1. 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)

2. 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

3. 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!

4. 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.

5. 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

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

7. 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

8. 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

9. 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:

10. 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

11. 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

12. 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

13. 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 ≈ (ncl = 106/cm3)
∆Qy ≈ (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

14. 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:

15. 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!
y(c/p)
Image charges(of pincbed
Cloud not beam!
-4 density(cm-3)

16. 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:

17. 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.

18. 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)
Y(c/p)
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
y(c/p)
These figures show that no matter how the beam is initially perturbed, the beam ends up having a similar
long term dynamics

19. 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

20. 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

21. 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

22. 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?

23. 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.

24. 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