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

Aleksandar Markovic 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 First tracking simulations Plans for e-cloud simulations

Aleksandar Markovic 3 3D Space charge fields of bunches in a beam pipe of elliptical cross section, part of MOEVE iterative solvers: BiCG, BiCGSTAB - step size: non-equidistant Beam Pipes with Elliptical Cross Section Poisson equation:

Aleksandar Markovic 4 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

Aleksandar Markovic 5 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 Space charge Simulation Results Space Charge Simulation Results Electric field E x along the x-axis of a square rectangular box a=b a = 1.5b

Aleksandar Markovic 6 Electric field E x along y = ±b/2 of a square rectangular box a=b a = 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 Space Charge Simulation Results

Aleksandar Markovic 7 Electric field E y along y = ±b/2 of a square rectangular box a=b a = 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 Space Charge Simulation Results

Aleksandar Markovic 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 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)

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

Aleksandar Markovic 10 First Tracking Simulations - relativistic bunch E kin =5 MeV cylindrical bunch 50,000 macro particles  x =  y =  z =1.0 mm uniform distribution charge –1 nC E kin =5 MeV tracking time 0.5 ns time step 5 ps r pipe =1cm Longitudinal plane z-y (m) Drift

Aleksandar Markovic 11 cylindrical bunch 50,000 macro particles  x =  y =  z =1.0 mm uniform distribution charge –1 nC E kin =5 MeV tracking time 0.5 ns time step 5 ps r pipe =1cm Transversal plane x-y First Tracking Simulations - relativistic bunch E kin =5 MeV Drift

Aleksandar Markovic 12 z y First Tracking Simulations - relativistic bunch E kin =5 MeV Distribution of  after 150ps (30 time steps) for a cylindrical bunch with 50,000 macro particles and initial E kin =5 MeV

Aleksandar Markovic 13 First Tracking Simulations / slow bunch E kin =10 eV Gaussian bunch 10,000 macro electrons  x =  y =1.5 mm  z =2.5 mm charge –1 nC E kin =10 eV tracking time 1ns time step 5 ps r pipe =1cm y x y z

Aleksandar Markovic 14 Modified CESR-c Period B y,peak Gap Width Poles Periods Length 400 mm 1.67 T 76 mm 238 mm m Modified CESR-c Wiggler

Aleksandar Markovic 15 Uniform bunch 40,000 macro electrons  x =  y =2 mm  z =20 mm charge –1 nC E kin =10 eV simulation time 4ns time step 20 ps ρ=199 nC/m 3 r pipe =1cm E-Cloud in a Period of the Modified CESR-c Wiggler(400 mm) wiggler field off z x x y

Aleksandar Markovic 16 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 E kin =10 eV simulation time 4ns time step 20 ps ρ=199 nC/m 3 r pipe =1cm y x x z

Aleksandar Markovic 17 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)