m.e. biagini, m. boscolo, t. demma (infn-lnf) a. chao, m.t.f. pivi (slac). status of multi-particle...
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M.E. Biagini, M. Boscolo, T. Demma (INFN-LNF)
A. Chao, M.T.F. Pivi (SLAC).
Status of Multi-particle simulation of IBS @ INFN
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• Introduction
• Conventional Calculation of IBS
• Multi-particles code structure
• Growth rates estimates and comparison with conventional theories
• Results of tracking simulations
• Conclusions
Plan of Talk
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IBS Calculations procedure
1. Evaluate equilibrium emittances i and radiation damping times i at low bunch charge
2. Evaluate the IBS growth rates 1/Ti(i) for the given emittances, averaged around the lattice (using Piwinski, BM, or K. Bane approximation*)
3. Calculate the "new equilibrium" emittance from:
• For the vertical emittance use* :
• where r varies from 0 (y generated from dispersion) to 1 (y generated from betatron coupling)
4. Iterate from step 2
i 1
1 i /Tii
y 1 r 1
1 y /Tyy r
1
1 x /Txy
* K. Kubo, S.K. Mtingwa, A. Wolski, "Intrabeam Scattering Formulas for High Energy Beams," Phys. Rev. ST Accel. Beams 8, 081001 (2005)
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IBS in SuperB LER (lattice V12)
Effect is reasonably small. Nonetheless, there are some interesting questions to answer:•What will be the impact of IBS during the damping process?•Could IBS affect the beam distribution, perhaps generating tails?
h=2.412 nm
@N=6.5e10
v=5.812 pm
@N=6.5e10
z=4.97 mm
@N=6.5e10
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Algorithm for Macroparticle Simulation of IBS
S
• The lattice is read from a MAD (X or 8) file containing the Twiss functions.
• A particular location of the ring is selected as an Interaction Point (S).
• 6-dim Coordinates of particles are generated (Gaussian distribution at S).
• At S location the scattering routine is called.
• Particles of the beam are grouped in cells.
• Particles inside a cell are coupled
• Momentum of particles is changed because of scattering.
• Invariants of particles and corresponding grow rate are recalculated.
• Radiation damping and excitation effects are evaluated
• Particles are tracked at S again through a one-turn 6-dim R matrix.
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For two particles colliding with each other, the changes in momentum for particle 1 can be expressed as:
with the equivalent polar angle eff and the azimuthal angle distributing uniformly in [0; 2], the invariant changes caused by the equivalent random process are the same as that of the IBS in the time interval ts
Zenkevich-Bolshakov Algorithm
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First Application: DANE
# of macroparticles: 104
Grid size: 5xx5yx5z
Cell size: x/2xy/2xz/2
DANE Crab Waist (Siddharta model)
x x x x y y y y
4.96 0.33 2.15 0.11 1.37 0.31 0 0
1/Th 1/Tv 1/Ts [s-1]
Multi-particle tracking code
Bane
CIMP
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Intrinsic Random Oscillations
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Emittances Evolution w/o IBS
z=12.0*10-3
p=4.8*10-4
x=(5.63*10-4)/y=(3.56*10-5)/ x = 1000-1 * 42.028822 * 10-3
y = 1000-1 * 37.161307 * 10-3
s = 1000-1 * 17.563599 * 10-3
MacroParticleNumber=40000 NTurn=1000 (≈10 damping times)
MC Simulation parameters
Longitudinal emittance
Horizontalemittance
Vertical emittance
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Emittances evolution w/ IBS
Nbunch=10000*2.1*1010
z=12.0*10-3
p=4.8*10-4
x=(5.63*10-4)/y=(3.56*10-5)/ x = 1000-1 * 42.028822 * 10-3
y = 1000-1 * 37.161307 * 10-3
s = 1000-1 * 17.563599 * 10-3
MacroParticleNumber=40000 NTurn=1000 (≈10 damping times)
Grid size: 6xx6yx6z Cell size: x/2xy/2xz/2
Longitudinal emittance
Horizontalemittance
Vertical emittance
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Scaling Law
Blue (100*dt):Nbunch=105*2.1*1010
x = 10-4 * 42.02 * 10-3
y = 10-4 * 37.16 * 10-3
s = 10-4 * 17.56 * 10-3
Magenta (10*dt):Nbunch=104*2.1*1010
x = 10-3 * 42.02 * 10-3
y = 10-3 * 37.16 * 10-3
s = 10-3 * 17.56 * 10-3
Gold (1*dt):Nbunch=103*2.1*1010
x = 10-3 * 42.02 * 10-3
y = 10-3 * 37.16 * 10-3
s = 10-3 * 17.56 * 10-3
z=12.0*10-3
p=4.8*10-4
x=(5.63*10-4)/y=(3.56*10-5)/
M.P. Number=40000 NTurn≈10 damping times
Grid size: 6xx6yx6z Cell size: x/2xy/2xz/2
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Bunch @ Last Turn (ppb10000_tau100_nt10000)
The Kolmogorov-Smirnov Normality Test gives a confidence level >99% in all cases
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)t()t(
Nb)t(
T
1
)t()t(
Na)t(
T
1
z4/3
xzeqz
revzz
z4/3
xxeqx
revxx
Radial and longitudinal emittance growths can be predicted by a model that takes the form of a coupled differential equations:
N number of particles per buncha and b coefficients characterizing IBS obtained once by fitting the tracking simulation data for a chosen benchmark case
Chao Model: differential equation system for x and z
zeqz
xeqx
0zz
0xx
)t(
)t(
)0t(
)0t(
M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010
Obtained by fitting the zero bunch intensity case (IBS =0)
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IBS = 0 (Nbunch=0)
M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010
x = xeq= 5.65 10-7
z = zeq= 5.72 10-6
z=12.0*10-3
p=4.8*10-4
x=(5.63*10-4)/y=(3.56*10-5)/ x = 1000-1 * 42.028822 * 10-3
y = 1000-1 * 37.161307 * 10-3
s = 1000-1 * 17.563599 * 10-3
NTurn=10000 (≈77damping times)
MC Simulation parameters MacroParticleNumber=40000
Cpu=20.10 hrs
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Benchmark I=10000*Inom
M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010
Na (BENCHMARK) = 4.7*10-20
Nb (BENCHMARK)=1.12* 10-18
Radial emittance
longitudinal emittance
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Nbunch=10000*2.1*1010
# lost macroparticles =0
z=12.0*10-3
p=4.8*10-4
x=(5.63*10-4)/y=(3.56*10-5)/ x = 1000-1 * 42.028822 * 10-3
y = 1000-1 * 37.161307 * 10-3
s = 1000-1 * 17.563599 * 10-3
MacroParticleNumber=40000 NTurn=1000 (≈10 damping times)
Grid size: 6xx6yx6z Cell size: x/2xy/2
MC Simulation parameters
Benchmark
M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010
dx = 129; dz = 54x = 5.65 10-7
z = 5.72 10-6
xeq= 5.65*10-7
zeq= 5.72 * 10-6
Scaling law parametersModified Model
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Monte Carlo vs rescaled Chao model for I=105*Inom
M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010
Na (BENCHMARK) = 4.7*10-20
Nb (BENCHMARK)=1.12* 10-18
Radial emittance
longitudinal emittance
z=12.0*10-3
p=4.8*10-4
x=(5.63*10-4)/y=(3.56*10-5)/ x = 1000-1 * 42.028822 * 10-3
y = 1000-1 * 37.161307 * 10-3
s = 1000-1 * 17.563599 * 10-3
NTurn=1000 (≈7.7damping times) MacroParticleNumber=40000
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Summary plots: DAFNE parameters
z=12.0*10-3
p=4.8*10-4
x=(5.63*10-4)/y=(3.56*10-5)/
x = 1000-1 * 0.042 y = 1000-1 * 0.037 s = 1000-1 * 0.017
MacroParticleNumber=40000 NTurn=1000 (≈10 damping times)
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Status of the FORTRAN version of the code
• The lattice is read from a MAD (X or 8) file containing the Twiss functions.
•6-dim Coordinates of particles are generated (Gaussian distribution at S).
• At each lattice element location the scattering routine is called.– Particles of the beam are grouped in cells.– Particles inside a cell are coupled – Momentum of particles is changed because of scattering.– Invariants of particles and corresponding growth rate are recalculated.
•Particles are tracked at next elemenet a 6-dim R matrix.
•Radiation damping and excitation effects are evaluated at each turn.
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Multi-particle tracking of IBS: SuperB LER
z=5.0*10-3 mp=6.3*10-4
x=1.8*10-9 my=0.25/100*x
x = 100-1 * 0.040 sec y = 100-1 * 0.040 sec s = 100-1 * 0.020 sec
MacroParticleNumber=10000 NTurn=10000 (≈10 damping times)
Mathematica vs Fortran implementation of the IBS multi-particle tracking code. The Fortran version is more then 1 order of magnitude faster!
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IBS Status
• The effect of IBS on the transverse emittances is about 30% in the LER and less then 5% in HER that is still reasonable if applied to lattice natural emittances values.
• Interesting aspects of the IBS such as its impact on damping process and on generation of non Gaussian tails may be investigated with a multiparticle algorithm.
•A code implementing the Zenkevich-Bolshakov algorithm to investigate IBS effects is being developed
– Benchmarking with conventional IBS theories gave good results.
• A preliminary FORTRAN version of the code has been produced:–Started collaborating with Mauro Pivi to include the IBS in CMAD (parallel-faster).
•Started studying SuperB full lattice (including coupling and errors?)