A 3D tracking algorithm for bunches in beam pipes with elliptical cross-section and a concept for simulation of the interaction with an e-cloud

Aleksandar Markovic E-Cloud 2007, Daegu, April 10, 2007

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Overview

Particle tracking and e-cloud simulations

• Poisson solver for beam pipes with elliptical cross section

• Space charge simulation results

• Implementation of the tracking algorithm

• Integration of particle equations

• First tracking simulations

• Plans for e-cloud simulations

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• 3D Space charge fields of bunches in a beam pipe of elliptical cross section, part of MOEVE 2.0

- iterative solvers: BiCG, BiCGSTAB

- step size: non-equidistant

Beam Pipes with Elliptical Cross Section

Poisson equation:

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Motivated from the paper “Simulation of transverse single bunch instabilities and emittance growth caused by electron cloud in LHC and SPS“.

E. Benedetto, D. Schulte, F. Zimmermann, CERN, Switzerland, G. Rumolo, GSI, Germany

Space Charge Simulation Results

• construction of a fast cosine transformation for the simulation of conducting boundaries

Numerical example:

• spherical bunch

• r<<a,b (r: radius of the bunch, a, b: the half axis of the elliptical beam pipe)

• uniformly distributed charge of 1nC

• bunch is located the centre of the beam pipe

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elliptic b.c. beam pipe with elliptical cross sectionw/o.b.c. open b. c. on a rectangularw.b.c. conducting b.c. on a rectangular

Space charge Simulation ResultsSpace Charge Simulation Results

Electric field Ex along the x-axis of a square rectangular box a=b a = 1.5b

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Electric field Ex along y = ±b/2 of a square rectangular box a=b a = 1.5b

elliptic b.c. beam pipe with elliptical cross sectionw/o.b.c. open b. c. on a rectangularw.b.c. conducting b.c. on a rectangular

Space Charge Simulation Results

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Electric field Ey along y = ±b/2 of a square rectangular box a=b a = 1.5b

elliptic b.c. beam pipe with elliptical cross sectionw/o.b.c. open b. c. on a rectangularw.b.c. conducting b.c. on a rectangular

Space Charge Simulation Results

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Particle Tracking

Tracking Algorithm

• Definition of macro particle distribution

• Deposition of charges in the mesh nodes

• Space charge computation in the center of mass system (3D Poisson solvers from MOEVE 2.0)

• Interpolation of the fields for each macro particle in the laboratory frame

• Time integration of the Newton-Lorentz equation for each macro particle (leap frog scheme)

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Time Integration of Particle Equations *

*Birdsall, C.K. and A.B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill, New York, 1985.

• relativistic generalisation of the Newton-Lorentz equation:

• electric impulse halves:

• Boris rotation for 3D fields:

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First Tracking Simulations - relativistic bunch Ekin=5 MeV

• cylindrical bunch

• 50,000 macro particles

• x=y=z=1.0 mm

• uniform distribution

• charge –1 nC

• Ekin=5 MeV

• tracking time 0.5 ns

• time step 5 ps

•rpipe=1cm Longitudinal plane z-y (m)

Drift

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• cylindrical bunch

• 50,000 macro particles

• x=y=z=1.0 mm

• uniform distribution

• charge –1 nC

• Ekin=5 MeV

• tracking time 0.5 ns

• time step 5 ps

•rpipe=1cm Transversal plane x-y

First Tracking Simulations - relativistic bunch Ekin=5 MeV

Drift

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z

y

First Tracking Simulations - relativistic bunch Ekin=5 MeV

Distribution of after 150ps (30 time steps) for a cylindrical bunch

with 50,000 macro particles and initial Ekin=5 MeV

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First Tracking Simulations / slow bunch Ekin=10 eV

• Gaussian bunch

• 10,000 macro electrons

• x=y=1.5 mm

• z=2.5 mm

• charge –1 nC

• Ekin=10 eV

• tracking time 1ns

• time step 5 ps

• rpipe=1cm

y

x

y

z

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Modified CESR-c

Period

By,peak

Gap

Width

Poles

Periods

Length

400 mm

1.67 T

76 mm

238 mm

14

7

2.5 m

Modified CESR-c Wiggler

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• Uniform bunch

• 40,000 macro electrons

• x=y=2 mm

• z=20 mm

• charge –1 nC

• Ekin=10 eV

• simulation time 4ns

• time step 20 ps

•ρ=199 nC/m3

•rpipe=1cm

E-Cloud in a Period of the Modified CESR-c Wiggler(400 mm)

wiggler field off

z

x

x

y

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E-Cloud in a Period of the Modified CESR-c Wiggler(400 mm)

wiggler field on

• Uniform bunch

• 40,000 macro electrons

• x=y=2 mm

• z=20 mm

• charge –1 nC

• Ekin=10 eV

• simulation time 4ns

• time step 20 ps

•ρ=199 nC/m3

•rpipe=1cm

y

x

x

z

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Plans: Interaction Beam - E-Cloud

• starting with uniform particle distribution for the e-cloud

• separate space charge computation for the relativistic positron beam and the e-cloud

• In each time step a new part of the computational domain gets particle distribution for the e-cloud (from the precomputed decay)