Beam-Beam Simulations Ji Qiang US LARP CM12 Collaboration Meeting Napa Valley, April 8-10, 2009 Lawrence Berkeley National Laboratory

Outline Strong-strong beam-beam simulation for crab cavity compensation at LHC Strong-strong beam-beam simulation for conducting wire compensation at LHC

Luminosity Loss from Crossing Angle Collision

90 degree Crab Cavity Compensation Scheme

BeamBeam3D: Parallel Strong-Strong / Strong-Weak Simulation Beam-Beam forces – integrated, shifted Green function method with FFT – O(N log(N)) computational cost Multiple-slice model for finite bunch length effects Parallel particle-based decomposition to achieve perfect load balance Lorentz boost to handle crossing angle collisions Arbitrary closed-orbit separation (static or time-dep) Multiple bunches, multiple collision points Linear transfer matrix + one turn chromaticity+thin lens sextupole kicks Conducting wire, crab cavity, and electron lens compensation

x y Particle Domain -R2RR 0 A Schematic Plot of the Geometry of Two Colliding Beams Field Domain Head-on collision Long-range collision Crossing angle collision

Green Function Solution of Poisson’s Equation ; r = (x, y) Direct summation of the convolution scales as N 4 !!!! N – grid number in each dimension

Green Function Solution of Poisson’s Equation (cont’d) Hockney’s Algorithm:- scales as (2N) 2 log(2N) - Ref: Hockney and Easwood, Computer Simulation using Particles, McGraw-Hill Book Company, New York, Shifted Green function Algorithm:

Comparison between Numerical Solution and Analytical Solution (Shifted Green Function) ExEx radius inside the particle domain

Green Function Solution of Poisson’s Equation (Integrated Green Function) Integrated Green function Algorithm for large aspect ratio: x (sigma) EyEy

IP Lab frame Moving frame: c cos(  ) 2  Head-on Beam-Beam Collision with Crossing Angle

Transform from the Lab Frame to the Boosted Moving Frame Refs: Hirata, Leunissen, et. al.

Thin Lens Approximation for Crab Cavity Deflection

B.Erdelyi and T.Sen, “Compensation of beam-beam effects in the Tevatron with wires,” (FNAL-TM-2268, 2004). Model of Conducting Wire Compensation (x p0,y p0 ) test particle

Beam energy (TeV) 7 Protons per bunch 10.5e10  */  crab (m) 0.5/4000 Rms spot size (mm) Betatron tunes (0.31,0.32) Rms bunch length (m) Synchrotron tune Momentum spread 0.111e-3 Crab cavity RF frequency MHz LHC Physical Parameters for Testing Crab Cavity

IP5 CAB 21 A Schematic Plot of LHC Collision at 1 IP and Crab Cavities IP

One Turn Transfer Map with Beam-Beam and Crab Cavity M = Ma M1 Mb M1 -1 M M2 -1 Mc M2 Ma: transfer map from head-on crossing angle beam-beam collision Mb,c: transfer maps from crab cavity deflection M1-2: transfer maps between crab cavity and collision point M: one turn transfer map of machine

Luminoisty Evolution with 0.15 mrad Half Crossing Angle with/without Crab Cavity turn

Luminosity vs. Beta* for LHC Crab Cavity Compensation with crab cavity no crab cavity

Correlated Random Error Time-dependent Error Effects of Phase Jitter

Emittance Growth/Per Hour vs. Random Offset Amplitude (beta*=0.25, preliminary results, voltage mismatch)

Emittance Growth/Per Hour vs. Time Modulated Amplitude (beta* = 0.25, preliminary results, voltage mismatch)

Emittance Growth with 0.85 um random offset without/with Correction

IP1 IP5 Strong-Strong Beam-Beam Simulation LHC Wire Compensation (2 Head-On + 64 Long Range)

peak luminosity evolution with conduting wire compensation and reduced separation

Strong-Strong Beam-Beam Simulation LHC Wire Compensation: effect of wire current fluctuation