U C L A Electron Cloud Effects on Long-Term Beam Dynamics in a Circular Accelerator By : A. Z. Ghalam, T. Katsouleas(USC) G. Rumolo, F.Zimmermann(CERN) C. Huang, V. Decyk, W.Mori(UCLA)

Outline Introduction- single kick approach for e-cloud modeling. What is QuickPIC and why using it for e-cloud problem. Results: Benchmark Results: Beam Dynamics in CERN-SPS and LHC Benchmark the results against HEAD-TAIL Why single kick is not a good approximation Physics Results I: Role of cloud image charges: Tilted beam equilibrium Cancellation of cloud compression tune shift Physics Result II: Dipole magnet effects on beam stability: Generally much more stable Vertical cloud motion can drive horizontal instability

Previous Models: Single Kick Approx. for Cloud Effect Reduces the computational costs . Assumes the cloud is concentrated at one thin slab in the ring, the cloud density at this slab is set so that the average cloud density is constant. Cloud Slab Harmonic Oscillator with Linear force (Linear force) The solutions are identical to: Where p(s) is the periodic impulse function with period 2R Valid when the perturbation due to the pinched cloud is linear!

Why are we using PIC Codes for Electron Cloud Modeling? E-cloud is a non-neutral plasma -- well suited to plasma accelerator PIC models used for plasma wakefield accelerator research Previous cloud models: Treat plasma as a single kick once per orbit New, relatively few benchmarks Cloud image forces were not considered. QuickPIC model: Continuous Modeling of the cloud electrons over the entire ring. Conducting boundary conditions included into the dynamics of the beam as it interacts with cloud. Capable of running over long range of beam propagation yet resolving betatron wavelengths with so many points owing to its parallel processing capability.

Description of the Simulation Model(QuickPIC) QuickPIC is a 3D PIC (particle In Cell) code with parallel processing capability It Uses Quasi- Static or frozen field approximation (>>z) 3D Maxwell equations are reduced to 2D equations The need for solving 2D equations results in larger time steps of the 3D push,enabling a time savings of 2-3 orders of magnitude compared to traditional PIC models

Description of the Simulation Model,(Continued) QuickPIC Cycle: 2-D Plasma Slab Wake (3-D) Beam (3-D)

Computational Challenges of applying a plasma PIC code to circular accelerator modeling Length of beam-plasma interaction to be modeled e- cloud is a 2000 turns * 27Km = 54000Km long plasma! 107 steps, 100m long * 106 particles = 1013 particle-steps! Solution: Parallel computing First order leap frog pusher introduces small numerical frequency shift but frequency (i.e., tune) matters to 4 decimal places! Solution : Advanced Pusher

beam Parallelization z y x Node 3 Node 2 Node 1 Node 0 plasma QuickPIC currently divides beam into evenly partitioned domains along z axis in 3D space; while uniform plasma in 2D space is divided into evenly partitioned domains along one of the transverse axes. MPI is used for communication between nodes z y x Node 3 Node 2 Node 1 Node 0 beam plasma

Overcoming Numerical Tune Shift Solving Harmonic Oscillator with Leap Frog pusher results in a tune shift (Quadratic in time step): Tune Shift ≈ 0.03(t)2 Modified Pusher:

Simulation Results : Benchmarking the Code Against HEAD-TAIL for LHC ring Red : QuickPIC Blue : HEAD-TAIL QuickPIC simulations are performed On USC’s Linux Cluster on 16 Processors for 28 days! Horizontal Spot Size of the beam predicted by the two codes

Snap shots of Beam Evolution on the ring(Horizontal Plane) After 250 turns Head-Tail Dephasing After 500 turns Emittance Growth y After 1000 turns After 2000 turns

Benchmarking: Single Kick QuickPIC vs. HEAD-TAIL Red : QuickPIC(Single Kick Mode) Blue : HEAD-TAIL For accurate benchmarking against single kick models, QuickPIC is modified to model Single Kick Approximation For comparison purposes, the LHC parameters have been used in the simulations

Physics Results I: Cloud Effects on the Frequency of Transverse beam oscillations(Tune Shift ) Vertical Tunes for different cloud densities by taking the FFT of the beam centroid motion over 150 turns of beam propagation on CERN-SPS: Thin : No cloud Dashed : ncl = 106cm-3 Thick : ncl = 107cm-3 Tune shift due to uniform cloud space charge is : ∆Qy ≈ 0.007 (ncl = 106/cm3) ∆Qy ≈ 0.07 (ncl = 107/cm3) (Unperturbed) Closely matched to results from the equation. But ncl> 40ncl(unperturbed) Why tune shift is not 40 times larger? Answer requires understanding image forces. Cloud Compression on the axis

Cloud image may be dominant contribution to coherent tune shift: Image Forces on the Beam Image forces: wall - - - + - - - - F - Beam Image - - F - + + - + + - - + - + + + Cloud Image Cloud image may be dominant contribution to coherent tune shift:

Different Forces from the Cloud on the beam 0.23 z(c/p) -0.23 Uniform Cloud + + Pinched Cloud Image charges(of pincbed Cloud not beam! -0.0075 y(c/p) 0.0075 Image charges(of pincbed Cloud not beam! -4107 density(cm-3) 0

Interaction between the forces A cartoon showing a tilted beam and the electric field on the tail of the beam due to cloud compression on the axis The oppositely Directed image and cloud forces create an equilibrium in which the beam is tilted. Lower bound on the tilt angle can be found by equating these two forces at the tail:

Verifying the Hypothesis with Simulations Displacement Pipe Axis Bean Axis We find total force on the tail of the beam for different tilt angles and different displacement amounts There is an equilibrium tilt angle (Net force=0) as predicted. Periodic boundary conditions do not correctly capture the physics.

Study of the Long Term Beam Dynamics 3 snapshots of beam over CERN-SPS ring. a) At t0=0.4ms (18 turns). b) At t1=t0+TB/4 c) At t2=t0+ TB/2, where TB is the nominal betatron period (0.9s),The beam is initially off centered 1mm from the axis of the pipe Initially Off-Centered Beam 0.2 Z(c/p) -0.2 (a) (b) (c) -0.004 Y(c/p) 0.004 3 snap shots of beam evolution over CERN-SPS a) At t=0 b) At t=136s (6 turns) c) At t=0.8ms (35 turns). The beam is initially tilted Initially Tilted Beam -0.75 z(c/p) 0.75 (a) (b) (c) D -0.015 y(c/p) 0.015 These figures show that no matter how the beam is initially perturbed, the beam ends up having a similar long term dynamics

Physical Results II:Effects of Dipole Magnets on Beam-cloud Interactions CERN_SPS Ring Specifications: 750 bending of length 6.26m. 70 percent bending sections. Straight sections 9m. Dipole Strength 0.12T. Modeling bending sections/Magnets on QuickPIC: The effect of magnetic field on cloud dynamics is significant Resolving the spatial profile of the magnets increases the run time by a factor of ten. Assume average B on the whole ring B = 0 B = 0.126T Vertical Plane Horizontal Plane Severe Cloud Compression Shallow Cloud Compression 3D cloud density with magnetic field Cloud Density in Horizontal Plane

Centroid Spot Size Dipole Magnets Reduce emittance growth The Beam is initially displaced in vertical direction There is less growth in both amplitude of centroid oscillation and spot size observed in the case with B field

Spot Size Growth in LHC with Dipole Magnet The peak dipole magnet strength is chosen(8.4T). The dipole is assumed all along the ring. No Spot Size Growth observed in any plane! Single Kick QuickPIC Red : Vertical Spot Size Blue: Horizontal Spot size Continuous QuickPIC Red : Vertical Spot Size Blue: Horizontal Spot size

Conclusions & Future Work Advanced plasma wakefield models in high performance computing platforms enable the most physically accurate simulations of beam propagation and stability in circular accelerators with electron clouds. Benchmark the code against experimental results, PEPII?

Open Question,Benchmarking the Code Against HEAD-TAIL from CERN QuickPIC( Cloud all over the ring) HEAD_TAIL(Single Kick) Emittance growth is much more severe in single kick models.

Why this approach is not appropriate for E-Cloud modeling? QuickPIC Results for LHC As the number of kicks is increased, less growth rate is observed. Single Kick model is not a reliable approach for Ecloud modeling. PEHTS(a single kick code from (Ohmi) for CERN-SPS