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
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