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

3 IBS Calculations procedure 1.Evaluate equilibrium emittances  i and radiation damping times  i at low bunch charge 2.Evaluate the IBS growth rates 1/T i (  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 * K. Kubo, S.K. Mtingwa, A. Wolski, "Intrabeam Scattering Formulas for High Energy Beams," Phys. Rev. ST Accel. Beams 8, (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  v =5.812  z =4.97

5 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.

6 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

7 First Application: DA  NE # of macroparticles: 10 4 Grid size: 5  x x5  y x5  z Cell size:  x /2x  y /2x  z /2 DA  NE Crab Waist (Siddharta model) 1/T h 1/T v 1/T s [s -1 ] Multi-particle tracking code Bane CIMP

8 Intrinsic Random Oscillations

9 Emittances Evolution w/o IBS  z =12.0*10 -3  p=4.8*10 -4 e x =(5.63*10 -4 )/g e y =(3.56*10 -5 )/g t x = * * t y = * * t s = * * MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) MC Simulation parameters Longitudinal emittance Horizontal emittance Vertical emittance

10 Emittances evolution w/ IBS N bunch =10000*2.1*10 10  z =12.0*10 -3  p=4.8*10 -4 e x =(5.63*10 -4 )/g e y =(3.56*10 -5 )/g t x = * * t y = * * t s = * * MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) Grid size: 6  x x6  y x6  z Cell size:  x /2x  y /2x  z /2 Longitudinal emittance Horizontal emittance Vertical emittance

11 Scaling Law Blue (100*dt): N bunch =10 5 *2.1*10 10 t x = * * t y = * * t s = * * Magenta (10*dt): N bunch =10 4 *2.1*10 10 t x = * * t y = * * t s = * * Gold (1*dt): N bunch =10 3 *2.1*10 10 t x = * * t y = * * t s = * *  z =12.0*10 -3  p=4.8*10 -4 e x =(5.63*10 -4 )/g e y =(3.56*10 -5 )/g M.P. Number=40000 NTurn≈10 damping times Grid size: 6  x x6  y x6  z Cell size:  x /2x  y /2x  z /2

12 Last Turn (ppb10000_tau100_nt10000) The Kolmogorov-Smirnov Normality Test gives a confidence level >99% in all cases

13 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 bunch a 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 M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010 Obtained by fitting the zero bunch intensity case (IBS =0)

14 IBS = 0 (N bunch =0) M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010  x  =  xeq  =  z  =  zeq  =  z =12.0*10 -3  p=4.8*10 -4 e x =(5.63*10 -4 )/g e y =(3.56*10 -5 )/g t x = * * t y = * * t s = * * NTurn=10000 (≈77damping times) MC Simulation parameters MacroParticleNumber=40000 Cpu=20.10 hrs

15 Benchmark I=10000*I nom M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010 Na (BENCHMARK) = 4.7* Nb (BENCHMARK)=1.12* Radial emittance longitudinal emittance

16 N bunch =10000*2.1*10 10 # lost macroparticles =0  z =12.0*10 -3  p=4.8*10 -4 e x =(5.63*10 -4 )/g e y =(3.56*10 -5 )/g t x = * * t y = * * t s = * * MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) Grid size: 6  x x6  y x6  z Cell size:  x /2x  y /2 MC Simulation parameters Benchmark M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010  dx = 129;  dz = 54  x  =  z  =  xeq  = 5.65*10 -7  zeq  = 5.72 * Scaling law parameters Modified Model

17 Monte Carlo vs rescaled Chao model for I=10 5 *I nom M. Boscolo, XIV SuperB Meeting, Sept. 29th 2010 Na (BENCHMARK) = 4.7* Nb (BENCHMARK)=1.12* Radial emittance longitudinal emittance  z =12.0*10 -3  p=4.8*10 -4 e x =(5.63*10 -4 )/g e y =(3.56*10 -5 )/g t x = * * t y = * * t s = * * NTurn=1000 (≈7.7damping times) MacroParticleNumber=40000

18 Summary plots: DAFNE parameters  z =12.0*10 -3  p=4.8*10 -4 e x =(5.63*10 -4 )/g e y =(3.56*10 -5 )/g t x = * t y = * t s = * MacroParticleNumber=40000 NTurn=1000 (≈10 damping times)

19 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.

20 Multi-particle tracking of IBS: SuperB LER  z =5.0*10 -3 m  p=6.3*10 -4 e x =1.8*10 -9 m e y =0.25/100*e x t x = * sec t y = * sec t s = * 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!

21 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?)