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Multi-particle simulation of IBS

Multi-particle simulation of IBS. M. Boscolo, T. Demma, INFN-LNF A. Chao, SLAC XIII SuperB General Meeting Isola d’Elba 31/05-05/06/2010. Introduction Conventional Calculation of IBS Multi-particles code structure Growth rates estimates and comparison with conventional theories

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Multi-particle simulation of IBS

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  1. Multi-particle simulation of IBS M. Boscolo, T. Demma, INFN-LNF A. Chao, SLAC XIII SuperB General MeetingIsola d’Elba 31/05-05/06/2010

  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 • Evaluate equilibrium emittances ei and radiation damping times ti at low bunch charge • Evaluate the IBS growth rates 1/Ti(ei) for the given emittances, averaged around the lattice, using K. Bane approximation* • 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) • Iterate from step 2 * 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) v=5.812 pm @N=6.5e10 h=2.412 nm @N=6.5e10 • Effect is reasonably small. Nonetheless, there are some interesting questions to answer: • What will be the impact of IBS during the damping process? We have calculated the equilibrium emittances in the presence of IBS, but the beam is extracted before it reaches equilibrium… • Could IBS affect the beam distribution, perhaps generating tails? • What happens when vertical emittance is very small as in SuperB? z=4.97 mm @N=6.5e10

  5. Algorithm for Macroparticle Simulation of IBS • 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. S

  6. Zenkevich-Bolshakov Algorithm For two particles colliding with each other, the changes in momentum for particle 1 can be expressed as: with the equivalent polar angle effand 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

  7. Code Benchmarking (Gaussian Distribution) DAFNE Optical Functions 1/Th1/Tv1/Ts [s-1] Bane Multi-particle # of macroparticles: 104 Grid size: 5xx5yx5z Cell size: x/2xy/2xz/2 CIMP

  8. Intrinsic Random Oscillations

  9. Emittances Evolution w/o IBS Horizontal emittance Longitudinal emittance MC Simulation parameters MacroParticleNumber=40000 Vertical emittance NTurn=1000 (≈10 damping times) sz=12.0*10-3 dp=4.8*10-4 ex=(5.63*10-4)/g ey=(3.56*10-5)/g tx = 1000-1 * 42.028822 * 10-3 ty = 1000-1 * 37.161307 * 10-3 ts = 1000-1 * 17.563599 * 10-3

  10. Emittances evolution w/ IBS Horizontal emittance Longitudinal emittance Nbunch=10000*2.1*1010 MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) sz=12.0*10-3 dp=4.8*10-4 Grid size: 6xx6yx6z Cell size: x/2xy/2xz/2 ex=(5.63*10-4)/g ey=(3.56*10-5)/g Vertical emittance tx = 1000-1 * 42.028822 * 10-3 ty = 1000-1 * 37.161307 * 10-3 ts = 1000-1 * 17.563599 * 10-3

  11. Scaling Law M.P. Number=40000 NTurn≈10 damping times sz=12.0*10-3 dp=4.8*10-4 ex=(5.63*10-4)/g ey=(3.56*10-5)/g Grid size: 6xx6yx6z Cell size: x/2xy/2xz/2 Gold (1*dt): Nbunch=103*2.1*1010 tx = 10-3 * 42.02 * 10-3 ty = 10-3 * 37.16 * 10-3 ts = 10-3 * 17.56 * 10-3 Magenta (10*dt): Nbunch=104*2.1*1010 tx = 10-3 * 42.02 * 10-3 ty = 10-3 * 37.16 * 10-3 ts = 10-3 * 17.56 * 10-3 Blue (100*dt): Nbunch=105*2.1*1010 tx = 10-4 * 42.02 * 10-3 ty = 10-4 * 37.16 * 10-3 ts = 10-4 * 17.56 * 10-3

  12. Differential equation system describing emittance evolution w/ IBS 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 by fitting the simulation data

  13. Fit with coupled diff. eq. system • Emittance evolution studies with MC requires very long CPU times • → Reliable MC predictions only on unrealistic cases • → Need to extrapolate MC results using a scaling law • Study of scaling law accuracy on MC simulations by varying relevant parameters in a wide range (within CPU time constraints)

  14. Benchmark Scaling law parameters MC Simulation parameters Nbunch=10000*2.1*1010 tdx= 129; tdz = 54 ex0 = 5.65 10-7 ez0 = 5.7 10-6 exeq= 5.7*10-7 ezeq= 5.75 * 10-6 Na = 1.2 10-21 Nb = 3. 10-20 MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) sz=12.0*10-3 dp=4.8*10-4 Grid size: 6xx6yx6z Cell size: x/2xy/2 # lost macroparticles =0 ex=(5.63*10-4)/g ey=(3.56*10-5)/g Obtained from fit to MC result tx = 1000-1 * 42.028822 * 10-3 ty = 1000-1 * 37.161307 * 10-3 ts = 1000-1 * 17.563599 * 10-3 Horizontal emittance longitudinal emittance

  15. N(benchmark) * 4 Scaling law parameters MC Simulation parameters Nbunch=40000*2.1*1010 tdx= 129; tdz = 54 ex0 = 5.65 10-7 ez0 = 5.7 10-6 exeq= 5.7*10-7 ezeq= 5.75 * 10-6 Na = Na(benchmark)*4 Nb = Nb(benchmark )*4 MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) sz=12.0*10-3 dp=4.8*10-4 Grid size: 6xx6yx6z Cell size: x/2xy/2 # lost macroparticles =2 ex=(5.63*10-4)/g ey=(3.56*10-5)/g tx = 1000-1 * 42.028822 * 10-3 ty = 1000-1 * 37.161307 * 10-3 ts = 1000-1 * 17.563599 * 10-3 Horizontal emittance longitudinal emittance

  16. N(benchmark) * 10 Scaling law parameters MC Simulation parameters Nbunch=100000*2.1*1010 tdx= 129; tdz = 54 ex0 = 5.65 10-7 ez0 = 5.7 10-6 exeq= 5.7*10-7 ezeq= 5.75 * 10-6 Na = Na(benchmark)*10 Nb = Nb(benchmark )*10 MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) sz=12.0*10-3 dp=4.8*10-4 Grid size: 6xx6yx6z Cell size: x/2xy/2 ex=(5.63*10-4)/g ey=(3.56*10-5)/g # lost macroparticles =8 tx = 1000-1 * 42.028822 * 10-3 ty = 1000-1 * 37.161307 * 10-3 ts = 1000-1 * 17.563599 * 10-3 Horizontal emittance longitudinal emittance

  17. N(benchmark) / 20 Scaling law parameters MC Simulation parameters Nbunch=500*2.1*1010 tdx= 1291; tdz = 54 ex0 = 5.65 10-7 ez0 = 5.7 10-6 exeq= 5.7*10-7 ezeq= 5.75 * 10-6 Na = Na(benchmark) /20 Nb = Nb(benchmark ) /20 MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) sz=12.0*10-3 dp=4.8*10-4 Grid size: 6xx6yx6z Cell size: x/2xy/2 # lost macroparticles = ex=(5.63*10-4)/g ey=(3.56*10-5)/g tx = 100-1 * 42.028822 * 10-3 ty = 100-1 * 37.161307 * 10-3 ts = 100-1 * 17.563599 * 10-3 Horizontal emittance longitudinal emittance M. Boscolo

  18. Comparison with Bane’s Algorithm Horizontal Emittance Nbunch=104*2.1*1010 Nbunch=105*2.1*1010 Nbunch=106*2.1*1010 Grid size: 6xx6yx6z Cell size: x/2xy/2xz/2 Longitudinal Emittance MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) sz=12.0*10-3 dp=4.8*10-4 ex=(5.63*10-4)/g tx = 1000-1 * 42.028822 * 10-3 ty = 1000-1 * 37.161307 * 10-3 ts = 1000-1 * 17.563599 * 10-3

  19. IBS effect vs initial vertical emittance Horizontal emittance Longitudinal emittance Nbunch=10000*2.1*1010 Vertical emittance MacroParticleNumber=40000 NTurn=1000 (≈10 damping times) sz=12.0*10-3 dp=4.8*10-4 Grid size: 6xx6yx6z Cell size: x/2xy/2xz/2 ex=(5.63*10-4)/g tx = 1000-1 * 42.028822 * 10-3 ty = 1000-1 * 37.161307 * 10-3 ts = 1000-1 * 17.563599 * 10-3

  20. 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. • Will continue paying attention to nonconventional effects as the vertical emittance continues to become smaller as in SuperB. • Produce the FORTRAN version of the code, maybe a parallel implementation (CMAD?) • Start studying SuperB full lattice (including coupling and errors?) • Study the effect of IBS on bunch distribution

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