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

x

zcLL

)2/tan( ;

1

12

0

Luminosity Loss from Crossing Angle Collision

c

*,

2 )tan(

:

xcrabx

sccE

V

voltageRF

90 degree 90 degree

90 degree90 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-R 2RR

2R

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) ')'()',()( drrrrGr

(ri) h G(rii '1

N

ri' )(ri' )

)log(2

1),( 22 yxyxG

Direct summation of the convolution scales as N4 !!!!N – grid number in each dimension

Green Function Solution of Poisson’s Equation (cont’d)

F(r) Gs(r,r')(r')dr'Gs(r,r') G(r rs,r')

c(ri) h Gc(rii '1

2N

ri' )c(ri' )

(ri) c(ri) for i = 1, N

Hockney’s Algorithm:- scales as (2N)2log(2N)- Ref: Hockney and Easwood, Computer Simulation using Particles, McGraw-Hill Book Company, New York, 1985.

Shifted Green function Algorithm:

Comparison between Numerical Solution and Analytical Solution (Shifted Green Function)

Ex

radius

inside the particle domain

Green Function Solution of Poisson’s Equation(Integrated Green Function)

c(ri) Gi(rii '1

2N

ri' )c(ri' )

Gi(r,r') Gs(r,r')dr'

Integrated Green function Algorithm for large aspect ratio:

x (sigma)

Ey

IP

Lab frame

Moving fram

e: c co s()

2

Head-on Beam-Beam Collision with Crossing Angle

Transform from the Lab Frame to the Boosted Moving Frame

Refs: Hirata, Leunissen, et. al.

nn

s

nn

nn

n

s

nn

nn

xczE

qVEE

zz

czE

qVPxPx

xx

)/cos(

)/sin(

1

1

1

1

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

(xp0,yp0)

test particle

Beam energy (TeV) 7Protons per bunch 10.5e10

*/crab (m) 0.5/4000

Rms spot size (mm) 0.01592

Betatron tunes (0.31,0.32)

Rms bunch length (m) 0.077

Synchrotron tune 0.0019

Momentum spread 0.111e-3

Crab cavity RF frequency 400.8 MHz

LHC Physical Parameters for Testing Crab Cavity

IP5

C A B2 1

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 collisionMb,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

rAxx

x

nn

1

)1

1(

0

1

0

)2sin( tfAx

Correlated Random Error

Time-dependent Error

Effects of Phase Jitter

rf

ccx

)2/tan(

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 offsetwithout/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