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CERN-ACC-SLIDES-2014-0066 HiLumi LHC FP7 High Luminosity Large Hadron Collider Design Study Presentation DELPHI: an Analytic Vlasov Solver for Impedance-Driven Modes Mounet, N (CERN) 07 May 2014 The HiLumi LHC Design Study is included in the High Luminosity LHC project and is partly funded by the European Commission within the Framework Programme 7 Capacities Specific Programme, Grant Agreement 284404. This work is part of HiLumi LHC Work Package 2: Accelerator Physics & Performance . The electronic version of this HiLumi LHC Publication is available via the HiLumi LHC web site <http://hilumilhc.web.cern.ch> or on the CERN Document Server at the following URL: <http://cds.cern.ch/search?p=CERN-ACC-SLIDES-2014-0066> CERN-ACC-SLIDES-2014-0066

DELPHI: an analytic Vlasov solver for impedance-driven modes N. Mounet Acknowledgments: X. Buffat, A. Burov, K. Li, E. Métral, G. Rumolo, B. Salvant, S. White

Outline  Introduction & motivation  Getting to Sacherer integral equation  How to solve Sacherer equation: Discrete Expansion over Laguerre Polynomials and HeadtaIl modes  Landau damping  Some benchmarks  What else could be done ? 2 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Introduction & motivation  For given machine and beam parameters, we often need to evaluate transverse beam stability w.r.t. impedance effects .  Also important to assess the efficiency of stabilization techniques: damper, non-linearities (Landau damping).  Time domain macroparticles tracking is often a very good tool and can give a complete vision, BUT: ➢ Too slow for certain large scale problems, e.g. typical LHC problem (~1400/2800 bunches with transverse damper → need fine modeling of intrabunch motion and more than 100000 turns to see an instability...), ➢ Too slow to perform large parameter space scans (chromaticity, non-linearities, intensity, damper gain, etc.), ➢ Very difficult to be sure the beam is stable in a certain configuration. 3 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Introduction & motivation Another possibility: Vlasov solver in ”mode domain”  → solver that looks for all modes that can develop, among which one can easily spot the most critical (i.e. unstable).  Idea is not new (non exhaustive list): ➢ Laclare formalism [J. L. Laclare, CERN-87-03-V-1, p. 264] , ➢ MOSES [Y. Chin, CERN/SPS/85-2 & CERN/LEP-TH/88-05] , ➢ NHTVS [A. Burov, Phys. Rev. ST AB 17, 021007 (2014)] . Another idea: write transfer map with azimuthal and radial mesh of  the bunch(es) (no macroparticles) [V. V. Danilov & E. A. Perevedentsev, Nucl. Instr. Meth. in Phys. Res. A 391 (1997) pp. 77-92] , recently extended & improved by S. White and X. Buffat. → all linear collective dynamics represented by a matrix; then eigenvalues = modes. 4 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Introduction & motivation  All current Vlasov solvers have limitations: ➢ Laclare cannot solve problems that involve too many betatron sidebands (give size of matrix to diagonalize), ➢ MOSES limited to single-bunch, resonator models, w/o damper, ➢ NHTVS does not automatically check convergence, relies on airbag rings for radial discretization & treats Landau damping in the framework of stability diagram theory (approximation). → Can we do better ? 5 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Getting to Sacherer integral equation  Outline: ➢ Vlasov equation ➢ Hamiltonian ➢ Perturbative approach adopted ➢ Impedance term → Sacherer integral equation for transverse modes.  We follow here an approach largely inspired from A. W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerators , John Wiley & Sons (1993), chap. 6.  Note: here, unlike Chao we use ”engineer” convention for the Fourier transform → e j ω t (unstable modes have imag. part<0). Also, SI units (c.g.s in Chao), and notations often different. 6 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Vlasov equation [A. A. Vlasov, J. Phys. USSR 9, 25 (1945)]  Vlasov equation expresses that the local phase space density does not change when one follows the flow of particles.  In other words: local phase space area is conserved in time: Courtesy A. W. Chao  Assumptions: ➢ conservative & deterministic system (governed by Hamiltonian) – no damping or diffusion from external sources (no synchrotron radiation), ➢ external forces (no discrete internal force or collision). → impedance seen as a collective field from ensemble of particles. 7 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Vlasov equation  Simplest expression: with ψ the general 6D phase space distribution density (and t the time),  In our case: ➢ independent variable chosen as s=v t (longitudinal position along accelerator orbit), ➢ particle coordinates (4D – no x/y coupling): - transverse: (y, p y ) ⇔ (J y , θ y ) (action/angle) - longitudinal: (z , δ ) → 8 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Vlasov equation  Simplest expression: with ψ the general 6D phase space distribution density (and t the time),  In our case: ➢ independent variable chosen as s=v t (longitudinal position along accelerator orbit), ➢ particle coordinates (4D – no x/y coupling): - transverse: (y, p y ) ⇔ (J y , θ y ) (action/angle) - longitudinal: (z , δ ) How to write them ? → 9 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Hamiltonian  In the presence of a dipolar vertical impedance resulting in a force F y (z,s) : with and 10 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Hamiltonian  In the presence of a dipolar vertical impedance resulting in a force F y (z,s) : synchrotron freq. slippage factor machine radius total with velocity= β c energy unperturbed tune chromaticity and 11 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Hamiltonian  In the presence of a dipolar vertical impedance resulting in a force F y (z,s) : with and 12 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Hamiltonian  In the presence of a dipolar vertical impedance resulting in a force F y (z,s) : dipolar wake fields transverse part longitudinal part (linear) with and 13 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Hamiltonian  In the presence of a dipolar vertical impedance resulting in a force F y (z,s) : dipolar wake fields transverse part longitudinal part (linear) with and unperturbed ! → important assumption : invariant (and action-angle variables) stay as in linear case... 14 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Hamilton's equations...  … will then give derivatives of J y , θ y , z and δ w.r.t. s: 15 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Hamilton's equations...  … will then give derivatives of J y , θ y , z and δ w.r.t. s: Neglected (not even mentioned in Chao's book) Neglected (from Chao: OK when far from synchro-betatron resonances & small transverse beam size) 16 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

How to solve Vlasov equation ?  Equation remains quite complicated: partial differential eq. for distribution function ψ (s, J y , θ y , z, δ ) :  To simplify the problem: ➢ Assume a mode is developping in the bunch along the revolutions, with a certain (complex) frequency Ω =Q c ω 0 , ➢ Assume we stay close to the stationary unperturbed r = √ 2 v 2 δ 2 2 + η distribution ψ 0 , function of invariants J y and z 2 ω s → perturbation formalism: 17 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

How to solve Vlasov equation ?  Equation remains quite complicated: partial differential eq. for distribution function ψ (s, J y , θ y , z, δ ) :  To simplify the problem: ➢ Assume a mode is developping in the bunch along the revolutions, with a certain (complex) frequency Ω =Q c ω 0 , ➢ Assume we stay close to the stationary unperturbed r =  2 v 2  2 2  distribution ψ 0 , function of invariants J y and z 2  s → perturbation formalism: ∆ψ 1 : self- consistent perturbation to be found 18 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Rewriting Vlasov equation  Use polar coordinates in longitudinal:  After some algebra, neglecting second order terms proportional to ∆ψ 1 F y (wake field force assumed to be small): 19 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

Rewriting Vlasov equation  Use polar coordinates in longitudinal:  After some algebra, neglecting second order terms proportional to ∆ψ 1 F y (wake field force assumed to be small): 20 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014

The trick is to find appropriate decompositions...  Writing f 1 as a Fourier series we can show that all f 1 k are zero except for k=-1 (this is exact except for k=1 for which it relies on |Q c -Q y |<<|Q c +Q y | ) →  For g 1 decomposition is more subtle: 21 The DELPHI Vlasov solver - N. Mounet - HSC meeting 09/04/2014