1. Recent developments of the HEADTAIL code and benchmarking
G. Rumolo,
G. Arduini, E. Benedetto, H. Burkhardt, E. Métral, D. Quatraro, B. Salvant, D. Schulte, E. Shaposhnikova, R. Tomás, F. Zimmermann
AB Seminar,
Giovanni Rumolo

2. Overview Description of the HEADTAIL code
history and model
recent upgrades
Examples of application and benchmarks
Coherent effects of the Electron Cloud
Transverse plane
Transverse Mode Coupling Instability in SPS and PS
Head-Tail modes in PS and SPS
Collimator induced tune shift
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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Giovanni Rumolo

3. Localized impedance source
"Electron cloud simulations: beam instabilities and wakefields" G. Rumolo and F. Zimmermann, PRST-AB 5, (2002)
Localized impedance source
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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Giovanni Rumolo

4. W Interaction with an impedance s z AB Seminar, 17.10.2008
W(z-z‘)r(z‘)dz‘ z W
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Giovanni Rumolo

5. 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) 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 withoptional 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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Giovanni Rumolo

6. What the HEADTAIL model includes (II)
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 weighed 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
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Giovanni Rumolo

7. 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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Giovanni Rumolo

8. Outputs of HEADTAIL (II)
 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 (full longitudinal motion, space charge)
It gives as an output the full bunch dynamics in the unstable regime.
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Giovanni Rumolo

9. 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
“Journal of HEADTAIL upgrades“ E. Benedetto, G. Rumolo and F. Zimmermann, (unpublished)
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Giovanni Rumolo

10. 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
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 format of ZBASE output.
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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Giovanni Rumolo

11. 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 2007, TAILHEAD mode)
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)
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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Giovanni Rumolo

12. 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 gtr-jump scheme
With and without higher order terms of h
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Giovanni Rumolo

13. Quasi-selfconsistent model of electron cloud
Electron cloud (I)
Quasi-selfconsistent model of electron cloud
 Electron distribution used in HEADTAIL generally was uniform in the beam pipe
 Model can be improved by using as an input the distribution of electrons at the beginning of a bunch passage, as it comes out of the build up ECLOUD code
beam
interbunch
Beam pipe x y
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Giovanni Rumolo

14. Electron cloud (II)
Dependence of the threshold of the electron cloud instability (ECI) on the beam energy
Using the upgraded HEADTAIL code, the dependence of the e-cloud instability threshold on energy has been studied (for constant bunch length and longitudinal emittance)
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Giovanni Rumolo

15. Electron cloud (III)
Dependence of the threshold of the electron cloud instability (ECI) on the beam energy
Vertical emittance evolution
Vertical centroid motion
Example: looking for the instability threshold at 40 GeV/c
 There is a coherent motion of the bunch with threshold between 5 and 7 x 1010
 simulations are in dipole field regions, the instability appears in the vertical plane.
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Giovanni Rumolo

16. Electron cloud (IV)
Dependence of the threshold of the electron cloud instability (ECI) on the beam energy
60 GeV/c
200 GeV/c
The coherent motion appears along the bunch with a peculiar intra-bunch pattern.
Outlook: Fourier analysis of these signals can show the band that a wide-band feedback system against single bunch ECI would need to cover (R. De Maria, LARP collaboration)
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Giovanni Rumolo

17. Electron cloud (V)
Dependence of the threshold of the electron cloud instability (ECI) on the beam energy
1/g
E-cloud build up threshold
HEADTAIL simulations
Full energy scan shows:
Bunch intensity threshold decreases with energy!
This is explained because of the smaller transverse beam size at higher energy
Above ~100 GeV/c the instability threshold levels off to the build-up threshold
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Giovanni Rumolo

18. Electron cloud (VI)
Dependence of the threshold of the electron cloud instability (ECI) on the beam energy
Experimental verification. Benchmark with experiment at the SPS shows:
The electron cloud instability appears at 55 GeV/c
It can be cured by blowing up the bunch transversely
"Dependence of the Electron Cloud Instability on the Beam Energy" G. Rumolo, G. Arduini, E. Métral, E. Shaposhnikova, E. Benedetto, R. Calaga, G. Papotti, B. Salvant
PRL 100, , 2008
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19. From the ECI to the Transverse Mode Coupling Instability.... (TMCI)
The e-cloud instability exhibits a different behaviour than the TMCI
„SPS Performance with PS2“, G. Rumolo, E. Métral, E. Shaposhnikova, Proceedings of LUMI‘06, Valencia
 Based on HEADTAIL simulations, shows a scaling law ||
This is in agreement with the analytically derived formula by B. Zotter (1981).
A fast instability was observed in the SPS in low longitudinal emittance single bunches
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Giovanni Rumolo

20. Experimental data from the SPS (2004): a suspect TMCI was observed
TMCI in the SPS
Experimental data from the SPS (2004): a suspect TMCI was observed
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Giovanni Rumolo

21. SPS TMCI simulated with HEADTAIL using a BB-impedance
TMCI in the SPS
SPS TMCI simulated with HEADTAIL using a BB-impedance
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Giovanni Rumolo

22. Benchmarking MOSES and HEADTAIL (I)
MOSES code (Y.H. Chin, CERN-LEP-Div-Rep TH)
E. Métral et al.
„Transverse Mode Coupling Instability in the CERN-SPS“
ICFA-HB2004, Bensheim, Germany, 18-22/10/2004
The agreement between thresholds predicted with HEADTAIL and MOSES is in general very good.
A deeper comparison between these two codes can be made by using the mode analysis technique
Transverse Mode Coupling Instability in the SPS:
HEADTAIL simulations and MOSES calculations
B. Salvant, E. Métral, G. Rumolo, R. Tomás
CARE-HHH APD BEAM’07
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Giovanni Rumolo

23. Benchmarking MOSES and HEADTAIL (II)
The fine structure of the coherent modes can be revealed from the HEADTAIL output of the centroid motion by applying SUSSIX to the complex BPM signal (x + j bxx‘)
Standard FFT or SUSSIX FFT
(*) Sussix code : R. Bartolini, F. Schmidt, SL Note AP, CERN 1998
Theory behind Sussix : R. Bartolini, F. Schmidt, LHC Project Report 132, CERN 1997
J. Laskar et al., Physica D 56, pp (1992)
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Giovanni Rumolo

24. Benchmarking MOSES and HEADTAIL (III)
Scanning in intensity we can observe how the mode in the spectrum shift.... x y
Round Chamber
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Giovanni Rumolo

25. Benchmarking MOSES and HEADTAIL (IV)
Round beam pipe / no chromaticity / no coupling - displaying Re[Q]=f(Ib)
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Giovanni Rumolo

26. Benchmarking MOSES and HEADTAIL (V)
Re[Q]=f(Ib) and comparison with MOSES
Results from MOSES coherent mode analysis and HEADTAIL are superimposed and the agreement is excellent
Most of the radial modes per azimuthal number can be seen from HEADTAIL
There are few ghost lines, probably due to a small initial mismatch of the bunch in the bucket
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Giovanni Rumolo

27. Benchmarking MOSES and HEADTAIL (VI)
Im[Q]=f(Ib) and comparison with MOSES
Also the agreement between the predicted instability growth rates is very good
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28. HEADTAIL with flat pipe
Flat (x)
Flat (y)
 The instability threshold in x is found to be about twice the threshold in y x y
Flat Chamber
Mode analysis with flat pipe:
 Modes shift differently in x and y
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Giovanni Rumolo

29. , R, V signals Time (10 ns/div) ~ 700 MHz
Observation of a fast vertical single-bunch instability near transition (~ 6 GeV) in the CERN-PS (nTOF)
, R, V signals
Time (10 ns/div)
~ 700 MHz
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Giovanni Rumolo

30. The case of the PS instability with HEADTAIL - I
PS TMCI simulated with HEADTAIL using a BB-impedance
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31. The case of the PS instability with HEADTAIL - II
HEADTAIL simulation - turn # 149
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Giovanni Rumolo

32. The PS resistive wall instability on the flat bottom
Head-Tail coherent modes (I)
The PS resistive wall instability on the flat bottom
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Giovanni Rumolo

33. The PS resistive wall instability on the flat bottom
Head-Tail coherent modes (II)
The PS resistive wall instability on the flat bottom
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Giovanni Rumolo

34. The SPS instability @26 GeV/c with negative chromaticity
Head-Tail coherent modes (III)
The SPS GeV/c with negative chromaticity
Measurement at the SPS ( ), xy=-0.25
Three traces of the D signal (vertical) from three subsequent turns during the evolution of a head-tail instability induced by negative chromaticity
 Broad-band impedance model with R=22 MW/m, wr=2p 1.3 GHz, Q=1
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Giovanni Rumolo

35. The SPS instability @26 GeV/c with negative chromaticity
Head-Tail coherent modes (IV)
The SPS GeV/c with negative chromaticity
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Giovanni Rumolo

36. The SPS instability @26 GeV/c with negative chromaticity
Head-Tail coherent modes (V)
The SPS GeV/c with negative chromaticity
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37. Tracking results could be fully benchmarked against experimental data.
The collimator impedance (I)
(resistive wall with inductive by-pass with nonlinear terms)
In measurements were done at the SPS with a prototype of an LHC collimator:
The collimator jaws were moved in and out to different gap aperture values.
The tune shift was measured for a few gap values.
The resistive wall model in HEADTAIL was improved to include both the inductive by-pass effect (impedance formula by L. Vos and Burov-Lebedev, wake field by A. Koschik) and the nonlinear coefficients for the wake at large (near-wall) amplitudes (Piwinski)
Tracking results could be fully benchmarked against experimental data.
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Giovanni Rumolo

38. The collimator impedance (II)
Classical resistive wall
Resistive wall with inductive by-pass
Inductive by-pass with distribution cut
Distribution cut with nonlinear wake terms
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39. The collimator impedance (III)
Comparing the tune shifts extrapolated from HEADTAIL simulations (left plot) with the experimental ones (right plot, red points) and those from analytical theory (right plot, green, magenta, blue points), the agreement is excellent.
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Giovanni Rumolo

40. Production of flat bunches: double rf-system in the SPS
Longitudinal plane (I)
Production of flat bunches: double rf-system in the SPS
0.7 MV
Idea from “Studies of beam behavior in a double RF system“, E. Shaposhnikova in APC Meeting
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Giovanni Rumolo

41. Production of flat bunches: double rf-system in the SPS
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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Giovanni Rumolo

42. Production of flat bunches: longitudinal dipole kick
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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Giovanni Rumolo

43. Bunch lengthening and microwave instability in the SPS
Longitudinal plane (IV)
Bunch lengthening and microwave instability in the SPS
Potential Well Bunch Lengthening
regime
Microwave
Instability
regime
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Giovanni Rumolo

44. Bunch lengthening and microwave instability in the SPS
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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45. Accelerating bucket and transition crossing
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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Giovanni Rumolo

46. Transition crossing in the PS
Longitudinal plane (VII)
Transition crossing in the PS
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47. Transition crossing in the PS
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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48. Transition crossing in the PS
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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49. Transition crossing in the PS
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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50. Transition crossing in the PS with a BB impedance
Longitudinal + Transverse....
Transition crossing in the PS with a BB impedance 
Relativistic gamma
Unstable at
 = 5.25
Linear scale <y>
Vertical position of centroid [m]
Log scale <y>
Growth rate
~ 60 µs
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51. Conclusions & outlook
HEADTAIL is a versatile tool that can be used to do particle tracking with a variety of collective interactions (electron cloud, broad-band impedances, resistive wall, space charge)
Benchmark of HEADTAIL against semi-analytical solutions, other codes and/or experiments has proved very successful
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 foresee:
A robust model for longitudinal space charge
Correctly include wake fields in the low energy range
Extension to multi-bunch simulations
AB Seminar,
Giovanni Rumolo