam Trajectories (1) Forward trajectories: Z -X, tree 2 D model

Parallel Beam Trajectories (2) Reverse trajectories: Z -X, tree 2 D model

Parallel Beam: tree 2 D X-Y (1) Difference between transverse positions at 0 mm of forward and reverse beams: X-Y, tree 2 D model

Parallel Beam: tree 2 D X-Y (2) Difference between transverse positions at 0 mm of forward and reverse beams: X-Y, tree 2 D model

Parallel Beam: tree 2 D X’-Y’ Difference between transverse angles at 0 mm of forward and reverse beams: X’-Y’, tree 2 D model

Parallel Beam Trajectories (3) Forward trajectories: Z -X, 2 Dline model

Parallel Beam: 2 Dline X-Y Difference between transverse positions at 0 mm of forward and reverse beams: X-Y, 2 Dline model

Parallel Beam: 2 Dline X’-Y’ Difference between transverse angles at 0 mm of forward and reverse beams: X’-Y’, 2 Dline model

Parallel Beam SC Models (1) Difference between forward trajectories (Z -X) for tree 2 D and 2 Dline space charge models

Parallel Beam SC Models (2) Difference between transverse positions at 0 mm (X-Y) for tree 2 D and 2 Dline space charge models

Gaussian Beam: Forward (1) Forward trajectories: Z -X, tree 2 D model

Gaussian Beam: Forward (2) Forward trajectories: Z -X, 2 Dline model

Gaussian Beam: Reverse trajectories: Z -X, 2 Dline model

Gaussian Beam: tree 2 D X-Y Difference between transverse positions at 0 mm of forward and reverse beams: X-Y, tree 2 D model

Gaussian Beam: 2 Dline X-Y Difference between transverse positions at 0 mm of forward and reverse beams: X-Y, 2 Dline model

Gaussian Beam: tree 2 D X’-Y’ Difference between transverse angles at 0 mm of forward and reverse beams: X’-Y’, tree 2 D model

Gaussian Beam: 2 Dline X’-Y’ Difference between transverse angles at 0 mm of forward and reverse beams: X’-Y’, 2 Dline model

Gaussian Beam: Z-X (1) Longitudinal particle position at 0 mm for reverse beam: Z-X, 2 Dline model

Gaussian Beam: Z-X (2) Longitudinal particle position at 0 mm for reverse beam (enhanced): Z -X, 2 Dline model

Gaussian Trajectory Diff (1) Difference between forward trajectories (Z -X) for tree 2 D and 2 Dline space charge models

Gaussian Trajectory Diff (2) Difference between forward trajectories (Z -X) for tree 2 D and 2 Dline space charge models (enhanced)

Gaussian Angle Diff (1) Difference between forward angles (Z-X’) for tree 2 D and 2 Dline space charge models

Gaussian Angle Diff (2) Difference between forward angles (Z-X’) for tree 2 D and 2 Dline space charge models (enhanced)

Gaussian Beam: 600 mm (1) Trajectories for reverse Gaussian beam tracked for 600 mm: Z -X, 2 Dline model

Gaussian Beam: 600 mm (2) Angle trajectories for reverse Gaussian beam tracked for 600 mm: Z-X’, 2 Dline model

Gaussian 2 Dline Results • Using Gaussian beam distribution gives larger variations between forward and reverse beams (2 Dline model, 0 mm): w Emittance: +0. 1% x, rms (0. 00833 to 0. 00834 mm mrad), +0. 3% y, rms (0. 00833 to 0. 00836 mm mrad). w Size: +1 nm xrms (1. 62326 to 1. 62327 mm), +1 nm yrms (1. 62346 to 1. 62347 mm). w Divergence: +280 nrad x’rms (1. 72808 to 1. 72836 mrad), +760 nrad x’rms (1. 72881 to 1. 72957 mrad).

Gaussian tree 2 D Results • Similar results for SCtree 2 D model (0 mm): w Emittance: +0. 1% x, rms (0. 00833 to 0. 00834 mm mrad), +0. 3% y, rms (0. 00833 to 0. 00836 mm mrad). w Size: +2 nm xrms (1. 62326 to 1. 62328 mm), -2 nm yrms (1. 62346 to 1. 62344 mm). w Divergence: +270 nrad x’rms (1. 72808 to 1. 72835 mrad), +760 nrad x’rms (1. 72881 to 1. 72957 mrad).

Conclusions • Space charge models are accurate enough to run “reverse” simulations in GPT. • Space charge models get worse with increasing angle: w From Pulsar: “We have no solid mathematical proof, but it seems to us that as long as the typical angle with respect to the z-axis times the 'thickness (in z)' of the bunch is less than the radius, all is fine. ” • Inaccuracies clear from simulation results, but not large enough to affect RMS beam parameters.