COherent Multibunch Beambeam Interactions COMBI COMBI Code COherent

COherent Multibunch Beam-beam Interactions COMBI

COMBI Code • COherent Multibunch Beam-beam Interactions • Meant to study beam-beam coherent modes in the multi bunch regime in all possible configurations(many Ips, asymmetric filling schemes, …) – T. Pieloni and W. Herr, Models to Study Multi-bunch Coupling through Head-on and Long-range beam-beam interactions, EPAC 2006, Edinburgh Scotland. – T. Pieloni, A study of beam-beam effects in Hadron Colliders with a Large number of bunches, EPFL Ph. D Thesis 2008. – F. W. Jones, W. Herr and T. Pieloni, Self-consistent Parallel Multibunch beam simulations using a grid-multipole method, PAC 2009 Vancouver, Canada. • Benchmarked against other models and data: – Beam 3 D (J. Qiang BNL) – RHIC BTF measurements – Observations of coherent modes in LHC and BTFs

Code workflow Master and slave approach 1 Master process: serves as a controller, moves the beams and sees who is where and what has to be done sends execution instructions to bunches (slave processors) 1 process per bunch (can have N bunches) Undergoes the instructions from Master can have: • 1 beam action impedance kick, linear transfer • 2 beam actions : beam-beam kick for which the two bunches (CPUs) exchange the relevant information re-calculated before each interaction 1 beam actions ( i. e. linear transfer, BPM, machine impedance…) 2 beam actions ( head-on and long-range beam-beam…)

Code workflow Particle distribution: each bunch is described by its particle distribution (can be Gaussian… anything) in 4 D or 6 D with N macro-particles each macro-particle has an intensity of Real Int/Nmacro-particles At each interaction the particle coordinates x’ and y’ are up-dated b 1(2, jj) = b 1(2, jj) + kick x b 1(4, jj) = b 1(4, jj) + kick y The arcs are described with a linear transfer rotation in phase space

INTEGER jj (number of macro particles) coordinates updated with head-on beam kick C write(10, *)'PKICK: ', Np, b 2 param do 370 jj=, 1 Np b 1(2, jj) = b 1(2, jj) + lrx 1 - x. Kick b 1(4, jj) = b 1(4, jj) + lry 1 - y. Kick lrx and lry are the kicks produced by the distribution of opposite bunch can be computed using Gaussian approximation or HFMM method We always need to subtract the constant part of the kick in case of a separation xkick and ykick In MADX this is done automatically

Beam-beam kick options lrx • Gaussian approximation for the distributions: – One computes the centroids of distribution (b 2 param (1), b 2 param(3)) and the sigmas and use them in the analytical incoherent beam-beam kick ax 1 =) b 1(1, jj) - b 2 param(1(( ay 1=) b 1(3, jj) - b 2 param(3(( hor 1 = sqrt(ax 1**2 + ay 1**2( horb 2 =) b 2 param(2)**2 + b 2 param(4)**2( horx 1 = - 1. 0) exp(1. 0*((hor 1**2)/horb 2)))*(ax 1 )/ & hor 1**2(( hory 1 = - 1. 0) exp(1. 0*((hor 1**2)/horb 2)))*(ay 1 )/ & hor 1**2(( lrx 1 =*2))) b 2 param(11)*rp) / ga) * horx 1( lry 1 =*2))) b 2 param(11)*rp) / ga) * hory 1( b 2 param(1) = centroid of distribution b 2 param(2) = (sx) b 2 param(3) = centroid of distribution b 2 param(4) = (sy)

Beam-beam kick options lrx • Hybrid Fast Multiple Method: no assumptions on particle distributions beyond Gaussian distribution: – One computs from particle distributions the electric field multiples Ex and Ey and applies the kick to opposite particles. do 370 jj=1, Np lrx 1 = (2*rp / ga) * EX(jj( lry 1 = (2*rp / ga) * EY(jj( x. Kick b 1(2, jj) = b 1(2, jj) + lrx 1 b 1(4, jj) = b 1(4, jj) + lry 1 - y. Kick We always need to subtract the constant part of the kick in case of a separation xkick and ykick In MADX this is done automatically

FMM and HFMM

Coherent effects Self-consistent treatment needed Static BB force Self-consistent method Self-consistent source of distortion changes as a result of the distortion Differences respect to Vladislav method

Gaussian Approximation: known limits when looking at distributions Number of macroparticle dependency The effect of particle parameter N macro-particles on the emittance evolution decreases when going to higher values reaching a saturated value for Nparticles > 106

Gaussian Approximation: known limits when looking at distributions Equilibrium emittance number of turns Since the model assumes always a Gaussian distribution of particles when a beam-beam interaction occurs the beam emittance show decoherence and filamentation effects before finding an equilibrium in collision mode. This is important when evaluating emittance growth effects since one should always use the equilibrium state as a starting point for any consideration. For this example equilibrium is found above 2 104 turns. The number of turns for equilibrium reduces if a HFMM model is used where no assumptions are made on the particle distributions.