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Aleksandar Markovic E -Cloud 2 007 , Daegu , April 10 , 2007

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 2 007 , Daegu , April 10 , 2007. Overview. Particle tracking and e-cloud simulations

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Aleksandar Markovic E -Cloud 2 007 , Daegu , April 10 , 2007

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  1. 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, April10, 2007

  2. 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 • Firsttracking simulations • Plans for e-cloud simulations

  3. Beam Pipes with Elliptical Cross Section • 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 Poisson equation:

  4. Space Charge Simulation Results 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 • 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

  5. Space Charge Simulation Results Electric field Exalong the x-axis of a square rectangular box a=ba = 1.5b Space charge Simulation Results elliptic b.c. beam pipe with elliptical cross section w/o.b.c. open b. c. on a rectangular w.b.c. conducting b.c. on a rectangular

  6. Space Charge Simulation Results Electric field Exalong y = ±b/2 of a square rectangular box a=ba = 1.5b elliptic b.c. beam pipe with elliptical cross section w/o.b.c. open b. c. on a rectangular w.b.c. conducting b.c. on a rectangular

  7. Space Charge Simulation Results Electric field Eyalong y = ±b/2 of a square rectangular box a=ba = 1.5b elliptic b.c. beam pipe with elliptical cross section w/o.b.c. open b. c. on a rectangular w.b.c. conducting b.c. on a rectangular

  8. 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 solversfrom 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)

  9. Time Integration of Particle Equations * • relativistic generalisation of the • Newton-Lorentz equation: • electric impulse halves: • Boris rotation for 3D fields: *Birdsall, C.K. and A.B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill, New York, 1985.

  10. FirstTracking Simulations - relativistic bunchEkin=5 MeV Drift • cylindrical bunch • 50,000 macro particles • sx=sy=sz=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)

  11. FirstTracking Simulations - relativistic bunchEkin=5 MeV Drift • cylindrical bunch • 50,000 macro particles • sx=sy=sz=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

  12. FirstTracking Simulations - relativistic bunchEkin=5 MeV y z Distribution of after 150ps (30 time steps) for a cylindrical bunch with 50,000 macro particles and initial Ekin=5 MeV

  13. FirstTracking Simulations / slow bunchEkin=10 eV • charge –1 nC • Ekin=10 eV • tracking time 1ns • time step 5 ps • rpipe=1cm • Gaussian bunch • 10,000 macro electrons • sx=sy=1.5 mm • sz=2.5 mm y y z x

  14. Modified CESR-c Wiggler 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

  15. E-Cloud in a Period of the Modified CESR-c Wiggler(400 mm) • charge –1 nC • Ekin=10 eV • simulation time 4ns • time step 20 ps • ρ=199 nC/m3 • rpipe=1cm • Uniform bunch • 40,000 macro electrons • sx=sy=2 mm • sz=20 mm wigglerfield off y x z x

  16. E-Cloud in a Period of the Modified CESR-c Wiggler(400 mm) • charge –1 nC • Ekin=10 eV • simulation time 4ns • time step 20 ps • ρ=199 nC/m3 • rpipe=1cm • Uniform bunch • 40,000 macro electrons • sx=sy=2 mm • sz=20 mm wigglerfield on y x z x

  17. Plans: Interaction Beam - E-Cloud • starting with uniform particle distribution for the e-cloud • separate space charge computation for the relativisticpositron 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)

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