M.E. Biagini, M. Boscolo, T. Demma (INFN-LNF) A. Chao, M.T.F. Pivi (SLAC). Status of Multi-particle simulation of IBS @ INFN

M.E. Biagini, M. Boscolo, T. Demma (INFN-LNF)

A. Chao, M.T.F. Pivi (SLAC).

Status of Multi-particle simulation of IBS @ INFN

2

• 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)

4

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

8

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


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)


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


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


Grid size: 6xx6yx6z Cell size: x/2xy/2

MC Simulation parameters

Benchmark


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


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


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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


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?)

Documents

M.E. Biagini, M. Boscolo, T. Demma (INFN-LNF) A. Chao, M.T.F. Pivi (SLAC). Status of Multi-particle simulation of IBS @ INFN