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ABP - LHC Injector Synchtrons Section GSI, Darmstadt, Giovanni Rumolo 1 Recent developments of the HEADTAIL code G. Rumolo, G. Arduini, E. Benedetto, E. Métral, D. Quatraro, B. Salvant, D. Schulte, R. Tomás, F. Zimmermann CERN/GSI Meeting, GSI, Darmstadt, 18-19/

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 2 Overview Description of the HEADTAIL code — history and model — features and motivations for upgrades Upgrades and applications –Transverse plane Transport based on maps Selection of interaction/observation points Application –Longitudinal plane: Bunch flattening with double rf system or rf dipole kick Bunch lengthening and microwave instability Accelerating bucket and transition crossing Outlook GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 3 Localized impedance source "Electron cloud simulations: beam instabilities and wakefields" G. Rumolo and F. Zimmermann, PRST-AB 5, (2002) The collective interaction is lumped in one or more points along the ring (kick points), where the subsequent slices of a bunch (macroparticles) interact with an electron cloud (macroelectrons) or an impedance (wake) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 4 Slice 1 W 1 N 1 +W 0 N 2 Σ W k N i-k Slice 2 K=0 i-1 Slice i Σ W k N i-k K=1 N s -1 Slice N s 1.Bunch macroparticles are transported across different interaction points through the sector matrices 2.At each interaction point macroparticles in each slice receive the kick from the wakes of the preceding slices 3.Slicing is refreshed at each turn taking into account the longitudinal motion W0N1W0N1 Longitudinal W i = W L (i z) Energy loss GSI, Darmstadt, i NsNs

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 5 Slice 1 N 1 (W 1d x 1 +W 1q x) Σ N k (W kd x k +W kq x) Slice 2 K=1 i-1 Slice i Σ N k (W kd x k +W kq x) K=1 N s -1 Slice N s 1.Bunch macroparticles are transported across different interaction points through the sector matrices 2.At each interaction point macroparticles in each slice receive the kick from the wakes of the preceding slices 3.Slicing is refreshed at each turn taking into account the longitudinal motion Transverse (x) dipolar: W id = W dx (i z) quadrupolar: W iq = W qx (i z) x i centroid of slice i x position of particle GSI, Darmstadt, i NsNs

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 6 Slice 1 Slice 2 Slice i Slice N s 1.Bunch macroparticles are transported across different interaction points through the sector matrices 2.At each interaction point macroparticles in each slice interact with the electron cloud, as it was modified by the interaction with the preceding slices 3.Slicing is updated Electrons step 1 Electrons step 0 Electrons step i-1 Electrons step N s -1 … … GSI, Darmstadt, i NsNs

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 7 What the HEADTAIL model includes (I) Synchrotron motion included Single bunch with longitdinal distribution that can be Gaussian or uniform (barrier bucket, 2002). Longitudinal dynamics is solved in a linear, sinusoidal (2004) voltage or no bucket ( debunching). Chromaticity and detuning with amplitude Dispersion at the kick section(s). Electron cloud kick(s): –Soft Gaussian approach with finite size electrons (used till 2001, obsolete) –PIC module on a grid inside the beam pipe (2001) –PIC solver with optional conducting boundary conditions (GR, D. Schulte, E. Benedetto, 2003) –Uniform or 1-2 stripes initial e-distributions (GR, E. Benedetto, 2005) –Kicks can be given at locations with different beta functions (2004) –Electrons can move in field free space or in certain magnetic field configurations, like dipole, solenoid, combined function magnet (2002) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 8 Short range wake field due to a broad band impedance or to classical thick resistive wall. x,y components (driving and detuning) of the wakes can be weighted by the Yokoya coefficients to include the effect of flat chamber. Space charge: each bunch particle can receive a transverse kick proportional to the local bunch density around the local centroid. Linear coupling between transverse planes What the HEADTAIL model includes (II) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 9 Outputs of HEADTAIL (I) The main direct output files of HEADTAIL give: –Bunch centroid positions, rms-sizes and emittances (horizontal, vertical and longitudinal) as a function of time –Slice by slice centroid positions and rms-sizes. Coherent intra-bunch patterns can be resolved using this information. –Transverse and longitudinal phase space of the bunch with a sub-sample of macroparticles and bunch longitudinal distributions Off line analysis of the HEADTAIL output allows evaluating tune shifts, growth rates, mode spectra (B. Salvant) Instability thresholds can be determined through massive simulation campaigns with different bunch intensities GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 10 Instability thresholds are inferred by HEADTAIL tracking when unstable coherent motion of the bunch centroid with exponential growth suddenly appears for a tiny change of bunch current. Advantages of HEADTAIL wrt analytical formulae that can be used to determine the instability thresholds: It allows for simulations with several types of impedance and with dipole and quadrupole components of the wake It allows for simulations in non-ideal conditions (correct longitudinal motion, chromaticity, amplitude detuning, linear coupling, space charge) It gives as an output the full bunch dynamics in the unstable regime. Outputs of HEADTAIL (II) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 11 HEADTAIL Predictions of tune shifts and instability thresholds (design of new machines) Comparison with beam-based measurements of collective effects (PSB, PS, SPS, LHC) MAD-X Z-Base Other EM codes… HFSS Particle Studio Gdfidl RESWALL Bench measurements Lattice GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 12 Recent upgrades of HEADTAIL (I) Since 2006 a number of modifications have been introduced into the HEADTAIL code, mainly in order to: –Broaden the range of problems that can be studied and understood using the code (see following slides) –Improve computation speed and accuracy of the results Revisit some parts of the code to optimize calculations over some loops or minimize conditional statements Introduce frozen models for electron cloud and wake fields (only applicable in some specific cases) –Make it more user-friendly and thus increase the number of potential users of the code inside and outside of CERN. “Practical User Guide for HEADTAIL“ G. Rumolo and F. Zimmermann, CERN-SL-Note AP “HEADTAIL upgrade“ D. Quatraro, G. Rumolo and B. Salvant (work in progress) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 13 Recent upgrades of HEADTAIL (II) Features which have been added to the HEADTAIL code Transverse plane:. –The initial distribution of electrons can be self-consistently loaded from a build-up code (ECLOUD) run –More wake field options have been included: The interaction with the resistive wall impedance has been extended to include the inductive by-pass effect and near-wall effects The wake field can be loaded from an external table, calculated as Fourier transform of a known impedance (e.g. kicker or resistive wall in low energy). It accepts an input having the Z-BASE output format. –Interaction of the bunch with several different resonators placed at locations with different beta functions (the list needs to be input on a separated file) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 14 Recent upgrades of HEADTAIL (III) Features which have been added to the HEADTAIL code Transverse plane (cont‘d): –The beam transport with a simple rotation one-turn matrix can be optionally replaced by transport using the correct lattice of the machine Sector maps generated by MAD-X between selected points of the ring are loaded and used for the transport between these points (R. Tomás 2006) The beta functions, as read from a MAD-X Twiss file, are used for building the linear transport matrices between kick points (D. Quatraro 2007, see next slides) This has the advantage of easier implementation of chromaticity through adjustment of the phase advance between kick points by the fractional part of the chromatic shift The signal from several BPMs can be saved and used for further analysis GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 15 Recent upgrades of HEADTAIL (IV) Features which have been added to the HEADTAIL code Longitudinal plane: –A higher harmonic rf system has been introduced with adjustable relative phase to the main rf system (e.g., Bunch Shortening and Bunch Lengthening modes) and a voltage ramp. –Full motion inside an accelerating bucket has been implemented (GR & B. Salvant 2008) Phenomena on the energy ramp can be simulated without approximations Transition crossing can be modeled in detail So far without tr -jump scheme With and without higher order terms of GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 16 Transverse plane (I) Transport matrices built from a MAD-X Twiss file GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 17 Transverse plane (II) Transport matrices built from a MAD-X Twiss file HEADTAIL reads tune and chromaticity values from the standard input file.cfg MAD-X is run internally and the lattice is matched to the given tune and chromaticity values Transport matrices are then built from the Twiss file output by MAD-X The local chromaticities j+1,j are also contained in the Twiss file, and they are used to give particles their correct phase advances at each turn according to their momenta (evolving according the synchrotron motion) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 18 Transverse plane (III) Interaction/observation points GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 19 Transverse plane (IV) Space charge kicks GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 20 Transverse plane (V) E-cloud/wake field/observation points GSI, Darmstadt, MBB 200 interaction points with space charge randomly chosen Interaction with electron cloud in all the MBB dipoles Interaction with wake fields at all the kickers Observation points at all the BPMs

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 22 Transverse plane (VI) TMCI in the SPS from the kicker impedance (mode shifts) GSI, Darmstadt, Mode shifting and coupling has been studied for an SPS bunch under the action of the wake fields from all the kickers. Kicks (20 per turn) were applied to the bunch particles exactly at the kickers’ locations.

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 23 Transverse plane (VII) TMCI in the SPS from the kicker impedance (mode shifts) GSI, Darmstadt, The red lines correspond to the one-kick approximation. The wake fields from the different kickers have been weighted by the beta’s in the kicker locations, added up and applied to the bunch once per turn.

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 24 Transverse plane (VIII) TMCI in the SPS from the kicker impedance (growth rates) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 24 Longitudinal plane (I) Production of flat bunches: double rf-system in the SPS MV Idea from “Studies of beam behavior in a double RF system“, E. Shaposhnikova in APC Meeting GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 26 Longitudinal plane (II) Production of flat bunches: double rf-system in the SPS Importance of this option: SPS: The 800 MHz cavity is used in BS mode in normal operation to keep the beam stable LHC upgrade: Stability studies for a beam in a double rf-system in BL mode (flat bunch) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 26 Longitudinal plane (III) Production of flat bunches: longitudinal dipole kick Importance of this option: LHC upgrade: Simulation studies of stability of flat hollow bunches GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 27 Longitudinal plane (IV) Bunch lengthening and microwave instability in the SPS Potential Well Bunch Lengthening regime Microwave Instability regime Broad-band, Z /n=10 f r =700 MHz GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 28 Longitudinal plane (V) Bunch lengthening and microwave instability in the SPS Bunch shape evolution in the regime of bunch lengthening (10 11 ppb, left movie) and just above the threshold for microwave instability (1.5 x ppb, right movie) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 29 Longitudinal plane (VI) Accelerating bucket and transition crossing Phenomena on the energy ramp can be simulated without approximations Transition crossing can be modeled in detail So far without tr -jump scheme With and without higher order terms of GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 30 Longitudinal plane (VII) Transition crossing in the PS GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 31 Longitudinal plane (VII) Transition crossing in the PS To have a better picture of the longitudinal phase space, only few particles at defined synchrotron amplitudes are plotted (10 subsequent turns for each particle) GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 32 Longitudinal plane (VIII) Transition crossing in the PS Analytical solution by Elias anticipated exactly the same type of evolution of the phase space ellipse when crossing transition GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 33 Longitudinal plane (IX) Transition crossing in the PS The agreement between the analytically calculated evolution and the one simulated with HEADTAIL is very good. GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 34 Longitudinal + Transverse.... Transition crossing in the PS with a BB impedance Relativistic gamma Vertical position of centroid [m] Linear scale Log scale Unstable at = 5.25 Growth rate ~ 60 µs GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 35 Conclusions & outlook HEADTAIL is a multi-purpose tool that can be used to do particle tracking with a variety of collective interactions (electron cloud, resonator impedances, resistive wall, space charge) HEADTAIL has been improved is interfaced with MAD-X and Z-BASE to track a single bunch in a real lattice with localized impedance sources can track a single bunch in a double harmonic rf system and in an accelerating bucket (also across transition) HEADTAIL is constantly under development To make it more performant and user-friendly To add features that can enlarge its range of applicability Near future upgrade plans: A robust model for longitudinal space charge Correctly include wake fields in the low energy range Extension to multi-bunch simulations GSI, Darmstadt,

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ABP - LHC Injector Synchtrons Section Giovanni Rumolo 21 Transverse plane (VI) E-cloud/wake field/observation points GSI, Darmstadt,

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