ABP - LHC Injector Synchtrons Section GSI, Darmstadt, 18.02.2009Giovanni Rumolo 1 Recent developments of the HEADTAIL code G. Rumolo, G. Arduini, E. Benedetto,

ABP - LHC Injector Synchtrons Section

GSI, Darmstadt, 18.02.2009

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. ZimmermannCERN/GSI Meeting, GSI, Darmstadt, 18-19/02.2009


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



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Localized impedance source

"Electron cloud simulations: beam instabilities and wakefields" G. Rumolo and F. Zimmermann, PRST-AB 5, 121002 (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)



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Slice 1

W1N1+W0N2 Σ WkNi-k

Slice 2

K=0

i-1

Slice i

Σ WkNi-kK=1

Ns-1

Slice Ns

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

W0N1

Longitudinal

Wi = WL(i Dz)

Energy loss


12iNs


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Slice 1

N1(W1dx1+W1qx) Σ Nk(Wkdxk+Wkqx)

Slice 2

K=1

i-1

Slice i

Σ Nk(Wkdxk+Wkqx)K=1

Ns-1

Slice Ns


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:Wid = Wdx(i Dz) quadrupolar:Wiq = Wqx(i Dz)xi centroid of slice ix position of particle


12iNs


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Slice 1 Slice 2 Slice i Slice Ns


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 1Electrons step 0 Electrons step i-1 Electrons step Ns-1

… …GSI, Darmstadt, 18.02.2009

12iNs


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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)



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• Short range wake field due to a broad band impedance

€

Z1⊥(ω) =ωR

ω

Z⊥

1+ iQ⊥

ωR

ω−

ω

ωR

⎛

⎝ ⎜

⎞

⎠ ⎟

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

€

xn +1

′ x n +1

⎛

⎝ ⎜

⎞

⎠ ⎟= M1(δp)M2(Ix,Iy ) Msc (z)

xn − ˆ x (z)

′ x n + Δ ′ x EC ,Z1⊥− ′ ˆ x (z)

⎛

⎝ ⎜

⎞

⎠ ⎟+

ˆ x (z)

ˆ ′ x (z)

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

What the HEADTAIL model includes (II)



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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



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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)



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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

GdfidlRESWALL

Bench measurements

Lattice



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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-2002-036-AP

“HEADTAIL upgrade“ D. Quatraro, G. Rumolo and B. Salvant (work in progress)



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Recent upgrades of HEADTAIL (II)

® Features which have been added to the HEADTAIL codeTransverse 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)



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Recent upgrades of HEADTAIL (III)

® Features which have been added to the HEADTAIL codeTransverse 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



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Recent upgrades of HEADTAIL (IV)

® Features which have been added to the HEADTAIL codeLongitudinal 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 gtr-jump scheme With and without higher order terms of h



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Transverse plane (I)Transport matrices built from a MAD-X Twiss file



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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 xj+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)



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Transverse plane (III)Interaction/observation points



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Transverse plane (IV)Space charge kicks



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Transverse plane (V)E-cloud/wake field/observation points


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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Transverse plane (VI)TMCI in the SPS from the kicker impedance (mode

shifts)


Mode shifting and coupling has been studied for an SPS bunch under the action of the wake fields fromall the kickers. Kicks (20 per turn) were applied to the bunch particles exactly at the kickers’ locations.


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Transverse plane (VII)TMCI in the SPS from the kicker impedance (mode

shifts)


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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Transverse plane (VIII)TMCI in the SPS from the kicker impedance (growth

rates)



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Longitudinal plane (I)Production of flat bunches: double rf-system in the SPS

0

0.7 MV

Idea from “Studies of beam behavior in a double RF system“, E. Shaposhnikova in APC Meeting 06.07.2007



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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)



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Longitudinal plane (III)Production of flat bunches: longitudinal dipole kick

Importance of this option:® LHC upgrade: Simulation studies of stability of flat hollow bunches



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Longitudinal plane (IV)Bunch lengthening and microwave instability in the SPS

Potential Well Bunch Lengthening

regime

MicrowaveInstability

regime

Broad-band, Z/n=10 , W fr=700 MHz



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Longitudinal plane (V)Bunch lengthening and microwave instability in the SPS

Bunch shape evolution in the regime of bunch lengthening (1011 ppb, left movie) and just above the threshold for microwave instability (1.5 x 1011 ppb, right movie)



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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 gtr-jump scheme With and without higher order

terms of h



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Longitudinal plane (VII)Transition crossing in the PS



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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)



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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



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Longitudinal plane (IX)Transition crossing in the PS

The agreement between the analytically calculated evolution and the one simulated with HEADTAIL is very good.



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Longitudinal + Transverse....Transition crossing in the PS with a BB impedance

Relativistic gamma

Vertical position of centroid [m]

Linear scale <y>

Log scale <y>

Unstable at = 5.25

Growth rate ~ 60 µs



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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 improvedÞ is interfaced with MAD-X and Z-BASE to track a single bunch in a real

lattice with localized impedance sourcesÞ can track a single bunch in a double harmonic rf system and in an

accelerating bucket (also across transition) • HEADTAIL is constantly under development

Þ To make it more performant and user-friendlyÞ To add features that can enlarge its range of applicability

• Near future upgrade plans:Þ A robust model for longitudinal space chargeÞ Correctly include wake fields in the low energy rangeÞ Extension to multi-bunch simulations



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Transverse plane (VI)E-cloud/wake field/observation points


Documents

ABP - LHC Injector Synchtrons Section GSI, Darmstadt, 18.02.2009Giovanni Rumolo 1 Recent developments of the HEADTAIL code G. Rumolo, G. Arduini, E. Benedetto,